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<!DOCTYPE html> <html lang="en" class="no-js"> <head> <meta charset="UTF-8"> <meta http-equiv="X-UA-Compatible" content="IE=edge"> <meta name="applicable-device" content="pc,mobile"> <meta name="viewport" content="width=device-width, initial-scale=1"> <title>On approximate solutions for two higher-order Caputo-Fabrizio fractional integro-differential equations | Advances in Continuous and Discrete Models | Full Text</title> <meta name="citation_abstract" content="We investigate the existence of solutions for two high-order fractional differential equations including the Caputo-Fabrizio derivative. In this way, we introduce some new tools for obtaining solutions for the high-order equations. Also, we discuss two illustrative examples to confirm the reported results. In this way one gets the possibility of utilizing some continuous or discontinuous mappings as coefficients in the fractional differential equations of higher order."/> <meta name="journal_id" content="13662"/> <meta name="dc.title" content="On approximate solutions for two higher-order Caputo-Fabrizio fractional integro-differential equations"/> <meta name="dc.source" content="Advances in Difference Equations 2017 2017:1"/> <meta name="dc.format" content="text/html"/> <meta name="dc.publisher" content="SpringerOpen"/> <meta name="dc.date" content="2017-08-03"/> <meta name="dc.type" content="OriginalPaper"/> <meta name="dc.language" content="En"/> <meta name="dc.copyright" content="2017 The Author(s)"/> <meta name="dc.rights" content="2017 The Author(s)"/> <meta name="dc.rightsAgent" content="reprints@biomedcentral.com"/> <meta name="dc.description" content="We investigate the existence of solutions for two high-order fractional differential equations including the Caputo-Fabrizio derivative. In this way, we introduce some new tools for obtaining solutions for the high-order equations. Also, we discuss two illustrative examples to confirm the reported results. In this way one gets the possibility of utilizing some continuous or discontinuous mappings as coefficients in the fractional differential equations of higher order."/> <meta name="prism.issn" content="1687-1847"/> <meta name="prism.publicationName" content="Advances in Difference Equations"/> <meta name="prism.publicationDate" content="2017-08-03"/> <meta name="prism.volume" content="2017"/> <meta name="prism.number" content="1"/> <meta name="prism.section" content="OriginalPaper"/> <meta name="prism.startingPage" content="1"/> <meta name="prism.endingPage" content="11"/> <meta name="prism.copyright" content="2017 The Author(s)"/> <meta name="prism.rightsAgent" content="reprints@biomedcentral.com"/> <meta name="prism.url" content="https://advancesincontinuousanddiscretemodels.springeropen.com/articles/10.1186/s13662-017-1258-3"/> <meta name="prism.doi" content="doi:10.1186/s13662-017-1258-3"/> <meta name="citation_pdf_url" content="https://advancesincontinuousanddiscretemodels.springeropen.com/counter/pdf/10.1186/s13662-017-1258-3"/> <meta name="citation_fulltext_html_url" content="https://advancesincontinuousanddiscretemodels.springeropen.com/articles/10.1186/s13662-017-1258-3"/> <meta name="citation_journal_title" content="Advances in Difference Equations"/> <meta name="citation_journal_abbrev" content="Adv Differ Equ"/> <meta name="citation_publisher" content="SpringerOpen"/> <meta name="citation_issn" content="1687-1847"/> <meta name="citation_title" content="On approximate solutions for two higher-order Caputo-Fabrizio fractional integro-differential equations"/> <meta name="citation_volume" content="2017"/> <meta name="citation_issue" content="1"/> <meta name="citation_publication_date" content="2017/12"/> <meta name="citation_online_date" content="2017/08/03"/> <meta name="citation_firstpage" content="1"/> <meta name="citation_lastpage" content="11"/> <meta name="citation_article_type" content="Research"/> <meta name="citation_fulltext_world_readable" content=""/> <meta name="citation_language" content="en"/> <meta name="dc.identifier" content="doi:10.1186/s13662-017-1258-3"/> <meta name="DOI" content="10.1186/s13662-017-1258-3"/> <meta name="size" content="744843"/> <meta name="citation_doi" content="10.1186/s13662-017-1258-3"/> <meta name="citation_springer_api_url" content="http://api.springer.com/xmldata/jats?q=doi:10.1186/s13662-017-1258-3&api_key="/> <meta name="description" content="We investigate the existence of solutions for two high-order fractional differential equations including the Caputo-Fabrizio derivative. 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In this way, we introduce some new tools for obtaining solutions for the high-order equations. Also, we discuss two illustrative examples to confirm the reported results. In this way one gets the possibility of utilizing some continuous or discontinuous mappings as coefficients in the fractional differential equations of higher order."/> <meta property="og:image" content="https://media.springernature.com/full/springer-static/cover-hires/journal/13662"/> <script type="application/ld+json">{"mainEntity":{"headline":"On approximate solutions for two higher-order Caputo-Fabrizio fractional integro-differential equations","description":"We investigate the existence of solutions for two high-order fractional differential equations including the Caputo-Fabrizio derivative. In this way, we introduce some new tools for obtaining solutions for the high-order equations. Also, we discuss two illustrative examples to confirm the reported results. In this way one gets the possibility of utilizing some continuous or discontinuous mappings as coefficients in the fractional differential equations of higher order.","datePublished":"2017-08-03T00:00:00Z","dateModified":"2017-08-03T00:00:00Z","pageStart":"1","pageEnd":"11","sameAs":"https://doi.org/10.1186/s13662-017-1258-3","keywords":["34A08","34A99","approximate fixed point","higher-order fractional differential equation","non-singular kernel","Caputo-Fabrizio derivation","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":"2017","@type":["Periodical","PublicationVolume"]},"publisher":{"name":"Springer International 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Iran","@type":"PostalAddress"},"@type":"Organization"}],"@type":"Person"},{"name":"Shahram Rezapour","affiliation":[{"name":"Azarbaijan Shahid Madani University","address":{"name":"Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran","@type":"PostalAddress"},"@type":"Organization"}],"@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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class="c-context-bar__container u-container" data-track-context="sticky banner"> <div class="c-context-bar__title"> On approximate solutions for two higher-order Caputo-Fabrizio fractional integro-differential equations </div> <div class="c-pdf-container"> <div class="c-pdf-download u-clear-both"> <a href="//advancesincontinuousanddiscretemodels.springeropen.com/counter/pdf/10.1186/s13662-017-1258-3.