Document Zbl 1540.34009

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Anal. Appl.</a> <a href="/?q=in%3A514035" title="Articles in this Issue">29, No. 2, 501-515 (2024)</a>. </div> <div class="abstract">Summary: The primary focus of this paper is to thoroughly examine and analyze a coupled system by a Hilfer-Hadamard-type fractional differential equation with coupled boundary conditions. To achieve this, we introduce an operator that possesses fixed points corresponding to the solutions of the problem, effectively transforming the given system into an equivalent fixed-point problem. The necessary conditions for the existence and uniqueness of solutions for the system are established using Banach’s fixed point theorem and Schaefer’s fixed point theorem. An illustrate example is presented to demonstrate the effectiveness of the developed controllability results.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7868119">Cited in <strong>1</strong> Document</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A34A08" title="MSC2020">34A08</a> </td> <td class="space"> Fractional ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34A12" title="MSC2020">34A12</a> </td> <td class="space"> Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A47N20" title="MSC2020">47N20</a> </td> <td class="space"> Applications of operator theory to differential and integral equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Afractional+differential+equations">fractional differential equations</a>; <a href="/?q=ut%3Afixed+point+theorems">fixed point theorems</a>; <a href="/?q=ut%3Aboundary+conditions">boundary conditions</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1540.34009" data-ciurl="/ci/07868119" data-biburl="/bibtex/07868119.bib" data-amsurl="/amsrefs/07868119.bib" data-xmlurl="/xml/07868119.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07868119.pdf" title="Zbl 1540.34009 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="http://nfaa.kyungnam.ac.kr/journal-nfaa/index.php/NFAA/article/view/1931" title="Full Text Link">Link</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">S. Abbas, M. Benchohra, N. Hamidi and Y. Zhou, Implicit coupled Hilfer-Hadamard fractional differential systems under weak topologies, Adv. Diff. Equ., 2018(1) (2018), 1-17. · <a href="/1448.34003" class="nowrap">Zbl 1448.34003</a></td> </tr><tr> <td>[2]</td> <td class="space">B.N. Abood, S.S. Redhwan, O. Bazighifan and K. Nonlaopon, Investigating a generalized fractional quadratic integral equation. Fractal and Fractional, 6(5) (2022), 251.</td> </tr><tr> <td>[3]</td> <td class="space">Y. Adjabi, F. Jarad, D. Baleanu and T. Abdeljawad, On Cauchy problems with Caputo Hadamard fractional derivatives, J. Comput. Anal. Appl., 21(1) (2016), 661-681. · <a href="/1336.34010" class="nowrap">Zbl 1336.34010</a></td> </tr><tr> <td>[4]</td> <td class="space">B. Ahmad, SK. Ntouyas and A. Alsaedi, On a coupled system of fractional differen-tial equations with coupled nonlocal and integral boundary conditions, Chaos Solitons Fractals, 83 (2016), 234-241. · <a href="/1355.34012" class="nowrap">Zbl 1355.34012</a></td> </tr><tr> <td>[5]</td> <td class="space">H.H. Alsulami, S.K. Ntouyas, R.P. Agarwal, B. Ahmad and A. Alsaedi, A study of fractional-order coupled systems with a new concept of coupled non-separated boundary conditions, Boundary Value Prob., 2017(1) (2017), 68. · <a href="/1382.34005" class="nowrap">Zbl 1382.34005</a></td> </tr><tr> <td>[6]</td> <td class="space">N. Anakira, Z. Chebana, T-E. Oussaeif, I. M. Batiha and A. Ouannas, A study of a weak solution of a diffusion problem for a temporal fractional differential equation , Nonlinear Funct. Anal. Appl., 27 (3) (2022), 679-689. · <a href="/1497.35491" class="nowrap">Zbl 1497.35491</a></td> </tr><tr> <td>[7]</td> <td class="space">D.B. Dhaigude and S.P. Bhairat, Local Existence and Uniqueness of Solution for Hilfer-Hadamard fractional differential problem, Nonlinear Dyn. Syst. Theory, 18(2) (2018), 144-153. · <a href="/1404.34004" class="nowrap">Zbl 1404.34004</a></td> </tr><tr> <td>[8]</td> <td class="space">Y. Gambo, F. Jarad, D. Baleanu and F. Jarad, On Caputo modification of the Hadamard fractional derivatives, Adv. Diff. Equ., 10 (2014), 1-12. · <a href="/1343.26002" class="nowrap">Zbl 1343.26002</a></td> </tr><tr> <td>[9]</td> <td class="space">A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York 2005.