S-asymptotically ω-periodic and asymptotically ω-periodic solutions to semi-linear Cauchy problems with non-dense domain

Nonlinear Analysis, Theory, Methods and Applications

Volumn 72, Issue 6, 2009, Pages 3190-3208

  de Andrade, Bruno ^a   Cuevas, Claudio ^a  

^a FEDERAL UNIVERSITY OF PERNAMBUCO   (Brazil)

Author keywords

Abstract Cauchy problem; Asymptotically periodic function; Fractional integro differential equations; Hille Yosida operator; Neutral differential equations; Non dense domain; Nonlocal conditions; S asymptotically periodic function

Indexed keywords

ABSTRACT CAUCHY PROBLEM; FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION; NEUTRAL DIFFERENTIAL EQUATION; NON-LOCAL CONDITIONS; PERIODIC FUNCTION;

ABSTRACTING; ASYMPTOTIC ANALYSIS; INTEGRODIFFERENTIAL EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; PROBLEM SOLVING;

MATHEMATICAL OPERATORS;

EID: 73249139305     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.na.2009.12.016     Document Type: Article

References (75)

1. Agarwal R.P., Diagana T., and Hernández E. Weighted pseudo almost periodic solutions to some partial neutral functional differential equations. J. Nonlinear Convex Anal. 8 3 (2007) 397-415
2. Corduneanu C. Almost Periodic Functions. second ed. (1989), Chelsea, New York
3. Cuevas C., and Lizama C. Almost automorphic solutions to a class of semilinear fractional differential equations. Appl. Math. Lett. 21 (2008) 1315-1319
4. Cuevas C., and Lizama C. Almost automorphic solutions to integral equations on the line. Semigroup Forum 78 (2009) 10.1007/s00233-009-9154-0
5. C. Cuevas, M. Rabelo, H. Soto, Pseudo-almost automorphic solutions to a class of semilinear fractional differential equations, Commun. Appl. Nonlinear Anal. (2010) (in press)
6. Fink A.M. Almost periodic differential equations. Lectures Notes in Mathematics vol. 377 (1974), Springer-Verlag
7. Grimmer R.C. Asymptotically almost periodic solutions of differential equations. SIAM J. Appl. Math. 17 (1969) 109-115
8. Liang Z.C. Asymptotically periodic solutions of a class of second order nonlinear differential equations. Proc. Amer. Math. Soc. 99 4 (1987) 693-699
9. N'Guérékata G.M. Almost Automorphic and Almost Periodic Functions in Abstract Spaces (2001), Kluwer Acad. Plenum, New York, Boston, Moscow, London
10. Utz W.R., and Waltman P. Asymptotic almost periodicity of solutions of a system of differential equations. Proc. Amer. Math. Soc. 18 (1967) 597-601
11. Wei F., and Wang K. Global stability and asymptotically periodic solutions for nonautonomous cooperative Lotka-Volterra diffusion system. Appl. Math. Comput. 182 (2006) 161-165
12. Wei F., and Wang K. Asymptoticaly periodic solutions of N-species cooperation system with time delay. Nonlinear Anal. RWA 7 (2006) 591-596
13. Adimy M., Elazzouzi A., and Ezzimbi K. Bohr-Neugebuer type theorem for some partial neutral functional differential equations. Nonlinear Anal. 66 5 (2007) 1145-1160
14. Amir B., and Maniar L. Composition of pseudo-almost periodic functions and Cauchy problem with operator of non dense domain. Ann. Math. Blaise Pascal 6 1 (1999) 1-11
15. de Andrade B., and Cuevas C. Almost automorphic and pseudo almost automorphic solutions to semilinear evolution equations with non dense domain. J. Inequal. Appl. 2009 (2009) 10.1155/2009/298207 Article ID 298207, 8 pages
16. de Andrade B., and Cuevas C. Compact almost automorphic solutions to semilinear Cauchy problems with non dense domain. Appl. Math. Comput. (2009) 10.1016/j.amc.2009.09.025
17. Cuevas C., and Hernández E. Pseudo-almost periodic solutions for abstract partial functional differential equations. Appl. Math. Lett. 22 (2009) 534-538
18. Cuevas C., and Pinto M. Existence and uniqueness of pseudo-almost periodic solutions of semilinear Cauchy problems with non dense domain. Nonlinear Anal. 45 1 (2001) 73-83