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> </div> <div 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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="2017-08-03">03 August 2017</time></li> </ul> <h1 class="c-article-title" data-test="article-title" data-article-title="">On approximate solutions for two higher-order Caputo-Fabrizio fractional integro-differential equations</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-S_Melike-Aydogan-Aff1" data-author-popup="auth-S_Melike-Aydogan-Aff1" data-author-search="Aydogan, S Melike">S Melike Aydogan</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-Dumitru-Baleanu-Aff2-Aff3" data-author-popup="auth-Dumitru-Baleanu-Aff2-Aff3" data-author-search="Baleanu, Dumitru" data-corresp-id="c1">Dumitru Baleanu<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><sup class="u-js-hide"><a href="#Aff2">2</a>,<a href="#Aff3">3</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-Asef-Mousalou-Aff4" data-author-popup="auth-Asef-Mousalou-Aff4" data-author-search="Mousalou, Asef">Asef Mousalou</a><sup class="u-js-hide"><a href="#Aff4">4</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-Shahram-Rezapour-Aff4" data-author-popup="auth-Shahram-Rezapour-Aff4" data-author-search="Rezapour, Shahram">Shahram Rezapour</a><sup class="u-js-hide"><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> 2017</b>, Article number: <span data-test="article-number">221</span> (<span data-test="article-publication-year">2017</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">2796 <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/s13662-017-1258-3/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>We investigate the existence of solutions for two high-order fractional differential equations including the Caputo-Fabrizio derivative. In this way, we introduce some new tools for obtaining solutions for the high-order equations. Also, we discuss two illustrative examples to confirm the reported results. In this way one gets the possibility of utilizing some continuous or discontinuous mappings as coefficients in the fractional differential equations of higher order.</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>Up to now, there have been defined some fractional derivations of which most used are the Caputo and Riemann-Liouville operators. The applications of the fractional calculus with these two main derivatives can be observed within a huge range of real world phenomena. In order to increase the power and applicability of the fractional calculus some researchers suggested a new type of fractional derivatives possessing different kernels. Thus, Caputo and Fabrizio defined recently a new fractional derivative possessing a singular kernel [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title=" Caputo, M, Fabrizzio, M: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 73-85 (2015) " href="/articles/10.1186/s13662-017-1258-3#ref-CR8" id="ref-link-section-d72216e387">8</a>] and the properties of it were discussed in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title=" Losada, J, Nieto, JJ: Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 87-92 (2015) " href="/articles/10.1186/s13662-017-1258-3#ref-CR11" id="ref-link-section-d72216e390">11</a>]. Some researchers have used distinct methods for solving some different equations including the Caputo-Fabrizio (CF) fractional derivative (see [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title=" Alsaedi, A, Baleanu, D, Etemad, S, Rezapour, S: On coupled systems of time-fractional differential problems by using a new fractional derivative. J. Funct. Spaces 2016, Article ID 4626940 (2016) " href="/articles/10.1186/s13662-017-1258-3#ref-CR1" id="ref-link-section-d72216e393">1</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title=" De La Sen, M, Hedayati, V, Gholizade Atani, Y, Rezapour, S: The existence and numerical solution for a k-dimensional system of multi-term fractional integro-differential equations. Nonlinear Anal., Model. Control 22(2), 188-209 (2017) " href="/articles/10.1186/s13662-017-1258-3#ref-CR7" id="ref-link-section-d72216e396">7</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title=" Gómez-Aguilar, JF, Yépez-Martínez, H, Calderón-Ramón, C, Cruz-Orduña, I, Escobar-Jiménez, RF, Olivares-Peregrino, VH: Modeling of a mass-spring-damper system by fractional derivatives with and without a singular kernel. Entropy 17(9), 6289-6303 (2015) " href="/articles/10.1186/s13662-017-1258-3#ref-CR9" id="ref-link-section-d72216e399">9</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title=" Goufo, EF, Doungmo, P, Morgan, K, Mwambakana, JN: Duplication in a model of rock fracture with fractional derivative without singular kernel. Open Math. 13, 839-846 (2015) " href="/articles/10.1186/s13662-017-1258-3#ref-CR10" id="ref-link-section-d72216e403">10</a>] and the references therein) and multi-singular pointwise defined equations [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 13" title=" Rezapour, S, Shabibi, M: A singular fractional differential equation with Riemann-Liouville integral boundary condition. J. Adv. Math. Stud. 8(1), 80-88 (2015) " href="/articles/10.1186/s13662-017-1258-3#ref-CR13" id="ref-link-section-d72216e406">13</a>–<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title=" Shabibi, M, Postolache, M, Rezapour, S: Positive solutions for a singular sum fractional differential system. Int. J. Anal. Appl. 13(1), 108-118 (2017) " href="/articles/10.1186/s13662-017-1258-3#ref-CR16" id="ref-link-section-d72216e409">16</a>]. Despite these original results, still several issues regarding this new fractional derivative have to be developed. Below we discuss the existence of approximate solutions analytically corresponding to two CF fractional differential equations (FDE).</p><p>The plan of the manuscript can be seen below. In the following section we recall the main results needed in this paper. Section <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1186/s13662-017-1258-3#Sec3">3</a> contains the original results and the illustrative examples. Finally, Section <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/articles/10.1186/s13662-017-1258-3#Sec4">4</a> summarizes our work.</p></div></div></section><section data-title="Basic tools"><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>Basic tools</h2><div class="c-article-section__content" id="Sec2-content"><p>Let <span class="mathjax-tex">$c>0$</span>, <span class="mathjax-tex">$u\in H^{1}(0,c)$</span> and <span class="mathjax-tex">$\alpha\in(0,1)$</span>. We recall that <span class="mathjax-tex">$^{CF}D^{\alpha}u(t)=\frac{(2-\alpha)M(\alpha)}{2(1-\alpha)}\int _{0}^{t}\exp(\frac{-\alpha}{1-\alpha}(t-r))u^{\prime}(r)\,dr$</span> means the CF fractional derivative, such that <span class="mathjax-tex">$t\geq0$</span> and <span class="mathjax-tex">$M(\alpha)$</span> denotes the normalization constant and fulfilling <span class="mathjax-tex">$M(0)=M(1)=1$</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title=" Caputo, M, Fabrizzio, M: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 73-85 (2015) " href="/articles/10.1186/s13662-017-1258-3#ref-CR8" id="ref-link-section-d72216e792">8</a>]. The corresponding fractional integral is written as <span class="mathjax-tex">$^{CF}I^{\alpha} u(t)=\frac{2(1-\alpha)}{(2-\alpha)M(\alpha)}u(t) +\frac{2\alpha}{(2-\alpha)M(\alpha)}\int_{0}^{t} u(r) \,dr$</span> for <span class="mathjax-tex">$0<\alpha<1$</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title=" Losada, J, Nieto, JJ: Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 87-92 (2015) " href="/articles/10.1186/s13662-017-1258-3#ref-CR11" id="ref-link-section-d72216e995">11</a>]. We recall that <span class="mathjax-tex">$M(\alpha)=\frac{2}{2-\alpha}$</span> for all <span class="mathjax-tex">$0\leq\alpha\leq 1$</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title=" Losada, J, Nieto, JJ: Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 87-92 (2015) " href="/articles/10.1186/s13662-017-1258-3#ref-CR11" id="ref-link-section-d72216e1070">11</a>]. As a result, the CF fractional derivative becomes <span class="mathjax-tex">$^{CF}D^{\alpha}u(t)=\frac{1}{1-\alpha}\int_{0}^{t}\exp(-\frac{\alpha }{1-\alpha}(t-r))u^{\prime}(r)\,dr$</span>, where <span class="mathjax-tex">$t\geq0$</span> and <span class="mathjax-tex">$0<\alpha<1$</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title=" Losada, J, Nieto, JJ: Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 87-92 (2015) " href="/articles/10.1186/s13662-017-1258-3#ref-CR11" id="ref-link-section-d72216e1258">11</a>]. Note that <span class="mathjax-tex">$^{CF}D^{\alpha}u\in H^{1}$</span> when <span class="mathjax-tex">$u\in H^{1}$</span>. This hints to a new idea about high-order derivations. If <span class="mathjax-tex">$n\geq1$</span> and <span class="mathjax-tex">$\alpha\in[0,1]$</span>, then the CF fractional derivative of order <span class="mathjax-tex">$n+\alpha$</span> is defined by <span class="mathjax-tex">${}^{CF}D^{\alpha+n}u:= {}^{CF}D^{\alpha }(^{CF}D^{n}u(t))$</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title=" Caputo, M, Fabrizzio, M: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 73-85 (2015) " href="/articles/10.1186/s13662-017-1258-3#ref-CR8" id="ref-link-section-d72216e1524">8</a>]. We need the following results.