Using techniques of measures of noncompactness, we prove existence, uniqueness, and dependence results for semilinear stochastic differential equations with infinite delay on an abstract phase space of Hilbert space valued functions defined axiomatically, where the unbounded linear part generates a noncompact semigroup and the nonlinear parts satisfies some growth condition and, with respect to the second variable, a condition weaker than the Lipschitz one. These results are applied to a stochastic parabolic partial differential equation with infinite delay.Mathematics Subject Classification. Primary 34K50; Secondary 47H08, 35K58, 60H15.
As an application, some results are given to the periodic boundary value problem for nonlinear differential inclusions governed by m-accretive operators generating not necessarily a compact semigroups.
In this paper we study the topological structure of the solution set of abstract inclusions, not necessarily linear, with infinite delay on a Banach space defined axiomatically. By using the techniques of the theory of condensing maps and multivalued analysis tools, we prove that the solution set is a compact R δ -set. Our approach makes possible to give a unified scheme in the investigation of the structure of the solution set of certain classes of differential inclusions with infinite delay.
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