Existence and uniqueness of mild solution for stochastic partial integro-differential equations with delays

A.Anguraj; K. Ramkumar

doi:10.26637/MJM0603/0013

Abstract

In this work, we study the existence results for stochastic partial integro-differential equations with delays. The mild solution of the problem is derived by using a different resolvent operator given in [18]. The existence of solution is proved by using contraction mapping principle.

Keywords:

Resolvent operator, mild solution, Stochastic partial integro-differential equations, Dynamic equation, nonmonotone delays, oscillatory solution, time scales.

Mathematics Subject Classification:

Mathematics

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A.Anguraj Department of Mathematics, PSG College of Arts and Science, Coimbatore-641 014 , Tamil Nadu, India.
K. Ramkumar Department of Mathematics, PSG College of Arts and Science, Coimbatore-641 014 , Tamil Nadu, India.

Pages: 542-546
Date Published: 01-07-2018
Vol. 6 No. 03 (2018): Malaya Journal of Matematik (MJM)

A. Anguraj and S. Kanjanadevi, Existence Results for Fractional Non- Instantaneous Impulsive IntegroDifferential Equations with Nonlocal Conditions, Journal of Applied Mathematics, vol.1, 2016, 44-58.

S. Aizicovici, M. Mckibben, Existence results for a class of abstract nonlocal cauchy problems, Nonlinear Anal., $39,2000,649-668$

P. Balasubramaiam, J. Y. Park, A. Vincent Antony Kumar, Existence of solution for semilinear neutral stochastic functional differential equations with nonlocal conditions, Nonlinear Anal., 2009, 1049-1058.

J. H. Bao, Z. T. Hou, Existence of mild solutions to stochastic neutral partial functional differential equations with non- Lipschitz coefficients. Comput. Math. Appl., $59,2010,207-214$

L. Byszewski, V. Lakshmikantham, Theorem about the existence and uniqueness of solution of a nonlocal Cauchy problem in a Banach space, Appl. Anal., 40, 1990, 11-19.

T. A. Burton, Stability by Fixed Point Theory For Functional Differential Equations, Dover Publications, Inc., New York, 2006.

T. Caraballo, Asymptotic exponential stability of stochastic partial differential equatios with delays, Stochastic, 33, 1990, 27-47.

G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University press, Cambridge, 1992.

E. Hernandez, D. O' Regan, K. Balachandran, On recent developments in the theory of abstract differential equations with fractional derivatives, Nonlinear Anal., 73, $2010,3462-3471$.

E. Hernandez, D. O' Regan, K. Balachandran, Existence results for abstract fractional differential equations with nonlocal conditions via resolvent operators, Indagationes mathematicae, 24, 2013, 68-82.

L. Y. Hu, Y. Ren. R. Sakthivel, Existence and uniqueness of mild solutions for semilinear integro- differential equations of fractional order with nonlocal initial conditions and delays. Semigroup Forum, 79, 2009, 507 514.

A. Ichikawa, Stability of semilinear stochastic evolution equations, J. Math. Anal., 90, 1982, 12-44.

I. Karatzas, S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer - Verlag, Berlin, 1991.

K. Liu, Lyapunov functional and asymptotic stability of stochastic delay evolution equations, Stochastic, 63,1998 , $1-26$.

J. Luo, Fixed points and stability of neutral stochastic delay differential equations, J. Math. Anal. Appl, 334, 2007, 431-440.

X. Mao, Stochastic Differential Equations and Applications, Ellis Horwood, Chichester, UK, 1997

G. M. Mophou, G.M. N' Guerekata, Existance of the mild solution for some fractional differential equations with nonlocal conditions, Semigroup Forum, 79, 2009, $315-322$

J. Pruss, Evolutionary integral equations and Applications, Monographs in Mathematics, Birkhauser verlag, Basel, 87, 1993.