</p> <h3 class="c-article__sub-heading" id="FPar1">Lemma 2.1</h3> <p>[<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title=" Losada, J, Nieto, JJ: Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 87-92 (2015) " href="/articles/10.1186/s13662-017-1258-3#ref-CR11" id="ref-link-section-d72216e1534">11</a>]</p> <p> <i>The unique solution of</i> <span class="mathjax-tex">$^{CF}D^{\alpha}u(t)=v(t)$</span> <i>with</i> <span class="mathjax-tex">$u(0)=c$</span> <i>and</i> <span class="mathjax-tex">$0<\alpha<1$</span> <i>is given by</i> <span class="mathjax-tex">$u(t)=c+a_{\alpha}(v(t)-v(0))+b_{\alpha}\int_{0}^{t} v(r)\,dr$</span>, <i>where</i> <span class="mathjax-tex">$a_{\alpha}=\frac{2(1-\alpha)}{(2-\alpha)M(\alpha)}=1-\alpha$</span> <i>and</i> <span class="mathjax-tex">$b_{\alpha}=\frac{2\alpha}{(2-\alpha)M(\alpha)}=\alpha$</span>. <i>In addition</i> <span class="mathjax-tex">$v(0)=0$</span>.</p> <p>For investigating the existence of solutions for most FDE, researchers utilized the well-defined fixed point results, <i>e.g.</i> the Banach contraction principle. We recall that there are many nonlinear FDE admitting no exact solutions [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title=" Area, I, Losada, J, Nieto, JJ: A note on the fractional logistic equation. Physica A 444, 182-187 (2016) " href="/articles/10.1186/s13662-017-1258-3#ref-CR2" id="ref-link-section-d72216e1976">2</a>]. In this case, numerical methods are utilized to get an approximation of exact solutions. In addition, <i>u</i> represents an approximate solution for FDE when we could get a sequence of functions <span class="mathjax-tex">$\{v_{n}\}_{n\geq1}$</span> such that <span class="mathjax-tex">$v_{n}\to u$</span>.</p><p>If an exact solution u is not obtained, then we use this approach. This case arises when we discuss the FDE within a non-complete metric space.</p><p>Below we present some basic notions needed in this manuscript. Let <span class="mathjax-tex">$(Y,d)$</span> denoting a metric space, <i>F</i> a self-map on <i>Y</i>, <span class="mathjax-tex">$\alpha: Y\times Y \to[0, \infty) $</span> a mapping and <i>ε</i> a positive number. We say that <i>F</i> is <i>α</i>-admissible when <span class="mathjax-tex">$\alpha(x,y) \geq1$</span> implies <span class="mathjax-tex">$\alpha(Fx, Fy )\geq1$</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title=" Miandaragh, MA, Postolache, M, Rezapour, S: Some approximate fixed point results for generalized α-contractive mappings. U.P.B. Sci. Bull., Ser. A 75(2), 3-10 (2013) " href="/articles/10.1186/s13662-017-1258-3#ref-CR12" id="ref-link-section-d72216e2228">12</a>]. When <span class="mathjax-tex">$d(F x_{0},x_{0}) \leq\varepsilon$</span>, <span class="mathjax-tex">$x_{0}\in Y $</span> is called an <i>ε</i>-fixed point of <i>F</i>.</p><p>We say that <i>F</i> possesses the approximate fixed point property when <i>F</i> possesses an <i>ε</i>-fixed point for all <span class="mathjax-tex">$\varepsilon> 0$</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title=" Miandaragh, MA, Postolache, M, Rezapour, S: Some approximate fixed point results for generalized α-contractive mappings. U.P.B. Sci. Bull., Ser. A 75(2), 3-10 (2013) " href="/articles/10.1186/s13662-017-1258-3#ref-CR12" id="ref-link-section-d72216e2349">12</a>]. We recall that some mappings admit approximate fixed points while possessing no fixed points [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title=" Miandaragh, MA, Postolache, M, Rezapour, S: Some approximate fixed point results for generalized α-contractive mappings. U.P.B. Sci. Bull., Ser. A 75(2), 3-10 (2013) " href="/articles/10.1186/s13662-017-1258-3#ref-CR12" id="ref-link-section-d72216e2353">12</a>]. Let <span class="mathjax-tex">$\mathcal{R}$</span> be the set of all continuous mappings <span class="mathjax-tex">$f: [0,\infty )^{5} \to [0,\infty)$</span> fulfilling <span class="mathjax-tex">$f(1,1,1,2,0)= f(1,1,1,0,2):=h_{1}\in(0,1)$</span>, <span class="mathjax-tex">$f(\mu x_{1},\mu x_{2},\mu x_{3},\mu x_{4},\mu x_{5}) \leq\mu f(x_{1}, x_{2},x_{3},x_{4},x_{5})$</span> for all <span class="mathjax-tex">$(x_{1},x_{2},x_{3},x_{4},x_{5}) \in[0,\infty)^{5} $</span> and <span class="mathjax-tex">$\mu\geq0 $</span> and also <span class="mathjax-tex">$f( x_{1},x_{2},x_{3},0,x_{4}) \leq f( y_{1},y_{2},y_{3},0,y_{4})$</span> and <span class="mathjax-tex">$f(x_{1},x_{2},x_{3},x_{4},0)\leq f(y_{1},y_{2},y_{3},y_{4},0)$</span> when <span class="mathjax-tex">$x_{1},\dots ,x_{4},y_{1},\dots,y_{4} \in[0,\infty)$</span> with <span class="mathjax-tex">$x_{j}< y_{j} $</span> for <span class="mathjax-tex">$j=1,2,3,4$</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title=" Miandaragh, MA, Postolache, M, Rezapour, S: Some approximate fixed point results for generalized α-contractive mappings. U.P.B. Sci. Bull., Ser. A 75(2), 3-10 (2013) " href="/articles/10.1186/s13662-017-1258-3#ref-CR12" id="ref-link-section-d72216e3160">12</a>]. <i>F</i> denotes a generalized <i>α</i>-contractive mapping when there exists <span class="mathjax-tex">$f \in\mathcal{R}$</span> obeying </p><div id="Equa" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\alpha(x,y)\,d(Fx,Fy)\leq f \bigl(d(x,y),d(x,Fx),d(y,Fy),d(x,Fy),d(y,Fx) \bigr) $$</span></div></div><p> for all <span class="mathjax-tex">$x,y \in Y$</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title=" Miandaragh, MA, Postolache, M, Rezapour, S: Some approximate fixed point results for generalized α-contractive mappings. U.P.B. Sci. Bull., Ser. A 75(2), 3-10 (2013) " href="/articles/10.1186/s13662-017-1258-3#ref-CR12" id="ref-link-section-d72216e3380">12</a>]. We need the following result.</p> <h3 class="c-article__sub-heading" id="FPar2">Theorem 2.2</h3> <p>[<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title=" Miandaragh, MA, Postolache, M, Rezapour, S: Some approximate fixed point results for generalized α-contractive mappings. U.P.B. Sci. Bull., Ser. A 75(2), 3-10 (2013) " href="/articles/10.1186/s13662-017-1258-3#ref-CR12" id="ref-link-section-d72216e3390">12</a>]</p> <p> <i>Let</i> <span class="mathjax-tex">$(Y,d)$</span> <i>denoting a metric space</i>, <span class="mathjax-tex">$\alpha: Y\times Y \to[0,\infty )$</span> <i>be a mapping and</i> <i>F</i> <i>denoting a generalized</i> <i>α</i>-<i>contractive and</i> <i>α</i>-<i>admissible self</i>-<i>map on</i> <i>Y</i>. <i>Let us suppose that there exists</i> <span class="mathjax-tex">$x_{0}\in Y$</span> <i>obeying</i> <span class="mathjax-tex">$\alpha(x_{0},Fx_{0}) \geq1$</span>. <i>Then</i> <i>F</i> <i>possesses an approximate fixed point</i>.</p> </div></div></section><section data-title="The results"><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>The results</h2><div class="c-article-section__content" id="Sec3-content"><p>We use the main idea of [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title=" Baleanu, D, Mousalou, A, Rezapour, S: A new method for investigating some fractional integro-differential equations involving the Caputo-Fabrizio derivative. Adv. Differ. Equ. 2017, 51 (2017) " href="/articles/10.1186/s13662-017-1258-3#ref-CR6" id="ref-link-section-d72216e3605">6</a>] for obtaining our results in this work.