## On type of periodicity and ergodicity to a class of fractional order differential equations.(English)Zbl 1194.34007

In this paper, the authors study some sufficient conditions for the existence and uniqueness of: a) pseudo almost periodic (in the sense of Zhang) mild solutions to some semilinear fractional differential equations, and b) asymptotically almost automorphic (in the sense of N’Guérékata) mild solutions to some semilinear fractional integro-differential equations; in all cases, the derivative $$D^{\alpha}_{t}$$ is considered in the sense of Riemann-Liouville with $$1<\alpha<2$$ and the operator $$A$$ is sectorial of negative type. The authors reach their goals using a theoretical operator theory approach and fixed point techniques. The results extend and complete several recent works by the authors and others (including C. Lizama, G. N’Guérékata, G. Mophou). An application to some fractional relaxation-oscillation equation is also given.

### MSC:

 34A08 Fractional ordinary differential equations 34G20 Nonlinear differential equations in abstract spaces 47N20 Applications of operator theory to differential and integral equations
Full Text:

### References:

 [1] Anh, VV; Mcvinish, R, Fractional differential equations driven by Lévy noise, Journal of Applied Mathematics and Stochastic Analysis, 16, 97-119, (2003) · Zbl 1042.60034 [2] Gorenflo, R; Mainardi, F; Carpinteri, A (ed.); Mainardi, F (ed.), Fractional calculus: integral and differential equations of fractional order, No. 378, 223-276, (1997), Vienna, Austria [3] Hilfer R (Ed): Applications of Fractional Calculus in Physics. World Scientific, River Edge, NJ, USA; 2000:viii+463. · Zbl 0998.26002 [4] Hu, T; Wang, Y, Numerical detection of the lowest “Efficient dimensions” for chaotic fractional differential systems, Open Mathematics Journal, 1, 11-18, (2008) · Zbl 1185.34006 [5] Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Volume 204. Elsevier Science B.V., Amsterdam, The Netherlands; 2006:xvi+523. [6] Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons, New York, NY, USA; 1993:xvi+366. · Zbl 0789.26002 [7] Podlubny I: Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of Their Application, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340. · Zbl 0924.34008 [8] Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach Science, Yverdon, France; 1993:xxxvi+976. · Zbl 0818.26003 [9] Diethelm, K; Freed, AD; Keil, F (ed.); Mackens, W (ed.); Voss, H (ed.); Werther, J (ed.), On the solution of nonlinear fractional order equations used in the modeling of viscoplasticity, 217-224, (1999), Heidelberg, Germany [10] Eidelman, SD; Kochubei, AN, Cauchy problem for fractional diffusion equations, Journal of Differential Equations, 199, 211-255, (2004) · Zbl 1068.35037 [11] Lin, W, Global existence theory and chaos control of fractional differential equations, Journal of Mathematical Analysis and Applications, 332, 709-726, (2007) · Zbl 1113.37016 [12] Oustalup A: Systéms Asservis D’ordre Fractionaire. Éditions Masson; 1983. [13] Oustalup A: La Dérivation non Entière: Théorie, Synthèse, Applications, Série Automatique. Editions Hermès; 1995. [14] Podlubny, I; Petráš, I; Vinagre, BM; O’Leary, P; Dorčák, L, Analogue realizations of fractional-order controllers. fractional order calculus and its applications, Nonlinear Dynamics, 29, 281-296, (2002) · Zbl 1041.93022 [15] Ross B (Ed): Fractional Calculus and Its Applications, Lecture Notes in Mathematics, vol. 457. Springer, Berlin, Germany; 1975:vi+381. [16] Benchohra, M; Henderson, J; Ntouyas, SK; Ouahab, A, Existence results for fractional order functional differential equations with infinite delay, Journal of Mathematical Analysis and Applications, 338, 1340-1350, (2008) · Zbl 