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Decay solutions and decay rate for a class of retarded abtract semilinear fractional evolution inclusions. (English) Zbl 1419.35215

Summary: In this paper, we prove the existence of decay integral solutions to a class of fractional differential inclusions with finite delays and estimate their decay rate. For these purposes, we have to construct a suitable regular measure of noncompactness on the space of solutions and then deploy the fixed point theory for condensing multivalued maps. An application to a class of fractional PDE with almost sectorial operator is also given.

MSC:

35R11 Fractional partial differential equations
35B35 Stability in context of PDEs
35R12 Impulsive partial differential equations
47H08 Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc.
47H10 Fixed-point theorems
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References:

[1] R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina and B. N. Sadovskiĭ, Measures of Noncompactness and Condensing Operators, Operator Theory: Advances and Applications 55, Birkhäuser Verlag, Basel, 1992. · Zbl 0748.47045
[2] N. T. Anh and T. D. Ke, Decay integral solutions for neutral fractional differential equations with infinite delays, Math. Methods Appl. Sci. 38 (2015), no. 8, 1601-1622. · Zbl 1322.34087 · doi:10.1002/mma.3172
[3] D. Bothe, Multivalued perturbations of \(m\)-accretive differential inclusions, Israel J. Math. 108 (1998), 109-138. · Zbl 0922.47048
[4] A. Cernea, On the existence of mild solutions for nonconvex fractional semilinear differential inclusions, Electron. J. Qual. Theory Differ. Equ. 2012 (2012), no. 64, 1-15. · Zbl 1340.34009
[5] S. Ji and S. Wen, Nonlocal Cauchy problem for impulsive differential equations in Banach spaces, Int. J. Nonlinear Sci. 10 (2010), no. 1, 88-95. · Zbl 1225.34082
[6] M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Series in Nonlinear Analysis and Applications 7, Walter de Gruyter, Berlin, 2001. · Zbl 0988.34001
[7] T. D. Ke and D. Lan, Decay integral solutions for a class of impulsive fractional differential equations in Banach spaces, Fract. Calc. Appl. Anal. 17 (2014), no. 1, 96-121. · Zbl 1312.34017 · doi:10.2478/s13540-014-0157-5
[8] ——–, Fixed point approach for weakly asymptotic stability of fractional differential inclusions involving impulsive effects, J. Fixed Point Theory Appl. 19 (2017), no. 4, 2185-2208. · Zbl 1376.34021
[9] A. A. Kilbas, H. M. Srivastava ande J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Elsevier Science B.V., Amsterdam, 2006. · Zbl 1092.45003
[10] Z. Liu, J. Lv and R. Sakthivel, Approximate controllability of fractional functional evolution inclusions with delay in Hilbert spaces, IMA J. Math. Control. Inform. 31 (2014), no. 3, 363-383. · Zbl 1297.93038 · doi:10.1093/imamci/dnt015
[11] S. Rathinasamy and R. Yong, Approximate controllability of fractional differential equations with state-dependent delay, Results Math. 63 (2013), no. 3-4, 949-963. · Zbl 1272.34105 · doi:10.1007/s00025-012-0245-y
[12] Y. Ren, L. Hu and R. Sakthivel, Controllability of impulsive neutral stochastic functional differential inclusions with infinite delay, J. Comput. Appl. Math. 235 (2011), no. 8, 2603-2614. · Zbl 1211.93025 · doi:10.1016/j.cam.2010.10.051
[13] R. Sakthivel, R. Ganesh and S. M. Anthoni, Approximate controllability of fractional nonlinear differential inclusions, Appl. Math. Comput. 225 (2013), 708-717. · Zbl 1334.93034 · doi:10.1016/j.amc.2013.09.068
[14] R. Sakthivel, R. Ganesh, Y. Ren and S. M. Anthoni, Approximate controllability of nonlinear fractional dynamical systems, Commun. Nonlinear Sci. Numer. Simul. 18 (2013), no. 12, 3498-3508. · Zbl 1344.93019 · doi:10.1016/j.cnsns.2013.05.015
[15] R.-N. Wang, D.-H. Chen and T.-J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations 252 (2012), no. 1, 202-235. · Zbl 1238.34015 · doi:10.1016/j.jde.2011.08.048
[16] J. Wang, A. G. Ibrahim and M. Fečkan, Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces, Appl. Math. Comput. 257 (2015), 103-118. · Zbl 1338.34027
[17] R.-N. Wang and Q.-H. Ma, Some new results for multi-valued fractional evolution equations, Appl. Math. Comput. 257 (2015), 285-294. · Zbl 1338.34111
[18] R. N. Wang, Q. M. Xiang and P. X. Zhu, Existence and approximate controllability for systems governed by fractional delay evolution inclusions, Optimization 63 (2014), no. 8, 1191-1204. · Zbl 1296.93029
[19] J. Wang and Y. Zhou, Existence and controllability results for fractional semilinear differential inclusions, Nonlinear Anal. Real World Appl. 12 (2011), no. 6, 3642-3653. · Zbl 1231.34108
[20] R.-N. Wang, P.-X. Zhu and Q.-H. Ma, Multi-valued nonlinear perturbations of time fractional evolution equations in Banach spaces, Nonlinear Dynam. 80 (2015), no. 4, 1745-1759. · Zbl 1345.47020
[21] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Hackensack, NJ, 2014. · Zbl 1336.34001
[22] Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl. 59 (2010), no. 3, 1063-1077. · Zbl 1189.34154 · doi:10.1016/j.camwa.2009.06.026
[23] Y. Zhou, V. Vijayakumar and R. Murugesu, Controllability for fractional evolution inclusions without compactness, Evol. Equ. Control Theory 4 (2015), no. 4, 507-524. · Zbl 1335.34096 · doi:10.3934/eect.2015.4.507
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