</p><p>As is well known, by using the Cauchy formula for repeated integration, we get </p><div id="Equb" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} J^{n}u(t)&= \underbrace{ \int_{0}^{t=t_{n}} \int_{0}^{s=t_{n-1}} \int_{0}^{t_{n-2}} \cdots \int_{0}^{t_{1}} u(t_{0}) \,dt_{0},dt_{1}, \ldots,d(t_{n-2}) \,ds}_{n} \\ &=\frac {1}{(n-1)!} \int_{0}^{t}u(s) (t-s)^{n-1}\,ds, \end{aligned}$$ </span></div></div><p> for all <span class="mathjax-tex">$n\geq1$</span>, <span class="mathjax-tex">$a, t\in\Bbb{R}$</span> and <span class="mathjax-tex">$t>0$</span>. If <i>n</i> is substituted by a positive real number <i>α</i> and <span class="mathjax-tex">$(n-1)!$</span> by its generalization <span class="mathjax-tex">$\Gamma(\alpha)$</span>, a formula for fractional integration is obtained for the fractional operator <span class="mathjax-tex">$J^{\alpha}u(t)=\frac{1}{ \Gamma(\alpha)}\int _{0}^{t}u(s)(t-s)^{\alpha-1}\,ds$</span>, which is called the Riemann-Liouville fractional integral of order <i>α</i>.</p><p>Let us to consider the following symbols: </p><div id="Equc" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} &{}^{CF }D^{\alpha^{[n]}}u(t):= \underbrace{^{CF }D^{\alpha} \bigl(^{CF }D^{\alpha} \bigl(^{CF }D^{\alpha}\cdots \bigl(^{CF }D^{\alpha}u(t) \bigr)\cdots \bigr) \bigr)}_{n},\qquad {}^{CF }D^{\alpha^{[0]}}u(t)=u(t),\hspace{26.5pt} (*) \\ & \int_{0}^{t^{[n]}} u(s)\,ds=\underbrace{ \int_{0}^{t=t_{n}} \int_{0}^{s=t_{n-1}} \int_{0}^{t_{n-2}} \cdots \int_{0}^{t_{1}} u(t_{0}) \,dt_{0},dt_{1}, \ldots ,d(t_{n-2}) \,ds}_{n}=J^{n}u(t),\hspace{18pt} (**) \end{aligned}$$ </span></div></div><p> and <span class="mathjax-tex">$J^{0}u(t)=\int_{0}^{t^{[0]}} u(s)\,ds:=u(t)$</span>. Also, we define </p><div id="Equd" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \bigl(a_{\alpha} +b_{\alpha}Ju(t) \bigr)^{[n]}={}& \biggl(a_{\alpha} +b_{\alpha} \int_{0}^{t} u(s)\,ds \biggr)^{[n]} \\ ={}& \begin{pmatrix}n\\0\end{pmatrix}a_{\alpha}^{n}b^{0}_{\alpha } \int_{0}^{t^{[0]}}u(s)\,ds+ \begin{pmatrix}n\\1\end{pmatrix}a_{\alpha}^{n-1} b_{\alpha}^{1} \int _{0}^{t^{[1]}}u(s)\,ds \\ &{}+\cdots+ \begin{pmatrix}n\\n-1\end{pmatrix} a_{\alpha}^{1} b_{\alpha}^{n-1} \int _{0}^{t^{[n-1]}}u(s)\,ds + \begin{pmatrix}n\\n\end{pmatrix} a_{\alpha}^{0} b_{\alpha}^{n} \int _{0}^{t^{[n]}}u(s)\,ds \\ ={}&\sum_{i=0} ^{n} \begin{pmatrix}n\\i\end{pmatrix} a_{\alpha}^{n-i} b_{\alpha}^{i} \int_{0}^{t^{[i]}}u(s)\,ds \\ ={}&\sum_{i=0} ^{n} \begin{pmatrix}n\\i\end{pmatrix} a_{\alpha}^{n-i} b_{\alpha}^{i}J^{i}u(t). \end{aligned}$$ </span></div></div> <p>Below we present the main results of the manuscript.</p> <h3 class="c-article__sub-heading" id="FPar3">Lemma 3.1</h3> <p> <i>Let</i> <span class="mathjax-tex">$u_{1}, v_{1}\in H^{1}(0,1)$</span> <i>and</i> <i>L</i> <i>a real number obeying</i> <span class="mathjax-tex">$\vert u_{1}(s)-v_{1}(s) \vert \leq L$</span> <i>for all</i> <span class="mathjax-tex">$s\in[0,1]$</span>. <i>Thus</i>, <span class="mathjax-tex">$\vert ^{CF}D^{\alpha^{[n]}}u_{1}(s)-{^{CF}}D^{\alpha^{[n]}}v_{1}(s) \vert \leq\frac{(2-\alpha)^{n}}{(1-\alpha)^{2n}}L$</span> <i>for all</i> <span class="mathjax-tex">$s\in[0,1]$</span>. <i>This result implies that</i> <span class="mathjax-tex">$\vert ^{CF}D^{\alpha^{[n]}}u_{1}(s) \vert \leq\frac {(2-\alpha)^{n}}{(1-\alpha)^{2n}}L$</span> <i>for all</i> <span class="mathjax-tex">$s\in[0,1]$</span> <i>whenever</i> <span class="mathjax-tex">$u_{1}\in H^{1}(0,1)$</span> <i>with</i> <span class="mathjax-tex">$\vert u_{1}(s) \vert \leq L$</span> <i>for some</i> <span class="mathjax-tex">$L\geq0$</span> <i>and all</i> <span class="mathjax-tex">$s\in[0,1]$</span>.</p> <h3 class="c-article__sub-heading" id="FPar4">Lemma 3.2</h3> <p> <i>Let</i> <span class="mathjax-tex">$u_{1}, v_{1}\in H^{1}(0,1)$</span> <i>with</i> <span class="mathjax-tex">$u_{1}(0)=v_{1}(0)$</span> <i>and</i> <i>La real number fulfilling</i> <span class="mathjax-tex">$\vert u_{1}(s)-v_{1}(s) \vert \leq L$</span> <i>for all</i> <span class="mathjax-tex">$s\in[0,1]$</span>. <i>Thus</i>, <span class="mathjax-tex">$\vert ^{CF}D^{\alpha^{[n]}}u_{1}(s)-{^{CF}}D^{\alpha^{[n]}}v_{1}(s) \vert \leq\frac{1}{(1-\alpha)^{2n}}L$</span> <i>for all</i> <span class="mathjax-tex">$s\in[0,1]$</span>. <i>This result implies that</i> <span class="mathjax-tex">$\vert ^{CF}D^{\alpha^{[n]}}u_{1}(s) \vert \leq\frac {1}{(1-\alpha)^{2n}}L$</span> <i>for all</i> <span class="mathjax-tex">$s\in[0,1]$</span> <i>whenever</i> <span class="mathjax-tex">$u_{1}\in H^{1}(0,1)$</span> <i>with</i> <span class="mathjax-tex">$u_{1}(0)=0$</span> <i>and</i> <span class="mathjax-tex">$\vert u_{1}(s) \vert \leq L$</span> <i>for some</i> <span class="mathjax-tex">$L\geq0$</span> <i>and all</i> <span class="mathjax-tex">$s\in[0,1]$</span>.</p> <h3 class="c-article__sub-heading" id="FPar5">Lemma 3.3</h3> <p> <i>Let</i> <span class="mathjax-tex">$u_{1},v_{1}\in C_{\mathbb{R}}[0,1]$</span> <i>and there is</i> <span class="mathjax-tex">$L\geq0$</span> <i>satisfying</i> <span class="mathjax-tex">$\vert u_{1}(s)-v_{1}(s) \vert \leq L$</span> <i>for all</i> <span class="mathjax-tex">$s\in[0,1]$</span>. <i>Thus</i>, <span class="mathjax-tex">$\vert ^{CF}I^{\alpha^{[n]}}u_{1}(s)-{}^{CF}I^{\alpha^{[n]}}v_{1}(s) \vert \leq L$</span> <i>for all</i> <span class="mathjax-tex">$s\in[0,1]$</span>.</p> <p>This result implies that <span class="mathjax-tex">$\vert ^{CF}I^{\alpha^{[n]}}u_{1}(s) \vert \leq L$</span> for all <span class="mathjax-tex">$s\in[0,1]$</span> whenever <span class="mathjax-tex">$u\in C_{\mathbb{R}}[0,1]$</span> with <span class="mathjax-tex">$\vert u_{1}(s) \vert \leq L$</span> for some <span class="mathjax-tex">$L\geq0$</span> and all <span class="mathjax-tex">$s\in[0,1]$</span>.</p> <h3 class="c-article__sub-heading" id="FPar6">Lemma 3.4</h3> <p> <i>Let</i> <span class="mathjax-tex">$0<\alpha<1$</span> <i>and</i> <span class="mathjax-tex">$u, v\in H^{1}(0,1)$</span>. <i>The problem</i> <span class="mathjax-tex">$^{CF}D^{\alpha^{[n]}}u(t)=v(t)$</span>, <span class="mathjax-tex">$u(0)=0$</span>, <i>possesses the following unique solution</i>: <span class="mathjax-tex">$u(t)=(a_{\alpha} +b_{\alpha}Jv(t))^{[n]}$</span>, <i>where</i> <span class="mathjax-tex">$^{CF}D^{\alpha^{[n]}}$</span> <i>is defined by </i>(<span class="stix">∗</span>).</p> <h3 class="c-article__sub-heading" id="FPar7">Proof</h3> <p>By using the Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-017-1258-3#FPar1">2.1</a> for <span class="mathjax-tex">$^{CF} D^{\alpha} u(t)=v(t)$</span>, we get <span class="mathjax-tex">$u(t)= a_{\alpha} v(t) +b_{\alpha} \int_{0}^{t} v(s)\,ds$</span>. Also by using Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-017-1258-3#FPar1">2.1</a> for <span class="mathjax-tex">$^{CF}D^{\alpha^{[2]}} u(t)=v(t)$</span>, we obtain <span class="mathjax-tex">$^{CF}D^{\alpha}u(t)= a_{\alpha} v(t) +b_{\alpha} \int_{0}^{t} v(s)\,ds$</span>. Hence, </p><div id="Eque" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} u(t)={}&a_{\alpha} \biggl( a_{\alpha} v(t) +b_{\alpha} \int_{0}^{t} v(s)\,ds \biggr)+b_{\alpha} \int_{0}^{t} \biggl( a_{\alpha} v(s) +b_{\alpha} \int_{0}^{s} v(r)\,dr \biggr) \,ds \\ ={}& a_{\alpha}^{2} v(t)+2a_{\alpha}b_{\alpha} \int_{0}^{t} v(s)\,ds+b_{\alpha}^{2} \int_{0}^{t} \int_{0}^{s} v(r)\,dr\,ds \\ ={}& \biggl(a_{\alpha} +b_{\alpha} \int_{0}^{t} v(s)\,ds \biggr)^{[2]}. \end{aligned}$$ </span></div></div><p> Suppose that <span class="mathjax-tex">$u(t)=(a_{\alpha} +b_{\alpha}Jv(t))^{[n]} $</span> is the solution of the equation <span class="mathjax-tex">$^{CF}D^{\alpha^{[n]}} u(t)=v(t)$</span>. We show that <span class="mathjax-tex">$u(t)=(a_{\alpha} +b_{\alpha}Jv(t))^{[n+1]} $</span> is the solution of the equation <span class="mathjax-tex">$^{CF}D^{\alpha^{[n+1]}} u(t)=v(t)$</span>.