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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
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[1] Anh VV, Mcvinish R: Fractional differential equations driven by Lévy noise.Journal of Applied Mathematics and Stochastic Analysis 2003,16(2):97-119. 10.1155/S1048953303000078 · Zbl 1042.60034 · doi:10.1155/S1048953303000078
[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 · Zbl 1438.26010 · doi:10.1007/978-3-7091-2664-6_5
[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 2008, 1: 11-18. 10.2174/1874117700801010011 · Zbl 1185.34006 · doi:10.2174/1874117700801010011
[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. · Zbl 1092.45003
[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 2004,199(2):211-255. 10.1016/j.jde.2003.12.002 · Zbl 1068.35037 · doi:10.1016/j.jde.2003.12.002
[11] Lin W: Global existence theory and chaos control of fractional differential equations.Journal of Mathematical Analysis and Applications 2007,332(1):709-726. 10.1016/j.jmaa.2006.10.040 · Zbl 1113.37016 · doi:10.1016/j.jmaa.2006.10.040
[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. · Zbl 0864.93004
[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 2002,29(1-4):281-296. 10.1023/A:1016556604320 · Zbl 1041.93022 · doi:10.1023/A:1016556604320
[15] Ross B (Ed): Fractional Calculus and Its Applications, Lecture Notes in Mathematics, vol. 457. Springer, Berlin, Germany; 1975:vi+381. · Zbl 0293.00010
[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 2008,338(2):1340-1350. 10.1016/j.jmaa.2007.06.021 · Zbl 1209.34096 · doi:10.1016/j.jmaa.2007.06.021
[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 2009,16(3):401-411. · 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 2008,69(10):3337-3343. 10.1016/j.na.2007.09.025 · Zbl 1162.34344 · doi:10.1016/j.na.2007.09.025
[21] Lakshmikantham V, Vatsala AS: Basic theory of fractional differential equations.Nonlinear Analysis: Theory, Methods & Applications 2008,69(8):2677-2682. 10.1016/j.na.2007.08.042 · Zbl 1161.34001 · doi:10.1016/j.na.2007.08.042
[22] Lakshmikantham V, Vatsala AS: Theory of fractional differential inequalities and applications.Communications in Applied Analysis 2007,11(3-4):395-402. · 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. · Zbl 1179.34002
[24] El-Borai MM: Some probability densities and fundamental solutions of fractional evolution equations.Chaos, Solitons and Fractals 2002,14(3):433-440. 10.1016/S0960-0779(01)00208-9 · Zbl 1005.34051 · doi:10.1016/S0960-0779(01)00208-9
[25] El-Borai MM: Semigroups and some nonlinear fractional differential equations.Applied Mathematics and Computation 2004,149(3):823-831. 10.1016/S0096-3003(03)00188-7 · Zbl 1046.34079 · doi:10.1016/S0096-3003(03)00188-7
[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 2009,79(2):315-322. 10.1007/s00233-008-9117-x · Zbl 1180.34006 · doi:10.1007/s00233-008-9117-x
[29] N’Guérékata GM: A Cauchy problem for some fractional abstract differential equation with non local conditions.Nonlinear Analysis: Theory, Methods & Applications 2009,70(5):1873-1876. 10.1016/j.na.2008.02.087 · Zbl 1166.34320 · doi:10.1016/j.na.2008.02.087
[30] El-Sayed AMA, Ibrahim A-G: Multivalued fractional differential equations.Applied Mathematics and Computation 1995,68(1):15-25. 10.1016/0096-3003(94)00080-N · Zbl 0830.34012 · doi:10.1016/0096-3003(94)00080-N
[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 2008,11(1):35-56. · Zbl 1149.26010
