Journal of Integral Equations and Applications

$L^p$-solutions for a class of fractional integral equations

Ravi P. Agarwal, Asma Asma, Vasile Lupulescu, and Donal O'Regan

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This paper considers the existence of $L^{p}$-solutions for a class of fractional integral equations involving abstract Volterra operators in a separable Banach space. Some applications for the existence of $L^{p}$-solutions for different classes of fractional differential equations are given.

Article information

J. Integral Equations Applications, Volume 29, Number 2 (2017), 251-270.

First available in Project Euclid: 17 June 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34A07: Fuzzy differential equations 34A08: Fractional differential equations

Evolution equation Caputo fractional derivative nonlocal Cauchy problem


Agarwal, Ravi P.; Asma, Asma; Lupulescu, Vasile; O'Regan, Donal. $L^p$-solutions for a class of fractional integral equations. J. Integral Equations Applications 29 (2017), no. 2, 251--270. doi:10.1216/JIE-2017-29-2-251.

Export citation


  • R.P. Agarwal and D. O'Regan, Infinite interval problems for differential, difference and integral equations, Kluwer Academic Publishers, Dordrecht, 2001.
  • J. Appell and P.P. Zabrejko, Nonlinear superposition operators, Cambr. Tracts Math. 95, Cambridge University Press, Cambridge, 1990.
  • A.O. Arancibia and M.P. Jimenez, $L^{p}$-solutions of nonlinear integral equations, Equad. 9 CD Brno, Masaryk University, 1997.
  • S. Arshad, V. Lupulescu and D. O'Regan, $L^{p}$-solutions for fractional integral equations, Fract. Calc. Appl. Anal. 17 (2014), 259–276.
  • D. Bǎleanu, O.G. Mustafa and R.P. Agarwal, On $Lp$-solutions for a class of sequential fractional differential equations, Appl. Math. Comp. 218 (2011), 2074–2081.
  • J. Banáš, Integrable solutions of Hammerstein and Urysohn integral equations, J. Austral. Math. Soc. 1 (1989), 61–68.
  • J. Banáš and A. Chlebowicz, On existence of integrable solutions of a functional integral equation under Caratheodory conditions, Nonlin. Anal. 70 (2009), 3172–3179.
  • J. Banáš and Z. Knap, Integrable solutions of a functional-integral equation, Rev. Math. Univ. Compl. Madrid 2 (1989), 31–38.
  • T.A. Barton and I.K. Purnaras, $L^{p}$-solutions of singular integro-differential equations, J. Math. Anal. Appl. 386 (2012), 830–841.
  • T.A. Barton and B. Zhang, $L^{p}$-solutions of fractional differential equations, Nonlin. Stud. 19 (2012), 161–177.
  • M. Bousselsal and S.H. Jah, Integrable solutions of a nonlinear integral equation via noncompactness measure and Krasnoselskii's fixed point theorem, Inter. J. Anal. 2014 (2014), Article ID 280709, 10 pages.
  • M. Cichoń and M.M.A. Metwali, On monotonic integrable solutions for quadratic functional integral equations, Mediterr. J. Math. 10 (2013), 909–926.
  • M. Cichoń and M.M.A. Metwali, On quadratic integral equations in Orlicz spaces, J. Math. Anal. Appl. 387 (2012), 419–432.
  • C. Corduneanu, Functional equations with causal operators, Taylor and Francis, London, 2002.
  • K. Diethelm, The analysis of fractional differential equations, Springer, Berlin, 2004.
  • R.E. Edwards, Functional analysis, theory and applications, Dover, New York, 1965.
  • A.M.A. El-Sayed and SH.A. Abd El-Salam, $L_{p}$ solution of weighted Cauchy type problem of a diffre-integral functional equation, Inter. J. Nonlin. Sci. 5 (2008), 281–288.