</p> <p>If <span class="mathjax-tex">$^{CF}D^{\alpha^{[n]}} (^{CF}D^{\alpha} u(t))=v(t)$</span>, then <span class="mathjax-tex">$^{CF}D^{\alpha} u(t)=(a_{\alpha} +b_{\alpha}Jv(t))^{[n]} $</span>. Thus, </p><div id="Equf" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} u(t)={}& a_{\alpha} \bigl(a_{\alpha} +b_{\alpha}Jv(t) \bigr)^{[n]} +b_{\alpha} \int_{0}^{t} \bigl(a_{\alpha} +b_{\alpha}Jv(s) \bigr)^{[n]}\,ds \\ ={}&a_{\alpha} \Biggl[ \begin{pmatrix}n\\0\end{pmatrix}a_{\alpha}^{n}b^{0}_{\alpha} \int _{0}^{t^{[0]}}v(s)\,ds+ \begin{pmatrix}n\\1\end{pmatrix} a_{\alpha}^{n-1} b_{\alpha}^{1} \int_{0}^{t^{[1]}}v(s)\,ds \\ &{}+\cdots+ \begin{pmatrix}n\\n-1\end{pmatrix} a_{\alpha}^{1} b_{\alpha}^{n-1} \int _{0}^{t^{[n-1]}}v(s)\,ds + \begin{pmatrix}n\\n\end{pmatrix} a_{\alpha}^{0} b_{\alpha}^{n} \int _{0}^{t^{[n]}}v(s)\,ds \Biggr] \\ &{}+b_{\alpha} \Biggl[ \begin{pmatrix}n\\0\end{pmatrix}a_{\alpha}^{n}b^{0}_{\alpha} \int _{0}^{t^{[1]}}v(s)\,ds+ \begin{pmatrix}n\\1\end{pmatrix} a_{\alpha}^{n-1} b_{\alpha}^{1} \int_{0}^{t^{[2]}}v(s)\,ds \\ &{}+\cdots+ \begin{pmatrix}n\\n-1\end{pmatrix} a_{\alpha}^{1} b_{\alpha}^{n-1} \int_{0}^{t^{[n]}}v(s)\,ds + \begin{pmatrix}n\\n\end{pmatrix} a_{\alpha}^{0} b_{\alpha}^{n} \int_{0}^{t^{[n+1]}}v(s)\,ds \Biggr] \\ ={}& \begin{pmatrix}n\\0\end{pmatrix}a_{\alpha}^{n+1}b^{0}_{\alpha} \int _{0}^{t^{[0]}}v(s)\,ds+ \left[ \begin{pmatrix}n\\1\end{pmatrix} + \begin{pmatrix}n\\0\end{pmatrix} \right]a_{\alpha}^{n} b_{\alpha}^{1} \int _{0}^{t^{[1]}}v(s)\,ds \\ &{}+\cdots+ \left[ \begin{pmatrix}n\\n\end{pmatrix} + \begin{pmatrix}n\\n-1\end{pmatrix} \right]a_{\alpha}^{1} b_{\alpha}^{n} \int _{0}^{t^{[n]}}v(s)\,ds + \begin{pmatrix}n\\n\end{pmatrix} a_{\alpha}^{0} b_{\alpha}^{n+1} \int _{0}^{t^{[n+1]}}v(s)\,ds \\ ={}& \begin{pmatrix}n+1\\0\end{pmatrix} a_{\alpha}^{n+1}b^{0}_{\alpha} \int_{0}^{t^{[0]}}v(s)\,ds+ \begin{pmatrix}n+1\\1\end{pmatrix} a_{\alpha}^{n} b_{\alpha}^{1} \int_{0}^{t^{[1]}}v(s)\,ds \\ &{}+\cdots+ \begin{pmatrix}n+1\\n\end{pmatrix} +a_{\alpha}^{1} b_{\alpha}^{n} \int_{0}^{t^{[n]}}v(s)\,ds \\ &{}+ \begin{pmatrix}n+1\\n+1\end{pmatrix} a_{\alpha}^{0} b_{\alpha}^{n+1} \int_{0}^{t^{[n+1]}}v(s)\,ds \\ ={}& \bigl(a_{\alpha} +b_{\alpha}Jv(t) \bigr)^{[n+1]} \end{aligned}$$ </span></div></div><p> and so <span class="mathjax-tex">$u(t)=(a_{\alpha} +b_{\alpha}\int_{0}^{t} v(s)\,ds)^{[n]}=(a_{\alpha} +b_{\alpha}Jv(t))^{[n]}$</span> holds for all <i>n</i>. □</p> <p>In the last result, we used some notation such as <span class="mathjax-tex">$\int _{0}^{t^{[n]}}u(s)\,ds$</span>, which was introduced by (<span class="stix">∗</span><span class="stix">∗</span>). We need the following result.</p> <h3 class="c-article__sub-heading" id="FPar8">Lemma 3.5</h3> <p>[<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title=" Tspkin, AG, Tsypkin, GG: Mathematical Formulas. Mir, Moscow (1985) " href="/articles/10.1186/s13662-017-1258-3#ref-CR17" id="ref-link-section-d72216e11407">17</a>]</p> <p> <i>Suppose</i> <span class="mathjax-tex">$t\in\Bbb{R}$</span>, <i>then</i> <span class="mathjax-tex">$e^{t}=\sum_{i=0} ^{\infty} \frac {t^{i}}{i!} $</span> <i>for</i> <span class="mathjax-tex">$0< \vert t \vert <\infty$</span>, <span class="mathjax-tex">$t\Pi_{i=1} ^{\infty}(1-\frac{t^{2}}{i^{2} \pi^{2}})=\operatorname{sin}t$</span> <i>and</i> <span class="mathjax-tex">$\Pi_{i=1} ^{\infty}(1-\frac{4t^{2}}{(2i-1)^{2} \pi^{2}})=\operatorname{cos}t$</span>.</p> <p>Let <span class="mathjax-tex">$\gamma,\lambda:[0,1] \times[0,1]\to[0,\infty)$</span> denoting two continuous maps with <span class="mathjax-tex">$\sup_{r\in I} \vert \int_{0}^{t} \lambda(r,s) \,ds \vert <\infty$</span> and <span class="mathjax-tex">$\sup_{r\in I} \vert \int_{0}^{r} \gamma(r,s) \,ds \vert <\infty$</span>, respectively.</p><p>Let <i>ϕ</i> and <i>φ</i> be two maps defined as <span class="mathjax-tex">$(\phi u)(r)= \int_{0}^{r} \gamma(r,s)u(s)\,ds $</span> and <span class="mathjax-tex">$(\varphi u)(r)= \int_{0}^{r} \lambda(r,s)u(s)\,ds $</span>, respectively. Let <span class="mathjax-tex">$\eta\in L^{\infty}(I)$</span> with <span class="mathjax-tex">$\eta^{\ast}=\sup_{t\in I} \vert \eta(t) \vert $</span> and <span class="mathjax-tex">$k,h$</span> and <i>g</i> be continuous on <span class="mathjax-tex">$[0,1]$</span> with <span class="mathjax-tex">$M_{1}=\sup_{t\in I} \vert k(t) \vert $</span>, <span class="mathjax-tex">$M_{2}=\sup_{t\in I} \vert h(t) \vert $</span> and <span class="mathjax-tex">$M_{3}=\sup_{t\in I} \vert g(t) \vert $</span>. Put <span class="mathjax-tex">$\gamma_{0}=\sup \vert \int_{0}^{t} \gamma(t,s) \,ds \vert $</span> and <span class="mathjax-tex">$\lambda_{0}=\sup \vert \int_{0}^{t} \lambda(t,s) \,ds \vert $</span>. Below we study the fractional-order integro-differential problem </p><div id="Equ1" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} ^{CF}D^{\alpha^{[n]}}z(s)={}& \mu k(s){}^{CF}D^{\beta^{[m]}} \bigl(z(s)+h(s){}^{CF}D^{\gamma^{[p]}} z(s) \bigr) \\ &{}+f \bigl(s,z(s),(\phi z) (s),(\varphi z) (s), {}^{CF}I^{\theta^{[q]}} z(s),g(s){}^{CF}D^{\delta^{[r]}} z(s) \bigr) \end{aligned}$$ </span></div><div class="c-article-equation__number"> (1) </div></div><p> with <span class="mathjax-tex">$z(0)=0$</span> under some conditions, where <span class="mathjax-tex">$\mu>1$</span> and <span class="mathjax-tex">$\alpha,\beta ,\gamma,\theta,\delta\in(0,1)$</span> as well as <span class="mathjax-tex">$n,m,p,q,r \geq1$</span>. Since <span class="mathjax-tex">$^{CF}D^{\alpha^{[n]}}u\in H^{1}$</span> for all <i>n</i>, the right hand is too.</p> <h3 class="c-article__sub-heading" id="FPar9">Theorem 3.6</h3> <p> <i>Let</i> <span class="mathjax-tex">$f:[0,1]\times\Bbb {R}^{5}\to\Bbb{R}$</span> <i>be a continuous function such that</i> </p><div id="Equg" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} & \bigl\vert f(t_{1},x_{1},y_{1},w_{1},u_{1},u_{2})-f \bigl(t_{1},x_{1}^{\prime},y_{1}^{\prime},w_{1}^{\prime},v_{1},v_{2} \bigr) \bigr\vert \\ &\quad\leq\eta(t_{1}) \bigl( \bigl\vert x_{1}-x_{1}^{\prime} \bigr\vert + \bigl\vert y_{1}-y_{1}^{\prime} \bigr\vert + \bigl\vert w_{1}-w_{1}^{\prime} \bigr\vert + \vert u_{1}-v_{1} \vert + \vert u_{2}-v_{2} \vert \bigr) \end{aligned}$$ </span></div></div><p> <i>for all</i> <span class="mathjax-tex">$t_{1}\in I$</span> <i>and</i> <span class="mathjax-tex">$x_{1},y_{1},w_{1},x_{1}^{\prime},y_{1}^{\prime},w_{1}^{\prime},u_{1}, u_{2},v_{1},v_{2} \in\Bbb{R}$</span>. <i>Then the stated problem</i> (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-017-1258-3#Equ1">1</a>) <i>possesses an approximate solution when</i> <span class="mathjax-tex">$\Delta=\eta^{*}(2+\gamma_{0} + \lambda_{0} + \frac{M_{3}}{(1-{\delta })^{2r}})+{ \mu} (\frac{M_{1}M_{2}}{(1-{\gamma})^{2p}(1-{\beta})^{2m}}+\frac {M_{1}}{(1-{\beta})^{2m}})<1$</span>.