Document Zbl 1181.45010

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A, Theory Methods</a> <a href="/?q=in%3A254077" title="Articles in this Issue">71, No. 11, 5491-5500 (2009)</a>. </div> <div class="abstract">The authors investigate the existence of a solution to the integral equation \[ x(t) = f_1(t,\, x(t)) + \frac{f_2(t,\, x(t))}{\Gamma(\alpha)}\int_0^t\frac{u(t,\,\tau,\,x(\tau))}{(t-\tau)^{1-\alpha}}\,d\tau,\;\;t\in [0,\,\infty). \] Here, $f_1,\, f_2$ and $u$ are some continuous functions, $\Gamma(\alpha)$ is the gamma function and $\alpha \in (0,\, 1)$. The equation is considered in the space of continuous bounded functions on $[0,\, \infty)$. Under some conditions imposed on the functions $f_1$, $f_2$ and $u$, it is shown that the equation admits at least one solution. An illustrative example is presented at the end of the paper.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A11294">Iulian Stoleriu (Iaşi)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5612975">Cited in <strong>21</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A45G05" title="MSC2020">45G05</a> </td> <td class="space"> Singular nonlinear integral equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A26A33" title="MSC2020">26A33</a> </td> <td class="space"> Fractional derivatives and integrals </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aintegral+equation+of+fractional+order">integral equation of fractional order</a>; <a href="/?q=ut%3ABanach+function+space">Banach function space</a>; <a href="/?q=ut%3Asuperposition+operator">superposition operator</a>; <a href="/?q=ut%3Ameasure+of+non-compactness">measure of non-compactness</a>; <a href="/?q=ut%3Afixed+point+theorem+of+Darbo+type">fixed point theorem of Darbo type</a>; <a href="/?q=ut%3Anumerical+examples">numerical examples</a>; <a href="/?q=ut%3Abounded+continuous+solution">bounded continuous solution</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 1181.45010" data-ciurl="/ci/05612975" data-biburl="/bibtex/05612975.bib" data-amsurl="/amsrefs/05612975.bib" data-xmlurl="/xml/05612975.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05612975.pdf" title="Zbl 1181.45010 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="https://doi.org/10.1016/j.na.2009.04.037" aria-label="DOI for “Solvability of a functional integral equation of fractional order in the class of functions having limits at infinity”" title="10.1016/j.na.2009.04.037">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">R.P. Agarwal, M. Benchohra, S. Hamani, Boundary value problems for functional differential equations, Georgian Math. J. (in press); R.P. Agarwal, M. Benchohra, S. Hamani, Boundary value problems for functional differential equations, Georgian Math. J. (in press) · <a href="/1179.26011" class="nowrap">Zbl 1179.26011</a></td> </tr><tr> <td>[2]</td> <td class="space">Appell, J.; Zabrejko, P. P., Nonlinear Superposition Operators, (Cambridge Tracts in Mathematics, vol. 95 (1990), Cambridge University Press) · <a href="/0970.47049" class="nowrap">Zbl 0970.47049</a></td> </tr><tr> <td>[3]</td> <td class="space">Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier Science B.V.: Elsevier Science B.V. Amsterdam · <a href="/1092.45003" class="nowrap">Zbl 1092.45003</a></td> </tr><tr> <td>[4]</td> <td class="space">Kilbas, A. A.; Trujillo, J. J., Differential equations of fractional order: Methods, results, problem I, Appl. Anal., 78, 153-192 (2001) · <a href="/1031.34002" class="nowrap">Zbl 1031.34002</a></td> </tr><tr> <td>[5]</td> <td class="space">Lakshmikantham, V.; Devi, J. V., Theory of fractional differential equations in a Banach space, Eur. J. Pure Appl. Math., 1, 38-45 (2008) · <a href="/1146.34042" class="nowrap">Zbl 1146.34042</a></td> </tr><tr> <td>[6]</td> <td class="space">Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Differential Equations (1993), John Wiley: John Wiley New York · <a href="/0789.26002" class="nowrap">Zbl 0789.26002</a></td> </tr><tr> <td>[7]</td> <td class="space">Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · <a href="/0918.34010" class="nowrap">Zbl 0918.34010</a></td> </tr><tr> <td>[8]</td> <td class="space">Samko, S.; Kilbas, A. A.; Marichev, O., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach: Gordon and Breach Amsterdam · <a href="/0818.26003" class="nowrap">Zbl 0818.26003</a></td> </tr><tr> <td>[9]</td> <td