S-asymptotically ω-periodic solutions for semilinear Volterra equations

Mathematical Methods in the Applied Sciences

Volumn 33, Issue 13, 2010, Pages 1628-1636

  Cuevas, Claudio ^a   Lizama, Carlos ^b  

^a FEDERAL UNIVERSITY OF PERNAMBUCO   (Brazil)

^b UNIVERSIDAD DE SANTIAGO DE CHILE   (Chile)

Author keywords

S asymptotically periodic function; Semilinear Volterra equations; Stable resolvent families

Indexed keywords

CAUCHY PROBLEMS; FIRST ORDER; FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION; MILD SOLUTION; PERIODIC FUNCTION; PERIODIC SOLUTION; RESOLVENT FAMILY; SEMILINEAR; VOLTERRA EQUATION;

ASYMPTOTIC ANALYSIS; BANACH SPACES; INTEGRODIFFERENTIAL EQUATIONS; TOPOLOGY;

VISCOELASTICITY;

EID: 77956932474     PISSN: 01704214     EISSN: 10991476     Source Type: Journal    
DOI: 10.1002/mma.1284     Document Type: Article

References (37)

1. Agarwal RP, de Andrade B, Cuevas C. On type of periodicity and ergodicity to a class of fractional order differential equations. Advances in Difference Equations. Hindawi Publ Corp: NY, U. S. A.
2. de Andrade B, Cuevas C. Almost automorphic and pseudo almost automorphic solutions to semilinear evolution equations with non dense domain. Journal of Inequalities and Applications. 2009;2009:8. Article ID 298207. DOI: 101155/2009/298207.
3. de Andrade B, Cuevas C. Compact almost automorphic solutions to semilinear Cauchy problems with nondense domain. Applied Mathematics and Computation 2009;215:2843-2849.
4. Araya D, Lizama C. Almost automorphic mild solutions to fractional differential equations. Nonlinear Analysis 2008;69(11):3692-3705.
5. Cuevas C, Pinto M. Existence and uniqueness of pseudo almost periodic solutions of semilinear Cauchy problems with non dense domain. Nonlinear Analysis 2001;45:73-83.
6. Cuevas C, Lizama C. Almost automorphic solutions to a class of semilinear fractional differential equations. Applied Mathematics Letters 2008;21:1315-1319.
7. Cuevas C, Lizama C. Almost automorphic solutions to integral equations on the line. Semigroup Forum 2009;79:461-472.
8. Diagana T, N'Guérékata GM. Almost automorphic solutions to semilinear evolution equations. Functional Differential Equations 2006;13(2):195-206.
9. Diagana T, N'Guérékata GM, Van Minh N. Almost automorphic solutions of evolution equations. Proceedings of the American Mathematical Society 2004;132:3289-3298.
10. Goldstein JA, N'Guérékata GM. Almost automorphic solutions of semilinear evolution equations. Proceedings of the American Mathematical Society 2005;133(8):2401-2408.
11. N'Guérékata GM. Existence and uniqueness of almost automorphic mild solutions of some semilinear abstract differential equations. Semigroup Forum 2004;69:80-86.
12. HenríquezH, Lizama C. Compact almost automorphic solutions to integral equations with infinite delay. Nonlinear Analysis 2009; DOI: 10.1016/j.na. 2009.05.042.
13. Hino Y, Naito T, Minh NV, Shin J. Almost Periodic Solutions of Differential Equations in Banach Spaces. Taylor & Francis: London, New York, 2002.
14. Wei F, Wang K. Global stability and asymptotically periodic solutions for nonautonomous cooperative Lotka-Volterra diffusion system. Applied Mathematics and Computation 2006;182:161-165.
15. Wei F, Wang K. Asymptotically periodic solutions of N-species cooperation system with time delay. Nonlinear Analysis: Real World and Applications 2006;7:591-596.
16. Henríquez H, Pierri M, Táboas P. On S-asymptotically ω-periodic functions on Banach spaces and applications. Journal of Mathematical Analysis and Applications 2008;343(2):1119-1130.
17. Grimmer RC. Asymptotically almost periodic solutions of differential equations. SIAM Journal on Applied Mathematics 1969;17:109-115.
18. Haiyin G, Ke W, Fengying W, Xiaohua D. Massera-type theorem and asymptotically periodic logistic equations. Nonlinear Analysis: Real World Applications 2006;7:1268-1283.
19. Liang ZC. Asymptotically periodic solutions of a class of second order nonlinear differential equations. Proceedings of the American Mathematical Society 1987;99(4):693-699.
20. Utz WR, Waltman P. Asymptotic almost periodicity of solutions of a system of differential equations. Proceedings of the American Mathematical Society 1967;18:597-601.
21. Wong W, Burton TA. Some properties of solutions of u′′ (t) +a (t) f (u) g (u′) = 0. II. Monatshefte für Mathematik 1965;69:368-374.
22. de Andrade B, Cuevas C. S-asymptotically omega;-periodic and asymptotically ω-periodic solutions to semilinear Cauchy problems with non dense domain. Nonlinear Analysis Series A: Theory, Methods and Applications 2010;72:3190-3208.
23. Cuevas C, De Souza JC. S-asymptotically omega;-periodic solutions of semilinear fractional integro-differential equations. Applied Mathematics Letters 2009;22:865-870.
24. Cuevas C, de Souza JC. Existence of S-asymptotically ω-periodic solutions for fractional order functional Integro-differential equations with infinite delay. Nonlinear Analysis 2010;72:1683-1689.
25. Caicedo A, Cuevas C. S-asymptotically ω-periodic solutions of abstract partial neutral integro-differential equations. Functional Differential Equations 2010;17(1-2).
26. Ahn VV, McVinish R. Fractional differential equations driven by Levy noise. Journal of Applied Mathematics and Stochastic Analysis 2003;16(2):97-119.
27. Eidelman SD, Kochubei AN. Cauchy problem for fractional diffusion equations. Journal of Differential Equations 2004;199:211-255.
28. Gorenflo R, Mainardi F. Fractional calculus: integral and differential equations of fractional order. Fractals and Fractional Calculus in Continuum Mechanics, Carpinteri A, Mainardi F (eds). Springer: Vienna and New York, 1997;223-276.
29. Prüss J. Evolutionary Integral Equations and Applications. Monographs in Mathematics, vol. 87. Birkhäuser: Basel, 1993.
30. Renardy M, Hrusa WJ, Nohel JA. Mathematical Problems in Viscoelasticity. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 35. Wiley: New York, 1987.
31. Lizama C. Regularized solutions for abstract Volterra equations. Journal of Mathematical Analysis and Applications 2000;243:278-292.
32. Gripenberg G, Londen S-O, Staffans O. Volterra integral functional equations. Encyclopedia of Mathematics and Applications, vol. 34. Cambridge University Press: Cambridge, New York, 1990.
33. Arendt W, Batty C, Hieber M, Neubrander F. Vector-valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics, vol. 96. Birkhäuser: Basel, 2001.
34. Fattorini HO. Second Order Linear Differential Equations in Banach Spaces. North-Holland Mathematics Studies, vol 108. North-Holland: Amsterdam, New York, Oxford, 1985.
35. Lizama C, Vergara V. Uniform stability of resolvent families. Proceedings of the American Mathematical Society 2004;132(1):175-181.
36. Henríquez H. Approximation of abstract functional differential equations with unbounded delay. Indian Journal of Pure and Applied Mathematics 1996;27(4):357-386.
37. Granas A, Dugundji J. Fixed Point Theory. Springer: New York, 2003.