1209.34096 [17] Agarwal RP, Benchohra M, Hamani S: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. to appear in Acta Applicandae Mathematicae · Zbl 1198.26004 [18] Agarwal, RP; Benchohra, M; Hamani, S, Boundary value problems for fractional differential equations, Georgian Mathematical Journal, 16, 401-411, (2009) · Zbl 1179.26011 [19] Agarwal, RP; Belmekki, M; Benchohra, M, A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, No. 2009, 47, (2009) · Zbl 1182.34103 [20] Lakshmikantham, V, Theory of fractional functional differential equations, Nonlinear Analysis: Theory, Methods & Applications, 69, 3337-3343, (2008) · Zbl 1162.34344 [21] Lakshmikantham, V; Vatsala, AS, Basic theory of fractional differential equations, Nonlinear Analysis: Theory, Methods & Applications, 69, 2677-2682, (2008) · Zbl 1161.34001 [22] Lakshmikantham, V; Vatsala, AS, Theory of fractional differential inequalities and applications, Communications in Applied Analysis, 11, 395-402, (2007) · Zbl 1159.34006 [23] Mophou GM, N’Guérékata GM: Mild solutions for semilinear fractional differential equations.Electronic Journal of Differential Equations 2009, (21):1-9. [24] El-Borai, MM, Some probability densities and fundamental solutions of fractional evolution equations, Chaos, Solitons and Fractals, 14, 433-440, (2002) · Zbl 1005.34051 [25] El-Borai, MM, Semigroups and some nonlinear fractional differential equations, Applied Mathematics and Computation, 149, 823-831, (2004) · Zbl 1046.34079 [26] El-Borai MM: The fundamental solutions for fractional evolution equations of parabolic type.Journal of Applied Mathematics and Stochastic Analysis 2004, (3):197-211. 10.1155/S1048953304311020 · Zbl 1081.34053 [27] Mophou GM, Nakoulima O, N’Guérékata GM: Existence results for some fractional differential equations with nonlocal conditions. submitted · Zbl 0941.34059 [28] Mophou, GM; N’Guérékata, GM, Existence of the mild solution for some fractional differential equations with nonlocal conditions, Semigroup Forum, 79, 315-322, (2009) · Zbl 1180.34006 [29] N’Guérékata, GM, A Cauchy problem for some fractional abstract differential equation with non local conditions, Nonlinear Analysis: Theory, Methods & Applications, 70, 1873-1876, (2009) · Zbl 1166.34320 [30] El-Sayed, AMA; Ibrahim, A-G, Multivalued fractional differential equations, Applied Mathematics and Computation, 68, 15-25, (1995) · Zbl 0830.34012 [31] Benchohra, M; Henderson, J; Ntouyas, SK; Ouahab, A, Existence results for fractional functional differential inclusions with infinite delay and applications to control theory, Fractional Calculus & Applied Analysis, 11, 35-56, (2008) · Zbl 1149.26010 [32] Henderson, J; Ouahab, A, Fractional functional differential inclusions with finite delay, Nonlinear Analysis: Theory, Methods & Applications, 70, 2091-2105, (2009) · Zbl 1159.34010 [33] Ouahab, A, Some results for fractional boundary value problem of differential inclusions, Nonlinear Analysis: Theory, Methods & Applications, 69, 3877-3896, (2008) · Zbl 1169.34006 [34] Cuevas, C; Lizama, C, Almost automorphic solutions to a class of semilinear fractional differential equations, Applied Mathematics Letters, 21, 1315-1319, (2008) · Zbl 1192.34006 [35] Cuevas, C; Lizama, C, Almost automorphic solutions to integral equations on the line, Semigroup Forum, 79, 461-472, (2009) · Zbl 1187.45005 [36] Zhang, CY, Pseudo-almost-periodic solutions of some differential equations, Journal of Mathematical Analysis and Applications, 181, 62-76, (1994) · Zbl 0796.34029 [37] Zhang, CY, Integration of vector-valued pseudo-almost periodic functions, Proceedings of the American Mathematical Society, 121, 167-174, (1994) · Zbl 0818.42003 [38] Zhang, CY, Pseudo almost periodic