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Our approach is based on Mönch’s fixed point theorem associated with the technique of measure of weak noncompactness."/> <meta property="og:image" content="https://media.springernature.com/full/springer-static/cover-hires/journal/13662"/> <script type="application/ld+json">{"mainEntity":{"headline":"Implicit coupled Hilfer–Hadamard fractional differential systems under weak topologies","description":"In this article, we present some existence of weak solutions for a coupled system of implicit fractional differential equations of Hilfer–Hadamard type. Our approach is based on Mönch’s fixed point theorem associated with the technique of measure of weak noncompactness.","datePublished":"2018-09-18T00:00:00Z","dateModified":"2018-09-18T00:00:00Z","pageStart":"1","pageEnd":"17","sameAs":"https://doi.org/10.1186/s13662-018-1787-4","keywords":["26A33","45D05","45G05","45M10","Coupled fractional differential system","Left-sided mixed Pettis–Hadamard integral of fractional order","Hilfer–Hadamard fractional derivative","Weak solution","Implicit","Fixed point","Difference and Functional Equations","Mathematics","general","Analysis","Functional Analysis","Ordinary Differential Equations","Partial Differential Equations"],"image":[],"isPartOf":{"name":"Advances in Difference Equations","issn":["1687-1847"],"volumeNumber":"2018","@type":["Periodical","PublicationVolume"]},"publisher":{"name":"Springer International Publishing","logo":{"url":"https://www.springernature.com/app-sn/public/images/logo-springernature.png","@type":"ImageObject"},"@type":"Organization"},"author":[{"name":"Saïd Abbas","affiliation":[{"name":"Tahar Moulay University of Saïda","address":{"name":"Laboratory of Mathematics, Geometry, Analysis, Control and Applications, Tahar Moulay University of Saïda, Saïda, Algeria","@type":"PostalAddress"},"@type":"Organization"}],"@type":"Person"},{"name":"Mouffak Benchohra","affiliation":[{"name":"Djillali Liabes University of Sidi Bel-Abbès","address":{"name":"Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbès, Sidi Bel-Abbès, Algeria","@type":"PostalAddress"},"@type":"Organization"}],"@type":"Person"},{"name":"Naima Hamidi","affiliation":[{"name":"Djillali Liabes University of Sidi Bel-Abbès","address":{"name":"Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbès, Sidi Bel-Abbès, Algeria","@type":"PostalAddress"},"@type":"Organization"}],"@type":"Person"},{"name":"Yong Zhou","url":"http://orcid.org/0000-0002-4099-8077","affiliation":[{"name":"Xiangtan University","address":{"name":"Faculty of Mathematics and Computational Science, Xiangtan University, Hunan, P.R. China","@type":"PostalAddress"},"@type":"Organization"},{"name":"King Abdulaziz University","address":{"name":"Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia","@type":"PostalAddress"},"@type":"Organization"}],"email":"yzhou@xtu.edu.cn","@type":"Person"}],"isAccessibleForFree":true,"@type":"ScholarlyArticle"},"@context":"https://schema.org","@type":"WebPage"}</script> </head> <body class="journal journal-fulltext" > <div class="ctm"></div>  <noscript> <iframe src="https://www.googletagmanager.com/ns.html?id=GTM-MRVXSHQ" height="0" width="0" style="display:none;visibility:hidden"></iframe> </noscript>  <div class="u-visually-hidden" aria-hidden="true"> <?xml version="1.0" encoding="UTF-8"?><!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd"><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="a" d="M0 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</div> <div class="c-pdf-button__container u-hide-at-lg js-context-bar-sticky-point-mobile"> <div class="c-pdf-container" data-track-context="article body"> <div class="c-pdf-download u-clear-both"> <a href="//advancesincontinuousanddiscretemodels.springeropen.com/counter/pdf/10.1186/s13662-018-1787-4.pdf" class="u-button u-button--full-width u-button--primary u-justify-content-space-between c-pdf-download__link" data-article-pdf="true" data-readcube-pdf-url="true" data-test="pdf-link" data-draft-ignore="true" data-track="content_download" data-track-type="article pdf download" data-track-action="download pdf" data-track-label="link" data-track-external download> <span class="c-pdf-download__text">Download PDF</span> <svg aria-hidden="true" focusable="false" width="16" height="16" class="u-icon"><use xlink:href="#icon-download"/></svg> </a> </div> </div> </div> <article lang="en"> <div class="c-article-header"> <ul class="c-article-identifiers" data-test="article-identifier"> <li class="c-article-identifiers__item" data-test="article-category">Research</li> <li class="c-article-identifiers__item"> <a href="https://www.springernature.com/gp/open-research/about/the-fundamentals-of-open-access-and-open-research" data-track="click" data-track-action="open access" data-track-label="link" class="u-color-open-access" data-test="open-access">Open access</a> </li> <li class="c-article-identifiers__item">Published: <time datetime="2018-09-18">18 September 2018</time></li> </ul> <h1 class="c-article-title" data-test="article-title" data-article-title="">Implicit coupled Hilfer–Hadamard fractional differential systems under weak topologies</h1> <ul class="c-article-author-list c-article-author-list--short" data-test="authors-list" data-component-authors-activator="authors-list"><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Sa_d-Abbas-Aff1" data-author-popup="auth-Sa_d-Abbas-Aff1" data-author-search="Abbas, Saïd">Saïd Abbas</a><sup class="u-js-hide"><a href="#Aff1">1</a></sup>, </li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Mouffak-Benchohra-Aff2" data-author-popup="auth-Mouffak-Benchohra-Aff2" data-author-search="Benchohra, Mouffak">Mouffak Benchohra</a><sup class="u-js-hide"><a href="#Aff2">2</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Naima-Hamidi-Aff2" data-author-popup="auth-Naima-Hamidi-Aff2" data-author-search="Hamidi, Naima">Naima Hamidi</a><sup class="u-js-hide"><a href="#Aff2">2</a></sup> & </li><li class="c-article-author-list__show-more" aria-label="Show all 4 authors for this article" title="Show all 4 authors for this article">…</li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Yong-Zhou-Aff3-Aff4" data-author-popup="auth-Yong-Zhou-Aff3-Aff4" data-author-search="Zhou, Yong" data-corresp-id="c1">Yong Zhou<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><span class="u-js-hide"> <a class="js-orcid" href="http://orcid.org/0000-0002-4099-8077"><span class="u-visually-hidden">ORCID: </span>orcid.org/0000-0002-4099-8077</a></span><sup class="u-js-hide"><a href="#Aff3">3</a>,<a href="#Aff4">4</a></sup> </li></ul><button aria-expanded="false" class="c-article-author-list__button"><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-down-medium"></use></svg><span>Show authors</span></button> <p class="c-article-info-details" data-container-section="info"> <a data-test="journal-link" href="/" data-track="click" data-track-action="journal homepage" data-track-category="article body" data-track-label="link"><i data-test="journal-title">Advances in Difference Equations</i></a> <b data-test="journal-volume"><span class="u-visually-hidden">volume</span> 2018</b>, Article number: <span data-test="article-number">328</span> (<span data-test="article-publication-year">2018</span>) <a href="#citeas" class="c-article-info-details__cite-as u-hide-print" data-track="click" data-track-action="cite this article" data-track-label="link">Cite this article</a> </p> <div class="c-article-metrics-bar__wrapper u-clear-both"> <ul class="c-article-metrics-bar u-list-reset"> <li class=" c-article-metrics-bar__item" data-test="access-count"> <p class="c-article-metrics-bar__count">1549 <span class="c-article-metrics-bar__label">Accesses</span></p> </li> <li class="c-article-metrics-bar__item"> <p class="c-article-metrics-bar__details"><a href="/articles/10.1186/s13662-018-1787-4/metrics" data-track="click" data-track-action="view metrics" data-track-label="link" rel="nofollow">Metrics <span class="u-visually-hidden">details</span></a></p> </li> </ul> </div> </div> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Abs1">Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>In this article, we present some existence of weak solutions for a coupled system of implicit fractional differential equations of Hilfer–Hadamard type. Our approach is based on Mönch’s fixed point theorem associated with the technique of measure of weak noncompactness.</p></div></div></section> <section data-title="Introduction"><div class="c-article-section" id="Sec1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec1"><span class="c-article-section__title-number">1 </span>Introduction</h2><div class="c-article-section__content" id="Sec1-content"><p>In recent years, fractional calculus and fractional differential equations are emerging as a useful tool in modeling the dynamics of many physical systems and electrical phenomena, which has been demonstrated by many researchers in the fields of mathematics, science, and engineering; see [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="
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				" href="/articles/10.1186/s13662-018-1787-4#ref-CR40" id="ref-link-section-d126833688e439">40</a>]. Recently, considerable attention has been given to the existence of solutions of initial and boundary value problems for fractional differential equations with Hilfer fractional derivative [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title="
					Furati, K.M., Kassim, M.D.: Non-existence of global solutions for a differential equation involving Hilfer fractional derivative. Electron. J. Differ. Equ. 2013, 235 (2013)
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					Furati, K.M., Kassim, M.D., Tatar, N.E.: Existence and uniqueness for a problem involving Hilfer fractional derivative. Comput. Math. Appl. 64, 1616–1626 (2012)
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					Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
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					Kamocki, R., Obczyński, C.: On fractional Cauchy-type problems containing Hilfer’s derivative. Electron. J. Qual. Theory Differ. Equ. 2016, 50 (2016)
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				" href="/articles/10.1186/s13662-018-1787-4#ref-CR34" id="ref-link-section-d126833688e458">34</a>] and other problems with Hilfer–Hadamard fractional derivative [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 28" title="
					Qassim, M.D., Furati, K.M., Tatar, N.-E.: On a differential equation involving Hilfer–Hadamard fractional derivative. Abstr. Appl. Anal. 2012, Article ID 391062 (2012)