class="space">Saxena, R. K.; Mathai, A. M.; Haubold, H. L., On generalized kinetic equations, Physica A, 344, 657-664 (2004)</td> </tr><tr> <td>[10]</td> <td class="space">Garh, M.; Rao, A.; Kalla, S. L., Fractional generalization of temperature fields problems in oil strata, Mat. Bilten, 30, 71-84 (2006) · <a href="/1164.26314" class="nowrap">Zbl 1164.26314</a></td> </tr><tr> <td>[11]</td> <td class="space">Gaul, L.; Klein, P.; Kempfle, S., Damping description involving fractional operators, Mech. Syst. Signal Process., 5, 81-88 (1991)</td> </tr><tr> <td>[12]</td> <td class="space">Hilfer, R., Applications of Fractional Calculus in Physics (2000), World Scientific: World Scientific Singapore · <a href="/0998.26002" class="nowrap">Zbl 0998.26002</a></td> </tr><tr> <td>[13]</td> <td class="space">Saxena, R. K.; Kalla, S. L., On a fractional generalization of free electron laser equation, Appl. Math. Comput., 143, 89-97 (2003) · <a href="/1110.45300" class="nowrap">Zbl 1110.45300</a></td> </tr><tr> <td>[14]</td> <td class="space">Srivastava, H. M.; Saxena, R. K., Operators of fractional integration and their applications, Appl. Math. Comput., 118, 1-52 (2001) · <a href="/1022.26012" class="nowrap">Zbl 1022.26012</a></td> </tr><tr> <td>[15]</td> <td class="space">Babakhani, A.; Daftar-Gejji, V., Existence of positive solutions of nonlinear functional differential equations, J. Math. Anal. Appl., 278, 434-442 (2003) · <a href="/1027.34003" class="nowrap">Zbl 1027.34003</a></td> </tr><tr> <td>[16]</td> <td class="space">Benchohra, M.; Graef, J. R.; Hamani, S., Existence results for boundary value problems with nonlinear fractional differential equations, Appl. Anal., 87, 851-863 (2008) · <a href="/1198.26008" class="nowrap">Zbl 1198.26008</a></td> </tr><tr> <td>[17]</td> <td class="space">Cichoń, M.; El-Sayed, A. M.A.; Salem, H. A.H., Existence theorem for nonlinear functional integral equations of fractional orders, Comment. Math., 41, 59-67 (2001) · <a href="/1011.45002" class="nowrap">Zbl 1011.45002</a></td> </tr><tr> <td>[18]</td> <td class="space">Darwish, M. A., On quadratic integral equation of fractional orders, J. Math. Anal. Appl., 311, 112-119 (2005) · <a href="/1080.45004" class="nowrap">Zbl 1080.45004</a></td> </tr><tr> <td>[19]</td> <td class="space">El-Sayed, A. M.A., Nonlinear functional differential equations of arbitrary order, Nonlinear Anal., 33, 181-186 (1998) · <a href="/0934.34055" class="nowrap">Zbl 0934.34055</a></td> </tr><tr> <td>[20]</td> <td class="space">Juradat, O. K.; Al-Omari, A.; Momani, S., Existence of mild solutions for fractional semilinear initial value problems, Nonlinear Anal., 69, 3153-3159 (2008) · <a href="/1160.34300" class="nowrap">Zbl 1160.34300</a></td> </tr><tr> <td>[21]</td> <td class="space">Banaś, J.; O’Regan, D., On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order, J. Math. Anal. Appl., 345, 573-582 (2008) · <a href="/1147.45003" class="nowrap">Zbl 1147.45003</a></td> </tr><tr> <td>[22]</td> <td class="space">Rzepka, B., On attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation of fractional order, Topol. Meth. Nonlin. Anal., 32, 89-102 (2008) · <a href="/1173.45003" class="nowrap">Zbl 1173.45003</a></td> </tr><tr> <td>[23]</td> <td class="space">Banaś, J.; Goebel, K., Measures of Noncompactness in Banach Spaces, (Lect. Notes in Pure and Appl. Math., vol. 60 (1980), Dekker: Dekker New York) · <a href="/0441.47056" class="nowrap">Zbl 0441.47056</a></td> </tr><tr> <td>[24]</td> <td class="space">R.P. Agarwal, J. Banaś, K. Banaś, D. O’Regan, Solvability of a quadratic Hammerstein integral equation in the class of functions having limits at infinity (preprint); R.P. Agarwal, J. Banaś, K. Banaś, D. O’Regan, Solvability of a quadratic Hammerstein integral equation in the class of functions having limits at infinity (preprint)</td> </tr><tr> <td>[25]</td> <td class="space">Banaś, J., Measures of noncompactness in the space of continuous tempered functions, Demonstratio Math., 14, 127-133 (1981) · <a href="/0462.47035" class="nowrap">Zbl 0462.47035</a></td> </tr><tr> <td>[26]</td> <td class="space">Fichtenholz, G. M., Differential and Integral Calculus, vol. II (1980), PWN: PWN Warsaw, (in Polish) · <a href="/0900.26001" class="nowrap">Zbl 0900.26001</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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