Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2012, Special Issue (2012), Article ID 901942, 18 pages.

A Nonlocal Cauchy Problem for Fractional Integrodifferential Equations

Fang Li, Jin Liang, Tzon-Tzer Lu, and Huan Zhu

Full-text: Open access


This paper is concerned with a nonlocal Cauchy problem for fractional integrodifferential equations in a separable Banach space X. We establish an existence theorem for mild solutions to the nonlocal Cauchy problem, by virtue of measure of noncompactness and the fixed point theorem for condensing maps. As an application, the existence of the mild solution to a nonlocal Cauchy problem for a concrete integrodifferential equation is obtained.

Article information

J. Appl. Math., Volume 2012, Special Issue (2012), Article ID 901942, 18 pages.

First available in Project Euclid: 3 January 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Li, Fang; Liang, Jin; Lu, Tzon-Tzer; Zhu, Huan. A Nonlocal Cauchy Problem for Fractional Integrodifferential Equations. J. Appl. Math. 2012, Special Issue (2012), Article ID 901942, 18 pages. doi:10.1155/2012/901942.

Export citation


  • S. Aizicovici and M. McKibben, “Existence results for a class of abstract nonlocal Cauchy problems,” Nonlinear Analysis, vol. 39, pp. 649–668, 2000.
  • L. Byszewski and V. Lakshmikantham, “Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space,” Applicable Analysis, vol. 40, no. 1, pp. 11–19, 1991.
  • E. P. Gatsori, “Controllability results for nondensely defined evolution differential inclusions with nonlocal conditions,” Journal of Mathematical Analysis and Applications, vol. 297, no. 1, pp. 194–211, 2004.
  • J. Liang and Z. Fan, “Nonlocal impulsive Cauchy problems for evolution equations,” Advances in Difference Equations, vol. 2011, Article ID 784161, 17 pages, 2011.
  • J. Liang, J. H. Liu, and T. J. Xiao, “Nonlocal impulsive problems for nonlinear differential equations in Banach spaces,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 798–804, 2009.
  • J. Liang, J. H. Liu, and T. J. Xiao, “Nonlocal Cauchy problems governed by compact operator families,” Nonlinear Analysis, vol. 57, no. 2, pp. 183–189, 2004.
  • J. Liang, J. van Casteren, and T. J. Xiao, “Nonlocal Cauchy problems for semilinear evolution equations,” Nonlinear Analysis, vol. 50, pp. 173–189, 2002.
  • J. Liang and T. J. Xiao, “Semilinear integrodifferential equations with nonlocal initial conditions,” Computers and Mathematics with Applications, vol. 47, no. 6-7, pp. 863–875, 2004.
  • Z. W. Lv, J. Liang, and T. J. Xiao, “Solutions to fractional differential equations with nonlocal initial condition in Banach spaces,” Advances in Difference Equations, vol. 2010, Article ID 340349, 10 pages, 2010.
  • T. J. Xiao and J. Liang, “Existence of classical solutions to nonautonomous nonlocal parabolic problems,” Nonlinear Analysis, vol. 63, pp. e225–e232, 2003.
  • L. Gaul, P. Klein, and S. Kempfle, “Damping description involving fractional operators,” Mechanical Systems and Signal Processing, vol. 5, pp. 81–88, 1991.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.
  • F. Li, “Solvability of nonautonomous fractional integrodifferential equations with infinite delay,” Advances in Difference Equations, vol. 2011, Article ID 806729, 18 pages, 2011.
  • F. Li, “An existence result for fractional differential equations of neutral type with infinite delay,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2011, no. 52, pp. 1–15, 2011.
  • F. Li, T. J. Xiao, and H. K. Xu, “On nonlinear neutral fractional integrodifferential inclusions with infinite delay,” Journal of Applied Mathematics, vol. 2012, Article ID 916543, 19 pages, 2012.
  • K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
  • I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  • R. N. Wang, D. H. Chen, and T. J. Xiao, “Abstract fractional Cauchy problems with almost sectorial operators,” Journal of Differential Equations, vol. 252, pp. 202–235, 2012.
  • J. Banaś and K. Goebel, Measures of noncompactness in Banach spaces, vol. 60 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980.
  • M. Kamenskii, V. Obukhovskii, and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, vol. 7 of de Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter, Berlin, Germany, 2001.
  • T. J. Xiao and J. Liang, The Cauchy Problem for Higher-Order Abstract Differential Equations, vol. 1701 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1998.
  • D. Henry, Geometric Theory of Semilinear Parabolic Partial Differential Equations, Springer, Berlin, Germany, 1989.