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					Qassim, M.D., Tatar, N.E.: Well-posedness and stability for a differential problem with Hilfer–Hadamard fractional derivative. Abstr. Appl. Anal. 2013, Article ID 605029 (2013)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR29" id="ref-link-section-d126833688e465">29</a>].</p><p>The measure of weak noncompactness was introduced by De Blasi [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="
					De Blasi, F.S.: On the property of the unit sphere in a Banach space. Bull. Math. Soc. Sci. Math. Roum. 21, 259–262 (1977)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR14" id="ref-link-section-d126833688e471">14</a>]. The strong measure of noncompactness was developed first by Banaś and Goebel [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="
					Banaś, J., Goebel, K.: Measures of Noncompactness in Banach Spaces. Marcel Dekker, New York (1980)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR8" id="ref-link-section-d126833688e474">8</a>] and subsequently developed and used in many papers; see, for example, Akhmerov et al. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="
					Akhmerov, R.R., Kamenskii, M.I., Patapov, A.S., Rodkina, A.E., Sadovskii, B.N.: Measures of Noncompactness and Condensing Operators. Birkhauser Verlag, Basel (1992)
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					Alvárez, J.C.: Measure of noncompactness and fixed points of nonexpansive condensing mappings in locally convex spaces. Rev. Real. Acad. Cienc. Exact. Fis. Natur. Madrid 79, 53–66 (1985)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR7" id="ref-link-section-d126833688e480">7</a>], Benchohra et al. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="
					Benchohra, M., Henderson, J., Seba, D.: Measure of noncompactness and fractional differential equations in Banach spaces. Commun. Appl. Anal. 12(4), 419–428 (2008)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR12" id="ref-link-section-d126833688e483">12</a>], Guo et al. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="
					Guo, D., Lakshmikantham, V., Liu, X.: Nonlinear Integral Equations in Abstract Spaces. Kluwer Academic, Dordrecht (1996)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR17" id="ref-link-section-d126833688e487">17</a>], and the references therein. In [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="
					Benchohra, M., Henderson, J., Seba, D.: Measure of noncompactness and fractional differential equations in Banach spaces. Commun. Appl. Anal. 12(4), 419–428 (2008)
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					O’Regan, D.: Weak solutions of ordinary differential equations in Banach spaces. Appl. Math. Lett. 12, 101–105 (1999)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR26" id="ref-link-section-d126833688e493">26</a>], the authors considered some existence results by the technique of measure of noncompactness. Recently, several researchers obtained other results by the technique of measure of weak noncompactness; see [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="
					Abbas, S., Benchohra, M., Graef, J.R., Henderson, J.: Implicit Fractional Differential and Integral Equations: Existence and Stability. De Gruyter, Berlin (2018)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR2" id="ref-link-section-d126833688e496">2</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="
					Abbas, S., Benchohra, M., N’Guérékata, G.M.: Advanced Fractional Differential and Integral Equations. Nova Science Publishers, New York (2015)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR4" id="ref-link-section-d126833688e499">4</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="
					Benchohra, M., Graef, J., Mostefai, F.Z.: Weak solutions for boundary-value problems with nonlinear fractional differential inclusions. Nonlinear Dyn. Syst. Theory 11(3), 227–237 (2011)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR10" id="ref-link-section-d126833688e502">10</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="
					Benchohra, M., Henderson, J., Mostefai, F.Z.: Weak solutions for hyperbolic partial fractional differential inclusions in Banach spaces. Comput. Math. Appl. 64, 3101–3107 (2012)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR11" id="ref-link-section-d126833688e506">11</a>] and the references therein.</p><p>Consider the following coupled system of implicit Hilfer–Hadamard fractional differential equations: </p><div id="Equ1" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \textstyle\begin{cases} ({}^{H}D_{1}^{\alpha,\beta }u_{1})(t)=f_{1}(t,u_{1}(t),u_{2}(t),({}^{H}D_{1}^{\alpha,\beta }u_{1})(t),({}^{H}D_{1}^{\alpha,\beta}u_{2})(t)), &\\ ({}^{H}D_{1}^{\alpha,\beta }u_{2})(t)=f_{2}(t,u_{1}(t),u_{2}(t),({}^{H}D_{1}^{\alpha,\beta }u_{1})(t),({}^{H}D_{1}^{\alpha,\beta}u_{2})(t)), \end{cases}\displaystyle t\in I, $$</span></div><div class="c-article-equation__number"> (1) </div></div><p> with the initial conditions </p><div id="Equ2" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \textstyle\begin{cases} ({}^{H}I_{1}^{1-\gamma}u_{i})(t)|_{t=1}=\phi_{1}, &\\ ({}^{H}I_{1}^{1-\gamma}u_{2})(t)|_{t=1}=\phi_{2}, \end{cases} $$</span></div><div class="c-article-equation__number"> (2) </div></div><p> where <span class="mathjax-tex">$I:=[1,T], T>1, \alpha\in(0,1), \beta\in[0,1], \gamma =\alpha+\beta-\alpha\beta, \phi_{i}\in E$</span>, <span class="mathjax-tex">$f_{i}:I\times E^{4}\to E, i=1,2$</span>, are given continuous functions, <i>E</i> is a real (or complex) Banach space with norm <span class="mathjax-tex">$\|\cdot\|_{E}$</span> and dual <span class="mathjax-tex">$E^{*}$</span>, such that <i>E</i> is the dual of a weakly compactly generated Banach space <i>X</i>, <span class="mathjax-tex">${}^{H}I_{1}^{1-\gamma}$</span> is the left-sided mixed Hadamard integral of order <span class="mathjax-tex">$1-\gamma$</span>, and <span class="mathjax-tex">${}^{H}D_{1}^{\alpha,\beta}$</span> is the Hilfer–Hadamard fractional derivative of order <i>α</i> and type <i>β</i>. In this paper, we prove the existence of weak solutions for a coupled system of implicit fractional differential equations of Hilfer–Hadamard type.</p></div></div></section><section data-title="Preliminaries"><div class="c-article-section" id="Sec2-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec2"><span class="c-article-section__title-number">2 </span>Preliminaries</h2><div class="c-article-section__content" id="Sec2-content"><p>Let C be the Banach space of all continuous functions <i>v</i> from <i>I</i> into <i>E</i> with the supremum (uniform) norm </p><div id="Equa" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\Vert v \Vert _{\infty}:= \sup _{t\in I} \bigl\Vert v(t) \bigr\Vert _{E}. $$</span></div></div><p> As usual, <span class="mathjax-tex">$\mathrm{AC}(I)$</span> denotes the space of absolutely continuous functions from <i>I</i> into <i>E</i>. We define the space </p><div id="Equb" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\mathrm{AC}^{1}(I):=\bigl\{ w:I\to E:w'\in\mathrm{AC}(I) \bigr\} , $$</span></div></div><p> where <span class="mathjax-tex">$w'(t)=\frac{\mathrm{d}}{\mathrm{d}t}w(t), t\in I$</span>. Let </p><div id="Equc" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\delta=t\frac{\mathrm{d}}{\mathrm{d}t},\qquad n=[q]+1, $$</span></div></div><p> where <span class="mathjax-tex">$[q]$</span> is the integer part of <span class="mathjax-tex">$q>0$</span>. Define the space </p><div id="Equd" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\mathrm{AC}_{\delta}^{n}:=\bigl\{ u:[1,T]\to E: \delta^{n-1}(u)\in\mathrm {AC}(I)\bigr\} . $$</span></div></div><p> Let <span class="mathjax-tex">$\gamma\in(0,1]$</span>. By <span class="mathjax-tex">$\mathrm{C}_{\gamma}(I), \mathrm{C}^{1}_{\gamma}(I)$</span>, and <span class="mathjax-tex">$\mathrm{C}_{\gamma,\ln}(I)$</span> we denote the weighted spaces of continuous functions defined by </p><div id="Eque" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\mathrm{C}_{\gamma}(I)=\bigl\{ w:(1,T]\to E: \in\bar{w}\in\mathrm{C}\bigr\} , $$</span></div></div><p> where <span class="mathjax-tex">$\bar{w}(t)= t^{1-\gamma}w(t), t\in(1,T]$</span>, with the norm </p><div id="Equf" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} &\Vert w \Vert _{\mathrm{C}_{\gamma}}:= \sup _{t\in I} \bigl\Vert \bar{w}(t) \bigr\Vert _{E}, \\ &\mathrm{C}^{1}_{\gamma}(I)=\bigl\{ w\in\mathrm{C}: w'\in\mathrm{C}_{\gamma}\bigr\} \end{aligned}$$ </span></div></div><p> with the norm </p><div id="Equg" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\Vert w \Vert _{\mathrm{C}^{1}_{\gamma}}:= \Vert w \Vert _{\infty}+ \bigl\Vert w' \bigr\Vert _{\mathrm{C}_{\gamma}}, $$</span></div></div><p> and </p><div id="Equh" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\mathrm{C}_{\gamma,\ln}(I)=\{w:I\to E:\widetilde{w}\in\mathrm{C}\}, $$</span></div></div><p> where <span class="mathjax-tex">$\widetilde{w}(t)= (\ln t)^{1-\gamma}w(t), t\in I$</span>, with the norm </p><div id="Equi" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\Vert w \Vert _{\mathrm{C}_{\gamma,\ln}}:= \sup _{t\in I} \bigl\Vert \widetilde{w}(t) \bigr\Vert _{E}. $$</span></div></div><p> We further denote <span class="mathjax-tex">$\|w\|_{\mathrm{C}_{\gamma,\ln}}$</span> by <span class="mathjax-tex">$\|w\|_{C}$</span>.</p><p>Define the weighted product space <span class="mathjax-tex">${\mathcal {C}}:=C_{\gamma,\ln}(I)\times C_{\gamma,\ln}(I)$</span> with the norm </p><div id="Equj" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl\Vert (w_{1},w_{2}) \bigr\Vert _{\mathcal{C}}:= \Vert w_{1} \Vert _{C}+ \Vert w_{2} \Vert _{C}. $$</span></div></div><p> In the same way, we can define the the weighted product space <span class="mathjax-tex">${\overline{C}}:=(C_{\gamma,\ln}(I))^{n}$</span> with the norm </p><div id="Equk" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl\Vert (w_{1},w_{2},\ldots,w_{n}) \bigr\Vert _{\overline{C}}:= \sum _{k=1}^{n} \Vert w_{k} \Vert _{C}. $$</span></div></div><p> Let <span class="mathjax-tex">$(E,w)=(E,\sigma(E,E^{*}))$</span> be the Banach space <i>E</i> with weak topology.</p> <h3 class="c-article__sub-heading" id="FPar1">Definition 2.1</h3> <p>A Banach space <i>X</i> is said to be weakly compactly generated (WCG) if it contains a weakly compact set whose linear span is dense in <i>X</i>.</p> <h3 class="c-article__sub-heading" id="FPar2">Definition 2.2</h3> <p>A function <span class="mathjax-tex">$h:E\rightarrow E$</span> is said to be weakly sequentially continuous if <i>h</i> takes each weakly convergent sequence in <i>E</i> to a weakly convergent sequence in <i>E</i> (i.e., for any <span class="mathjax-tex">$(u_{n})$</span> in <i>E</i> with <span class="mathjax-tex">$u_{n}\rightarrow u$</span> in <span class="mathjax-tex">$(E,w)$</span>, we have <span class="mathjax-tex">$h(u_{n})\rightarrow h(u)$</span> in <span class="mathjax-tex">$(E,w)$</span>).</p> <h3 class="c-article__sub-heading" id="FPar3">Definition 2.3</h3> <p>([<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 27" title="
