Almost periodic and pseudo-almost periodic solutions to fractional differential and integro-differential equations

Applied Mathematics and Computation

Volumn 218, Issue 5, 2011, Pages 1735-1745

  Cuevas, Claudio ^a   Sepúlveda, Alex ^b   Soto, Herme ^b  

^a FEDERAL UNIVERSITY OF PERNAMBUCO   (Brazil)

^b UNIVERSIDAD DE LA FRONTERA   (Chile)

Author keywords

Fractional order differential equations; Integro differential equations; Mild solutions; Stepanov almost periodic functions; Stepanov like pseudo almost periodic functions

Indexed keywords

ALMOST PERIODIC; ALMOST PERIODIC FUNCTION; FORCING TERMS; FRACTIONAL DIFFERENTIAL; FRACTIONAL ORDER DIFFERENTIAL EQUATIONS; MILD SOLUTION; PERIODIC FUNCTION; PERIODIC SOLUTION;

INTEGRODIFFERENTIAL EQUATIONS;

DIFFERENTIAL EQUATIONS;

EID: 80052268859     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2011.06.054     Document Type: Article

References (55)

1. R.P. Agarwal, B. de Andrade, and C. Cuevas Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations Nonlinear Anal. RWA 11 2010 3532 3554
2. R.P. Agarwal, B. de Andrade, and C. Cuevas On type of periodicity and ergodicity to a class of integral equations with infinite delay J. Nonlinear Convex Anal. 11 2 2010 309 335
3. R.P. Agarwal, B. de Andrade, and C. Cuevas On type of periodicity and ergodicity to a class of fractional order differential equations Adv. Differ. Equ. 2010 1 25 Article ID 179750
4. L. Amerio, and G. Prouse Almost-periodic Functions and Functional Equations 1971 Van Nostrand Reinhold Co.
5. J. Andres, and A.M. Bersani Almost-periodicity problem as a fixed-point problem for evolution inclusions Topol. Methods Nonlinear Anal. 18 2 2001 337 349
6. D. Araya, and C. Lizama Almost automorphic mild solutions to fractional differential equations Nonlinear Anal. 69 11 2008 3692 3705
7. D.A. Benson, The fractional advection-dispersion equation, Ph.D. thesis, University of Nevada, Reno, NV, 1998.
8. D. Bugajewski, and G.M. N'Guérékata Almost periodicity in Frechet spaces J. Math. Anal. Appl. 299 2 2004 534 549
9. S. Calzadillas, and C. Lizama Bounded mild solutions of perturbed Volterra equations with infinite delay Nonlinear Anal. Ser. A, Theory Methods Appl. 72 11 2010 3976 3983
10. Ph. Clément On abstract Volterra equations with completely positive kernels in infinite dimensional systems Lectures Notes in Mathematics Vol. 1076 1984 Springer 32 40
11. Ph. Clément, and G. Da Prato Existence and regularity results for an integral equation with infinite delay in a Banach space Integral Equ. Oper. Theory 11 1988 480 500
12. B.D. Coleman, and M.E. Gurtin Equipresence and constitutive equation for rigid heat conductors Z. Angew. Math. Phys. 18 1967 199 208
13. E. Cuesta Asymptotic behavior of the solutions of fractional integro-differential equations and some time discretizations Discrete Contin. Dyn. Syst. (Supplement) 2007 277 285
14. C. Cuevas, and C. Lizama Almost automorphic solutions to integral equations on the line Semigroup Forum 79 2009 461 472
15. C. Cuevas, and C. Lizama S-asymptotically ω-periodic solutions for semilinear Volterra equations Math. Methods Appl. Sci. 33 2010 1628 1636
16. G. Da Prato, and A. Lunardi Periodic solutions for linear integrodifferential equations with infinite delay in Banach space Differ. Equ. Banach spaces Lect. Notes Math. 1223 1985 49 60
17. T. Diagana, C.M. Mahop, G.M. N'Guérékata, and B. Toni Existence and uniqueness of pseudo-almost periodic solutions to some classes of semilinear differential equations and applications Nonlinear Anal. 64 11 2006 2442 2453
18. T. Diagana Stepanov-like pseudo almost periodic functions and their applications to differential equations Commun. Math. Anal. 3 1 2007 9 18
19. T. Diagana Stepanov-like pseudo almost periodicty and its applications to some nonautonomous differential equations Nonlinear Anal. TMA 69 2008 4277 4285
20. H.S. Ding, J. Liang, G.M. N'Guérékata, and T.J. Xiao Mild pseudo-almost periodic solutions of nonautonomous semilinear evolution equations Math. Comput. Model. 45 5-6 2007 579 584 (Pubitemid 44820794)
21. H.S. Ding, J. Liang, G.M. N'Guérékata, and T.J. Xiao Pseudo-almost periodicity of some nonautonomous evolution equations with delay Nonlinear Anal. 67 5 2007 1412 1418 (Pubitemid 46634810)
22. H.S. Ding, W. Long, and G.M. N'Guérékata Almost periodic solutions to abstract semilinear evolution equations with Stepanov almost periodic coefficients J. Comput. Anal. Appl. 3 2 2011 231 242