</p> <h3 class="c-article__sub-heading" id="FPar10">Proof</h3> <p>Let <span class="mathjax-tex">$H^{1}$</span> equipped with <span class="mathjax-tex">$d(z,v)= \Vert z-v \Vert $</span> on <i>X</i>, such that <span class="mathjax-tex">$\Vert z \Vert =\sup_{t\in I} \vert z(t) \vert $</span>. Let <span class="mathjax-tex">$F:H^{1}\to H^{1}$</span> be a map defined as follows: </p><div id="Equh" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} (Fz) (t) ={}& \biggl(a_{ \alpha} +b_{\alpha} \int_{0}^{t} \bigl[ \mu k(s){}^{CF}D^{\beta^{[m]}} \bigl(z(s)+h(s){}^{CF}D^{\gamma^{[p]}}z(s) \bigr) \\ &{}+ f \bigl(s,z(s),(\phi z) (s),(\varphi z) (s), {}^{CF}I^{\theta^{[q]}} z(s),g(s){}^{CF}D^{\delta^{[r]}} z(s) \bigr) \bigr]\,ds \biggr)^{[n]}, \end{aligned}$$ </span></div></div><p> where <span class="mathjax-tex">$a_{\alpha}$</span> and <span class="mathjax-tex">$b_{\alpha}$</span> are introduced in Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-017-1258-3#FPar1">2.1</a> and the notation <span class="mathjax-tex">$^{CF}I^{\theta^{[q]}} z(s)$</span> and <span class="mathjax-tex">$^{CF}D^{\gamma^{[p]}}z(s)$</span> is introduced by (<span class="stix">∗</span>) and (<span class="stix">∗</span><span class="stix">∗</span>). By using Lemmas <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-017-1258-3#FPar4">3.2</a> and <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-017-1258-3#FPar5">3.3</a>, we get </p><div id="Equi" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} & \bigl\vert \mu k(s){}^{CF}D^{\beta^{[m]}} \bigl( z(s)+h(s){}^{CF}D^{\gamma ^{[p]}}z(s) \bigr) \\ &\qquad{}+f \bigl(s,z(s),(\phi z) (s),(\varphi z) (s), {}^{CF}I^{\theta^{[q]}} z(s),g(s){}^{CF}D^{\delta^{[r]}}z(s) \bigr) \\ &\qquad{}-\mu k(s){}^{CF}D^{\beta^{[m]}} \bigl( v(s)+h(s){}^{CF}D^{\gamma^{[p]}}v(s) \bigr) \\ &\qquad{}-f \bigl(s,v(s),(\phi v) (s),(\varphi v) (s), {}^{CF}I^{\theta^{[q]}} v(s),g(s){}^{CF}D^{\delta^{[r]}}v(s) \bigr) \bigr\vert \\ &\quad\leq\mu \bigl\vert \bigl\vert k(s) \bigr\vert ^{CF}D^{\beta^{[m]}} \bigl(z(s)+h(s){}^{CF}D^{\gamma ^{[p]}}z(s) \bigr)-{}^{CF}D^{\beta^{[m]}} \bigl(v(s)+h(s){}^{CF}D^{\gamma^{[p]}}v(s) \bigr) \bigr\vert \\ &\qquad{}+ \bigl\vert f \bigl(s,z(s),(\phi z) (s),(\varphi z) (s), {}^{CF}I^{\theta^{[q]}} z(s),g(s){}^{CF}D^{\delta^{[r]}} z(s) \bigr) \\ &\qquad{} -f \bigl(s,v(s),(\phi v) (s),(\varphi v) (s), {}^{CF}I^{\theta^{[q]}} v(s),g(s){}^{CF}D^{\delta^{[r]}} v(s) \bigr) \bigr\vert \\ &\quad\leq\mu \bigl[ \bigl\vert k(s) \bigr\vert \bigl\vert ^{CF}D^{\beta^{[m]}} \bigl(z(s)-v(s) \bigr)\bigr\vert \\ &\qquad{}+ \bigl\vert k(s) \bigr\vert \bigl\vert h(s) \bigr\vert ^{CF}D^{\beta^{[m]}} \bigl(^{CF}D^{\gamma ^{[p]}} \bigl(z(s)-v(s) \bigr) \bigr) \bigr] \\ &\qquad{}+ \bigl\vert \eta(s) \bigr\vert \bigl[ \bigl\vert z(s)-v(s) \bigr\vert + \bigl\vert (\phi z) (s)-(\phi v) (s)\bigr\vert \\ &\qquad{}+ \bigl\vert (\varphi z) (s)-(\varphi v) (s)\bigr\vert + \bigl\vert ^{CF}I^{\theta^{[q]}} z(s)- {}^{CF}I^{\theta^{[q]}} v(s) \bigr\vert + \bigl\vert g(s) \bigr\vert \bigl\vert ^{CF}D^{\delta^{[r]}} z(s)-{}^{CF}D^{\delta^{[r]}} v(s) \bigr\vert \bigr] \\ &\quad\leq \biggl[\eta^{*} \biggl(2+\gamma_{0} + \lambda_{0} + \frac{M_{3}}{(1-{\delta })^{2r}} \biggr)+{ \mu} \biggl( \frac{M_{1}M_{2}}{(1-{\gamma})^{2p}(1-{\beta})^{2m}}+\frac {M_{1}}{(1-{\beta})^{2m}} \biggr) \biggr]\Vert z-v\Vert . \end{aligned}$$ </span></div></div><p> Since <span class="mathjax-tex">$z(0)=0$</span>, we obtain </p><div id="Equj" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} & \bigl\vert (Fz) (t)-(Fv) (t) \bigr\vert \\ &\quad\leq \biggl(a_{ \alpha} +b_{\alpha} \int_{0}^{t } \biggl[\eta^{*} \biggl(2+ \gamma_{0} + \lambda_{0} + \frac{M_{3}}{(1-{\delta})^{2r}} \biggr) \\ &\qquad{}+{ \mu} \biggl( \frac {M_{1}M_{2}}{(1-{\gamma})^{2p} (1-{\beta})^{2m}}+\frac{M_{1}}{(1-{\beta})^{2m}} \biggr) \biggr] \Vert z-v \Vert \,ds \biggr)^{[n]}, \end{aligned}$$ </span></div></div><p> and so </p><div id="Equk" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} &\Vert Fz-Fv \Vert \\ &\quad\leq(a_{\alpha} + b_{\alpha} )^{n} \biggl[ \eta^{*} \biggl(2+\gamma_{0} + \lambda_{0} + \frac{M_{3}}{(1-{\delta})^{2r}} \biggr) \\ &\qquad{}+ { \mu} \biggl( \frac{M_{1}M_{2}}{(1-{\gamma})^{2p}(1-{\beta})^{2m}}+\frac {M_{1}}{(1-{\beta})^{2m}} \biggr) \biggr] \Vert z-v \Vert \end{aligned}$$ </span></div></div><p> for all <span class="mathjax-tex">$t\in I$</span> and <span class="mathjax-tex">$z,v\in H^{1}$</span>. Define the mappings <span class="mathjax-tex">$g:[0,\infty)^{5} \to [0,\infty)$</span> and <span class="mathjax-tex">$\alpha:H^{1}\times H^{1}\to[0,\infty)$</span> by <span class="mathjax-tex">$g( x_{1}, x_{2}, x_{3},x_{4},x_{5}) = \frac{\Delta}{9}( 3x_{1} +2x_{2}+4x_{3})$</span> and <span class="mathjax-tex">$\alpha(x,y) = 1$</span> for all <span class="mathjax-tex">$x,y\in H^{1}$</span> and <span class="mathjax-tex">$x_{1},\dots,x_{5}\in[0,\infty)$</span>. One can easily check that <span class="mathjax-tex">$g \in\mathcal{R}$</span>. In addition we conclude that <i>F</i> denotes a generalized <i>α</i>-contractive map. From Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-017-1258-3#FPar2">2.2</a>, we conclude that <i>F</i> possesses an approximate fixed point which is an approximate solution of (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-017-1258-3#Equ1">1</a>). □</p> <p>Suppose that functions <i>k</i>, <i>s</i>, <i>h</i>, <i>g</i> and <i>q</i> are bounded on <span class="mathjax-tex">$[0,1]$</span> with <span class="mathjax-tex">$M_{1}=\sup_{t\in I} \vert k(t) \vert <\infty,M_{2}=\sup_{t\in I} \vert s(t) \vert <\infty, M_{3}=\sup_{t\in I} \vert h(t) \vert <\infty, M_{4}=\sup_{t\in I} \vert g(t) \vert <\infty$</span> and <span class="mathjax-tex">$M_{5}= \sup_{t\in I} \vert q(t) \vert <\infty$</span>. Now we discus the following problem: </p><div id="Equ2" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} ^{CF}D^{\alpha^{[n]}}u(r)={}&\lambda k(r){}^{CF}D^{\beta^{[m]}}u(r)+{ \mu }s(r){}^{CF}I^{\rho^{[p]}}u(r) \\ &{}+ \sum_{i=0} ^{\infty}\frac{^{CF}D^{ \theta^{[i]}}f_{1}(r,u(r),(\phi u)(r), h(r){}^{CF}I^{\nu^{[q]}} u(r),g(r){}^{CF}D^{ \delta^{[r]}}u(r) )}{i!} \\ &{}+ \int_{0}^{r} f_{2} \Biggl(s,u(s),(\varphi u) (s),q(s)\sum_{i=0} ^{\infty} \frac {^{CF}D^{ \gamma^{[i]}}u(s) }{d^{i}} \Biggr)\,ds \end{aligned}$$ </span></div><div class="c-article-equation__number"> (2) </div></div><p> with <span class="mathjax-tex">$u(0)=0$</span> under some conditions, where <span class="mathjax-tex">$\lambda,\mu\geq0$</span>, <span class="mathjax-tex">$\alpha , \beta,\rho,\theta,\nu,\delta\in(0,1)$</span>, <span class="mathjax-tex">$\vert \frac{1}{d(1-\gamma)^{2}} \vert <1 $</span> and <span class="mathjax-tex">$n,m,p,q,r,k \geq1$</span>. The functions <i>k</i>, <i>s</i>, <i>h</i>, <i>g</i> and <i>q</i> maybe are not continuous, but the right hand of (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-017-1258-3#Equ2">2</a>) should be a member of <span class="mathjax-tex">$H^{1}$</span> because <span class="mathjax-tex">$^{CF}D^{\alpha^{[n]}}u\in H^{1}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar11">Theorem 3.7</h3> <p> <i>Suppose that</i> <span class="mathjax-tex">$f_{1}:[0,1]\times\Bbb {R}^{4}\rightarrow\Bbb{R}$</span> <i>and</i> <span class="mathjax-tex">$f_{2}:[0,1]\times\Bbb {R}^{3}\rightarrow\Bbb{R}$</span> <i>are integrable functions such that</i> </p><div id="Equl" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} & \bigl\vert f_{1}(t_{1},x_{1},y_{1},w_{1},v_{1})-f_{1} \bigl(t,x_{1}^{\prime},y_{1}^{\prime},w_{1}^{\prime},v_{1}^{\prime} \bigr) \bigr\vert \\ &\quad \leq\xi _{1} \bigl\vert x_{1} -x_{1}^{\prime} \bigr\vert +\xi_{2} \bigl\vert y_{1} -y_{1}^{\prime} \bigr\vert +\xi_{3} \bigl\vert w_{1}-w_{1}^{\prime} \bigr\vert + \xi_{4} \bigl\vert v_{1} -v_{1}^{\prime} \bigr\vert , \\ & \bigl\vert f_{2}(t_{1},x_{1},y_{1},w_{1})-f_{2} \bigl(t_{1},x_{1}^{\prime},y_{1}^{\prime},w_{1}^{\prime} \bigr) \bigr\vert \\ &\quad\leq\xi_{1} ^{\prime} \bigl\vert x_{1} -x_{1}^{\prime} \bigr\vert + \xi_{2}^{\prime} \bigl\vert y_{1} -y_{1}^{\prime} \bigr\vert + \xi^{\prime} _{3} \bigl\vert w_{1}-w_{1}^{\prime} \bigr\vert \end{aligned}$$ </span></div></div><p> <i>for some nonnegative real numbers</i> <span class="mathjax-tex">$\lambda, \mu,\xi_{1},\xi_{2},\xi _{3},\xi_{4},\xi^{\prime}_{1},\xi^{\prime}_{2},\xi^{\prime}_{3} $</span> <i>and all</i> <span class="mathjax-tex">$x_{1},y_{1},w_{1},v_{1},x_{1}^{\prime},y_{1}^{\prime},w_{1}^{\prime}, v_{1}^{\prime}\in \Bbb{R}$</span> <i>and</i> <span class="mathjax-tex">$t \in I$</span>. <i>If</i> <span class="mathjax-tex">$\Delta= [\lambda\frac{M_{1}}{(1-\beta)^{2m}}+\mu M_{2} + e^{\frac {1}{(1-\theta)^{2}}} (\xi_{1} + \xi_{2} \gamma_{0} +\xi_{3} M_{3} +\xi_{4} \frac {M_{4}}{(1-\delta)^{2r}} ) + \xi_{1}^{\prime} +\xi_{2}^{\prime} \lambda_{0} + \frac{\xi_{3}^{\prime}\,d(1-\gamma)^{2} M_{5}}{d(1-\gamma )^{2}-1}]<1$</span>, <i>then the stated problem</i> (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-017-1258-3#Equ2">2</a>) <i>possesses an approximate solution</i>.