On a class of retarded integro-differential equations with nonlocal initial conditions

Computers and Mathematics with Applications

Volumn 59, Issue 12, 2010, Pages 3700-3709

  Wang, Rong Nian ^a   Chen, De Han ^a  

^a NANCHANG UNIVERSITY   (China)

Author keywords

Fractional integral; Nonlocal initial conditions; Solution operator; Time delays

Indexed keywords

CAUCHY PROBLEMS; CONVOLUTION INTEGRALS; EXISTENCE RESULTS; FRACTIONAL INTEGRALS; FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION; GLOBAL SOLUTIONS; LOCAL EXISTENCE; MILD SOLUTION; NON-LOCAL CONDITIONS; NONLOCAL INITIAL CONDITIONS;

BANACH SPACES; CONVOLUTION; INTEGRODIFFERENTIAL EQUATIONS; TIME DELAY; TOPOLOGY;

MATHEMATICAL OPERATORS;

EID: 77953233394     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2010.04.003     Document Type: Article

References (38)

1. Cuevas C., and de Souza J.C. S-asymptotically ω-periodic solutions of semilinear fractional integro-differential equations. Appl. Math. Lett. 22 (2009) 865-870
2. Cuevas C., and de Souza J.C. Existence of S-asymptotically ω-periodic solutions for fractional order functional integro-differential equations with infinite delay. Nonlinear Anal. 72 (2010) 1683-1689
3. E. Bazhlekova, Fractional evolution equations in Banach spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001.
4. Podlubny I., Petras ̌ I., Vinagre B.M., O'Leary P., and Dorc ̌ak L. Analogue realizations of fractional-order controllers: fractional order calculus and its applications. Nonlinear Dynam. 29 (2002) 281-296
5. Anh V.V., and Leonenko N.N. Spectral analysis of fractional kinetic equations with randomdata. J. Stat. Phys. 104 (2001) 1349-1387
6. Anh V.V., and Mcvinish R. Fractional differential equations driven by Levy noise. J. Appl. Stoch. Anal. 16 2 (2003) 97-119
7. Miller K.S., and Ross B. An introduction to the Fractional Calculus and Differential Equations (1993), John Wiley, New York
8. Agarwal R.P., de Andrade B., and Cuevas C. On type of periodicity and ergodicity to a class of fractional order differential equations. Adv. Differential Equations 2010 (2010) 25 pages 10.1155/2010/179750 Article ID 179750
9. R.P. Agarwal, B. de Andrade, C. Cuevas, Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations, Nonlinear Anal. RWA, in press (doi:10.1016/j.nonrwa.2010.01.002).
10. Agarwal R.P., Belmekki M., and Benchohra M. A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative. Adv. Difference Equ. 2009 (2009) 47 pages 10.1155/2009/981728 Article ID 981728
11. Agarwal R.P., Benchohra M., and Hamani S. A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109 (2010) 973-1033
12. Agarwal R.P., Lakshmikantham V., and Nieto J.J. On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. 72 (2010) 2859-2862
13. Cuevas C., and Lizama C. Almost automorphic solutions to a class of semilinear fractional differential equations. Appl. Math. Lett. 21 (2008) 1315-1319
14. Cuevas C., Rabelo M., and Soto H. Pseudo almost automorphic solutions to a class of semilinear fractional differential equations. Commun. Appl. Nonlinear Anal. 17 (2010) 33-48
15. Lakshmikantham V. Theory of fractional differential equations. Nonlinear Anal. 69 10 (2008) 3337-3343
16. Lakshmikantham V., and Vatsala A. Basic theory of fractional differential equations. Nonlinear Anal. 69 8 (2008) 2677-2682
17. Podlubny I. Fractional Differential Equations (1999), San Diego Academic Press
18. Hilfer H. Applications of Fractional Calculus in Physics (2000), World Scientific Publ. Co., Singapore
19. Henriquez H.R. Approximation of abstract functional differential equations with unbounded delay. Indian J. Pure Appl. Math. 27 4 (1996) 357-386
20. Hino Y., Murakami S., and Naito T. Functional-Differential Equations with Infinite Delay. Lectures Notes in Mathematics vol. 1473 (2002), Springer, Berlin
21. Byszewski L. Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162 (1991) 494-505
22. Balachandran K., and Park J.Y. Sobolev type integrodifferential equation with nonlocal condition in Banach spaces. Taiwanese J. Math. 7 (2003) 155-163
23. N'Guérékata G.M. Existence and uniqueness of an integral solution to some Cauchy problem with nonlocal conditions. Differential and Difference Equations and Applications (2006), Hindawi, New York 843-849
24. N'Guérékata G.M. A Cauchy Problem for some fractional abstract differential equation with nonlocal conditions. Nonlinear Anal. 70 (2009) 1873-1876
25. Liang J., and Xiao T.J. Semilinear integrodifferential equations with nonlocal initial conditions. Comput. Math. Appl. 47 (2004) 863-875
26. Mophou G.M., and N'Guérékata G.M. Existence of mild solution for some fractional differential equations with nonlocal conditions. Semigroup Forum 79 (2009) 315-322
27. Deng K. Exponential decay of solutions of semilinear parabolic equations with non-local initial conditions. J. Math. Anal. Appl. 179 (1993) 630-637
28. Henriquez H.R., Hernandez E., and Akca H. Global solutions for an abstract Cauchy problem with nonlocal conditions. Internat. J. Math. Manuscripts 1 (2007)
29. Liang J., Liu J.H., and Xiao T.J. Nonlocal Cauchy problems governed by compact operator families. Nonlinear Anal. 57 (2004) 183-189
30. Dubey S.A., and Bahuguna D. Existence and regularity of solutions to nonlocal retarded differential equations. Appl. Math. Comput. 215 (2009) 2413-2424
31. Henriquez H.R., Pierri M., and Goncalves G. Existence results for an impulsive abstract partial differential equation with state-dependent delay. Comput. Math. Appl. 52 (2006) 411-420
32. Kolmanovskii V., and Myshkis A. Introduction to the Theory and Applications of Functional Differential Equations (1999), Kluwer Academic, Dordrecht
33. Byszewski L., and Akca H. On a mild solution of a semilinear functional-differential evolution nonlocal problem. J. Appl. Math. Stoch. Anal. 10 (1997) 265-271
34. Cuesta E. Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations. Discrete Contin. Dyn. Syst. (2007) 277-285
35. Lizama C. Regularized solutions for abstract Volterra equations. J. Math. Anal. Appl. 243 (2000) 278-292
36. Engel K.J., and Nagel R. One Parameter Semigroups for Linear Evolution Equations (1995), Springer-Verlag
37. Agarwal S., and Bahuguna D. Existence of solutions to Sobolev-type partial neutral differential equations. J. Appl. Math. Stoch. Anal. 2006 (2006) Article ID 16308
38. Sadovskii B.N. On a fixed point principle. Funct. Anal. Appl. 1 (1967) 74-76