Abstract
We discuss existence results of mild solutions for stochastic differential inclusions subject to nonlocal conditions. We provide sufficient conditions in order to obtain a priori bounds on possible solutions of a one-parameter family of problems related to the original one. We, then, rely on fixed point theorems for multivalued operators to prove our main results.
Highlights
We investigate nonlocal stochastic differential inclusions SDIns of the form dx t ∈ Ax t f t, xt dt G t, xt dw t, t ∈ J 0, T, m x0 γix ti, 1.1
The importance of nonlocal conditions and their applications in different field have been discussed in 1–3
Existence results for semilinear evolution equations with nonlocal conditions were investigated in 4–7, and the case of semilinear evolution inclusions with nonlocal conditions and a nonconvex right-hand side was discussed in 8
Summary
We investigate nonlocal stochastic differential inclusions SDIns of the form dx t ∈ Ax t f t, xt dt G t, xt dw t , t ∈ J 0, T , m x0 γix ti , i1 x t φ t , t ∈ J1 −∞, 0 , where T > 0, 0 < t1 < t2 < · · · < tm < T , γi are real numbers, f is a single-valued function, and G is multivalued map. The importance of nonlocal conditions and their applications in different field have been discussed in 1–3. Very few articles have been devoted to the study of stochastic differential inclusions with nonlocal conditions, see 13–15 and the references therein.
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