Research Article
Asymptotic periodicity and almost automorphy for a class of Volterra integro-differential equations
Bruno de Andrade,
Claudio Cuevas,
Erwin Henríquez,
Bruno de Andrade
Departamento de Matemática, ICMC, USP-São Carlos, São Carlos-SP, CEP. 13569-970 Brazil
Search for more papers by this authorCorresponding Author
Claudio Cuevas
Departamento de Matemática, Universidade Federal de Pernambuco, Recife-PE, CEP. 50540-740 Brazil
Claudio Cuevas, Departamento deMatemática, Universidade Federal de Pernambuco, Recife-PE, CEP. 50540-740, Brazil.
E-mail: [email protected]
Search for more papers by this authorErwin Henríquez
Departamento de Matemática y Estadística, Universidad de La Frontera, Casilla 54D, Temuco, Chile
Search for more papers by this authorBruno de Andrade,
Claudio Cuevas,
Erwin Henríquez,
Bruno de Andrade
Departamento de Matemática, ICMC, USP-São Carlos, São Carlos-SP, CEP. 13569-970 Brazil
Search for more papers by this authorCorresponding Author
Claudio Cuevas
Departamento de Matemática, Universidade Federal de Pernambuco, Recife-PE, CEP. 50540-740 Brazil
Claudio Cuevas, Departamento deMatemática, Universidade Federal de Pernambuco, Recife-PE, CEP. 50540-740, Brazil.
E-mail: [email protected]
Search for more papers by this authorErwin Henríquez
Departamento de Matemática y Estadística, Universidad de La Frontera, Casilla 54D, Temuco, Chile
Search for more papers by this authorAbstract
This work deals with the existence of asymptotically periodic (resp. asymptotically almost automorphic) solutions for a class of Volterra integro-differential equations. As application, we examine sufficient conditions for the existence of asymptotically periodic (resp. asymptotically almost automorphic) mild solutions of an equation arising in the study of heat conduction in materials with memory. Copyright © 2012 John Wiley & Sons, Ltd.
References
- 1 Coleman B, Gurtin M. Equipresence and constitutive equations for rigid heat conductors. Zeitschrift fr angewandte Mathematik und Physik ZAMP 1967; 18: 199–208.
- 2 Grimmer RC. Resolvent operators for integral equations in a Banach space. Transactions of the American Mathematical Society 1982; 273: 333–349.
- 3 Liang J, Xiao TJ. Semilinear integrodifferential equations with nonlocal initial conditions. Computers & Mathematics with Applications 2004; 47: 863–875.
- 4 Miller RK. An integrodifferential equation for rigid heat conductors with memory. Journal of Mathematical Analysis and Applications 1978; 66: 313–332.
- 5 Desch W, Grimmer R, Schappacher W. Well-posedness and wave propagation for a class of integrodifferential equations in Banach space. Journal of Differential Equations 1988; 74(2): 391–411.
- 6 Grimmer R, Prüss J. On linear Volterra equations in Banach spaces, hyperbolic partial differential equations II. Computers & Mathematics with Applications 1985; 11(1-3): 189–205.
- 7 Aizicovici S, McKibben M. Existence results for a class of abstract nonlocal Cauchy problems. Nonlinear Analysis, Theory, Methods & Applications 2000; 39: 649–668.
- 8 Aizicovici S, Lee H. Nonlinear nonlocal Cauchy problems in Banach spaces. Applied Mathematics Letters 2005; 18: 401–407.
- 9 Liang J, van Casteren J, Xiao TJ. Nonlocal Cauchy problems for semilinear evolution equations. Nonlinear Analysis, Theory, Methods & Applications 2002; 50: 173–189.
- 10 Hernandez E, dos Santos JPC. Asymptotically almost periodic and almost periodic solutions for a class of partial integrodifferential equations. Electronic Journal of Differential Equations 2006; 38: 1–8.
- 11 Ding HS, Xiao TJ, Liang J. Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions. Journal of Mathematical Analysis and Applications 2008; 338: 141–151.
- 12 de Andrade B, Cuevas C. Almost automorphic and pseudo-almost automorphic solutions to semilinear evolution equations with nondense domain. Journal of Inequalities and Applications 2009; 2009, DOI: 10.1155/2009/298207. Article ID 298207, 8 pages.
- 13 Cuevas C, Pinto M. Existence and uniqueness of pseudo-almost periodic solutions of semilinear Cauchy problems with non dense domain. Nonlinear Analysis, Theory, Methods & Applications 2001; 45(1): 73–83.