[32] Henderson J, Ouahab A: Fractional functional differential inclusions with finite delay.Nonlinear Analysis: Theory, Methods & Applications 2009,70(5):2091-2105. 10.1016/j.na.2008.02.111 · Zbl 1159.34010 · doi:10.1016/j.na.2008.02.111
[33] Ouahab A: Some results for fractional boundary value problem of differential inclusions.Nonlinear Analysis: Theory, Methods & Applications 2008,69(11):3877-3896. 10.1016/j.na.2007.10.021 · Zbl 1169.34006 · doi:10.1016/j.na.2007.10.021
[34] Cuevas C, Lizama C: Almost automorphic solutions to a class of semilinear fractional differential equations.Applied Mathematics Letters 2008,21(12):1315-1319. 10.1016/j.aml.2008.02.001 · Zbl 1192.34006 · doi:10.1016/j.aml.2008.02.001
[35] Cuevas C, Lizama C: Almost automorphic solutions to integral equations on the line.Semigroup Forum 2009,79(3):461-472. 10.1007/s00233-009-9154-0 · Zbl 1187.45005 · doi:10.1007/s00233-009-9154-0
[36] Zhang CY: Pseudo-almost-periodic solutions of some differential equations.Journal of Mathematical Analysis and Applications 1994,181(1):62-76. 10.1006/jmaa.1994.1005 · Zbl 0796.34029 · doi:10.1006/jmaa.1994.1005
[37] Zhang CY: Integration of vector-valued pseudo-almost periodic functions.Proceedings of the American Mathematical Society 1994,121(1):167-174. 10.1090/S0002-9939-1994-1186140-8 · Zbl 0818.42003 · doi:10.1090/S0002-9939-1994-1186140-8
[38] Zhang CY: Pseudo almost periodic solutions of some differential equations. II.Journal of Mathematical Analysis and Applications 1995,192(2):543-561. 10.1006/jmaa.1995.1189 · Zbl 0826.34040 · doi:10.1006/jmaa.1995.1189
[39] Zhang C: Almost Periodic Type Functions and Ergodicity. Science Press, Beijing, China; 2003:xii+355. · Zbl 1068.34001 · doi:10.1007/978-94-007-1073-3
[40] Ait Dads E, Arino O: Exponential dichotomy and existence of pseudo almost-periodic solutions of some differential equations.Nonlinear Analysis: Theory, Methods & Applications 1996,27(4):369-386. 10.1016/0362-546X(95)00027-S · Zbl 0855.34055 · doi:10.1016/0362-546X(95)00027-S
[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 1997,28(7):1141-1155. 10.1016/S0362-546X(97)82865-9 · Zbl 0874.34041 · doi:10.1016/S0362-546X(97)82865-9
[42] Amir B, Maniar L: Composition of pseudo-almost periodic functions and Cauchy problems with operator of nondense domain.Annales Mathématiques Blaise Pascal 1999,6(1):1-11. 10.5802/ambp.110 · Zbl 0941.34059 · doi:10.5802/ambp.110
[43] Cuevas C, Hernández M E: Pseudo-almost periodic solutions for abstract partial functional differential equations.Applied Mathematics Letters 2009,22(4):534-538. 10.1016/j.aml.2008.06.026 · Zbl 1170.35551 · doi:10.1016/j.aml.2008.06.026
[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 2001,45(1):73-83. 10.1016/S0362-546X(99)00330-2 · Zbl 0985.34052 · doi:10.1016/S0362-546X(99)00330-2
[45] Diagana T: Pseudo almost periodic solutions to some differential equations.Nonlinear Analysis: Theory, Methods & Applications 2005,60(7):1277-1286. 10.1016/j.na.2004.11.002 · Zbl 1061.34040 · doi:10.1016/j.na.2004.11.002
[46] Diagana T, Mahop CM, N’Guérékata GM: Pseudo-almost-periodic solutions to some semilinear differential equations.Mathematical and Computer Modelling 2006,43(1-2):89-96. 10.1016/j.mcm.2005.04.013 · Zbl 1096.34038 · doi:10.1016/j.mcm.2005.04.013
[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 2006,64(11):2442-2453. 10.1016/j.na.2005.08.024 · Zbl 1102.34043 · doi:10.1016/j.na.2005.08.024
[48] Diagana T, Mahop CM: Pseudo almost periodic solutions to a neutral delay integral equation.Cubo 2007,9(1):47-55. · 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 2009,22(6):865-870. 10.1016/j.aml.2008.07.013 · Zbl 1176.47035 · doi:10.1016/j.aml.2008.07.013