23. M. Dzherbashian Harmonic Analysis and Boundary Value Problems in the Complex Domain 1993 Birkhäuser Verlag Basel
24. A. Erdélyi Higher transcendental functions Baterman Project vol. 3 1955 McGraw-Hill New York
25. A.M. Fink Almost automorphic and almost periodic solutions which minimize functionals Tôhoku Math. J. 20 2 1968 323 332
26. E. Fasǎngová, and J. Prüss Asymptotic behavior of a semilinear viscoelastic beam model Arch. Math. (Basel) 77 2001 488 497 (Pubitemid 33402790)
27. G. Gripenberg, S-O Londen, and O. Staffans Volterra Integral and Functional Equations Encyclopedia Mathematics and Applications vol. 34 1990 Cambridge University Press Cambridge-New York
28. A. Haraux Asymptotic behavior for two-dimensional quasi-autonomous almost-periodic evolution equations J. Differ. Equ. 66 1987 62 70
29. B.I. Henry, and S.L. Wearne Existence of turing instabilities in a two-species fractional reaction-diffusion system SIAM J. Appl. Math. 62 2002 870 887 (Pubitemid 34750573)
30. H. Henríquez, and C. Lizama Compact almost automorphic solutions to integral equations with infinite delay Nonlinear Anal. 71 2 2009 6029 6037
31. H. Henríquez On Stepanov-almost periodic semigroup and cosine functions of operator J. Math. Anal. Appl. 160 1991 157 175
32. H. Henríquez Asymptotically almost-periodic solutions of abstract differential equations J. Math. Anal. Appl. 146 1990 420 433
33. Z. Hu, and Z. Jin Stepanov-like pseudo almost periodic mild solutions to perturbed nonautonomous evolution equations with infinite delay Nonlinear Anal. 71 11 2009 5381 5391
34. Z. Hu, and A.B. Mingarelli Bochner's theorem and Stepanov almost periodic functions Annali di Math. 187 2008 719 736
35. J. Kaczorowski, and O. Ramaré Almost periodicity of some error terms in prime number theory Acta Arithmetica 106 3 2003 278 297
36. B.M. Levitan, and V.V. Zhikov Almost Periodic Functions and Differential Equations 1982 Cambridge University Press
37. C. Lizama Regularized solutions for abstract Volterra equations J. Math. Anal. Appl. 243 2000 278 292
38. C. Lizama, and G.M. N'Guérékata Bounded mild solutions for semilinear integro-differential equations in Banach spaces Integral Equ. Oper. Theory 68 2 2010 207 227
39. A. Lunardi Analytic semigroup and optimal regularity in parabolic problems Progress Nonlinear Differential Equations Applications vol. 16 1995 Birkhäuser-Verlag Besel
40. G.M. N'Guérékata Almost Automorphic and Almost Periodic Functions in Abstract Spaces 2001 Kluwer Academic/ Plenum Publishers New York, Boston, Dordrecht Moscow, London
41. F. Mainardi, and R. Gorenflo On Mittag-Leffler-type functions in fractional evolution processes J. Comput. Appl. Math. 118 2000 283 299
42. J.W. Nunziato On heat conduction in materials with memory Quart. Appl. Math. 29 1971 187 204
43. A. Pankov Bounded solutions, almost periodic in time, of a class of nonlinear evolutions equations Math. USSR. Sbornik 49 1 1984 73 86
44. A. Pankov Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations Equations 1990 Kluwer Dordrecht
45. J. Prüss Evolutionary integral equations and applications Monographs. Math. vol. 87 1993 Birkhäuser
46. A.S. Rao, and W. Hengertner On the existence of a unique almost periodic solution of an abstract differential equation J. London Math. Soc. 12 8 1974 577 581
47. A.S. Rao On the Stepanov-almost periodic solution of a second-order differential equation Proc. Edin. Math. Soc. 19 1975 261 263
48. A.S. Rao On the weakly almost periodic solutions of certain abstract differential equations Czechoslovak Math, J. 26 101 1976 65 70
49. A.S. Rao On the Stepanov almost periodic solution of a second-order infinitesimal generator differential equation Int. J. Math. Math. Sci. 14 4 1991 757 762
50. G. Sansone, J. Gerretsen, Lectures on the theory of functions of a complex variable, Holomorphic Functions, vol. I, Nordhoff, Groningen 1960, pp. 345-349.
51. R. Schumer, and D.A. Benson Eulerian derivative of the fractional advection-dispersion equation J. Contamin. 48 2001 69 88 (Pubitemid 32141059)
52. M. Tarallo A Stepanov version for Farvard theory Arch. Math. 90 2008 53 59
53. C. Zhang Pseudo almost periodic solutions of some differential equations J. Math. Anal. Appl. 181 1994 62 76
54. C. Zhang Integration of vector-valued pseudo almost periodic functions Proc. Amer. Math. Soc. 121 1994 167 174
55. C. Zhang Pseudo almost periodic solutions of some differential equations II J. Math. Anal. Appl. 192 1995 543 561