Topological Methods in Nonlinear Analysis

On attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation of fractional order

Beata Rzepka

Full-text: Open access

Abstract

We study the existence of solutions of a nonlinear quadratic Volterra integral equation of fractional order. In our considerations we apply the technique of measures of noncompactness in conjunction with the classical Schauder fixed point principle. The mentioned equation is considered in the Banach space of real functions defined, continuous and bounded on an unbounded interval. We will show that solutions of the investigated integral equation are locally attractive.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 32, Number 1 (2008), 89-102.

Dates
First available in Project Euclid: 13 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1463150464

Mathematical Reviews number (MathSciNet)
MR2466804

Zentralblatt MATH identifier
1173.45003

Citation

Rzepka, Beata. On attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation of fractional order. Topol. Methods Nonlinear Anal. 32 (2008), no. 1, 89--102. https://projecteuclid.org/euclid.tmna/1463150464


Export citation

References

  • A. Babakhani and V. Daftardar-Gejii, Existence of positive solutions of nonlinear fractional differential equations , J. Math. Anal. Appl., 278 (2003), 434–442 \ref
  • J. Banaś and K. Goebel, Measures of noncompactness in Banach spaces , Lecture Notes in Pure and Applied Mathematics, 60 , Marcel Dekker, New York (1980) \ref
  • J. Banaś and D. O'Regan, On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order , preprint \ref
  • J. Banaś and B. Rzepka, An application of a measure of noncompactness in the study of asymptotic stability , Appl. Math. Letters, 16 (2003), 1–6 \ref ––––, On existence and asymptotic stability of solutions of a nonlinear integral equation , J. Math. Anal. Appl., 284 (2003), 165–173 \ref
  • T. A. Burton and B. Zhang, Fixed points and stability of an integral equation: nonuniqueness , Appl. Math. Letters, 17 (2004), 839–846 \ref
  • M. Cichoń, A. M. A. El-Sayed and H. A. H. Salem, Existence theorem for nonlinear integral equations of fractional orders , Comment. Math., 41 (2001), 59–67 \ref
  • C. Corduneanu, Integral Equations and Applications, Cambridge Univ. Press, Cambridge (1991) \ref
  • A. M. A. El-Sayed, Nonlinear functional differential equations of arbitrary order , Nonlinear Anal., 33 (1998), 181–186 \ref
  • G. H. Fichtenholz, Differential and Integral Calculus II, PWN, Warsaw (1980), (Polish) \ref
  • R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore (2000) \ref
  • X. Hu, J. Yan, The global attractivity and asymptotic stability of solution of a nonlinear integral equation , J. Math. Anal. Appl., 321 (2006), 147–156 \ref
  • A. A. Kilbas and J. J. Trujillo, Differential equations of fractional order: Methods, results, problems I , Appl. Anal., 78 (2001), 153–192 \ref
  • K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York (1993) \ref
  • D. O'Regan and M. Meehan, Existence Theory for Nonlinear Integral and Integrodifferential Equations, Kluwer Academic Publishers, Dordrecht (1998) \ref
  • I. Podlubny, Fractional Differencial Equations, Academic Press, San Diego (1999) \ref
  • S. Samko, A. A. Kilbas and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Amsterdam (1993)