[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 2010,72(3-4):1683-1689. 10.1016/j.na.2009.09.007 · Zbl 1197.47063 · doi:10.1016/j.na.2009.09.007
[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 · doi:10.1007/3-7643-7698-8
[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. · Zbl 0784.45006 · doi:10.1007/978-3-0348-8570-6
[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 · doi:10.1017/CBO9780511662805
[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 · doi:10.1007/978-3-0348-5075-9
[57] Fattorini O: Second Order Differential Equations in Banach Spaces, North-Holland Mathematics Studies. Volume 108. North-Holland, Amsterdam, The Netherlands; 1985. · Zbl 0564.34063
[58] Lizama C:On approximation and representation of[InlineEquation not available: see fulltext.]-regularized resolvent families.Integral Equations and Operator Theory 2001,41(2):223-229. 10.1007/BF01295306 · Zbl 1011.45006 · doi:10.1007/BF01295306
[59] Lizama C, Prado H: Rates of approximation and ergodic limits of regularized operator families.Journal of Approximation Theory 2003,122(1):42-61. 10.1016/S0021-9045(03)00040-6 · Zbl 1032.47024 · doi:10.1016/S0021-9045(03)00040-6
[60] Lizama C, Sánchez J:On perturbation of[InlineEquation not available: see fulltext.]-regularized resolvent families.Taiwanese Journal of Mathematics 2003,7(2):217-227. · 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 2006,10(2):531-542. · 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. · Zbl 0325.34039
[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 2001,255(2):436-446. 10.1006/jmaa.2000.7225 · Zbl 1047.47030 · doi:10.1006/jmaa.2000.7225
[66] Henríquez HR, Lizama C: Compact almost automorphic solutions to integral equations with infinite delay.Nonlinear Analysis: Theory, Methods & Applications 2009,71(12):6029-6037. 10.1016/j.na.2009.05.042 · Zbl 1179.43004 · doi:10.1016/j.na.2009.05.042
[67] Cuevas C, Henríquez H: Solutions of second order abstract retarded functional differential equations on the line. submitted · Zbl 1231.34135
[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 1964, 52: 907-910. 10.1073/pnas.52.4.907 · Zbl 0134.30102 · doi:10.1073/pnas.52.4.907
[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 · doi:10.1007/978-1-4757-4482-8
[71] N’Guérékata GM: Quelques remarques sur les fonctions asymptotiquement presque automorphes.Les Annales des Sciences Mathématiques du Québec 1983,7(2):185-191. · 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 2004,59(8):1333-1345. · Zbl 1071.34055 · doi:10.1016/S0362-546X(04)00329-3
[73] Diagana T, N’Guérékata GM: Almost automorphic solutions to some classes of partial evolution equations.Applied Mathematics Letters 2007,20(4):462-466. 10.1016/j.aml.2006.05.015 · Zbl 1169.35300 · doi:10.1016/j.aml.2006.05.015
[74] Diagana T, N’Guérékata GM, Minh NV: Almost automorphic solutions of evolution equations.Proceedings of the American Mathematical Society 2004,132(11):3289-3298. 10.1090/S0002-9939-04-07571-9 · Zbl 1053.34050 · doi:10.1090/S0002-9939-04-07571-9
[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 2008,338(1):141-151. 10.1016/j.jmaa.2007.05.014 · Zbl 1142.45005 · doi:10.1016/j.jmaa.2007.05.014
[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 2009,71(1-2):248-257. 10.1016/j.na.2008.10.046 · Zbl 1172.45002 · doi:10.1016/j.na.2008.10.046
[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 1987, 146: 65-96. · Zbl 0629.46031 · doi:10.1007/BF01762360
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