- 14 Cuevas C, Hernández E. Pseudo-almost periodic solutions for abstract partial functional differential equations. Applied Mathematics Letters 2009; 22: 534–538.
- 15 Cuevas C, Henríquez H, Lizama C. On the existence of almost automorphic solutions of Volterra difference equations. Journal of Difference Equations and Applications 2011: 1–16.
- 16 Cuevas C, Lizama C. Almost automorphic solutions to integral equations on the line. Semigroup Forum 2009; 79: 461–472.
- 17 Cuevas C, Lizama C. Almost automorphic solutions to a class of semilinear fractional differential equations. Applied Mathematics Letters 2008; 21: 1315–1319.
- 18 Cuevas C, N'Guérékata GM, Sepulveda A. Pseudo almost automorphic solutions to fractional differential and integro-differential equations. Communications in Applied Analysis to appear.
- 19 Grimmer RC. Asymptotically almost periodic solutions of differential equations. SIAM Journal on Applied Mathematics 1969; 17: 109–115.
- 20 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.
- 21 Lizama C, N'Guérékata GM. Bounded mild solutions for semilinear integro-differential equations in Banach spaces. Integral Equations and Operator Theory 2010; 68: 207–227.
- 22 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.
- 23 Zhao H. Existence and global attractivity of almost periodic solutions for cellular neutral network with distributed delays. Applied Mathematics and Computation 2004; 154(3): 683–695.
- 24
Caicedo A,
Cuevas C,
Mophou GM,
N'Guérékata GM. Asymptotic behavior of solutions of some semilinear functional differential and integro-differential equations with infinite delay in Banach spaces. Journal of The Franklin Institute. DOI: 10.1016/j.jfranklin.2011.02.001. (in press).
10.1016/j.jfranklin.2011.02.001. Google Scholar
- 25 Diagana T, Hernández E, dos Santos JPC. Existence of asymptotically almost automorphic solutions to some abstract partial neutral integro-differential equations. Nonlinear Analysis, Theory, Methods & Applications 2009; 71: 248–257.
- 26 Goldstein JA, N'Guérékata GM. Almost automorphic solutions of semilinear evolution equations. Proceedings of the American Mathematical Society 2005; 133(8): 2401–2408.
- 27 N'Guérékata GM. Topics in Almost Automorphy. Springer-Verlag: New York, 2005.
- 28 N'Guérékata GM. Almost Automorphic and Almost Periodic Functions in Abstract Spaces. Kluwer Acad/Plenum: New York–Boston–Moscow–London, 2001.
- 29 N'Guérékata GM. Quelques remarques sur les functions asymptotiquement presque automorphes. Annales Des Sciences Mathmatiques Du Qubec 1983; 7: 185–191.
- 30 Mophou G, N'Guérékata GM. On some classes of almost automorphic functions and applications to fractional differential equations. Computers & Mathematics with Applications 59(3): 1310–1317.
- 31 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.
- 32 Cushing JM. Forced asymptotically periodic solutions of predator–prey systems with or without hereditary effects. SIAM Journal on Applied Mathematics 1976; 30(4): 665–674.
- 33 Wei F, Wang K. Asymptoticaly periodic solutions of N-species cooperation system with time delay. Nonlinear Analysis: Real World and Applications 2006; 7: 591–596.
- 34 de Andrade B, Cuevas C. S-asymptotically ω-periodic and asymptotically ω-periodic solutions to semilinear Cauchy problems with non dense domain. Nonlinear Analysis, Theory, Methods & Applications 2010; 72: 3190–3208.
- 35 Agarwal RP, de Andrade B, Cuevas C, Henríquez E. Asymptotic periodicity for some classes of integro-differential equations and applications. Advances in Mathematical Sciences and Applications 2011; 21(1): 1–31.
- 36 Henríquez HR, 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.
- 37
Bochner S,
von Neumann J. On compact solutions of operational-differential equations. I. The Annals of Mathematics 1935; 36: 255–291.
10.2307/1968678 Google Scholar
- 38 Bochner S. A new approach to almost-periodicity. Proceedings of the National Academy of Sciences of the United States of America 1962; 48: 2039–2043.
- 39 Bochner S. Continuous mappings of almost automorphic and almost periodic functions. Proceedings of the National Academy of Sciences of the United States of America 1964; 52: 907–910.