					Pettis, B.J.: On integration in vector spaces. Trans. Am. Math. Soc. 44, 277–304 (1938)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR27" id="ref-link-section-d126833688e3540">27</a>])</p> <p>The function <span class="mathjax-tex">$u:I\rightarrow E$</span> is said to be Pettis integrable on <i>I</i> if and only if there is an element <span class="mathjax-tex">$u_{J}\in E$</span> corresponding to each <span class="mathjax-tex">$J\subset I$</span> such that <span class="mathjax-tex">$\phi(u_{J})=\int_{J} \phi(u(s))\,\mathrm{d}s$</span> for all <span class="mathjax-tex">$\phi\in E^{\ast}$</span>, where the integral on the right-hand side is assumed to exist in the Lebesgue sense (by definition <span class="mathjax-tex">$u_{J}=\int_{J}u(s)\,\mathrm{d}s)$</span>.</p> <p>Let <span class="mathjax-tex">$\mathrm{P}(I,E)$</span> be the space of all <i>E</i>-valued Pettis-integrable functions on <i>I</i>, and let <span class="mathjax-tex">$L^{1}(I,E)$</span> be the Banach space of Bochner-integrable measurable functions <span class="mathjax-tex">$u:I\to E$</span>. Define the class </p><div id="Equl" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\mathrm{P}_{1}(I,E)=\bigl\{ u\in P(I,E): \varphi(u)\in L^{1}(I,{\mathbb {R}}) \text{ for every } \varphi\in E^{*}\bigr\} . $$</span></div></div><p> The space <span class="mathjax-tex">$\mathrm{P}_{1}(I,E) $</span> is normed by </p><div id="Equm" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\Vert u \Vert _{\mathrm{P}_{1}}= \sup _{\varphi\in E^{*}, \Vert \varphi \Vert \leq1} \int _{1}^{T} \bigl\vert \varphi\bigl(u(x)\bigr) \bigr\vert \,\mathrm{d}\lambda x, $$</span></div></div><p> where <i>λ</i> is the Lebesgue measure on <i>I</i>.</p><p>The following result is due to Pettis [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 27" title="
					Pettis, B.J.: On integration in vector spaces. Trans. Am. Math. Soc. 44, 277–304 (1938)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR27" id="ref-link-section-d126833688e4181">27</a>, Thm. 3.4 and Cor. 3.41].</p> <h3 class="c-article__sub-heading" id="FPar4">Proposition 2.4</h3> <p>([<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 27" title="
					Pettis, B.J.: On integration in vector spaces. Trans. Am. Math. Soc. 44, 277–304 (1938)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR27" id="ref-link-section-d126833688e4191">27</a>])</p> <p><i>If</i> <span class="mathjax-tex">$u\in\mathrm{P}_{1}(I,E)$</span> <i>and</i> <i>h</i> <i>is a measurable and essentially bounded</i> <i>E</i>-<i>valued function</i>, <i>then</i> <span class="mathjax-tex">$uh\in\mathrm{P}_{1}(I,E)$</span>.</p> <p>In what follows, the symbol “∫” denotes the Pettis integral.</p><p>Now, we give some results and properties of fractional calculus.</p> <h3 class="c-article__sub-heading" id="FPar5">Definition 2.5</h3> <p>([<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="
					Abbas, S., Benchohra, M., N’Guérékata, G.M.: Topics in Fractional Differential Equations. Springer, New York (2012)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR3" id="ref-link-section-d126833688e4319">3</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="
					Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science B.V., Amsterdam (2006)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR22" id="ref-link-section-d126833688e4322">22</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title="
					Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Amsterdam (1987) Engl. Trans. from the Russian
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR30" id="ref-link-section-d126833688e4325">30</a>])</p> <p>The left-sided mixed Riemann–Liouville integral of order <span class="mathjax-tex">$r>0$</span> of a function <span class="mathjax-tex">$w\in L^{1}(I)$</span> is defined by </p><div id="Equn" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl( I_{1}^{r}w\bigr) (t) =\frac{1}{\Gamma(r)} \int_{1}^{t}( t-s) ^{r-1}w(s)\,\mathrm{d}s\quad \text{for a.e. } t\in I, $$</span></div></div><p> where Γ is the (Euler) gamma function defined by </p><div id="Equo" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\Gamma(\xi)= \int_{0}^{\infty}t^{\xi-1}e^{-t}\,{ \mathrm {d}}t,\quad \xi>0. $$</span></div></div> <p>Notice that, for all <span class="mathjax-tex">$r,r_{1},r_{2}>0$</span> and <span class="mathjax-tex">$w\in\mathrm{C}$</span>, we have <span class="mathjax-tex">$I_{0}^{r}w\in\mathrm{C}$</span> and </p><div id="Equp" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl(I_{1}^{r_{1}}I_{1}^{r_{2}}w\bigr) (t)= \bigl(I_{1}^{r_{1}+r_{2}}w\bigr) (t);\quad \mbox{for a.e. } t\in I. $$</span></div></div> <h3 class="c-article__sub-heading" id="FPar6">Definition 2.6</h3> <p>([<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="
					Abbas, S., Benchohra, M., N’Guérékata, G.M.: Topics in Fractional Differential Equations. Springer, New York (2012)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR3" id="ref-link-section-d126833688e4847">3</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="
					Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science B.V., Amsterdam (2006)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR22" id="ref-link-section-d126833688e4850">22</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title="
					Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Amsterdam (1987) Engl. Trans. from the Russian
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR30" id="ref-link-section-d126833688e4853">30</a>])</p> <p>The Riemann–Liouville fractional derivative of order <span class="mathjax-tex">$r>0$</span> of a function <span class="mathjax-tex">$w\in L^{1}(I)$</span> is defined by </p><div id="Equq" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \bigl(D^{r}_{1} w\bigr) (t)&=\biggl(\frac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}I_{1}^{n-r}w \biggr) (t) \\ &=\frac{1}{\Gamma(n-r)}\frac{\mathrm{d}^{n}}{\mathrm{d}t^{n}} \int _{1}^{t}(t-s)^{n-r-1}w(s)\,\mathrm{d}s\quad \text{for a.e. } t\in I, \end{aligned}$$ </span></div></div><p> where <span class="mathjax-tex">$n=[r]+1$</span>, and <span class="mathjax-tex">$[r]$</span> is the integer part of <i>r</i>.</p> <p>In particular, if <span class="mathjax-tex">$r\in(0,1]$</span>, then </p><div id="Equr" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \bigl(D^{r}_{1} w\bigr) (t)&= \biggl(\frac{\mathrm{d}}{\mathrm{d}t}I_{1}^{1-r}w \biggr) (t) \\ &=\frac{1}{\Gamma(1-r)}\frac{\mathrm{d}}{\mathrm{d}t} \int _{1}^{t}(t-s)^{-r}w(s)\,\mathrm{d}s\quad \text{for a.e. } t\in I. \end{aligned}$$ </span></div></div><p> Let <span class="mathjax-tex">$r\in(0,1], \gamma\in[0,1)$</span>, and <span class="mathjax-tex">$w\in\mathrm{C}_{1-\gamma }(I)$</span>. Then the following expression leads to the left inverse operator: </p><div id="Equs" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl(D_{1}^{r}I_{1}^{r}w\bigr) (t)=w(t) \quad\text{for all } t\in(1,T]. $$</span></div></div><p> Moreover, if <span class="mathjax-tex">$I_{1}^{1-r}w\in C^{1}_{1-\gamma}(I)$</span>, then the following composition is proved in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title="
					Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Amsterdam (1987) Engl. Trans. from the Russian
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR30" id="ref-link-section-d126833688e5742">30</a>]: </p><div id="Equt" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl(I_{1}^{r}D_{1}^{r}w\bigr) (t)=w(t)-\frac{(I_{1}^{1-r}w)(1^{+})}{\Gamma (r)}t^{r-1} \quad\text{for all } t\in(1,T]. $$</span></div></div> <h3 class="c-article__sub-heading" id="FPar7">Definition 2.7</h3> <p>([<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="
					Abbas, S., Benchohra, M., N’Guérékata, G.M.: Topics in Fractional Differential Equations. Springer, New York (2012)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR3" id="ref-link-section-d126833688e5916">3</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="
					Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science B.V., Amsterdam (2006)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR22" id="ref-link-section-d126833688e5919">22</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title="
					Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Amsterdam (1987) Engl. Trans. from the Russian
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR30" id="ref-link-section-d126833688e5922">30</a>])</p> <p>The Caputo fractional derivative of order <span class="mathjax-tex">$r>0$</span> of a function <span class="mathjax-tex">$w\in L^{1}(I)$</span> is defined by </p><div id="Equu" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \bigl({}^{c}D^{r}_{1}w\bigr) (t)&= \biggl(I_{1}^{n-r}\frac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}w \biggr) (t) \\ &=\frac{1}{\Gamma(n-r)} \int_{1}^{t}(t-s)^{n-r-1}\frac {\mathrm{d}^{n}}{\mathrm{d}s^{n}}w(s)\,{ \mathrm {d}}s \quad\text{for a.e. } t\in I. \end{aligned}$$ </span></div></div> <p>In particular, if <span class="mathjax-tex">$r\in(0,1]$</span>, then </p><div id="Equv" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \bigl({}^{c}D^{r}_{1}w\bigr) (t)&= \biggl(I_{1}^{1-r}\frac{\mathrm{d}}{\mathrm{d}t}w \biggr) (t) \\ &=\frac{1}{\Gamma(1-r)} \int_{1}^{t}(t-s)^{-r}\frac{d}{\mathrm {d}s}w(s)\,{ \mathrm {d}}s \quad\text{for a.e. } t\in I. \end{aligned}$$ </span></div></div><p> Let us recall some definitions and properties of Hadamard fractional integration and differentiation. We refer to [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="
					Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science B.V., Amsterdam (2006)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR22" id="ref-link-section-d126833688e6508">22</a>] for more details.</p> <h3 class="c-article__sub-heading" id="FPar8">Definition 2.8</h3> <p>([<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="
					Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science B.V., Amsterdam (2006)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR22" id="ref-link-section-d126833688e6519">22</a>])</p> <p>The Hadamard fractional integral of order <span class="mathjax-tex">$q>0$</span> for a function <span class="mathjax-tex">$g\in L^{1}(I,E)$</span> is defined as </p><div id="Equw" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl({}^{H}I_{1}^{q}g\bigr) (x)= \frac{1}{\Gamma(q)} \int_{1}^{x} \biggl(\ln\frac {x}{s} \biggr)^{q-1}\frac{g(s)}{s}\,\mathrm{d}s, $$</span></div></div><p> provided that the integral exists.</p> <h3 class="c-article__sub-heading" id="FPar9">Example 2.9</h3> <p>Let <span class="mathjax-tex">$0< q<1$</span>. Then </p><div id="Equx" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$^{H}I_{1}^{q} \ln t=\frac{1}{\Gamma(2+q)}(\ln t)^{1+q} \quad\text{for a.e. } t\in[0,e]. $$</span></div></div> <h3 class="c-article__sub-heading" id="FPar10">Remark 2.10</h3> <p>Let <span class="mathjax-tex">$g\in\mathrm{P}_{1}(I, E)$</span>. For every <span class="mathjax-tex">$\varphi\in E^{*}$</span>, we have </p><div id="Equy" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\varphi\bigl({}^{H}I_{1}^{q}g\bigr) (t)= \bigl({}^{H}I_{1}^{q}\varphi g\bigr) (t) \quad\text{for a.e. } t\in I. $$</span></div></div> <p>Similarly to the Riemann–Liouville fractional calculus, the Hadamard fractional derivative is defined in terms of the Hadamard fractional integral as follows.</p> <h3 class="c-article__sub-heading" id="FPar11">Definition 2.11</h3> <p>([<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="
					Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science B.V., Amsterdam (2006)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR22" id="ref-link-section-d126833688e7072">22</a>])</p> <p>The Hadamard fractional derivative of order <span class="mathjax-tex">$q>0$</span> applied to a function <span class="mathjax-tex">$w\in\mathrm{AC}_{\delta}^{n}$</span> is defined as </p><div id="Equz" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl({}^{H}D_{1}^{q}w\bigr) (x)= \delta^{n} \bigl({}^{H}I_{1}^{n-q}w\bigr) (x). $$</span></div></div> <p>In particular, if <span class="mathjax-tex">$q\in(0,1]$</span>, then </p><div id="Equaa" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl({}^{H}D_{1}^{q}w\bigr) (x)=\delta \bigl({}^{H}I_{1}^{1-q}w\bigr) (x). $$</span></div></div> <h3 class="c-article__sub-heading" id="FPar12">Example 2.12</h3> <p>Let <span class="mathjax-tex">$0< q<1$</span>. Then </p><div id="Equab" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$^{H}D_{1}^{q} \ln t=\frac{1}{\Gamma(2-q)}(\ln t)^{1-q} \quad\text{for a.e. } t\in[0,e]. $$</span></div></div> <p>It has been proved (see, e.g., Kilbas [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="
					Kilbas, A.A.: Hadamard-type fractional calculus. J. Korean Math. Soc. 38(6), 1191–1204 (2001)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR21" id="ref-link-section-d126833688e7525">21</a>, Thm. 4.8]) that, in the space <span class="mathjax-tex">$L^{1}(I,E)$</span>, the Hadamard fractional derivative is the left-inverse operator to the Hadamard fractional integral, that is, </p><div id="Equac" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl({}^{H}D_{1}^{q}\bigr) \bigl({}^{H}I_{1}^{q}w \bigr) (x)=w(x). $$</span></div></div><p> From [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="
					Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science B.V., Amsterdam (2006)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR22" id="ref-link-section-d126833688e7657">22</a>, Thm. 2.3] we have </p><div id="Equad" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl({}^{H}I_{1}^{q}\bigr) \bigl({}^{H}D_{1}^{q}w \bigr) (x)=w(x)-\frac {({}^{H}I_{1}^{1-q}w)(1)}{\Gamma(q)}(\ln x)^{q-1}. $$</span></div></div><p>Similarly to the Hadamard fractional calculus, the Caputo–Hadamard fractional derivative is defined as follows.</p> <h3 class="c-article__sub-heading" id="FPar13">Definition 2.13</h3> <p>The Caputo–Hadamard fractional derivative of order <span class="mathjax-tex">$q>0$</span> applied to a function <span class="mathjax-tex">$w\in\mathrm{AC}_{\delta}^{n}$</span> is defined as </p><div id="Equae" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl({}^{Hc}D_{1}^{q}w\bigr) (x)= \bigl({}^{H}I_{1}^{n-q}\delta^{n}w\bigr) (x). $$</span></div></div> <p>In particular, if <span class="mathjax-tex">$q\in(0,1]$</span>, then </p><div id="Equaf" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl({}^{Hc}D_{1}^{q}w\bigr) (x)= \bigl({}^{H}I_{1}^{1-q}\delta w\bigr) (x). $$</span></div></div><p>Hilfer [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="
					Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR18" id="ref-link-section-d126833688e8159">18</a>] studied applications of the generalized fractional operator having the Riemann–Liouville and the Caputo derivatives as particular cases (see also [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 20" title="
					Kamocki, R., Obczyński, C.: On fractional Cauchy-type problems containing Hilfer’s derivative. Electron. J. Qual. Theory Differ. Equ. 2016, 50 (2016)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR20" id="ref-link-section-d126833688e8162">20</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 32" title="
					Tomovski, Ž., Hilfer, R., Srivastava, H.M.: Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions. Integral Transforms Spec. Funct. 21(11), 797–814 (2010)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR32" id="ref-link-section-d126833688e8165">32</a>]).</p> <h3 class="c-article__sub-heading" id="FPar14">Definition 2.14</h3> <p>Let <span class="mathjax-tex">$\alpha\in(0,1), \beta\in[0,1], w\in L^{1}(I)$</span> and <span class="mathjax-tex">$I_{1}^{(1-\alpha)(1-\beta)}w\in\mathrm{AC}^{1}(I)$</span>. The Hilfer fractional derivative of order <i>α</i> and type <i>β</i> of <i>w</i> is defined as </p><div id="Equ3" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \bigl(D_{1}^{\alpha,\beta}w\bigr) (t)= \biggl(I_{1}^{\beta(1-\alpha)}\frac{d}{\mathrm {d}t} I_{1}^{(1-\alpha)(1-\beta)}w \biggr) (t) \quad\text{for a.e. } t\in I. $$</span></div><div class="c-article-equation__number"> (3) </div></div> <h3 class="c-article__sub-heading" id="FPar15">Properties</h3> <p>Let <span class="mathjax-tex">$\alpha\in(0,1), \beta\in[0,1], \gamma =\alpha+\beta-\alpha\beta$</span>, and <span class="mathjax-tex">$w\in L^{1}(I)$</span>. </p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">1.</span> <p>The operator <span class="mathjax-tex">$(D_{1}^{\alpha,\beta}w)(t)$</span> can be written as </p><div id="Equag" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl(D_{1}^{\alpha,\beta}w\bigr) (t)= \biggl(I_{1}^{\beta(1-\alpha)} \frac{d}{\mathrm {d}t} I_{1}^{1-\gamma}w \biggr) (t)= \bigl(I_{1}^{\beta(1-\alpha)} D_{1}^{\gamma}w \bigr) (t)\quad \text{for a.e. } t\in I. $$</span></div></div><p> Moreover, the parameter <i>γ</i> satisfies </p><div id="Equah" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\gamma\in(0,1],\qquad \gamma\geq\alpha,\qquad \gamma>\beta,\qquad 1-\gamma < 1-\beta(1-\alpha). $$</span></div></div> </li> <li> <span class="u-custom-list-number">2.</span> <p>For <span class="mathjax-tex">$\beta=0$</span>, generalization (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-018-1787-4#Equ3">3</a>) coincides with the Riemann–Liouville derivative and for <span class="mathjax-tex">$\beta=1$</span>, with the Caputo derivative: </p><div id="Equai" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$D_{1}^{\alpha,0}=D_{1}^{\alpha}\quad \mbox{and}\quad D_{1}^{\alpha,1}= ^{c}D_{1}^{\alpha}. $$</span></div></div> </li> <li> <span class="u-custom-list-number">3.</span> <p>If <span class="mathjax-tex">$D_{1}^{\beta(1-\alpha)}w$</span> exists and is in <span class="mathjax-tex">$L^{1}(I)$</span>, then </p><div id="Equaj" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl(D_{1}^{\alpha,\beta}I_{1}^{\alpha}w\bigr) (t)= \bigl(I_{1}^{\beta(1-\alpha )}D_{1}^{\beta(1-\alpha)}w\bigr) (t)\quad \text{for a.e. } t\in I. $$</span></div></div><p> Furthermore, if <span class="mathjax-tex">$w\in C_{\gamma}(I)$</span> and <span class="mathjax-tex">$I_{1}^{1-\beta(1-\alpha )}w\in C^{1}_{\gamma}(I)$</span>, then </p><div id="Equak" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl(D_{1}^{\alpha,\beta}I_{1}^{\alpha}w\bigr) (t)=w(t) \quad\text{for a.e. } t\in I. $$</span></div></div> </li> <li> <span class="u-custom-list-number">4.</span> <p>If <span class="mathjax-tex">$D_{1}^{\gamma}w$</span> exists and is in <span class="mathjax-tex">$L^{1}(I)$</span>, then </p><div id="Equal" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl(I_{1}^{\alpha}D_{1}^{\alpha,\beta}w\bigr) (t)= \bigl(I_{1}^{\gamma}D_{1}^{\gamma}w\bigr) (t) =w(t)-\frac{I_{1}^{1-\gamma}(1^{+})}{\Gamma(\gamma)}t^{\gamma-1} \quad\text{for a.e. } t\in I. $$</span></div></div> </li> </ol> <p>Based on the Hadamard fractional integral, the Hilfer–Hadamard fractional derivative (introduced for the first time in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 28" title="
					Qassim, M.D., Furati, K.M., Tatar, N.-E.: On a differential equation involving Hilfer–Hadamard fractional derivative. Abstr. Appl. Anal. 2012, Article ID 391062 (2012)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR28" id="ref-link-section-d126833688e9802">28</a>]) is defined as follows.</p> <h3 class="c-article__sub-heading" id="FPar16">Definition 2.15</h3> <p>Let <span class="mathjax-tex">$\alpha\in(0,1), \beta\in[0,1]$</span>, <span class="mathjax-tex">$\gamma=\alpha+\beta -\alpha\beta, w\in L^{1}(I)$</span>, and <span class="mathjax-tex">${}^{H}I_{1}^{(1-\alpha)(1-\beta)}w\in\mathrm{AC}^{1}(I)$</span>. The Hilfer–Hadamard fractional derivative of order <i>α</i> and type <i>β</i> applied to a function <i>w</i> is defined as </p><div id="Equ4" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \bigl({}^{H}D_{1}^{\alpha,\beta}w \bigr) (t)&= \bigl({}^{H}I_{1}^{\beta(1-\alpha )} \bigl({}^{H}D_{1}^{\gamma}w\bigr) \bigr) (t) \\ &= \bigl({}^{H}I_{1}^{\beta(1-\alpha)}\delta\bigl({}^{H}I_{1}^{1-\gamma }w \bigr) \bigr) (t)\quad \text{for a.e. } t\in I. \end{aligned}$$ </span></div><div class="c-article-equation__number"> (4) </div></div> <p>This new fractional derivative (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-018-1787-4#Equ4">4</a>) may be viewed as interpolation of the Hadamard and Caputo–Hadamard fractional derivatives. Indeed, for <span class="mathjax-tex">$\beta=0$</span>, this derivative reduces to the Hadamard fractional derivative, and, for <span class="mathjax-tex">$\beta=1$</span>, we recover the Caputo–Hadamard fractional derivative: </p><div id="Equam" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$^{H}D_{1}^{\alpha,0}= ^{H}D_{1}^{\alpha}\quad \mbox{and}\quad ^{H}D_{1}^{\alpha,1}= ^{Hc}D_{1}^{\alpha}. $$</span></div></div><p>From [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 29" title="
