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Study on fractional non-autonomous evolution equations with delay. (English) Zbl 1375.34115

This paper is concerned with the existence of mild solutions for initial value problem to nonlinear fractional non-autonomous evolution equations with delay in a Banach space.

MSC:

34K37 Functional-differential equations with fractional derivatives
34K30 Functional-differential equations in abstract spaces
37C60 Nonautonomous smooth dynamical systems
Full Text: DOI

References:

[1] Metzler, R.; Klafter, J., The random walks guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, 1-77 (2000) · Zbl 0984.82032
[2] Agarwal, R. P.; Benchohra, M.; Hamani, S., A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109, 973-1033 (2010) · Zbl 1198.26004
[3] Bajlekova, E. G., Fractional Evolution Equations in Banach Spaces (2001), Department of Mathematics, Eindhoven University of Technology, (Ph.D. thesis) · Zbl 0989.34002
[4] Chen, P.; Li, Y., Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions, Z. Angew. Math. Phys., 65, 4, 711-728 (2014) · Zbl 1304.34006
[5] Chen, P.; Li, Y.; Zhang, X., On the initial value problem of fractional stochastic evolution equations in Hilbert spaces, Commun. Pure Appl. Anal., 14, 5, 1817-1840 (2015) · Zbl 1322.34007
[6] El-Borai, M. M.; El-Nadi, K. E.; El-Akabawy, E. G., On some fractional evolution equations, Comput. Math. Appl., 59, 1352-1355 (2010) · Zbl 1189.45009
[7] Li, M.; Chen, C.; Li, F. B., On fractional powers of generators of fractional resolvent families, J. Funct. Anal., 259, 2702-2726 (2010) · Zbl 1203.47021
[8] Li, K.; Peng, J.; Jia, J., Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives, J. Funct. Anal., 263, 476-510 (2012) · Zbl 1266.47066
[9] Shu, X.; Shi, Y., A study on the mild solution of impulsive fractional evolution equations, Appl. Math. Comput., 273, 465-476 (2016) · Zbl 1410.34031
[10] Wang, R. N.; Chen, D. H.; Xiao, T. J., Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations, 252, 202-235 (2012) · Zbl 1238.34015
[11] Wang, J.; Zhou, Y., A class of fractional evolution equations and optimal controls, Nonlinear Anal. RWA, 12, 262-272 (2011) · Zbl 1214.34010
[12] Zhou, Y.; Jiao, F., Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59, 1063-1077 (2010) · Zbl 1189.34154
[13] Tanabe, H., Functional Analytic Methods for Partial Differential Equations (1997), Marcel Dekker: Marcel Dekker New York, USA · Zbl 0867.35003
[14] El-Borai, M. M., The fundamental solutions for fractional evolution equations of parabolic type, J. Appl. Math. Stoch. Anal., 3, 197-211 (2004) · Zbl 1081.34053
[15] Ouyang, Z., Existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay, Comput. Math. Appl., 61, 860-870 (2011) · Zbl 1217.35206
[16] Zhu, B.; Liu, L.; Wu, Y., Existence and uniqueness of global mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay, Comput. Math. Appl. (2016)
[17] Henry, D., (Geometric Theory of Semilinear Parabolic Equations. Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840 (1981), Springer-verlag: Springer-verlag New York) · Zbl 0456.35001
[18] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Springer-verlag: Springer-verlag Berlin · Zbl 0516.47023
[19] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., (Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204 (2006), Elsevier Science B.V: Elsevier Science B.V Amsterdam) · Zbl 1092.45003
[20] Gorenflo, R.; Mainardi, F., Fractional calculus and stable probability distributions, Arch. Mech., 50, 3, 377-388 (1998) · Zbl 0934.35008
[21] Banas̀, J.; Goebel, K., (Measures of Noncompactness in Banach Spaces. Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, vol. 60 (1980), Marcel Dekker: Marcel Dekker New York) · Zbl 0441.47056
[22] Deimling, K., Nonlinear Functional Analysis (1985), Springer-Verlag: Springer-Verlag New York · Zbl 0559.47040
[23] Chen, P.; Li, Y., Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math., 63, 731-744 (2013) · Zbl 1279.34072
[24] Li, Y., Existence of solutions of initial value problems for abstract semilinear evolution equations, Acta Math. Sinica, Engl. Ser. Mar., 48, 1089-1094 (2005), (in Chinese) · Zbl 1124.34341
[25] Heinz, H. P., On the behaviour of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal., 7, 1351-1371 (1983) · Zbl 0528.47046
[26] Xiao, T. J.; Liang, J., Existence of classical solutions to nonautonomous nonlocal parabolic problems, Nonlinear Anal., 63, e225-e232 (2005) · Zbl 1159.35383
[27] Friedman, A., Partial Differential Equations (1969), Holt, Rinehart and Winston: Holt, Rinehart and Winston New York, NY, USA · Zbl 0224.35002
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