- 40 Zaidman S. Almost automorphic solutions of some abstract evolution equations. Istituto lombardo di scienze e lettere 1976; 110: 578–588.
- 41 Zaidman S. Existence of asymptotically almost periodic and of almost automorphic solutions for some classes of abstract differential equations. Annales Des Sciences Mathmatiques Du Qubec 1989; 13: 79–88.
- 42
Veech WA. Almost automorphic functions on group. American Journal of Mathematics 1965; 87: 719–751.
10.2307/2373071 Google Scholar
- 43 Gal SG, N'Guérékata GM. Almost automorphic fuzzy-number-valued functions. Journal of Fuzzy Mathematics 2005; 13(1): 185–208.
- 44 Gal CG, Gal SG, N'Guérékata GM. Almost automorphic functions in Fréchet spaces and applications to differential equations. Semigroup Forum 2005; 71(2): 23–48.
- 45 Gal CG, Gal SG, N'Guérékata GM. Almost automorphic functions with values in a p − Fréchet space. Electronic Journal of Differential Equations 2008; 21: 1–18.
- 46 Agarwal RP, de Andrade B, Cuevas C. On type of periodicity and ergodicity to a class of integral equations with infinite delay. Journal of Nonlinear and Convex Analysis 2010; 11(2): 309–333.
- 47 Bugajewski D, N'Guérékata GM. On the topological structure of almost automorphic and asymptotically almost automorphic solutions of differential and integral equations in abstract spaces. Nonlinear Analysis, Theory, Methods & Applications 2004; 59: 1333–1345.
- 48 dos Santos JPC, Cuevas C. Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral equations, Applied. Applied Mathematics Letters 2010; 23(9): 960–965.
- 49 Cuevas C, Henríquez H. Solutions of second order abstract retarded functional differential equations on the line. Journal of Nonlinear and Convex Analysis 2011; 12(2): 225–240.
- 50 Desch W, Grimmer R, Schappacher W. Some considerations for linear integro-differential equations. Journal of Mathematical Analysis and Applications 1984; 104: 219–234.
- 51 Granas A, Dugundji J. Fixed Point Theory. Springer-Verlag: New York, 2003.
- 52 Deng K. Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions. Journal of Mathematical Analysis and Applications 1993; 179: 630–637.
- 53 Aizicovici S, McKibben M. Existence results for a class of abstract nonlocal Cauchy problems. Nonlinear Analysis, Theory, Methods & Applications 2000; 39: 649–668.
- 54 Byszewski L. Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. Journal of Mathematical Analysis and Applications 1991; 162: 494–505.
- 55 N'Guérékata GM. A Cauchy problem for some fractional abstract differential equation with nonlocal conditions. Nonlinear Analysis, Theory, Methods & Applications 2009; 70: 1873–1876.
- 56 Chen G. Control and stabilization for the wave equation in a bounded domain. SIAM Journal on Control and Optimization 1979; 17: 66–81.
- 57 Brézis H. Analyse Fonctionnelle. Masson: Paris, 1993.
- 58
Caicedo A,
Cuevas C,
Henríquez H. Asymptotic periodicity for a class of partial integro-differential equations. ISRN Mathematical Analysis 2011; 2011, DOI: 10.5402/2011/537890. Article ID 537890, 18p.
10.5402/2011/537890 Google Scholar
- 59 Agarwal RP, Cuevas C, Soto H, El-Gebeily M. Asymptotic periodicity for some evolution equations in Banach spaces. Nonlinear Analysis, Theory, Methods & Applications 2011; 74: 1769–1798.
- 60
Prüss J. Evolutionary Integral Equations and Applications. Monographs in Mathematics, Vol. 87. Birkhäuser Verlag: Basel, 1993.
10.1007/978-3-0348-8570-6 Google Scholar
- 61
Ahn VV,
Mcvinish R. Fractional differential equations driven by Levy noise. Journal of Applied Stochastic Analysis 2003; 16(2): 97–119.
10.1155/S1048953303000078 Google Scholar
- 62 Eidelman SD, Kochubei AN. Cauchy problem for fractional diffusion equations. Journal of Differential Equations 2004; 199: 211–255.
- 63 Cuevas C, Lizama C. S-asymptotically ω-periodic solutions for semilinear Volterra equations. Mathematical Methods in Applied Sciences 2010; 33: 1628–1636.
- 64 Martin RH. Nonlinear Operators and Differential Equations in Banach Spaces. Robert E. Krieger Publishing Company: Florida, 1987.
