Existence and Stability of Mixed Stochastic Fractional Order Differential Inclusion Equations via Cosine Dynamical System

Salah H Abid¹, Sameer Q Hasan^1, and Zainab A Khudhur¹

¹Department of Mathematics, College of Education Almustansryah University

Cite this paper:
Salah H Abid, Sameer Q Hasan and Zainab A Khudhur. Existence and Stability of Mixed Stochastic Fractional Order Differential Inclusion Equations via Cosine Dynamical System. American Journal of Systems and Software. 2016; 4(2):57-68. doi: 10.12691/ajss-4-2-5

Abstract

In this paper, we shall consider the existence and stability of stochastic fractional order differential inclusion nonlinear equations in infinite dimensional space by mixed fractional Brownian motion in Hilbert space H.

Keywords:
Neutral mixed stochastic fractional order differential inclusion equations existence stability via cosine dynamical system with fractional derivative as component in nonlinear functions 0<α β<1

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]	Benchohra. M and Ntouyas .S. K. “Existence of mild solutions of second order initial value problems for delay Integro-differential inclusions with nonlocal conditions”, Mathematica Bohemica, 4 (127) (2002), 613-622.
[2]	Benchohra. M and Ntouyas. S. K,. “Existence results for the semi-infinite interval for first and second order Integro-differential equations in Banach spaces with nonlocal condition”s, Acta Univ. Palacki. Olomuc, Fac. Rer. Nat. Mathematica 41 (2002), 13-19.
[3]	Byszewski. L and Laksmikantham. V,. “Theorems about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space”, Appl. Anal. 40 (1) (1991), 11-19.
[4]	Balasubramaniam. P,. “Existence of solution of functional stochastic differential inclusions”, Tamkang J. Math. 33 (2002) 35-43.
[5]	Balasubramaniam. P, Vinayagam. D,. “Existence of solutions of nonlinear neutral stochastic differential inclusions in a Hilbert space”, Stochastic Anal. Appl. 23 (2005) 137-151.
[6]	Balakrishnan, A. V., “Applications of Mathematics: Applied Functional Analysis”, by Springer -verlag New York, Inc., (1976(.
[7]	Chang. Y. K,. “Controllability of impulsive functional differential systems with infinite delay in Banach spaces”, Chaos Solitons Fractals 33 (2007) 1601-1609.
[8]	Chang. Y.K, Anguraj. A, Mallika Arjunan. M. M, “Existence results for impulsive neutral functional differential equations with infinite delay”, Nonlinear, Anal. Hybrid Syst. 2 (2008) 209-218.
[9]	Chang. Y .K, Nieto. J. J,. “Existence of solutions for impulsive neutral integro-differential inclusions with nonlocal initial conditions via fractional operators”, Numer. Funct. Anal. Optim. 30 (2009) 227-244.
[10]	Cao. Y, Fu. X, “Existence for neutral impulsive differential inclusions with nonlocal conditions”, Nonlinear Anal. 68 (2008) 3707-3718.
[11]	Dhage. B. C,. “Multi-valued mappings and fixed points II”, Tamkang J. Math. 37 (2006) 27-46.
[12]	Da Prato. G, J. Zabczyk. J,. “Stochastic Equations in Infinite Dimensions”, Cambridge University Press, Cambridge, 1992.
[13]	Diagana, T., “An Introduction to Classical and P-ADIC Theory of Linear Operators and Application”, Nova Science Publishers, (2006).
[14]	Goldstein J. A.,. “Semigroup of linear operators and applications”, Oxford Univ. Press, New York, 1985.
[15]	Grimmer. R, Pritchard. A. J,. “Analytic resolvent operators for integral equations in a Banach space”. J. Differential Equations 50 (1983) 234-259.
[16]	Hernandez. M. E,. “Existence results for a second order abstract Cauchy problem with nonlocal conditions”, Electr. J. Diff. Eqs, 2005 No. 73 (2005), 1-17.
[17]	Hino. Y, Murakam. S, Naito. T, “Functional-differential equations with infinite delay”, in: Lecture Notes in Mathematics, vol. 1473, Springer-Verlag, Berlin, 1991.
[18]	Lin, A., Hu. L., “Existence results for Impulsive Neutral Stochastic Functional Integro-differential Inclusions with Nonlocal Initial Conditions”, J. Computers and Mathematics with Applications, 59(2010). 64-73.
[19]	Li K.. “Stochastic Delay Fractional Evolut- ions Driven by Fractional Brownian Motion”, Mathematical Method in the Applied Sciences, 2014.
[20]	Lasikcka, I. “Feedback semigroups and cosine operators for boundary feedback parabolic and hyperbolic equations”, J. Deferential Equation, 47, pp. 246-272, (1983).
[21]	Nouyas. S .K, “Existence results for impulsive partial neutral functional differential inclusions”, Electron. J. Differential Equations 30 (2005) 1-11.
[22]	Nasser. E. T., “Existence of Mild Solustions For a Neutral Fractional Equation With Fractional Nonlocal Conditions”, E. J, of Differential Equations, Vol. 2012(2012), No.153, pp.1-12.
[23]	Opial., A. Lasota, Z., “Application of the kakutani-Ky-Fan theorem in the theory of ordinary differential equations or noncompact acyclic-valued map”, Bull. Acad. olon. Sci. Ser.Sci.Math.Astronom.phys. 13(1965 )781-786.
[24]	Pazy. A,. “Semigroups of linear operators and applications to partial differential equations”, in: Applied Methematical Sciences, vol. 44, Springer Verlag, New York, 1983.
[25]	Park. J .Y, Balachandtran. K, Annapooran. N, “Existence results for impulsive neutral functional integro-differential equations with infinite delay”, Nonlinear Anal. (2009).
[26]	Ren. Y, Hu. L, “Existence results for impulsive neutral stochastic functional integro-differential equations with infinite delays”, Acta Appl. Math. (2009).
[27]	Travis, C. C. and Webb, G. F., “Compactness, regularity and uniform continuity properties of strongly continuous cosine families “, Houston J.Math.3(4) (1977), 555-567.
[28]	Travis, C. C. and Webb, G.F.. “Cosine families and abstract nonlinear second order differential equtions”, Acta Math. Acad. Sci. Hungaricae, 32(1978), 76-96.