19. Maniar L. Equations différentielles à retard par la méthode d'extrapolation. Port. Math. 54 1 (1997) 101-113
20. Maniar L. Extrapolation theory and some applications. Cubo 2 (2000) 287-295
21. Da Prato G., and Sinestrari E. Differential operators with non dense domain. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 14 2 (1989) 285-344
22. Henríquez H.R., Pierri M., and Táboas P. On S-asymptotically ω-periodic functions on Banach spaces and applications. J. Math. Anal. Appl. 343 2 (2008) 1119-1130
23. Haiyin G., Ke W., Fengying W., and Xiaohua D. Massera-type theorem and asymptotically periodic logistic equations. Nonlinear Anal. RWA 7 (2006) 1268-1283
24. Nicola S., and Pierri M. A note on S-asymptotically ω-periodic functions. Nonlinear Anal. RWA (2008) 10.1016/j. nonrwa.2008.09.011
25. Henríquez H., Pierri M., and Táboas P. Existence of S-asymptotically ω-periodic for abstract neutral equations. Bull. Austral. Math. Soc. 78 (2008) 365-382
26. 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
27. A. Caicedo, C. Cuevas, S-asymptotically ω-periodic solutions of abstract partial neutral integro-differential equations, Funct. Differ. Equ., 2010 (in press)
28. 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
29. Nagel R., and Sinestrari E. Inhomogeneous Volterra integrodifferential equations for Hille-Yosida operators. In: Bierstedt K.D., Pietsch A., Ruess W.M., and Vogt D. (Eds). Functional Analysis. Lectures Notes Pure Appl. Math. vol. 150 (1994), Marcel Dekker 51-70
30. Da Prato G., and Grisvard P. On extrapolation spaces. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 72 6 (1982) 330-332
31. R. Nagel, Sobolev spaces and Semigroup, Semesterbericht Funktionalanalysis Tübinger, (Sommersemester 1983), pp. 1-19
32. Rhandi A. Extrapolation methods to solve non-autonomous retarded partial differential equations. Studia Math. 126 3 (1997)
33. Nagel R., and Sinestrari E. Extrapolation spaces and minimal regularity for evolution equations. J. Evol. Equ. 6 (2006) 287-303
34. Kato T. Remarks on pseudo-resolvents and infinitesimal generators of semigroup. Proc. Japan Acad. 35 (1959) 467-468
35. Engel K.J., and Nagel R. One-parameter semigroup for linear evolution equations. Graduate Texts in Mathematics (2001), Springer-Verlag 195
36. C. Cuevas, H. Henríquez, Solutions of second order abstract retarded functional differential equations on the line (submitted for publication)
37. Goldstein G.A. Semigroup of Operators and Applications (1985), Oxford Evolution Press
38. Deng K. Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions. J. Math. Anal. Appl. 179 (1993) 630-637
39. Aizicovici S., and McKibben M. Existence results for a class of abstract nonlocal Cauchy problems. Nonlinear Anal. TMA 39 (2000) 649-668
40. 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
41. H. Liu, J.C. Chang, Existence for a class of partial differential equations with nonlocal conditions, Nonlinear Anal., in press (doi:10.1016/j.na.2008.04.009)
42. G.M. Mophou, G.M. N'Guérékata, Existence of the mild solutions for some fractional differential equations with nonlocal conditions, Semigroup Forum. doi:10.1007/100233-008-9117-x
43. Henríquez H., and Lizama C. Compact almost automorphic solutions to integral equations with infinite delay. Nonlinear Anal. (2009) 10.1016/j.na.2009.05.042
44. Granas A., and Dugundji J. Fixed Point Theory (2003), Springer-Verlag, New York
45. Henríquez H. Approximation of abstract functional differential equations with unbounded delay. Indian J. Pure Appl. Math. 27 4 (1996) 357-386
46. Cushing J.M. Forced asymptotically periodic solutions of predator-prey systems with or without hereditary effects. SIAM J. Appl. Math. 30 4 (1976) 665-674
47. Ahn V.V., and Mcvinish R. Fractional differential equations driven by Levy noise. J. Appl. Math. Stoch. Anal. 16 2 (2003) 97-119
48. Eidelman S.D., and Kochubei A.N. Cauchy problem for fractional diffusion equations. J. Differential Equations 199 (2004) 211-255
