Xuping Zhang

Study on fractional non-autonomous evolution equations with delay

Abstract In this paper, we deal with a class of nonlinear time fractional non-autonomous evolution equations with delay by introducing the operators ψ ( t , s ) , φ ( t , η ) and U ( t ) , which are generated by the operator − A ( t ) and probability density function. The definition of mild solutions for studied problem was given based on these operators. Combining the techniques of fractional calculus, operator semigroups, measure of noncompactness and fixed point theorem with respect to k -set-contractive, we obtain new existence result of mild solutions with the assumptions that the nonlinear term satisfies some growth condition and noncompactness measure condition and the closed linear operator − A ( t ) generates an analytic semigroup for every t > 0 . The results obtained in this paper improve and extend some related conclusions on this topic. At last, by utilizing the abstract result obtained in this paper, the existence of mild solutions for a class of nonlinear time fractional reaction–diffusion equation introduced in Ouyang (2011) is obtained.

Nonlocal problem for fractional stochastic evolution equations with solution operators

Abstract In this article, we are concerned with a class of fractional stochastic evolution equations with nonlocal initial conditions in Hilbert spaces. The existence of mild solutions is obtained under the situation that the nonlinear term satisfies some appropriate growth conditions by using fractional calculations, Schauder fixed point theorem, stochastic analysis theory, α-order fractional solution operator theory and α-resolvent family theory. The results obtained in this paper improve and extend some related conclusions on this topic. An example is also given to illustrate the feasibility of our abstract result.

Fractional stochastic evolution equations with nonlocal initial conditions and noncompact semigroups

Abstract This paper deals with the existence of mild solutions for a class of fractional stochastic evolution equations with nonlocal initial conditions and and noncompact semigroups in a real separable Hilbert space. We assume that the linear part generates an equicontinuous -semigroup, and the nonlinear functions satisfy some appropriate growth conditions and noncompactness measure conditions. Our results generalize and improve some previous results on this topic to the case that the corresponding solution semigroup is not compact. An example is also given to demonstrate the applications of our abstract result.

On the Initial Value Problem of Stochastic Evolution Equations in Hilbert Spaces

The present paper studies the initial value problem of stochastic evolution equations with compact semigroup in real separable Hilbert spaces. The existence of saturated mild solution and global mild solution is obtained under the situation that the nonlinear term satisfies some appropriate growth conditions. The results obtained in this paper improve and extend some related conclusions on this topic. An example is also given to illustrate that our results are valuable.