					Qassim, M.D., Tatar, N.E.: Well-posedness and stability for a differential problem with Hilfer–Hadamard fractional derivative. Abstr. Appl. Anal. 2013, Article ID 605029 (2013)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR29" id="ref-link-section-d126833688e10432">29</a>, Thm. 21] we have the following lemma.</p> <h3 class="c-article__sub-heading" id="FPar17">Lemma 2.16</h3> <p><i>Let</i> <span class="mathjax-tex">$f_{i}:I\times E^{4}\rightarrow E, i=1,2$</span>, <i>be such that</i> <span class="mathjax-tex">$f_{i}(\cdot ,u,v,\bar{u},\bar{v})\in\mathrm{C}_{\gamma,\ln}(I)$</span> <i>for any</i> <span class="mathjax-tex">$u,v,\bar{u},\bar{v}\in\mathrm{C}_{\gamma,\ln}(I)$</span>. <i>Then system</i> (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-018-1787-4#Equ1">1</a>)<i>–</i>(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-018-1787-4#Equ2">2</a>) <i>is equivalent to the problem of obtaining the solution of the coupled system</i> </p><div id="Equan" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\textstyle\begin{cases} g_{1}(t)=f_{1} (t,\frac{\phi_{1}}{\Gamma(\gamma)}(\ln t)^{\gamma -1}+({}^{H}I_{1}^{\alpha}g_{1})(t), \frac{\phi_{2}}{\Gamma(\gamma)}(\ln t)^{\gamma-1}+({}^{H}I_{1}^{\alpha }g_{2})(t),g_{1}(t),g_{2}(t) ),\\ g_{2}(t)=f_{2} (t,\frac{\phi_{1}}{\Gamma(\gamma)}(\ln t)^{\gamma -1}+({}^{H}I_{1}^{\alpha}g_{1})(t), \frac{\phi_{2}}{\Gamma(\gamma)}(\ln t)^{\gamma-1}+({}^{H}I_{1}^{\alpha }g_{2})(t),g_{1}(t),g_{2}(t) ), \end{cases} $$</span></div></div><p> <i>and if</i> <span class="mathjax-tex">$g_{i}(\cdot)\in\mathrm{C}_{\gamma,\ln}$</span> <i>are the solutions of this system</i>, <i>then</i> </p><div id="Equao" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\textstyle\begin{cases} u_{1}(t)=\frac{\phi_{1}}{\Gamma(\gamma)}(\ln t)^{\gamma -1}+({}^{H}I_{1}^{\alpha}g_{1})(t),\\ u_{2}(t)=\frac{\phi_{2}}{\Gamma(\gamma)}(\ln t)^{\gamma -1}+({}^{H}I_{1}^{\alpha}g_{2})(t). \end{cases} $$</span></div></div> <h3 class="c-article__sub-heading" id="FPar18">Definition 2.17</h3> <p>([<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="
					De Blasi, F.S.: On the property of the unit sphere in a Banach space. Bull. Math. Soc. Sci. Math. Roum. 21, 259–262 (1977)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR14" id="ref-link-section-d126833688e11549">14</a>])</p> <p>Let <i>E</i> be a Banach space, let <span class="mathjax-tex">$\Omega_{E}$</span> be the set of bounded subsets of <i>E</i>, and let <span class="mathjax-tex">$B_{1}$</span> be the unit ball of <i>E</i>. The De Blasi measure of weak noncompactness is the map <span class="mathjax-tex">$\mu:\Omega_{E}\rightarrow[0, \infty)$</span> defined by </p><div id="Equap" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\mu(X)=\inf\{\varepsilon>0: \text{there exists a weakly compact set } \Omega \subset E \text{ such that } X\subset\varepsilon B_{1}+ \Omega\}. $$</span></div></div> <p>The De Blasi measure of weak noncompactness satisfies the following properties: </p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(a)</span> <p><span class="mathjax-tex">$A\subset B\Rightarrow\mu(A)\leq\mu(B)$</span>,</p> </li> <li> <span class="u-custom-list-number">(b)</span> <p><span class="mathjax-tex">$\mu(A)= 0 \Leftrightarrow A $</span> is weakly relatively compact,</p> </li> <li> <span class="u-custom-list-number">(c)</span> <p><span class="mathjax-tex">$\mu(A\cup B)=\max\{\mu(A), \mu(B)\}$</span>,</p> </li> <li> <span class="u-custom-list-number">(d)</span> <p><span class="mathjax-tex">$\mu(\overline{A}^{\omega})=\mu(A)$</span>, where <span class="mathjax-tex">$\overline{A}^{\omega}$</span> denotes the weak closure of <i>A</i>,</p> </li> <li> <span class="u-custom-list-number">(e)</span> <p><span class="mathjax-tex">$\mu(A+B)\leq\mu(A)+\mu(B)$</span>,</p> </li> <li> <span class="u-custom-list-number">(f)</span> <p><span class="mathjax-tex">$\mu(\lambda A)=|\lambda| \mu(A)$</span>,</p> </li> <li> <span class="u-custom-list-number">(g)</span> <p><span class="mathjax-tex">$\mu(\operatorname{conv}(A))=\mu(A)$</span>,</p> </li> <li> <span class="u-custom-list-number">(h)</span> <p><span class="mathjax-tex">$\mu(\bigcup_{|\lambda|\leq h} \lambda A)= h \mu(A)$</span>.</p> </li> </ol><p>The next result follows directly from the Hahn–Banach theorem.</p> <h3 class="c-article__sub-heading" id="FPar19">Proposition 2.18</h3> <p><i>If</i> <i>E</i> <i>is a normed space and</i> <span class="mathjax-tex">$x_{0}\in E-\{0\}$</span>, <i>then there exists</i> <span class="mathjax-tex">$\varphi\in E^{\ast}$</span> <i>with</i> <span class="mathjax-tex">$\|\varphi\|=1$</span> <i>and</i> <span class="mathjax-tex">$\varphi(x_{0})=\|x_{0}\|$</span>.</p> <p>For a given set <i>V</i> of functions <span class="mathjax-tex">$v: I\to E$</span>, let us denote </p><div id="Equaq" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$V(t)=\bigl\{ v(t): v\in V\bigr\} ; \quad t\in I \quad\textit{and}\quad V(I)=\bigl\{ v(t):v\in V, t\in I\bigr\} . $$</span></div></div> <h3 class="c-article__sub-heading" id="FPar20">Lemma 2.19</h3> <p>([<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="
					Guo, D., Lakshmikantham, V., Liu, X.: Nonlinear Integral Equations in Abstract Spaces. Kluwer Academic, Dordrecht (1996)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR17" id="ref-link-section-d126833688e12628">17</a>] )</p> <p><i>Let</i> <span class="mathjax-tex">$H\subset C$</span> <i>be a bounded equicontinuous subset</i>. <i>Then the function</i> <span class="mathjax-tex">$t\to\mu(H(t))$</span> <i>is continuous on</i> <i>I</i>, </p><div id="Equar" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\mu_{C}(H)= \max _{t\in I}\mu\bigl(H(t)\bigr), $$</span></div></div><p> <i>and</i> </p><div id="Equas" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\mu \biggl( \int_{I}u(s)\,\mathrm{d}s \biggr)\leq \int_{I}\mu\bigl(H(s)\bigr)\,\mathrm{d}s, $$</span></div></div><p> <i>where</i> <span class="mathjax-tex">$H(t)=\{u(t):u\in H\}, t\in I$</span>, <i>and</i> <span class="mathjax-tex">$\mu_{C}$</span> <i>is the De Blasi measure of weak noncompactness defined on the bounded sets of</i> <i>C</i>.</p> <p>For our purpose, we will need the following fixed point theorem.</p> <h3 class="c-article__sub-heading" id="FPar21">Theorem 2.20</h3> <p>([<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 25" title="
					O’Regan, D.: Fixed point theory for weakly sequentially continuous mapping. Math. Comput. Model. 27(5), 1–14 (1998)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR25" id="ref-link-section-d126833688e12996">25</a>])</p> <p><i>Let</i> <i>Q</i> <i>be a nonempty</i>, <i>closed</i>, <i>convex</i>, <i>and equicontinuous subset of a metrizable locally convex vector space</i> <span class="mathjax-tex">$C(I,E)$</span> <i>such that</i> <span class="mathjax-tex">$0\in{Q}$</span>. <i>Suppose</i> <span class="mathjax-tex">$T:Q\rightarrow Q$</span> <i>is weakly sequentially continuous</i>. <i>If the implication</i> </p><div id="Equ5" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \overline{V}=\overline{\operatorname{conv}}\bigl(\{0\}\cup T(V)\bigr) \Rightarrow V \quad\textit{is relatively weakly compact} $$</span></div><div class="c-article-equation__number"> (5) </div></div><p> <i>holds for every subset</i> <span class="mathjax-tex">$V\subset Q$</span>, <i>then the operator</i> <i>T</i> <i>has a fixed point</i>.</p> </div></div></section><section data-title="Existence of weak solutions"><div class="c-article-section" id="Sec3-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec3"><span class="c-article-section__title-number">3 </span>Existence of weak solutions</h2><div class="c-article-section__content" id="Sec3-content"><p>Let us start by the definition of a weak solution of problem (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-018-1787-4#Equ1">1</a>).</p> <h3 class="c-article__sub-heading" id="FPar22">Definition 3.1</h3> <p>By a weak solution of the coupled system (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-018-1787-4#Equ1">1</a>)–(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-018-1787-4#Equ2">2</a>) we mean a coupled measurable functions <span class="mathjax-tex">$(u_{1},u_{2})\in{\mathcal{C}}$</span> such that <span class="mathjax-tex">$({}^{H}I_{1}^{1-\gamma }u_{i})(1^{+})=\phi_{i}, i=1,2$</span>, and the equations <span class="mathjax-tex">$({}^{H}D_{1}^{\alpha,\beta }u_{i})(t)=f_{i}(t,u_{1}(t),u_{2}(t),({}^{H}D_{1}^{\alpha,\beta }u_{1})(t),({}^{H}D_{1}^{\alpha,\beta}u_{2})(t))$</span> are satisfied on <i>I</i>.</p> <p>We further will use the following hypotheses. </p><dl class="c-abbreviation_list"><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn> <span class="mathjax-tex">$(H_{1})$</span> :</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>The functions <span class="mathjax-tex">$v\to f_{i}(t,v,w,\bar{v},\bar{w}), w\to f_{i}(t,v,w,\bar{v},\bar{w}), \bar{v}\to f_{i}(t,v,w,\bar{v},\bar{w})$</span>, and <span class="mathjax-tex">$\bar{w}\to f_{i}(t,v,w,\bar{v},\bar{w}), i=1,2$</span>, are weakly sequentially continuous for a.e. <span class="mathjax-tex">$t\in I$</span>,</p> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn> <span class="mathjax-tex">$(H_{2})$</span> :</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>For all <span class="mathjax-tex">$v,w,\bar{v},\bar{w}\in E$</span>, the functions <span class="mathjax-tex">$t\to f_{i}(t,v,w,\bar{v},\bar{w}), i=1,2$</span>, are Pettis integrable a.e. on <i>I</i>,</p> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn> <span class="mathjax-tex">$(H_{3})$</span> :</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>There exist <span class="mathjax-tex">$p_{i},q_{i}\in C(I,[0,\infty))$</span> such that, for all <span class="mathjax-tex">$\varphi\in E^{*}$</span>, </p><div id="Equat" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl\vert \varphi\bigl(f_{i}(t,u,v,\bar{u},\bar{v})\bigr) \bigr\vert \leq\frac{p_{i}(t) \Vert u \Vert _{E}+q_{i}(t) \Vert v \Vert _{E}}{1+ \Vert \varphi \Vert + \Vert u \Vert _{E}+ \Vert v \Vert _{E}+ \Vert \bar{u} \Vert _{E}+ \Vert \bar{v} \Vert _{E}} $$</span></div></div><p> <span class="mathjax-tex">$\text{ for a.e. } t\in I \text{ and all } u,v,\bar{u},\bar{v}\in E$</span>,</p> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn> <span class="mathjax-tex">$(H_{4})$</span> :</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>For all bounded measurable sets <span class="mathjax-tex">$B_{i}\subset E, i=1,2$</span>, and all <span class="mathjax-tex">$t\in I$</span>, we have </p><div id="Equau" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\mu\bigl(f_{1}\bigl(t,B_{1},B_{2},^{H}D_{1}^{\alpha,\beta}B_{1},^{H}D_{1}^{\alpha,\beta }B_{2} \bigr),0\bigr)\leq p_{1}(t)\mu(B_{1})+q_{1}(t) \mu(B_{2}) $$</span></div></div><p> and </p><div id="Equav" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\mu\bigl(0,f_{2}\bigl(t,B_{1},B_{2},^{H}D_{1}^{\alpha,\beta}B_{1},^{H}D_{1}^{\alpha,\beta }B_{2} \bigr)\bigr)\leq p_{2}(t)\mu(B_{1})+q_{2}(t) \mu(B_{2}), $$</span></div></div><p> where <span class="mathjax-tex">$^{H}D_{1}^{\alpha,\beta}B_{i}=\{^{H}D_{1}^{\alpha,\beta}w:w\in B_{i}\}, i=1,2$</span>.