Abstract
We first present several existence results and compactness of solutions set for the following Volterra type integral inclusions of the form: , where , is the infinitesimal generator of an integral resolvent family on a separable Banach space , and is a set-valued map. Then the Filippov's theorem and a Filippov-Wazewski result are proved.
Highlights
In the past few years, several papers have been devoted to the study of integral equations on real compact intervals under different conditions on the kernel see, e.g., 1–4 and references therein
Advances in Difference Equations where a ∈ L1 0, b, R and A : D A ⊂ E → E is the generator of an integral resolvent family defined on a complex Banach space E, and F : 0, b × E → P E is a multivalued map
In 1980, Da Prato and Iannelli introduced the concept of resolvent families, which can be regarded as an extension of C0-semigroups in the study of a class of integrodifferential equations 9
Summary
In the past few years, several papers have been devoted to the study of integral equations on real compact intervals under different conditions on the kernel see, e.g., 1–4 and references therein. Topological structure of the solution set of integral inclusions of Volterra type is studied in 8. In this paper we present some results on the existence of solutions, the compactness of set of solutions, Filippov’s theorem, and relaxation for linear and semilinear integral inclusions of Volterra type of the form t y t ∈ a t − s Ay s F s, y s ds, a.e. t ∈ J : 0, b , Advances in Difference Equations where a ∈ L1 0, b , R and A : D A ⊂ E → E is the generator of an integral resolvent family defined on a complex Banach space E, and F : 0, b × E → P E is a multivalued map.
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