Analysis of nonlinear integral equations with Erdélyi-Kober fractional operator

Communications in Nonlinear Science and Numerical Simulation

Volumn 17, Issue 8, 2012, Pages 3129-3139

  Wang, JinRong ^a   Dong, XiWang ^a   Zhou, Yong ^b  

^a GUIZHOU UNIVERSITY   (China)

^b XIANGTAN UNIVERSITY   (China)

Author keywords

Erd lyi Kober fractional operator; Existence and uniqueness; Local stability; Nonlinear integral equations

Indexed keywords

FIXED POINT ARITHMETIC; INTEGRAL EQUATIONS; NONLINEAR EQUATIONS;

ERDÉLYI-KOBER FRACTIONAL OPERATOR; EXISTENCE AND UNIQUENESS; EXISTENCE AND UNIQUENESS RESULTS; FRACTIONAL OPERATORS; INTEGRAL INEQUALITIES; LOCAL STABILITY; NONLINEAR INTEGRAL EQUATIONS; SCHAUDER FIXED-POINT THEOREM; SOLUTIONS SETS; WEAKLY SINGULAR INTEGRALS;

COMPUTATION THEORY;

EID: 84857642558     PISSN: 10075704     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cnsns.2011.12.002     Document Type: Article

References (28)

1. Diethelm K. The analysis of fractional differential equations, Lecture Notes in Mathematics, 2010.
2. Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and applications of fractional differential equations. North-Holland mathematics studies 2006, vol. 204. Elsevier Science B.V., Amsterdam.
3. Lakshmikantham V., Leela S., Vasundhara Devi J. Theory of fractional dynamic systems 2009, Cambridge Scientific Publishers.
4. Miller K.S., Ross B. An introduction to the fractional calculus and differential equations 1993, John Wiley, New York.
5. Podlubny I. Fractional differential equations 1999, Academic Press, San Diego.
6. Tarasov V.E. Fractional dynamics: application of fractional calculus to dynamics of particles, fields and media 2010, Springer, HEP.
7. Agarwal R.P., Benchohra M., Hamani S. A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl Math 2010, 109:973-1033.
8. Ahmad B., Nieto J.J. Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory. Topol Methods Nonlinear Anal 2010, 35:295-304.
9. Bai Z. On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal: TMA 2010, 72:916-924.
10. Benchohra M., Henderson J., Ntouyas S.K., Ouahab A. Existence results for fractional order functional differential equations with infinite delay. J Math Anal Appl 2008, 338:1340-1350.
11. Mophou G.M., N'Guérékata G.M. Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay. Appl Math Comput 2010, 216:61-69.
12. Wang J., Zhou Y. A class of fractional evolution equations and optimal controls. Nonlinear Anal: RWA 2011, 12:262-272.
13. Wang J., Zhou Y., Wei W. A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces. Commun Nonlinear Sci Numer Simulat 2011, 16:4049-4059.
14. Wang J., Dong X., Zhou Y., Existence attractiveness and stability of solutions for quadratic Urysohn fractional integral equations. Commun Nonlinear Sci Numer Simulat 2012, 17:545-554.
15. Zhang S. Existence of positive solution for some class of nonlinear fractional differential equations. J Math Anal Appl 2003, 278:136-148.
16. Zhou Y., Jiao F. Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal: RWA 2010, 11:4465-4475.
17. Zhou Y., Jiao F., Li J. Existence and uniqueness for p-type fractional neutral differential equations. Nonlinear Anal: TMA 2009, 71:2724-2733.
18. Zhou Y., Jiao F., Li J. Existence and uniqueness for fractional neutral differential equations with infinite delay. Nonlinear Anal: TMA 2009, 71:3249-3256.
19. Ibrahim R.W., Momani S. On the existence and uniqueness of solutions of a class of fractional differential equations. J Math Anal Appl 2007, 334:1-10.
20. Ma Q.H., Pečarić J. Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations. J Math Anal Appl 2008, 341:894-905.
21. Sneddon I.N. The use in mathematical analysis of Erdélyi-Kober operators and some of their applications. Lecture Notes in Math 1975, vol. 457:37-79. Springer, New York. B. Ross (Ed.).
22. Prudnikov A.P., Brychkov Y.A., Marichev O.I. Integral and series. Elementary functions 1981, vol. 1. Nauka, Moscow.
23. Kiryakova V. Generalized fractional calculus and applications. Pitman Res Notes Math Ser 1994, vol. 301. Longman, New York.
24. Deng W. Smoothness and stability of the solutions for nonlinear fractional differential equations. Nonlinear Anal: TMA 2010, 72:1768-1777.
25. Li Y., Chen Y., Podlubny I. Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 2009, 45:1965-1969.
26. Li Y., Chen Y., Podlubny I. Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput Math Appl 2010, 59:1810-1821.
27. Ma Q.H., Yang E.H. Estimates on solutions of some weakly singular Volterra integral inequalities. Acta Math Appl Ser A 2002, 25:505-515.
28. Deimling K. Nonlinear functional analysis 1985, Springer-Verlag.