</p> </dd></dl><p> Set </p><div id="Equaw" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$p_{i}^{*}= \sup _{t\in I}p_{i}(t) \quad\mbox{and}\quad q_{i}^{*}= \sup _{t\in I}q_{i}(t),\quad i=1,2. $$</span></div></div> <h3 class="c-article__sub-heading" id="FPar23">Theorem 3.2</h3> <p><i>Assume that the hypotheses</i> <span class="mathjax-tex">$(H_{1})$</span><i>–</i><span class="mathjax-tex">$(H_{4}) $</span> <i>hold</i>. <i>If</i> </p><div id="Equ6" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ L:=\frac{(p_{1}^{*}+p_{2}^{*}+q_{1}^{*}+q_{2}^{*})(\ln T)^{\alpha}}{\Gamma (1+\alpha)}< 1, $$</span></div><div class="c-article-equation__number"> (6) </div></div><p> <i>then the coupled system</i> (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-018-1787-4#Equ1">1</a>)<i>–</i>(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-018-1787-4#Equ2">2</a>) <i>has at least one weak solution defined on</i> <i>I</i>.</p> <h3 class="c-article__sub-heading" id="FPar24">Proof</h3> <p>Consider the operators <span class="mathjax-tex">$N_{i}:C_{\gamma,\ln }\rightarrow C_{\gamma,\ln}, i=1,2$</span>, defined by </p><div id="Equax" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$(N_{i}u_{i}) (t)=\frac{\phi_{i}}{\Gamma(\gamma)}(\ln t)^{\gamma -1}+ \bigl({}^{H}I_{1}^{\alpha}g_{i}\bigr) (t), $$</span></div></div><p> where <span class="mathjax-tex">$g_{i}\in C_{\gamma,\ln}, i=1,2$</span>, are defined as </p><div id="Equay" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$g_{i}(t)=f_{i} \biggl(t,\frac{\phi_{1}}{\Gamma(\gamma)}(\ln t)^{\gamma -1}+\bigl({}^{H}I_{1}^{\alpha}g_{1} \bigr) (t), \frac{\phi_{2}}{\Gamma(\gamma)}(\ln t)^{\gamma-1}+\bigl({}^{H}I_{1}^{\alpha }g_{2} \bigr) (t),g_{1}(t),g_{2}(t) \biggr). $$</span></div></div><p> Consider the operator <span class="mathjax-tex">$N:{\mathcal{C}}\to{\mathcal{C}}$</span> such that, for any <span class="mathjax-tex">$(u_{1},u_{2})\in{\mathcal{C}}$</span>, </p><div id="Equ7" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \bigl(N(u_{1},u_{2})\bigr) (t)= \bigl((N_{1}u_{1}) (t),(N_{2}u_{2}) (t) \bigr). $$</span></div><div class="c-article-equation__number"> (7) </div></div><p> First, notice that the hypotheses imply that, for each <span class="mathjax-tex">$g_{i}\in C_{\gamma,\ln}, i=1,2$</span>, the function </p><div id="Equaz" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$t\mapsto \biggl(\ln\frac{t}{s} \biggr)^{\alpha-1}g_{i}(s) $$</span></div></div><p> is Pettis integrable over <i>I</i>, and </p><div id="Equba" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$t\mapsto f_{i} \biggl(t,\frac{\phi_{1}}{\Gamma(\gamma)}(\ln t)^{\gamma -1}+ \bigl({}^{H}I_{1}^{\alpha}g_{1}\bigr) (t), \frac{\phi_{2}}{\Gamma(\gamma)}(\ln t)^{\gamma-1}+\bigl({}^{H}I_{1}^{\alpha }g_{2} \bigr) (t),g_{1}(t),g_{2}(t) \biggr) $$</span></div></div><p> <span class="mathjax-tex">$\text{for a.e. } t\in I$</span> is Pettis integrable. Thus, the operator <i>N</i> is well defined. Let <span class="mathjax-tex">$R>0$</span> be such that <span class="mathjax-tex">$R>L_{1}+L_{2}$</span>, where </p><div id="Equbb" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$L_{i}:=\frac{(p_{i}^{*}+q_{i}^{*})(\ln T)^{1-\gamma+\alpha}}{\Gamma (1+\alpha)},\quad i=1,2, $$</span></div></div><p> and consider the set </p><div id="Equbc" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} Q={}& \biggl\{ (u_{1},u_{2})\in{ \mathcal{C}}: \bigl\Vert (u_{1},u_{2}) \bigr\Vert _{\mathcal{C}}\leq R \text{ and } \bigl\Vert (\ln t_{2})^{1-\gamma}u_{i}(t_{2})-( \ln t_{1})^{1-\gamma}u_{i}(t_{1}) \bigr\Vert _{E} \\ \leq{}& L_{i} \biggl(\ln\frac{t_{2}}{t_{1}} \biggr)^{\alpha} \\ &{} +\frac{p_{i}^{*}+q_{i}^{*}}{\Gamma(\alpha)} \int_{1}^{t_{1}} \biggl\vert (\ln t_{2})^{1-\gamma} \biggl(\ln\frac{t_{2}}{s} \biggr)^{\alpha-1} -(\ln t_{1})^{1-\gamma} \biggl(\ln \frac{t_{1}}{s} \biggr)^{\alpha -1} \biggr\vert \,\mathrm{d}s, i=1,2 \biggr\} . \end{aligned}$$ </span></div></div><p> Clearly, the subset <i>Q</i> is closed, convex, and equicontinuous. We will show that the operator <i>N</i> satisfies all the assumptions of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-018-1787-4#FPar21">2.20</a>. The proof will be given in several steps.</p> <p><i>Step</i> 1. <i>N</i> <i>maps</i> <i>Q</i> <i>into itself.</i> Let <span class="mathjax-tex">$(u_{1},u_{2})\in Q, t\in I$</span>, and assume that <span class="mathjax-tex">$(N(u_{1},u_{2}))(t)\neq(0.0)$</span>. Then there exists <span class="mathjax-tex">$\varphi\in E^{*}$</span> such that <span class="mathjax-tex">$\|(\ln t)^{1-\gamma}(N_{i}u_{i})(t)\|_{E}=|\varphi((\ln t)^{1-\gamma }(N_{i}u_{i})(t))|, i=1,2$</span>. Thus, for any <span class="mathjax-tex">$i\in\{1,2\}$</span>, we have </p><div id="Equbd" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl\Vert (\ln t)^{1-\gamma}(N_{i}u_{i}) (t) \bigr\Vert _{E}=\varphi \biggl(\frac{\phi _{i}}{\Gamma(\gamma)}+\frac{(\ln t)^{1-\gamma}}{\Gamma(\alpha)} \int_{1}^{t} \biggl(\ln\frac{t}{s} \biggr)^{\alpha-1}g_{i}(s)\frac {\mathrm{d}s}{s} \biggr), $$</span></div></div><p> where <span class="mathjax-tex">$g_{i}\in C_{\gamma,\ln}$</span> are defined as </p><div id="Eqube" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$g_{i}(t)=f_{i} \biggl(t,\frac{\phi_{1}}{\Gamma(\gamma)}(\ln t)^{\gamma -1}+\bigl({}^{H}I_{1}^{\alpha}g_{1} \bigr) (t), \frac{\phi_{2}}{\Gamma(\gamma)}(\ln t)^{\gamma-1}+\bigl({}^{H}I_{1}^{\alpha }g_{2} \bigr) (t),g_{1}(t),g_{2}(t) \biggr). $$</span></div></div><p> Then from <span class="mathjax-tex">$(H_{3})$</span> we get </p><div id="Equbf" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl\vert \varphi\bigl(g_{i}(t)\bigr) \bigr\vert \leq p_{i}^{*}+q_{i}^{*}. $$</span></div></div><p> Thus </p><div id="Equbg" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} &\bigl\Vert (\ln t)^{1-\gamma}(N_{i}u_{i}) (t) \bigr\Vert _{E}\\ &\quad\leq\frac{(\ln t)^{1-\gamma }}{\Gamma(\alpha)} \int_{1}^{t} \biggl(\ln\frac{t}{s} \biggr)^{\alpha-1} \bigl\vert \varphi \bigl(g_{i}(s)\bigr) \bigr\vert \frac{\mathrm{d}s}{s} \\ &\quad\leq \frac{(p_{i}^{*}+q_{i}^{*})(\ln T)^{1-\gamma}}{\Gamma(\alpha)} \int_{1}^{t} \biggl(\ln\frac{t}{s} \biggr)^{\alpha-1}\frac{\mathrm {d}s}{s} \\ &\quad\leq \frac{(p_{i}^{*}+q_{i}^{*})(\ln T)^{1-\gamma+\alpha}}{\Gamma (1+\alpha)} \\ &\quad= L_{i}. \end{aligned}$$ </span></div></div><p> Hence we get </p><div id="Equbh" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl\Vert N(u_{1},u_{1}) \bigr\Vert _{\mathcal{C}}\leq L_{1}+L_{2}< R. $$</span></div></div><p> Next, let <span class="mathjax-tex">$t_{1},t_{2}\in I$</span> be such that <span class="mathjax-tex">$t_{1}< t_{2}$</span>, and let <span class="mathjax-tex">$u\in Q$</span> be such that </p><div id="Equbi" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$(\ln t_{2})^{1-\gamma}(N_{i}u_{i}) (t_{2})-(\ln t_{1})^{1-\gamma }(N_{i}u_{i}) (t_{1})\neq0. $$</span></div></div><p> Then there exists <span class="mathjax-tex">$\varphi\in E^{*}$</span> such that </p><div id="Equbj" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} &\bigl\Vert (\ln t_{2})^{1-\gamma}(N_{i}u_{i}) (t_{2})-(\ln t_{1})^{1-\gamma }(N_{i}u_{i}) (t_{1}) \bigr\Vert _{E}\\ &\quad= \bigl\vert \varphi\bigl((\ln t_{2})^{1-\gamma}(N_{i}u_{i}) (t_{2})-(\ln t_{1})^{1-\gamma }(N_{i}u_{i}) (t_{1})\bigr) \bigr\vert \end{aligned}$$ </span></div></div><p> and <span class="mathjax-tex">$\|\varphi\|=1$</span>. Then, for any <span class="mathjax-tex">$i\in\{1,2\}$</span>, we have </p><div id="Equbk" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} & \bigl\Vert (\ln t_{2})^{1-\gamma}(N_{i}u_{i}) (t_{2})-(\ln t_{1})^{1-\gamma }(N_{i}u_{i}) (t_{1}) \bigr\Vert _{E} \\ &\quad = \bigl\vert \varphi\bigl((\ln t_{2})^{1-\gamma}(N_{i}u_{i}) (t_{2})-(\ln t_{1})^{1-\gamma }(N_{i}u_{i}) (t_{1})\bigr) \bigr\vert \\ &\quad\leq\varphi \biggl((\ln t_{2})^{1-\gamma} \int_{1}^{t_{2}} \biggl(\ln \frac{t_{2}}{s} \biggr)^{\alpha-1}\frac{g_{i}(s)}{s\Gamma(\alpha)}\,\mathrm{d}s- (\ln t_{1})^{1-\gamma} \int_{1}^{t_{1}} \biggl(\ln\frac{t_{1}}{s} \biggr)^{\alpha-1}\frac{g_{i}(s)}{s\Gamma(\alpha)}\,\mathrm{d}s \biggr), \end{aligned}$$ </span></div></div><p> where <span class="mathjax-tex">$g_{i}\in\mathrm{C}_{\gamma,\ln}$</span> are defined as </p><div id="Equbl" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$g_{i}(t)=f_{i} \biggl(t,\frac{\phi_{1}}{\Gamma(\gamma)}(\ln t)^{\gamma -1}+\bigl({}^{H}I_{1}^{\alpha}g_{1} \bigr) (t),\frac{\phi_{1}}{\Gamma(\gamma)}(\ln t)^{\gamma-1}+\bigl({}^{H}I_{1}^{\alpha}g_{2} \bigr) (t),g_{1}(t),g_{2}(t) \biggr). $$</span></div></div><p> Then </p><div id="Equbm" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} & \bigl\Vert (\ln t_{2})^{1-\gamma}(N_{i}u_{i}) (t_{2})-(\ln t_{1})^{1-\gamma }(N_{i}u_{i}) (t_{1}) \bigr\Vert _{E} \\ &\quad\leq(\ln t_{2})^{1-\gamma} \int_{t_{1}}^{t_{2}} \biggl(\ln\frac {t_{2}}{s} \biggr)^{\alpha-1}\frac{ \vert \varphi(g_{i}(s)) \vert }{s\Gamma(\alpha )}\,\mathrm{d}s \\ &\qquad{} + \int_{1}^{t_{1}} \biggl\vert (\ln t_{2})^{1-\gamma} \biggl(\ln\frac {t_{2}}{s} \biggr)^{\alpha-1} -(\ln t_{1})^{1-\gamma} \biggl(\ln \frac{t_{1}}{s} \biggr)^{\alpha -1} \biggr\vert \frac{ \vert \varphi(g_{i}(s)) \vert }{s\Gamma(\alpha)}\, \mathrm{d}s \\ &\quad\leq(\ln t_{2})^{1-\gamma} \int_{t_{1}}^{t_{2}} \biggl(\ln\frac {t_{2}}{s} \biggr)^{\alpha-1}\frac{p_{i}(s)+q_{i}(s)}{s\Gamma(\alpha)}\,\mathrm {d}s \\ & \qquad{}+ \int_{1}^{t_{1}} \biggl\vert (\ln t_{2})^{1-\gamma} \biggl(\ln\frac {t_{2}}{s} \biggr)^{\alpha-1} -(\ln t_{1})^{1-\gamma} \biggl(\ln \frac{t_{1}}{s} \biggr)^{\alpha -1} \biggr\vert \frac{p_{i}(s)+q_{i}(s)}{s\Gamma(\alpha)}\, \mathrm{d}s. \end{aligned}$$ </span></div></div><p> Thus, we get </p><div id="Equbn" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} & \bigl\Vert (\ln t_{2})^{1-\gamma}(N_{i}u_{i}) (t_{2})-(\ln t_{1})^{1-\gamma }(N_{i}u_{i}) (t_{1}) \bigr\Vert _{E} \\ &\quad\leq L_{i} \biggl(\ln \frac{t_{2}}{t_{1}} \biggr)^{\alpha} \\ & \qquad{}+\frac{p_{i}^{*}+q_{i}^{*}}{\Gamma(\alpha)} \int_{1}^{t_{1}} \biggl\vert (\ln t_{2})^{1-\gamma} \biggl(\ln\frac{t_{2}}{s} \biggr)^{\alpha-1} -(\ln t_{1})^{1-\gamma} \biggl(\ln \frac{t_{1}}{s} \biggr)^{\alpha -1} \biggr\vert \,\mathrm{d}s. \end{aligned}$$ </span></div></div><p> Hence <span class="mathjax-tex">$N(Q)\subset Q$</span>.