  • G. Emmanuele, Integrable solutions of a functional-integral equation, J. Integral Equations Appl. 4 (1992), 89–94.
  • L. Gasiński and N.S. Papageorgiou, Nonlinear analysis, Taylor & Francis, New York, 2005.
  • D. Guo, V. Lakshmikantham and X. Liu, Nonlinear integral equations in abstract spaces, Kluwer Academic Publishers, Dordrecht, 1996.
  • H.P. Heinz, On the behaviour of measures of noncompactness with respect to differentiation and integration of vector valued-functions, Nonlin. Anal. 7 (1983), 1351–1371.
  • D. Henry, Geometric theory of semilinear parabolic equations, Springer-Verlag, Berlin, 1981.
  • M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing multivalued maps and semilinear differential inclusions in Banach spaces, de Gruyter Nonlin. Anal. Appl. 7, Walter de Gruyter, Berlin, 2001.
  • A. Karoui and A. Jawahdou, Existence and approximate $L^{p}$ and continuous solutions of nonlinear integral equations of the Hammerstein and Volterra types, Appl. Math. Comp. 216 (2010), 2077–2091.
  • L. Kexue, P. Jigen and G. Jinghuai, Existence results for semilinear fractional differential equations via Kuratowski measure of noncompactness, Fract. Calc. Appl. Anal. 15, (2012) 591–610.
  • A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, North-Holland Math. Stud., Elsevier, New York, 2006.
  • C. Kuratowski, Sur les espaces complets, Fund. Math. 51 (1930), 301–309.
  • V. Lakshmikantham, S. Leela and J. Vasundhara Devi, Theory of fractional dynamic systems, Cambridge Academic Publishers, Cambridge, 2009.
  • Y. Li and M. Mahdavi, Linear and quasilinear equations with abstract Volterra operators, in Volterra equations and applications, Gordon & Breach, Reading, UK, 2000.
  • J. Liang, S. Yan, R.P. Agarwal and T. Huang, Integral solution of a class of nonlinear integral equations, Appl. Math. Comp. 219 (2013), 4950–4957.
  • D. Mamrilla, On $L^{p}$-solutions of $n$th order nonlinear differential equations, Casop. Mat. 113 (1988), 363–368.
  • M. Meehan and D. O'Regan, Positive $L^{p}$ solutions of Hammerstein integral equations, Arch. Math. (Basel) 76 (2001), 366–376.
  • L. Olszowy, On solutions of functional-integral equations of Urysohn type on an unbounded interval, Math. Comp. Model. 47 (2008), 1125–1133.
  • D. O'Regan and M. Meehan, Existence theory for nonlinear integral and integrodifferential equations, Kluwer Academic Publishers, Dordrecht, 1998.
  • R. Precup, Methods in nonlinear integral equations, Kluwer Academic Publishers, Dordrecht, 2002.
  • H.A.H. Salem and M. Väth, An abstract Gronwall lemma and application to global existence results for functional differential and integral equations of fractional order, J. Integral Equat. Appl. 16 (2004), 441–439.
  • N. Salhi and M.A. Taoudi, Existence of integrable solutions of an integral equation of Hammerstein type on an unbounded interval, Mediterr. J. Math. 9 (2012), 729–739.
  • S. Szufla, Existence for $L^{p}$-solutions of integral equations in Banach spaces, Publ. Inst. Math. 54 (1986), 99–105.
  • M.A. Taoudi, Integrable solutions of a nonlinear functional integral equation on an unbounded interval, Nonlin. Anal. 71 (2009), 4131–4136.
  • M. Väth, Volterra and integral equations of vector functions, Marcel Dekker, New York, 2000.
  • X. Xue, $L^{p}$ theory for semilinear nonlocal problems with measure of noncompactness in separable Banach spaces, J. Fixed Point Th. Appl. 5 (2009), 129–144.
  • Y. Zhou, Basic theory of fractional differential equations, World Scientific Publishing, Singapore, 2014.