</p> <h3 class="c-article__sub-heading" id="FPar12">Proof</h3> <p>Let <span class="mathjax-tex">$H^{1}$</span> equipped with <span class="mathjax-tex">$d(z,v)= \Vert z-v \Vert $</span>, such that <span class="mathjax-tex">$\Vert z \Vert =\sup_{t\in I} \vert z(t) \vert $</span>. Define the map <span class="mathjax-tex">$F:H^{1}\to H^{1} $</span> by </p><div id="Equm" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} (Fz) (t)={}& \Biggl(a_{\alpha} + b_{\alpha} \int_{0}^{t } \Biggl[\lambda k(s){}^{CF}D^{\beta^{[m]}}z(s)+{ \mu}s(s){}^{CF}I^{\rho^{[p]}}z(s) \\ &{}+ \sum_{i=0} ^{\infty}\frac{^{CF}D^{ \theta^{[i]}}f_{1}(s,z(s),(\phi z)(s), h(s){}^{CF}I^{\nu^{[q]}} z(s),g(s){}^{CF}D^{ \delta^{[r]}}z(s) )}{i!} \\ &{}+ \int_{0}^{s} f_{2} \bigl(r,z(r),(\varphi z) (r),q(r) \bigr)\sum_{i=0} ^{\infty} \frac {^{CF}D^{ \gamma^{[i]}}z(r) }{d^{i}} \,dr \Biggr]\,ds \Biggr)^{[n]}. \end{aligned}$$ </span></div></div><p> By using Lemmas <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-017-1258-3#FPar4">3.2</a>, <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-017-1258-3#FPar5">3.3</a> and <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-017-1258-3#FPar8">3.5</a>, we get </p><div id="Equn" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} & \Biggl\vert \Biggl[\lambda k(s){}^{CF}D^{\beta^{[m]}}z(s)+{ \mu}s(s){}^{CF}I^{\rho^{[p]}}z(s) \\ &\qquad{}+\sum_{i=0} ^{\infty} \frac{^{CF}D^{ \theta^{[i]}}f_{1}(s,z(s),(\phi z)(s), h(s){}^{CF}I^{\nu^{[q]}} z(s),g(s){}^{CF}D^{ \delta^{[r]}}z(s) )}{i!} \\ &\qquad{}+ \int_{0}^{s} f_{2}(r,z(r),(\varphi z) (r),q(r)\sum_{i=0} ^{\infty}\frac {^{CF}D^{ \gamma^{[i]}}z(r) }{2^{i}} \,dr \Biggr] \\ &\qquad{} - \Biggl[\lambda k(s){}^{CF}D^{\beta^{[m]}}v(s)+{ \mu}s(s){}^{CF}I^{\rho^{[p]}}v(s) \\ &\qquad{}+\sum_{i=0} ^{\infty} \frac{^{CF}D^{ \theta^{[i]}}f_{1}(s,v(s),(\phi v)(s), h(s){}^{CF}I^{\nu^{[q]}} v(s),g(s){}^{CF}D^{ \delta^{[r]}}v(s) )}{i!} \\ &\qquad{}+ \int_{0}^{r} f_{2}(r,v(r),(\varphi v) (r),q(r)\sum_{i=0} ^{\infty}\frac {^{CF}D^{ \gamma^{[i]}}v(r) }{d^{i}} \,dr \Biggr] \Biggr\vert \\ &\quad \leq\lambda \bigl\vert k(s) \bigr\vert \bigl\vert ^{CF}D^{\beta^{[m]}} \bigl(z(s)-v(s) \bigr) \bigr\vert +{\mu } \bigl\vert s(s) \bigr\vert \bigl\vert ^{CF}I^{\rho^{[p]}} \bigl(z(s)-v(s) \bigr) \bigr\vert \\ &\qquad{}+\Biggl\vert \sum_{i=0} ^{\infty} \frac{1}{i!(1-\theta)^{2i}} \bigl[f_{1} \bigl(s,z(s),(\phi z) (s), h(s){}^{CF}I^{\nu^{[q]}} z(s),g(s){}^{CF}D^{ \delta^{[r]}}z(s) \bigr) \\ &\qquad{}-f_{1} \bigl(s,v(s),(\phi v) (s), h(s){}^{CF}I^{\nu^{[q]}} v(s),g(s){}^{CF}D^{ \delta ^{[r]}}v(s) \bigr) \bigr] \Biggr\vert \\ &\qquad{}+ \int_{0}^{s} \Biggl\vert f_{2}(r,z(r),( \varphi z) (r),q(r)\sum_{i=0} ^{\infty} \frac {^{CF}D^{ \gamma^{[i]}}z(r) }{d^{i}} \\ &\qquad{}- f_{2}(r,v(r),(\varphi v) (r),q(r)\sum _{i=0} ^{\infty} \frac{^{CF}D^{ \gamma^{[i]}}v(r) }{d^{i}} \Biggr\vert \,dr \\ &\quad\leq\lambda\frac{M_{1}}{(1-\beta)^{2m}} \Vert z-v \Vert +\mu M_{2} \Vert z-v \Vert \\ &\qquad{} + \sum_{i=0} ^{\infty} \frac{1}{i!(1-\theta)^{2i}} \biggl( \xi_{1} \Vert z-v \Vert + \xi_{2} \gamma_{0} \Vert z-v \Vert +\xi_{3} M_{3} \Vert z-v \Vert +\xi_{4} \frac{M_{4}}{(1-\delta)^{2r}} \Vert z-v \Vert \biggr) \\ &\qquad{}+ \xi_{1}^{\prime} \Vert z-v \Vert +\xi _{2}^{\prime} \lambda_{0} \Vert z-v \Vert +\sum _{i=0} ^{\infty} \frac{\xi_{3}^{\prime}M_{5}}{d^{i}(1-\gamma)^{2i}} \Vert z-v \Vert \\ &\quad = \biggl[\lambda \frac{M_{1}}{(1-\beta)^{2m}}+\mu M_{2} + e^{\frac{1}{(1-\theta)^{2}}} \biggl(\xi_{1} + \xi_{2} \gamma_{0} +\xi_{3} M_{3} +\xi _{4} \frac{M_{4}}{(1-\delta)^{2r}} \biggr) \\ &\qquad{}+ \xi_{1}^{\prime} + \xi_{2}^{\prime} \lambda_{0} + \frac{\xi_{3}^{\prime}\,d(1-\gamma)^{2} M_{5}}{d(1-\gamma)^{2}-1} \biggr] \Vert z-v \Vert \\ &\quad =\Delta \Vert z-v \Vert . \end{aligned}$$ </span></div></div><p> Since <span class="mathjax-tex">$z(0)=v(0)$</span>, <span class="mathjax-tex">$\vert (Fz)(t)-(Fv)(t) \vert \leq(a_{\alpha} + b_{\alpha} \int_{0}^{t }\Delta \Vert z-v \Vert \,ds)^{[n]}$</span>. Hence, <span class="mathjax-tex">$\Vert Fz-Fv \Vert \leq \Delta \Vert z-v \Vert $</span> for all <span class="mathjax-tex">$t\in I$</span> and <span class="mathjax-tex">$z,v\in H^{1}$</span>. Define the mappings <span class="mathjax-tex">$g:[0,\infty)^{5} \to[0,\infty)$</span> and <span class="mathjax-tex">$\alpha:H^{1}\times H^{1}\to[0,\infty)$</span> by <span class="mathjax-tex">$g( x_{1}, x_{2}, x_{3},x_{4},x_{5}) = \frac{\Delta}{9}( 3x_{1} +2x_{2}+4x_{3})$</span> and <span class="mathjax-tex">$\alpha(x,y) = 1$</span> for all <span class="mathjax-tex">$z,v\in H^{1}$</span> and <span class="mathjax-tex">$x_{1},\dots,x_{5}\in [0,\infty)$</span>. One can easily check that <span class="mathjax-tex">$g \in\mathcal{R}$</span>. By simple calculations we prove that <i>F</i> represents a generalized <i>α</i>-contractive map. Besides, from Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-017-1258-3#FPar2">2.2</a>, we conclude that <i>F</i> possesses an approximate fixed point which represents an approximate solution for (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-017-1258-3#Equ2">2</a>). □</p> <p>Below we show two illustrative examples.</p> <h3 class="c-article__sub-heading" id="FPar13">Example 3.1</h3> <p>Let the maps <span class="mathjax-tex">$\eta_{1}\in L^{\infty}([0,1])$</span> and <span class="mathjax-tex">$\gamma _{1},\lambda_{1}:[0,1] \times[0,1]\to[0,\infty)$</span> be <span class="mathjax-tex">$\eta_{1}(t)=\frac {e^{-(\pi t +16)}}{2}$</span>, <span class="mathjax-tex">$\gamma_{1}(t,s )=e^{t-s}$</span> and <span class="mathjax-tex">$\lambda_{1}(t,s)=e^{ \operatorname{ln}( \vert 2t-s \vert +1)}$</span>. Then <span class="mathjax-tex">$\eta^{*}=\frac {1}{2e^{16}},\gamma_{0}\leq e$</span> and <span class="mathjax-tex">$\lambda_{0}\leq e^{ \operatorname{ln}( 3)}$</span>. Now, put <span class="mathjax-tex">$\mu=\frac{1}{e^{16}}$</span>, <span class="mathjax-tex">$\alpha=\frac{1}{3}$</span>, <span class="mathjax-tex">$\beta=\frac {1}{4},\gamma=\frac{3}{5}$</span>, <span class="mathjax-tex">$\theta=\frac{6}{7}$</span>, <span class="mathjax-tex">$\delta=\frac{1}{2}$</span>, <span class="mathjax-tex">$n=61$</span>, <span class="mathjax-tex">$m=3$</span>, <span class="mathjax-tex">$p=2$</span>, <span class="mathjax-tex">$q=57$</span> and <span class="mathjax-tex">$r=73$</span>. Let <span class="mathjax-tex">$k_{1}(t)=\operatorname{sin}(t)$</span>, <span class="mathjax-tex">$h_{1}(t)=\frac{t-2}{2t+1}$</span> and <span class="mathjax-tex">$g_{1}(t)=\frac {1}{2^{141}}\operatorname{tan}^{-1}(t)$</span> be two functions. Then <span class="mathjax-tex">$M_{1}=\sup_{t\in I} \vert k_{1}(t) \vert =1,M_{2}=\sup_{t\in I} \vert h_{1}(t) \vert =2$</span> and <span class="mathjax-tex">$M_{3}=\sup_{t\in I} \vert g_{1}(t) \vert =\frac{\pi}{2^{142}}$</span>. Let us discus </p><div id="Equ3" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} ^{CF}D^{\frac{1}{3}^{[61]}}u(t)= {}&\frac{1}{e^{16}} \operatorname{sin}t^{CF}D^{\frac {1}{4}^{[3]}} \biggl( u(t)+\frac{t-2}{2t+1} ^{CF}D^{\frac{3}{5}^{[2]}} u(t) \biggr) \\ &{}+ \frac{e^{-(\pi t +16)}}{2} \biggl[2t+\frac{1}{8}u(t) +\frac{5}{11} \int_{0}^{t} e^{t-s} u(s)\,ds+ \int_{0}^{t}e^{ \operatorname{ln}( \vert 2t-s \vert +1)}u(s)\,ds \\ &{}+ \frac{1}{e^{6}}{}^{CF}I^{\frac{6}{7}^{[57]}} u(t)+ \frac {1}{2^{141}}\operatorname{tan}^{-1}(t){}^{CF}D^{\frac{1}{2}^{[73]}} u(t)) \biggr] \end{aligned}$$ </span></div><div class="c-article-equation__number"> (3) </div></div><p> with <span class="mathjax-tex">$u(0)=0$</span>. Let <span class="mathjax-tex">$f(t_{1},x_{1},y_{1},w_{1},u_{1},u_{2})= \frac{e^{-(\pi t_{1} +6)}}{2}(2t+\frac {1}{8} x_{1}+\frac{5}{11} y_{1}+ w_{1}+ e^{-6} u_{1}+ u_{2})$</span>. In our case <span class="mathjax-tex">$\Delta=[\eta^{*}(2+\gamma_{0} + \lambda_{0} + \frac{M_{3}}{(1-{\delta })^{2r}})+{ \mu}( \frac{M_{1}M_{2}}{(1-{\gamma})^{2p}(1-{\beta})^{2m}}+\frac {M_{1}}{(1-{\beta})^{2m}})]< 0.0374<1$</span>. Now by using Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-017-1258-3#FPar9">3.6</a>, (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-017-1258-3#Equ3">3</a>) has an approximate solution.