solutions of some differential equations. II, Journal of Mathematical Analysis and Applications, 192, 543-561, (1995) · Zbl 0826.34040 [39] Zhang C: Almost Periodic Type Functions and Ergodicity. Science Press, Beijing, China; 2003:xii+355. · Zbl 1068.34001 [40] Ait Dads, E; Arino, O, Exponential dichotomy and existence of pseudo almost-periodic solutions of some differential equations, Nonlinear Analysis: Theory, Methods & Applications, 27, 369-386, (1996) · Zbl 0855.34055 [41] Ait Dads, E; Ezzinbi, K; Arino, O, Pseudo almost periodic solutions for some differential equations in a Banach space, Nonlinear Analysis: Theory, Methods & Applications, 28, 1141-1155, (1997) · Zbl 0874.34041 [42] Amir, B; Maniar, L, Composition of pseudo-almost periodic functions and Cauchy problems with operator of nondense domain, Annales Mathématiques Blaise Pascal, 6, 1-11, (1999) · Zbl 0941.34059 [43] Cuevas, C; Hernández M, E, Pseudo-almost periodic solutions for abstract partial functional differential equations, Applied Mathematics Letters, 22, 534-538, (2009) · Zbl 1170.35551 [44] 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, 45, 73-83, (2001) · Zbl 0985.34052 [45] Diagana, T, Pseudo almost periodic solutions to some differential equations, Nonlinear Analysis: Theory, Methods & Applications, 60, 1277-1286, (2005) · Zbl 1061.34040 [46] Diagana, T; Mahop, CM; N’Guérékata, GM, Pseudo-almost-periodic solutions to some semilinear differential equations, Mathematical and Computer Modelling, 43, 89-96, (2006) · Zbl 1096.34038 [47] Diagana, T; Mahop, CM; N’Guérékata, GM; Toni, B, Existence and uniqueness of pseudo-almost periodic solutions to some classes of semilinear differential equations and applications, Nonlinear Analysis: Theory, Methods & Applications, 64, 2442-2453, (2006) · Zbl 1102.34043 [48] Diagana, T; Mahop, CM, Pseudo almost periodic solutions to a neutral delay integral equation, Cubo, 9, 47-55, (2007) · Zbl 1122.45002 [49] Diagana T: Existence of pseudo almost periodic solutions to some classes of partial hyperbolic evolution equations.Electronic Journal of Qualitative Theory of Differential Equations 2007, (3):12. · Zbl 1108.35122 [50] Cuevas, C; de Souza, JC, [inlineequation not available: see fulltext.]-asymptotically [inlineequation not available: see fulltext.]-periodic solutions of semilinear fractional integro-differential equations, Applied Mathematics Letters, 22, 865-870, (2009) · Zbl 1176.47035 [51] Cuevas, C; de Souza, JC, Existence of [inlineequation not available: see fulltext.]-asymptotically [inlineequation not available: see fulltext.]-periodic solutions for fractional order functional integro-differential equations with infinite delay, Nonlinear Analysis: Theory, Methods & Applications, 72, 1683-1689, (2010) · Zbl 1197.47063 [52] Haase M: The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications. Volume 169. Birkhäuser, Basel, Switzerland; 2006:xiv+392. · Zbl 1101.47010 [53] Cuesta, E, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, 277-285, (2007) · Zbl 1163.45306 [54] Prüss J: Evolutionary Integral Equations and Applications, Monographs in Mathematics. Volume 87. Birkhäuser, Basel, Switzerland; 1993:xxvi+366. [55] Gripenberg G, Londen S-O, Staffans O: Volterra Integral and Functional Equations, Encyclopedia of Mathematics and Its Applications. Volume 34. Cambridge University Press, Cambridge, UK; 1990:xxii+701. · Zbl 0695.45002 [56] Arendt W, Batty CJK, Hieber M, Neubrander F: Vector-Valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics. Volume 96. Birkhäuser, Basel, Switzerland; 2001:xii+523. · Zbl 0978.34001 [57] Fattorini O: Second Order Differential Equations in Banach Spaces, North-Holland Mathematics Studies. Volume 108. North-Holland, Amsterdam, The Netherlands; 1985. [58] Lizama, C, On approximation and representation of [inlineequation not available: see fulltext.]