GLOBALLY ATTRACTING SOLUTIONS TO IMPULSIVE FRACTIONAL DIFFERENTIAL INCLUSIONS OF SOBOLEV TYPE

doi:10.1016/S0252-9602(17)30074-7

Abstract

Abstract:

We study a generalized Cauchy problem associated with a class of impulsive fractional differential inclusions of Sobolev type in Banach spaces. Our aim is to prove the existence of a compact set of globally attracting solutions to the problem in question. An application to fractional partial differential equations subject to impulsive effects is given to illustrate our results.

Key words: globally attracting solution, impulsive condition, nonlocal condition, condensing map, measure of non-compactness, MNC-estimate

Van Hien LE, Dinh Ke TRAN, Trong Kinh CHU. GLOBALLY ATTRACTING SOLUTIONS TO IMPULSIVE FRACTIONAL DIFFERENTIAL INCLUSIONS OF SOBOLEV TYPE[J].Acta mathematica scientia,Series B, 2017, 37(5): 1295-1318.

Trendmd

References

[1] Akhmerov R R, Kamenskii M I, Potapov A S, Rodkina A E, Sadovskii B N. Measures of Noncompactness and Condensing Operators. Boston-Basel-Berlin:Birkhäuser, 1992
[2] Anh N T, Ke T D. Decay integral solutions for neutral fractional differential equations with infinite delays. Math Methods Appl Sci, 2015, 38:1601-1622
[3] Balachandran K, Anandhi E R, Dauer J P. Boundary controllability of Sobolev-type abstract nonlinear integrodifferential systems. J Math Anal Appl, 2003, 277:446-464
[4] Balachandran K, Kiruthika S, Trujillo J J. On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces. Comput Math Appl, 2011, 62:1157-1165
[5] Barenblat G, Zheltor J, Kochiva I. Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J Appl Math Mech, 1960, 24:1286-1303
[6] Bothe D. Multivalued perturbations of m-accretive differential inclusions. Israel J Math, 1998, 108:109-138
[7] Brill H. A semilinear Sobolev evolution equation in Banach space. J Diff Equ, 1977, 24:412-425
[8] Burton T A. Stability by Fixed Point Theory for Functional Differential Equations. New York:Dover Publications, 2006
[9] Burton T A, Furumochi T. Fixed points and problems in stability theory for ordinary and functional differential equations. Dyn Syst Appl, 2001, 10:89-116
[10] Byszewski L. Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J Math Anal Appl, 1991, 162:494-505
[11] Carvalho A N, Langa J A, Robinson J C. Attractors for Infinite-Dimensional non-Autonomous Dynamical Systems. New York:Springer, 2013
[12] Chuong N M, Ke T D, Quan N N. Stability for a class of fractional partial integro-differential equations. J Integral Equations Appl, 2014, 26:145-170
[13] Diestel J, Ruess W M, Schachermayer S. Weak compactness in L^l(μ, X). Proc Amer Math Soc, 1993, 118:447-453
[14] Ekeland I, Temam R. Convex Analysis and Variational Problems. Philadelphia PA:SIAM, 1999
[15] Feckan M, Wang J R, Zhou Y. Controllability of fractional functional evolution equations of sobolev type via characteristic solution operators. J Optim Theory Appl, 2013, 156:79-95
[16] Filippov A F. Differential Equations with Discontinuous Righthand Sides//Translated from the Russian. Dordrecht:Kluwer Academic Publishers Group, 1988
[17] Hilfer R. Applications of Fractional Calculus in Physics. Singapore:World Scientific, 2000
[18] Ibrahim R W. On the existence for diffeo-integral inclusion of Sobolev-type of fractional order with applications. Anziam J, 2010, 52(E):E1-E21
[19] Ji S, Wen S. Nonlocal cauchy problem for impulsive differential equations in Banach spaces. Int J Nonlinear Sci, 2010, 10:88-95
[20] Kamenskii M, Obukhovskii V, Zecca P. Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. Berlin-New York:Walter de Gruyter, 2001
[21] Ke T D, Kinh CT. Generalized Cauchy problem involving a class of degenerate fractional differential equations. Dyn Contin Discrete Impuls Syst Ser A Math Anal, 2014, 21:449-472
[22] Ke T D, Lan D. Decay integral solutions for a class of impulsive fractional differential equations in Banach spaces. Fract Calc Appl Anal, 2014, 17:96-121
[23] Kilbas A A, Srivastava H M, Trujillo J J. Theory and Applications of Fractional Differential Equations. Amsterdam:Elsevier, 2006
[24] Li F, Liang J, Xu H K. Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions. J Math Anal Appl, 2012, 391:510-525
[25] Lightbourne J H, Rankin S M. A partial functional differential equation of Sobolev type. J Math Anal Appl, 1983, 93:328-337
[26] Monje C A, Chen Y Q, Blas M V, Xue D Y, Feliu V. Fractional-Order Systems and Controls. Fundamentals and Applications. London:Springer, 2010
[27] Seidman T I. Invariance of the reachable set under nonlinear perturbations. SIAM J Control Optim, 1987, 25(5):1173-1191
[28] Pang J S, Stewart D E. Differential variational inequalities. Math Program Ser A, 2008, 113:345-424
[29] Wang J R, Feckan M, Zhou Y. On the new concept of solutions and existence results for impulsive fractional evolution equations. Dyn Partial Differ Equ, 2011, 8:345-361
[30] Wang J R, Feckan M, Zhou Y. A survey on impulsive fractional differential equations. Fract Calc Appl Anal, 2016, 19:806-831
[31] Wang J R, Ibrahim A G, Feckan M. Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces. Appl Math Comput, 2015, 257:103-118
[32] Wang J R, Zhou Y, Feckan M. On recent developments in the theory of boundary value problems for impulsive fractional differential equations. Comput Math Appl, 2012, 64:3008-3020
[33] Wang J R, Zhou Y, Feckan M. Nonlinear impulsive problems for fractional differential equations and Ulam stability. Comp Math Appl, 2012, 64:3389-3405
[34] Wang J R, Zhang Y. On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives. Appl Math Lett, 2015, 39:85-90
[35] Zhou Y, Jiao F. Existence of mild solutions for fractional neutral evolution equations. Comput Math Appl, 2010, 59:1063-1077