Journal of Integral Equations and Applications

Asymptotic behavior of fractional order Riemann-Liouville Volterra-Stieltjes integral equations

Saïd Abbas, Mouffak Benchohra, Boualem A. Slimani, and Juan J. Trujillo

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Abstract

In this paper, we present some results concerning the existence and global asymptotic stability of solutions for a functional integral equation of fractional order. We use Schauder's fixed point theorem for the existence of solutions, and we prove that all these solutions are globally asymptotically stable.

Article information

Source
J. Integral Equations Applications, Volume 27, Number 3 (2015), 311-323.

Dates
First available in Project Euclid: 17 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1450388937

Digital Object Identifier
doi:10.1216/JIE-2015-27-3-311

Mathematical Reviews number (MathSciNet)
MR3435802

Zentralblatt MATH identifier
06535146

Subjects
Primary: 26A33: Fractional derivatives and integrals 45G05: Singular nonlinear integral equations 45M10: Stability theory

Keywords
Volterra-Stieltjes integral equation left-sided mixed Riemann-Liouville integral of fractional order solution global asymptotic stability fixed point

Citation

Abbas, Saïd; Benchohra, Mouffak; Slimani, Boualem A.; Trujillo, Juan J. Asymptotic behavior of fractional order Riemann-Liouville Volterra-Stieltjes integral equations. J. Integral Equations Applications 27 (2015), no. 3, 311--323. doi:10.1216/JIE-2015-27-3-311. https://projecteuclid.org/euclid.jiea/1450388937


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References

  • S. Abbas and M. Benchohra, On the set of solutions of fractional order Riemann–Liouville integral inclusions, Demons. Math. 46 (2013), 271–281.
  • ––––, Fractional order Riemann–Liouville integral equations with multiple time delay, Appl. Math. E-Notes 12 (2012), 79–87.
  • ––––, On the existence and local asymptotic stability of solutions of fractional order integral equations, Comm. Math. 52 (2012), 91–100.
  • S. Abbas, M. Benchohra and J. Henderson, On global asymptotic stability of solutions of nonlinear quadratic Volterra integral equations of fractional order, Comm. Appl. Nonlin. Anal. 19 (2012), 79–89.
  • S. Abbas, M. Benchohra and G.M. N'Guérékata, Topics in fractional differential equations, Springer, New York, 2012.
  • ––––, Advanced fractional differential and integral equations, Nova Science Publishers, New York, 2015.
  • S. Abbas, M. Benchohra, M. Rivero and J. Trujillo, Existence and stability results for nonlinear fractional order Riemann–Liouville Volterra–Stieltjes quadratic integral equations, Appl. Math. Comp. 247 (2014), 319–328.
  • S. Abbas, M. Benchohra and A.N. Vityuk, On fractional order derivatives and Darboux problem for implicit differential equations, Frac. Calc. Appl. Anal. 15 (2012), 168–182.
  • J. Appell, J. Banas and N. Merentes, Variation and around, Ser. Nonlin. Anal. Appl. 17, Walter de Gruyter, Berlin, 2014.
  • D. Baleanu, K. Diethelm, E. Scalas and J.J. Trujillo, Fractional calculus models and numerical methods, World Scientific Publishing, New York, 2012.
  • J. Bana\`s and K. Goebel, Measures of noncompactness in Banach spaces, Marcel Dekker, New York, 1980.
  • J. Banaś and B. Rzepka, On existence and asymptotic stability of solutions of a nonlinear integral equation, J. Math. Anal. Appl. 284 (2003), 165–173.
  • ––––, Monotonic solutions of a quadratic integral equation of fractional order, J. Math. Anal. Appl. 332 (2007), 1371–1379.
  • C. Corduneanu, Integral equations and stability of feedback systems, Academic Press, New York, 1973.
  • M.A. Darwish, J. Henderson and D. O'Regan, Existence and asymptotic stability of solutions of a perturbed fractional functional integral equations with linear modification of the argument, Bull. Kor. Math. Soc. 48 (2011), 539–553.
  • A. Granas and J. Dugundji, Fixed point theory, Springer-Verlag, New York, 2003.
  • R. Hilfer, Applications of fractional calculus in physics, World Scientific, Singapore, 2000.
  • A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science B.V., Amsterdam, 2006.
  • V. Lakshmikantham, S. Leela and J. Vasundhara, Theory of fractional dynamic systems, Cambridge Academic Publishers, Cambridge, 2009.
  • K.S. Miller and B. Ross, An Introduction to the fractional calculus and differential equations, John Wiley, New York, 1993.
  • I.P. Natanson, Theory of functions of a real variable, Ungar, New York, 1960.
  • I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.
  • R. Sikorski, Real functions, PWN, Warsaw, 1958 (in Polish).
  • V.E. Tarasov, Fractional dynamics. Applications of fractional calculus to dynamics of particles, fields and media, Springer, Heidelberg, 2010.
  • Y. Zhou, Basic theory of fractional differential equations, World Scientific, Singapore, 2014.