-regularized resolvent families, Integral Equations and Operator Theory, 41, 223-229, (2001) · Zbl 1011.45006 [59] Lizama, C; Prado, H, Rates of approximation and ergodic limits of regularized operator families, Journal of Approximation Theory, 122, 42-61, (2003) · Zbl 1032.47024 [60] Lizama, C; Sánchez, J, On perturbation of [inlineequation not available: see fulltext.]-regularized resolvent families, Taiwanese Journal of Mathematics, 7, 217-227, (2003) · Zbl 1051.45009 [61] Shaw, S-Y; Chen, J-C, Asymptotic behavior of [inlineequation not available: see fulltext.]-regularized resolvent families at zero, Taiwanese Journal of Mathematics, 10, 531-542, (2006) · Zbl 1106.45004 [62] Zaidman S: Almost-Periodic Functions in Abstract Spaces, Research Notes in Mathematics. Volume 126. Pitman, Boston, Mass, USA; 1985:iii+133. · Zbl 0648.42006 [63] Fink AM: Almost Periodic Differential Equations, Lecture Notes in Mathematics, vol. 377. Springer, Berlin, Germany; 1974:viii+336. [64] Zhang C: Pseudo almost periodic functions and their applications, thesis. The University of Western Ontario; 1992. [65] Li, H-X; Huang, F-L; Li, J-Y, Composition of pseudo almost-periodic functions and semilinear differential equations, Journal of Mathematical Analysis and Applications, 255, 436-446, (2001) · Zbl 1047.47030 [66] Henríquez, HR; Lizama, C, Compact almost automorphic solutions to integral equations with infinite delay, Nonlinear Analysis: Theory, Methods & Applications, 71, 6029-6037, (2009) · Zbl 1179.43004 [67] Cuevas C, Henríquez H: Solutions of second order abstract retarded functional differential equations on the line. submitted · Zbl 1185.34006 [68] Bochner, S, Continuous mappings of almost automorphic and almost periodic functions, Proceedings of the National Academy of Sciences of the United States of America, 52, 907-910, (1964) · Zbl 0134.30102 [69] N’Guérékata GM: Topics in Almost Automorphy. Springer, New York, NY, USA; 2005:xii+168. · Zbl 1073.43004 [70] N’Guérékata GM: Almost Automorphic and Almost Periodic Functions in Abstract Spaces. Kluwer Academic Publishers/Plenum Press, New York, NY, USA; 2001:x+138. · Zbl 1001.43001 [71] N’Guérékata, GM, Quelques remarques sur LES fonctions asymptotiquement presque automorphes, Les Annales des Sciences Mathématiques du Québec, 7, 185-191, (1983) · Zbl 0524.34064 [72] 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, 59, 1333-1345, (2004) · Zbl 1071.34055 [73] Diagana, T; N’Guérékata, GM, Almost automorphic solutions to some classes of partial evolution equations, Applied Mathematics Letters, 20, 462-466, (2007) · Zbl 1169.35300 [74] Diagana, T; N’Guérékata, GM; Minh, NV, Almost automorphic solutions of evolution equations, Proceedings of the American Mathematical Society, 132, 3289-3298, (2004) · Zbl 1053.34050 [75] Ding, H-S; Xiao, T-J; Liang, J, Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions, Journal of Mathematical Analysis and Applications, 338, 141-151, (2008) · Zbl 1142.45005 [76] Diagana, T; Hernández, EM; dos Santos, JPC, Existence of asymptotically almost automorphic solutions to some abstract partial neutral integro-differential equations, Nonlinear Analysis: Theory, Methods & Applications, 71, 248-257, (2009) · Zbl 1172.45002 [77] Martin, RH Jr.: Nonlinear Operators and Differential Equations in Banach Spaces. Robert E. Krieger, Melbourne, Fla, USA; 1987:xiv+440. [78] Simon, J, Compact sets in the space [inlineequation not available: see fulltext.], Annali di Matematica Pura ed Applicata. Serie Quarta, 146, 65-96, (1987) · Zbl 0629.46031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.