</p> <p><i>Step</i> 2. <i>N</i> <i>is weakly sequentially continuous.</i> Let <span class="mathjax-tex">$\{(u_{n},v_{n})\}_{n}$</span> be a sequence in <i>Q</i>, and let <span class="mathjax-tex">$(u_{n}(t),v_{n}(t)\to (u(t),v(t)) $</span> in <span class="mathjax-tex">$(E,\omega)\times(E,\omega)$</span> for each <span class="mathjax-tex">$t\in I$</span>. Fix <span class="mathjax-tex">$t\in I$</span>. Since for any <span class="mathjax-tex">$i\in{1,2}$</span>, the function <span class="mathjax-tex">$f_{i}$</span> satisfies assumption <span class="mathjax-tex">$(H_{1})$</span>, we have that <span class="mathjax-tex">$f_{i}(t,u_{n}(t),v_{n}(t),({}^{H}D_{1}^{\alpha,\beta }u_{n})(t), ({}^{H}D_{1}^{\alpha,\beta}v_{n})(t))$</span> converges weakly uniformly to <span class="mathjax-tex">$f_{i}(t,u(t),v(t),(D_{0}^{\alpha,\beta}u)(t),(D_{0}^{\alpha ,\beta}v)(t))$</span>. Hence the Lebesgue dominated convergence theorem for Pettis integral implies that <span class="mathjax-tex">$(N(u_{n},v_{n}))(t)$</span> converges weakly uniformly to <span class="mathjax-tex">$(N(u,v))(t)$</span> in <span class="mathjax-tex">$(E,\omega)$</span> for each <span class="mathjax-tex">$t\in I$</span>. Thus <span class="mathjax-tex">$N(u_{n},v_{n})\to N(u,v)$</span>. Hence <span class="mathjax-tex">$N:Q\to Q$</span> is weakly sequentially continuous.</p> <p><i>Step</i> 3. <i>Implication (</i><a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-018-1787-4#Equ5"> <i>5</i> </a><i>) holds.</i> Let <i>V</i> be a subset of <i>Q</i> such that <span class="mathjax-tex">$\overline{V}=\overline{ {\operatorname{conv}}}(N(V)\cup\{(0,0)\})$</span>. Obviously, </p><div id="Equbo" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$V(t)\subset\overline{\operatorname{conv}}(NV) (t))\cup\bigl\{ (0,0)\bigr\} ),\quad t\in I. $$</span></div></div><p> Further, as <i>V</i> is bounded and equicontinuous, by [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 13" title="
					Bugajewski, D., Szufla, S.: Kneser’s theorem for weak solutions of the Darboux problem in a Banach space. Nonlinear Anal. 20(2), 169–173 (1993)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR13" id="ref-link-section-d126833688e23067">13</a>, Lemma 3] the function <span class="mathjax-tex">$t\to\mu(V(t))$</span> is continuous on <i>I</i>. From <span class="mathjax-tex">$(H_{3}), (H_{4})$</span>, Lemma <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1186/s13662-018-1787-4#Sec2">2.19</a>, and the properties of the measure <i>μ</i>, for any <span class="mathjax-tex">$t\in I$</span>, we have </p><div id="Equbp" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} &\mu\bigl((\ln t)^{1-\gamma}V(t)\bigr)\\ &\quad\leq\mu\bigl((\ln t)^{1-\gamma}(NV) (t)\cup \bigl\{ (0,0)\bigr\} \bigr) \\ &\quad\leq\mu\bigl((\ln t)^{1-\gamma}(NV) (t)\bigr) \\ &\quad\leq\mu(\bigl\{ \bigl((\ln t)^{1-\gamma}(N_{1}v_{1}) (t),(\ln t)^{1-\gamma }(N_{2}v_{2}) (t):(v_{1},v_{2}) \in V\bigr\} \bigr) \\ &\quad \leq\frac{1}{\Gamma(\alpha)} \int_{1}^{t} \biggl(\ln\frac {t}{s} \biggr)^{\alpha-1}\mu\bigl(\bigl\{ (\ln s)^{1-\gamma }\bigl(f_{1} \bigl(s,v_{1}(s),v_{2}(s), \\ &\qquad\bigl({}^{H}D_{1}^{\alpha,\beta}v_{1}\bigr) (t), \bigl({}^{H}D_{1}^{\alpha,\beta }v_{2}\bigr) (t) \bigr),0\bigr):(v_{1},v_{2})\in V\bigr\} \bigr) \frac{\mathrm{d}s}{s} \\ &\qquad{}+\frac{1}{\Gamma(\alpha)} \int_{1}^{t} \biggl(\ln\frac {t}{s} \biggr)^{\alpha-1}\mu\bigl(\bigl\{ (\ln s)^{1-\gamma }\bigl(0,f_{2} \bigl(s,v_{1}(s),v_{2}(s), \\ &\qquad\bigl({}^{H}D_{1}^{\alpha,\beta}v_{1}\bigr) (t), \bigl({}^{H}D_{1}^{\alpha,\beta }v_{2}\bigr) (t)\bigr) \bigr):(v_{1},v_{2})\in V\bigr\} \bigr)\frac{\mathrm{d}s}{s} \\ &\quad\leq\frac{1}{\Gamma(\alpha)} \int_{1}^{t} \biggl(\ln\frac {t}{s} \biggr)^{\alpha-1}\bigl[p_{1}(s)\mu\bigl(\bigl\{ (\ln s)^{1-\gamma }\bigl(v_{1}(s),0\bigr):(v_{1},0)\in V\bigr\} \bigr) \\ &\qquad{}+q_{1}(s)\mu\bigl(\bigl\{ (\ln s)^{1-\gamma}\bigl(0,v_{2}(s) \bigr):(0,v_{2})\in V\bigr\} \bigr)\bigr]\frac {\mathrm{d}s}{s} \\ &\qquad{}+\frac{1}{\Gamma(\alpha)} \int_{1}^{t} \biggl(\ln\frac {t}{s} \biggr)^{\alpha-1}\bigl[p_{2}(s)\mu\bigl(\bigl\{ (\ln s)^{1-\gamma }\bigl(v_{1}(s),0\bigr):(v_{1},0)\in V\bigr\} \bigr) \\ &\qquad{}+q_{2}(s)\mu\bigl(\bigl\{ (\ln s)^{1-\gamma}\bigl(0,v_{2}(s) \bigr):(0,v_{2})\in V\bigr\} \bigr)\bigr]\frac {\mathrm{d}s}{s}. \end{aligned}$$ </span></div></div><p> Thus </p><div id="Equbq" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} &\mu\bigl((\ln t)^{1-\gamma}V(t)\bigr) \\ &\quad \leq\frac{1}{\Gamma(\alpha)} \int_{1}^{t} \biggl(\ln\frac {t}{s} \biggr)^{\alpha-1}\bigl(p_{1}(s)+q_{1}(s)+p_{2}(s)+q_{2}(s) \bigr) \\ &\qquad{}\times\mu\bigl((\ln s)^{1-\gamma}V(s)\bigr)\frac{\mathrm{d}s}{s} \\ &\quad \leq\frac{1}{\Gamma(\alpha)} \int_{1}^{t} \biggl(\ln\frac{t}{s} \biggr)^{\alpha -1}\bigl(p_{1}(s)+q_{1}(s)+p_{2}(s)+q_{2}(s) \bigr) \\ &\qquad{}\times \sup _{s\in I}\mu\bigl((\ln s)^{1-\gamma}V(s)\bigr) \frac{\mathrm {d}s}{s} \\ &\quad\leq\frac{(p_{1}^{*}+p_{2}^{*}+q_{1}^{*}+q_{2}^{*})(\ln T)^{\alpha }}{\Gamma(1+\alpha)} \sup _{t\in I}\mu\bigl((\ln t)^{1-\gamma}V(t) \bigr). \end{aligned}$$ </span></div></div><p> Hence </p><div id="Equbr" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\sup _{t\in I}\mu\bigl((\ln t)^{1-\gamma}V(t)\bigr)\leq L \sup _{t\in I}\mu \bigl((\ln t)^{1-\gamma}V(t)\bigr). $$</span></div></div><p> From (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-018-1787-4#Equ6">6</a>) we get <span class="mathjax-tex">$\sup _{t\in I}\mu((\ln t)^{1-\gamma }V(t))=0$</span>, that is, <span class="mathjax-tex">$\mu(V(t))=0$</span> for each <span class="mathjax-tex">$t\in I$</span>. Then by [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 24" title="
					Mitchell, A.R., Smith, C.: Nonlinear equations in abstract spaces. In: Lakshmikantham, V. (ed.) An Existence Theorem for Weak Solutions of Differential Equations in Banach Spaces, pp. 387–403. Academic Press, New York (1978)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR24" id="ref-link-section-d126833688e25912">24</a>, Thm. 2] <i>V</i> is weakly relatively compact in <span class="mathjax-tex">${\mathcal{C}}$</span>. From Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-018-1787-4#FPar21">2.20</a> we conclude that <i>N</i> has a fixed point, which is a weak solution of the coupled system (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-018-1787-4#Equ1">1</a>)–(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-018-1787-4#Equ2">2</a>). □</p> <p>As a consequence of the theorem, we get the following corollary.</p> <h3 class="c-article__sub-heading" id="FPar25">Corollary 3.3</h3> <p><i>Consider the following system of implicit Hilfer–Hadamard fractional differential equations</i>: </p><div id="Equ8" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} &\textstyle\begin{cases} ({}^{H}D_{1}^{\alpha,\beta}u_{1})(t)\\ \quad=f_{1}(t,u_{1}(t),u_{2}(t),\ldots,u_{n}(t),\\ \qquad({}^{H}D_{1}^{\alpha,\beta}u_{1})(t),({}^{H}D_{1}^{\alpha,\beta }u_{2})(t),\ldots,({}^{H}D_{1}^{\alpha,\beta}u_{n})(t)), \\ ({}^{H}D_{1}^{\alpha,\beta}u_{2})(t)\\ \quad=f_{2}(t,u_{1}(t),u_{2}(t),\ldots,u_{n}(t),\\ \qquad({}^{H}D_{1}^{\alpha,\beta}u_{1})(t),({}^{H}D_{1}^{\alpha,\beta }u_{2})(t),\ldots,({}^{H}D_{1}^{\alpha,\beta}u_{n})(t)), \\ \vdots\\ ({}^{H}D_{1}^{\alpha,\beta}u_{n})(t)\\ \quad=f_{n}(t,u_{1}(t),u_{2}(t),\ldots,u_{n}(t),\\ \qquad({}^{H}D_{1}^{\alpha,\beta}u_{1})(t),({}^{H}D_{1}^{\alpha,\beta }u_{2})(t),\ldots,({}^{H}D_{1}^{\alpha,\beta}u_{n})(t)), \end{cases}\displaystyle t\in I, \end{aligned}$$ </span></div><div class="c-article-equation__number"> (8) </div></div> <div id="Equ9" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} & \bigl({}^{H}I_{1}^{1-\gamma}u_{i} \bigr) (t)|_{t=1}=\phi_{i},\quad i=1,2,\dots,n, \end{aligned}$$ </span></div><div class="c-article-equation__number"> (9) </div></div><p> <span class="mathjax-tex">$I:=[1,T], T>1, \alpha\in(0,1), \beta\in[0,1], \gamma=\alpha +\beta-\alpha\beta, \phi_{i}\in E, f_{i}:I\times E^{2n}\to E, i=1,2,\dots,n$</span>, <i>are given continuous functions</i>, <i>E</i> <i>is a real</i> (<i>or complex</i>) <i>Banach space with norm</i> <span class="mathjax-tex">$\|\cdot\|_{E}$</span> <i>and dual</i> <span class="mathjax-tex">$E^{*}$</span>, <i>such that</i> <i>E</i> <i>is the dual of a weakly compactly generated Banach space</i> <i>X</i>, <span class="mathjax-tex">${}^{H}I_{1}^{1-\gamma}$</span> <i>is the left</i>-<i>sided mixed Hadamard integral of order</i> <span class="mathjax-tex">$1-\gamma$</span>, <i>and</i> <span class="mathjax-tex">${}^{H}D_{1}^{\alpha,\beta}$</span> <i>is the Hilfer–Hadamard fractional derivative of order</i> <i>α</i> <i>and type</i> <i>β</i>.</p> <p><i>Assume that the following hypotheses hold</i>: </p><dl class="c-abbreviation_list"><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn> <span class="mathjax-tex">$(H_{01})$</span> :</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p><i>The functions</i> <span class="mathjax-tex">$v_{j}\to f_{i}(t,v_{1},v_{2},\dots,v_{j},\dots ,v_{2n}), i=1,\dots,n, j=1,\dots,2n$</span>, <i>are weakly sequentially continuous for a</i>.<i>e</i>. <span class="mathjax-tex">$t\in I$</span>,</p> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn> <span class="mathjax-tex">$(H_{02})$</span> :</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p><i>For each</i> <span class="mathjax-tex">$v_{j}\in E, j=1,\dots,2n$</span>, <i>the functions</i> <span class="mathjax-tex">$t\to f_{i}(t,v_{1},v_{2},\dots,v_{j},\dots,v_{2n}), i=1,2$</span>, <i>are Pettis integrable a</i>.<i>e</i>. <i>on</i> <i>I</i>,</p> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn> <span class="mathjax-tex">$(H_{03})$</span> :</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p><i>There exist</i> <span class="mathjax-tex">$p_{ij}\in C(I,[0,\infty))$</span> <i>such that</i>, <i>for all</i> <span class="mathjax-tex">$\varphi\in E^{*}$</span>, <i>we have</i> </p><div id="Equbs" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl\vert \varphi\bigl(f_{i}(t,v_{1},v_{2}, \dots,v_{2n})\bigr) \bigr\vert \leq\frac{ \sum _{i=1}^{n} \sum _{j=1}^{n}p_{ij}(t) \Vert v_{j} \Vert _{E}}{1+ \Vert \varphi \Vert + \sum _{j=1}^{n} \Vert v_{i} \Vert _{E}} $$</span></div></div><p> <span class="mathjax-tex">$\textit{for a.e. }t\in I\textit{ and each }v_{i}\in E, i=1,2,\dots,n$</span>,</p> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn> <span class="mathjax-tex">$(H_{04})$</span> :</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p><i>For all bounded measurable sets</i> <span class="mathjax-tex">$B_{i}\subset E, i=1,\dots,n$</span>, <i>and for each</i> <span class="mathjax-tex">$t\in I$</span>, <i>we have</i> </p><div id="Equbt" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} &\mu\bigl(0,\dots,f_{j}\bigl(t,B_{1},B_{2}, \dots,B_{n},^{H}D_{1}^{\alpha,\beta }B_{1},^{H}D_{1}^{\alpha,\beta}B_{2}, \dots, ^{H}D_{1}^{\alpha,\beta}B_{n}\bigr),\dots,0 \bigr) \\ &\quad\leq \sum _{i=1}^{n}p_{ij}(t)\mu(B_{i}),\quad j=1,\dots,n, \end{aligned}$$ </span></div></div><p> <i>where</i> <span class="mathjax-tex">$^{H}D_{1}^{\alpha,\beta}B_{i}=\{^{H}D_{1}^{\alpha,\beta}w:w\in B_{i}\}, i=1,\dots,n$</span>.