49. Gorenflo R., and Mainardi F. Fractional calculus: Integral and differential equations of fractional order. In: Carpinteri A., and Mainardi F. (Eds). Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer Verlag, Vienna, New York 223-276
50. Hilfer H. Applications of Fractional Calculus in Physics (2000), World Scientific Publ. Co., Singapure
51. Cuesta E., Lubich Ch., and Palancia C. Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comp. 75 (2006) 673-696
52. Cuesta E., and Palencia C. A numerical method an integro-differential equation with memory in Banach spaces: Qualitative properties. SIAM J. Numer. Anal. 41 (2003) 1232-1241
53. Clément Ph., and Da Prato G. Existence and regularity results for an integral equation with infinite delay in a Banach space. Integral Equations Operator Theory 11 (1988) 480-500
54. Cuesta E. Asymptotic behavior of the solutions of fractional integro-differential equations and some time discretizations. Discrete Contin. Dyn. Syst. suppl. (2007) 277-285
55. Prüss J. Evolutionary Integral Equations and Applications. Monographs Math. vol. 87 (1993), Birkhäuser Verlag
56. Arendt W., Batty C., Hieber M., and Neubrander F. Vector-valued Laplace Transforms and Cauchy problems. Monographs in Mathematics vol. 96 (2001), Birkhäuser, Basel
57. p (0, T ; B). Ann. Mat. Pura Appl. CXLVI (1987) 65-96
58. Hale J.K., and Verduyn Lunel S.M. Introduction to Functional-Differential Equations. Applied Mathematical Sciences vol. 99 (1993), Springer-Verlag, New York
59. Lakshmikanthan V., Wen L., and Zhang B. Theory of Differential Equations with Unbounded Delay (1994), Kluwer Acad. Publ., Dordrecht
60. Kolmanovskii V., and Myshkis A. Introduction to the Theory and Applications of Functional Differential Equations (1999), Kluwer Acad. Publ., Dordrecht
61. Hale J.K. Partial neutral functional-differential equations. Rev. Roumaine Math. Pures Appl. 39 4 (1994) 339-344
62. Wu J., and Xia H. Rotating waves in neutral partial functional-differential equations. J. Dynam. Differential Equations 11 2 (1999) 209-238
63. Hernández E., and Henríquez H.R. Existence of periodic solutions of partial neutral functional-differential equations with unbounded delay. J. Math. Anal. Appl. 221 2 (1998) 499-522
64. Hernández E., and Henríquez H.R. Existence results for partial neutral functional equations with unbounded delay. J. Math. Anal. Appl. 221 2 (1998) 452-475
65. Martin R.H. Nonlinear Operators and Differential Equations in Banach Spaces (1987), Robert E. Krieger Publ. Co., Florida
66. Appell J. Measure of noncompactness, condensing operators and fixed points: An application-oriented survey. Fixed Point Theory 6 2 (2005) 157-229
67. Pazy A. Semigroup of Linear Operators and Applications to Partial Differential Equations (1983), Springer-Verlag, New York
68. Dalietzkii Y.L., and Krein M.G. Stability of Solutions of Differential Equations in Banach Spaces (1978), AMS, Providence, RI
69. Massera J.L., and Schäffer J.J. Linear Differential Equations and Functions Spaces (1966), Academic Press, New York
70. Coppel W.A. Dichotomies in Stability Theory. Lectures Notes in Mathematics vol. 629 (1978), Springer-Verlag
71. Massera J.L., and Schäffer J.J. Linear differential equations and functional analysis I. Ann. Math. 67 (1958) 517-573
72. Cardoso F., and Cuevas C. Exponential dichotomy and boundedness for retarded functional difference equations. J. Difference Equ. Appl. 15 (2009) 261-290
73. Ezzimbi K., and Liu J.H. Nondensely defined evolution equations with nonlocal conditions. Math. Comput. Modelling 36 (2002) 1027-1038
74. Schnaubelt R. Asymptotic Behavior of Parabolic Nonautonomous Evolution Equations. Lectures Notes in Math. vol. 1855 (2004), Springer-Verlag, Berlin 401-472
75. N'Guérékata G.M. A Cauchy problem for some fractional abstract differential equation with nonlocal conditions. Nonlinear Anal. TMA 70 (2009) 1873-1876