</td> </tr><tr> <td>[10]</td> <td class="space">M. M. A. Hasan and S. A. Alkhatibd, Analytic Solution of High Order Fractional Bound-ary Value Problems , Nonlinear Funct. Anal. Appl., 28 (3) (2023), 601-612. · <a href="/1538.34096" class="nowrap">Zbl 1538.34096</a></td> </tr><tr> <td>[11]</td> <td class="space">A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier, Amsterdam 2006. · <a href="/1092.45003" class="nowrap">Zbl 1092.45003</a></td> </tr><tr> <td>[12]</td> <td class="space">S. Liang and J. Zhang, Existence of multiple positive solutions for m-point fractional boundary value problems on an infinite interval, Math. Comput. Model., 54 (2011), 1334-1346. · <a href="/1235.34023" class="nowrap">Zbl 1235.34023</a></td> </tr><tr> <td>[13]</td> <td class="space">K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. · <a href="/0789.26002" class="nowrap">Zbl 0789.26002</a></td> </tr><tr> <td>[14]</td> <td class="space">L. Palve, M.S. Abdo and S.K. Panchal, Some existence and stability results of Hilfer-Hadamard fractional implicit differential fractional equation in a weighted space, preprint: arXiv, arXiv:1910.08369v1, math.GM, 2019.</td> </tr><tr> <td>[15]</td> <td class="space">I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. · <a href="/0924.34008" class="nowrap">Zbl 0924.34008</a></td> </tr><tr> <td>[16]</td> <td class="space">M.D. Qassim, K.M. Furati and N.E. Tatar, On a differential equation involving Hilfer-Hadamard fractional derivative, Abstr. Appl. Anal., 2012, Article ID 391062 (2012). · <a href="/1255.34007" class="nowrap">Zbl 1255.34007</a></td> </tr><tr> <td>[17]</td> <td class="space">M.D. Qassim and N.E. Tatar, Well-posedness and stability for a differential problem with Hilfer-Hadamard fractional derivative. Abstr. Appl. Anal., 2013, Article ID 605029 (2013). · <a href="/1470.34017" class="nowrap">Zbl 1470.34017</a></td> </tr><tr> <td>[18]</td> <td class="space">S. Redhwan and S. Shaikh, Implicit fractional differential equation with nonlocal integral-multipoint boundary conditions in the frame of Hilfer fractional derivative, J. Math. Anal. Model., 2(1) (2021), 62-71.</td> </tr><tr> <td>[19]</td> <td class="space">S. Redhwan and S. Shaikh, Analysis of implicit type of a generalized fractional differen-tial equations with nonlinear integral boundary conditions, J. Math. Anal. Model., 1(1) (2020), 64-76.</td> </tr><tr> <td>[20]</td> <td class="space">S. Redhwan, S. Shaikh and A.B.D.O. Mohammed, Caputo-Katugampola-type implicit fractional differential equation with anti-periodic boundary conditions, Results Nonlinear Anal., 5(1), (2022), 12-28.</td> </tr><tr> <td>[21]</td> <td class="space">S. Redhwan, A.M. Suad, S. Shaikh and A.B.D.O. Mohammed, A coupled non-separated system of Hadamard-type fractional differential equations, Adv. Theory Nonlinear Anal. Appl., 6(1) (2021), 33-44.</td> </tr><tr> <td>[22]</td> <td class="space">J. Sabatier, O.P. Agrawal and JAT (eds.) Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dor-drecht 2007. · <a href="/1116.00014" class="nowrap">Zbl 1116.00014</a></td> </tr><tr> <td>[23]</td> <td class="space">W. Saengthong, E. Thailert and S.K. Ntouyas, Existence and uniqueness of solutions for system of Hilfer-Hadamard sequential fractional differential equations with two point boundary conditions, Adv. Diff. Equ., 2019(1) (2019), 525. · <a href="/1487.34037" class="nowrap">Zbl 1487.34037</a></td> </tr><tr> <td>[24]</td> <td class="space">H. Wang, Existence of solutions for fractional anti-periodic BVP, Results Math., 68 (2015), 227-245. · <a href="/1327.34010" class="nowrap">Zbl 1327.34010</a></td> </tr><tr> <td>[25]</td> <td class="space">J.R. Wang, Y. Zhou and M. Feckan, On recent developments in the theory of boundary value problems for impulsive fractional differential equations, Comput. Math. Appl., 64 (2012), 3008-3020. · <a href="/1268.34032" class="nowrap">Zbl 1268.34032</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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