2012
DOI: 10.1016/j.jkss.2011.10.001
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Nonlocal Cauchy problem for some stochastic integro-differential equations in Hilbert spaces

Abstract: a b s t r a c tIn this paper, we study the existence results of mild solutions for a class of stochastic integro-differential equations with nonlocal conditions and stochastic impulsive integrodifferential equations with nonlocal conditions in Hilbert spaces. Sufficient conditions for the existence of mild solutions are derived by means of Leray-Schauder nonlinear alternative. An example is provided to illustrate the theory.

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Cited by 13 publications
(14 citation statements)
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“…Theorem 3.1 can be applied to a class of nonautonomous stochastic partial differential equations of evolution type with nonlocal initial conditions, in which the corresponding evolution family is not compact. Therefore, Theorem 3.1 in this paper is supplement to the papers [8], [9], [6] and [14]. This distinguishes the present paper from earlier works on stochastic evolution equations with nonlocal initial conditions.…”
mentioning
confidence: 71%
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“…Theorem 3.1 can be applied to a class of nonautonomous stochastic partial differential equations of evolution type with nonlocal initial conditions, in which the corresponding evolution family is not compact. Therefore, Theorem 3.1 in this paper is supplement to the papers [8], [9], [6] and [14]. This distinguishes the present paper from earlier works on stochastic evolution equations with nonlocal initial conditions.…”
mentioning
confidence: 71%
“…In recent years, stochastic evolution equations with nonlocal initial conditions have also been investigated extensively and some interesting results have been obtained. In 2012, Cui et al [14] studied the existence results of mild solutions for a class of stochastic integro-differential evolution equations with nonlocal initial conditions in Hilbert spaces assuming that the nonlocal item is only continuous but without imposing some compactness and convexity. Later, Chen and Li [8] obtained the existence of α-mild solutions for a class of fractional stochastic integro-differential evolution equations with nonlocal initial conditions in a real separable Hilbert space by using a new strategy which relies on the compactness of the operator semigroup, Schauder fixed point theorem and approximating techniques in 2015.…”
mentioning
confidence: 99%
“…One of the branches of stochastic differential equations is the theory of stochastic evolution equations. Since semilinear stochastic evolution equations are abstract formulations for many problems arising in the domain of engineering technology, biology, economic system, and so forth, stochastic evolution equations have attracted increasing attention in recent years and the existence, uniqueness, and asymptotic behavior of mild solutions to stochastic evolution equations have been considered by many authors; see [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and the references therein. Taniguchi et al [6] discussed the existence, uniqueness, th moment, and almost sure Lyapunov exponents of mild solutions to a class of stochastic partial functional differential equations with finite delays by using semigroup methods.…”
Section: Introductionmentioning
confidence: 99%
“…Chang et al [12][13][14] studied the existence and uniqueness of Stepanovlike almost automorphic mild solutions, the existence of square-mean almost automorphic mild solutions, and the existence and uniqueness of quadratic mean almost periodic mild solutions to nonlinear stochastic evolution equations in real separable Hilbert spaces, respectively. Moreover, the existence of mild solutions of stochastic evolution equations in Hilbert spaces has also been discussed in [15][16][17][18][19][20].…”
Section: Introductionmentioning
confidence: 99%
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