</p> <h3 class="c-article__sub-heading" id="FPar14">Example 3.2</h3> <p>Let <span class="mathjax-tex">$\gamma_{1},\lambda_{1}:[0,1] \times[0,1]\to[0,\infty)$</span> be <span class="mathjax-tex">$\gamma_{1}(t,s )=\frac{t-s}{1+2t}$</span> and <span class="mathjax-tex">$\lambda_{1}(t,s)=\operatorname{sin}(t-s)e^{ \operatorname{ln}( \vert 2t-s \vert +1)}$</span>, respectively. Then <span class="mathjax-tex">$\gamma_{0}\leq1$</span> and <span class="mathjax-tex">$\lambda_{0}\leq e^{ \operatorname{ln}( 3)}$</span>. Put <span class="mathjax-tex">$\lambda= \frac{2}{2{,}037}$</span>, <span class="mathjax-tex">$\mu=\frac{2}{421}$</span>, <span class="mathjax-tex">$\alpha=\frac {1}{4}$</span>, <span class="mathjax-tex">$\beta=\frac{1}{2}$</span>, <span class="mathjax-tex">$\rho=\frac{1}{2}$</span>, <span class="mathjax-tex">$\theta=\frac{1}{2}$</span>, <span class="mathjax-tex">$\nu=\frac{1}{4}$</span>, <span class="mathjax-tex">$\delta=\frac{1}{4}$</span>, <span class="mathjax-tex">$\gamma=\frac{1}{2}$</span>, <span class="mathjax-tex">$n=7$</span>, <span class="mathjax-tex">$m=3$</span>, <span class="mathjax-tex">$p=2$</span>, <span class="mathjax-tex">$q=3$</span>, <span class="mathjax-tex">$r=3$</span>, <span class="mathjax-tex">$d=12$</span>, <span class="mathjax-tex">$\xi_{1}=\frac{3}{6{,}041}$</span>, <span class="mathjax-tex">$\xi_{2}=\frac{1}{5{,}920}$</span>, <span class="mathjax-tex">$\xi_{3}=\frac{1}{803}$</span>, <span class="mathjax-tex">$\xi_{4}=\frac{1}{e^{18}}$</span>, <span class="mathjax-tex">$\xi^{\prime}_{1}=\frac {2}{10^{6}}$</span>, <span class="mathjax-tex">$\xi^{\prime}_{2}=\frac{e}{\pi^{11}}$</span> and <span class="mathjax-tex">$\xi^{\prime }_{3}=\frac{1}{600}$</span>. Now, consider the functions <span class="mathjax-tex">$k(t)=\ln(2+t)$</span>, <span class="mathjax-tex">$s(t)=1$</span>, <span class="mathjax-tex">$h(t)=1$</span>, <span class="mathjax-tex">$g(t)=e^{\operatorname{sin}\pi t}$</span>, <span class="mathjax-tex">$q(t)=\frac{1}{432}$</span> when <span class="mathjax-tex">$x\in{\mathbb {Q}}\cap [0,1]$</span> and <span class="mathjax-tex">$q(t)=0$</span> when <span class="mathjax-tex">$x\in{\mathbb {Q}}^{c} \cap[0,1]$</span>. Then we have <span class="mathjax-tex">$M_{1}=\sup_{t\in I} \vert k(t) \vert =\ln3$</span>, <span class="mathjax-tex">$M_{2}=\sup_{t\in I} \vert s(t) \vert =1$</span>, <span class="mathjax-tex">$M_{3}=\sup_{t\in I} \vert h(t) \vert =1$</span>, <span class="mathjax-tex">$M_{4}=\sup_{t\in I} \vert g(t) \vert =e$</span> and <span class="mathjax-tex">$M_{5}=\sup_{t\in I} \vert q(t) \vert =\frac{1}{432}$</span>. Now, consider the integro-differential problem </p><div id="Equ4" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} ^{CF}D^{\frac{1}{4}^{[7]}}z(t)={}&\frac{2}{2{,}037} \operatorname{ln}(2+t){}^{CF}D^{\frac {1}{2}^{[3]}}z(t)+ \frac{2}{421} {}^{CF}I^{\frac{1}{2}^{[2]}}z(t) \\ &{}+\sum_{i=0} ^{\infty}\frac{1}{i!} {}^{CF}D^{ \frac{1}{2}^{[i]}} \biggl(\frac{2}{91}t+\frac{3}{6{,}041}z(t)+ \frac{1}{5{,}920} \int_{0}^{t} \frac{t-s}{1+2t}\,ds \\ &{}+ \frac{1}{803}{}^{CF}I^{ \frac{1}{4}^{[3]}}z(t)+\frac{1}{e^{18}}e^{\operatorname{sin}\pi t}(t){}^{CF}D^{ \frac{1}{4}^{[3]}}z(t) \biggr) \\ &{}+ \int_{0}^{t} \Biggl[ s +\frac{2z(s)}{10^{6}}+ \frac{e}{\pi^{11}} \int_{0}^{s} \operatorname{sin}(t-s)e^{ \operatorname{ln}( \vert 2s-r \vert +1)}z(r) \,dr \\ &{}+ \frac{1}{600}q(s)\sum_{i=0} ^{\infty }\frac{^{CF}D^{ \frac{1}{2}^{[i]}}z(s) }{12^{i}} \Biggr]\,ds \end{aligned}$$ </span></div><div class="c-article-equation__number"> (4) </div></div><p> with <span class="mathjax-tex">$z(0)=0$</span>. Let <span class="mathjax-tex">$f_{1}(t_{1},x_{1},y_{1},w_{1},v_{1})=\frac{2}{91}t_{1}+\frac{3}{6{,}041}x_{1}+\frac {1}{5{,}920}y_{1}+\frac{1}{803}w_{1}+\frac{1}{e^{18}}v_{1}$</span> and <span class="mathjax-tex">$f_{2}(t_{1},x_{1},y_{1}, w_{1}) = t_{1}+\frac{2}{10^{6}}x_{1}+\frac{e}{\pi^{11}}y_{1}+\frac{1}{600}w_{1}1$</span>. In addition, </p><div id="Equo" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex"> $$\begin{aligned} \Delta={}& \biggl[\lambda\frac{M_{1}}{(1-\beta)^{2m}}+\mu M_{2} + e^{\frac {1}{(1-\theta)^{2}}} \biggl(\xi_{1} + \xi_{2} \gamma_{0} + \xi_{3} M_{3} +\xi_{4} \frac {M_{4}}{(1-\delta)^{2r}} \biggr) \\ &{} + \xi_{1}^{\prime} + \xi_{2}^{\prime} \lambda_{0} + \frac{\xi _{3}^{\prime}\,d(1-\gamma)^{2} M_{5}}{d(1-\gamma)^{2}-1} \biggr] \\ < {}& 0.179< 1. \end{aligned}$$ </span></div></div><p> Now by using Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/articles/10.1186/s13662-017-1258-3#FPar11">3.7</a>, the problem (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/10.1186/s13662-017-1258-3#Equ4">4</a>) possesses an approximate solution.</p> </div></div></section><section data-title="Conclusions"><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>Conclusions</h2><div class="c-article-section__content" id="Sec4-content"><p>The higher-order FDE play an important role in modelling the dynamics of complex systems. This direction is an important topic in modelling the dissipative phenomena especially by fractional derivatives as Riemann-Liouville and Caputo. However, the CF derivative is equipped with a non-singular kernel, therefore it was found attractive and very suitable for several types of models possessing a memory effect. Thus, finding suitable numerical techniques and their approximate solutions for some complicated models containing a CF higher-order derivative are subjects of current interest. Along this line of thought in this manuscript we show the existence of approximate solutions analytically for two higher-order Caputo-Fabrizio FDE. We check our results by providing two examples. 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id="citeas">Cite this article</h3><p class="c-bibliographic-information__citation">Aydogan, S.M., Baleanu, D., Mousalou, A. <i>et al.</i> On approximate solutions for two higher-order Caputo-Fabrizio fractional integro-differential equations. <i>Adv Differ Equ</i> <b>2017</b>, 221 (2017). https://doi.org/10.1186/s13662-017-1258-3</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/s13662-017-1258-3?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="2017-03-20">20 March 2017</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="2017-06-29">29 June 2017</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="2017-08-03">03 August 2017</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/s13662-017-1258-3</span></p></li></ul><div data-component="share-box"><div class="c-article-share-box u-display-none" 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href="/search?query=approximate%20fixed%20point&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">approximate fixed point</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=higher-order%20fractional%20differential%20equation&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">higher-order fractional differential equation</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=non-singular%20kernel&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">non-singular kernel</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=Caputo-Fabrizio%20derivation&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">Caputo-Fabrizio derivation</a></span></li></ul><div 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