</p> </dd></dl><p> <i>If</i> </p><div id="Equbu" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$L^{*}:=\frac{ \sum _{i=1}^{n} \sum _{j=1}^{n}p_{ij}^{*}(\ln T)^{\alpha }}{\Gamma(1+\alpha)}< 1, $$</span></div></div><p> <i>where</i> </p><div id="Equbv" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$p_{ij}^{*}= \sup _{t\in I}p_{ij}(t),\quad i,j=1,\dots,n, $$</span></div></div><p> <i>then the coupled system</i> (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-018-1787-4#Equ8">8</a>)<i>–</i>(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-018-1787-4#Equ9">9</a>) <i>has at least one weak solution defined on</i> <i>I</i>.</p> </div></div></section><section data-title="An example"><div class="c-article-section" id="Sec4-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec4"><span class="c-article-section__title-number">4 </span>An example</h2><div class="c-article-section__content" id="Sec4-content"><p>Let </p><div id="Equbw" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$E=l^{1}= \bigl\{ u =(u_{1}, u_{2},\ldots, u_{n},\ldots), \sum ^{\infty }_{n=1} \vert u_{n} \vert < \infty \bigr\} $$</span></div></div><p> be the Banach space with the norm </p><div id="Equbx" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\Vert u \Vert _{E}= \sum _{n=1}^{\infty} \vert u_{n} \vert . $$</span></div></div><p> As an application of our results, we consider the coupled system of Hilfer–Hadamard fractional differential equations </p><div id="Equ10" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} &\textstyle\begin{cases} ({}^{H}D_{1}^{\frac{1}{2},\frac {1}{2}}u_{n})(t)=f_{n}(t,u(t),v(t),({}^{H}D_{1}^{\frac{1}{2},\frac{1}{2}}u_{n})(t), ({}^{H}D_{1}^{\frac{1}{2},\frac{1}{2}}v_{n})(t)),\\ ({}^{H}D_{1}^{\frac{1}{2},\frac {1}{2}}v_{n})(t)=g_{n}(t,u(t),v(t),({}^{H}D_{1}^{\frac{1}{2},\frac{1}{2}}u_{n})(t), ({}^{H}D_{1}^{\frac{1}{2},\frac{1}{2}}v_{n})(t)), \end{cases}\displaystyle t\in[1,e], \end{aligned}$$ </span></div><div class="c-article-equation__number"> (10) </div></div> <div id="Equ11" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} & \bigl({}^{H}I_{1}^{\frac{1}{4}}u\bigr) (t)|_{t=1}=\bigl({}^{H}I_{1}^{\frac {1}{4}}v\bigr) (t)|_{t=1}=(0,0,\ldots,0,\ldots), \end{aligned}$$ </span></div><div class="c-article-equation__number"> (11) </div></div><p> where </p><div id="Equby" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$f_{n}\bigl(t,u(t),v(t)\bigr)=\frac{ct^{2}}{1+ \Vert u(t) \Vert _{E}+ \Vert v(t) \Vert _{E}+ \Vert \bar{u}(t) \Vert _{E}+ \Vert \bar{v}(t) \Vert _{E}}\frac{u_{n}(t)}{e^{t+4}},\quad t \in[1,e], $$</span></div></div><p> and </p><div id="Equbz" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$g_{n}\bigl(t,u(t),v(t)\bigr)=\frac{ct^{2}}{1+ \Vert v(t) \Vert _{E}+ \Vert v(t) \Vert _{E}+ \Vert \bar{u}(t) \Vert _{E}+ \Vert \bar{v}(t) \Vert _{E}}\frac{u_{n}(t)}{e^{t+4}},\quad t \in[1,e], $$</span></div></div><p> with </p><div id="Equca" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$u=(u_{1},u_{2},\ldots,u_{n},\ldots),\qquad v=(v_{1},v_{2},\ldots,v_{n},\ldots) \quad\mbox{and}\quad c:=\frac{e^{3}}{16}\sqrt{\pi}. $$</span></div></div><p> Set </p><div id="Equcb" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$f=(f_{1},f_{2},\ldots,f_{n},\ldots) \quad\mbox{and}\quad g=(g_{1},g_{2},\ldots ,g_{n},\ldots). $$</span></div></div><p> Clearly, the functions <i>f</i> and <i>g</i> are continuous.</p><p>For all <span class="mathjax-tex">$u,v,\bar{u},\bar{v}\in E$</span> and <span class="mathjax-tex">$t\in[1,e]$</span>, we have </p><div id="Equcc" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\bigl\Vert f\bigl(t,u(t),v(t),\bar{u}(t),\bar{v}(t)\bigr) \bigr\Vert _{E}\leq ct^{2}\frac{1}{e^{t+4}} \quad\mbox{and}\quad \bigl\Vert g \bigl(t,u(t),v(t),\bar{u}(t),\bar{v}(t)\bigr) \bigr\Vert _{E}\leq ct^{2}\frac {1}{e^{t+4}}. $$</span></div></div><p> Hence, hypothesis <span class="mathjax-tex">$(H_{3})$</span> is satisfied with <span class="mathjax-tex">$p_{i}^{*}=ce^{-3}$</span> and <span class="mathjax-tex">$q_{i}^{*}=0, i=1,2$</span>. We will show that condition (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-018-1787-4#Equ6">6</a>) holds with <span class="mathjax-tex">$T=e$</span>. Indeed, </p><div id="Equcd" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\frac{(p_{1}^{*}+q_{1}^{*}+p_{2}^{*}+q_{2}^{*})(\ln T)^{\alpha}}{\Gamma (1+\alpha)} =\frac{4ce^{-3}}{\sqrt{\pi}}=\frac{1}{4}< 1. $$</span></div></div><p> Simple computations show that all conditions of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-018-1787-4#FPar23">3.2</a> are satisfied. It follows that the coupled system (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-018-1787-4#Equ10">10</a>)–(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-018-1787-4#Equ11">11</a>) has at least one weak solution defined on <span class="mathjax-tex">$[1,e]$</span>.</p></div></div></section><section data-title="Conclusion"><div class="c-article-section" id="Sec5-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec5"><span class="c-article-section__title-number">5 </span>Conclusion</h2><div class="c-article-section__content" id="Sec5-content"><p>In the recent years, implicit functional differential equations have been considered by many authors [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="
					Abbas, S., Benchohra, M., Bohner, M.: Weak solutions for implicit differential equations of Hilfer–Hadamard fractional derivative. Adv. Dyn. Syst. Appl. 12(1), 1–16 (2017)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR1" id="ref-link-section-d126833688e31391">1</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="
					Abbas, S., Benchohra, M., Vityuk, A.N.: On fractional order derivatives and Darboux problem for implicit differential equations. Fract. Calc. Appl. Anal. 15(2), 168–182 (2012)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR5" id="ref-link-section-d126833688e31394">5</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title="
					Benavides, T.D.: An existence theorem for implicit differential equations in a Banach space. Ann. Mat. Pura Appl. 4, 119–130 (1978)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR9" id="ref-link-section-d126833688e31397">9</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 33" title="
					Vityuk, A.N., Mykhailenko, A.V.: The Darboux problem for an implicit fractional-order differential equation. J. Math. Sci. 175(4), 391–401 (2011)
				" href="/articles/10.1186/s13662-018-1787-4#ref-CR33" id="ref-link-section-d126833688e31400">33</a>]. In this work, we give some existence results for coupled implicit Hilfer–Hadamard fractional differential systems. This paper initiates the application of the measure of weak noncompactness to such a class of problems.</p></div></div></section> <div id="MagazineFulltextArticleBodySuffix"><section aria-labelledby="Bib1" data-title="References"><div class="c-article-section" id="Bib1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Bib1">References</h2><div class="c-article-section__content" id="Bib1-content"><div data-container-section="references"><ol class="c-article-references" data-track-component="outbound reference" data-track-context="references section"><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="1."><p class="c-article-references__text" id="ref-CR1"> Abbas, S., Benchohra, M., Bohner, M.: Weak solutions for implicit differential equations of Hilfer–Hadamard fractional derivative. Adv. Dyn. Syst. 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id="citeas">Cite this article</h3><p class="c-bibliographic-information__citation">Abbas, S., Benchohra, M., Hamidi, N. <i>et al.</i> Implicit coupled Hilfer–Hadamard fractional differential systems under weak topologies. <i>Adv Differ Equ</i> <b>2018</b>, 328 (2018). https://doi.org/10.1186/s13662-018-1787-4</p><p class="c-bibliographic-information__download-citation u-hide-print"><a data-test="citation-link" data-track="click" data-track-action="download article citation" data-track-label="link" data-track-external="" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1186/s13662-018-1787-4?format=refman&flavour=citation">Download citation<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p><ul class="c-bibliographic-information__list" data-test="publication-history"><li class="c-bibliographic-information__list-item"><p>Received<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2018-07-19">19 July 2018</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Accepted<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2018-09-05">05 September 2018</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Published<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2018-09-18">18 September 2018</time></span></p></li><li class="c-bibliographic-information__list-item c-bibliographic-information__list-item--full-width"><p><abbr title="Digital Object Identifier">DOI</abbr><span class="u-hide">: </span><span class="c-bibliographic-information__value">https://doi.org/10.1186/s13662-018-1787-4</span></p></li></ul><div data-component="share-box"><div class="c-article-share-box u-display-none" 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data-track-label="link">45G05</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=45M10&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">45M10</a></span></li></ul><h3 class="c-article__sub-heading">Keywords</h3><ul class="c-article-subject-list"><li class="c-article-subject-list__subject"><span><a href="/search?query=Coupled%20fractional%20differential%20system&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">Coupled fractional differential system</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=Left-sided%20mixed%20Pettis%E2%80%93Hadamard%20integral%20of%20fractional%20order&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">Left-sided mixed Pettis–Hadamard integral of fractional order</a></span></li><li 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