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Existence of solutions of Sobolev-type semilinear mixed integrodifferential inclusions in Banach spaces. (English) Zbl 1037.45004
This paper discusses the existence of mild solutions to the following Sobolev-type semilinear mixed integro-differential inclusion \[ (Eu(t))'+Au\in G\left(t,u,\int_{0}^{t}k(t,s,u)\,ds, \int_{0}^{a}b(t,s,u) \,ds\right), \quad t\in I=[0,\infty), \] \[ u(0)=u_{0}, \] where \(G:I\times X\times X\times X\to 2^{Y}\) is a bounded, closed, convex valued multivalued map, \(k: \Delta\times X\to X, \;b: \Delta\times X\to X, \) where \(\Delta=\{(t,s)\in I\times I: t\geq s\}, \;u_{0}\in X, \;a>0\) and \(X, \;Y\) are Banach spaces. The proofs rely on the use of a fixed point theorem due to T.-W. Ma [Topological degrees of set-valued compact fields in locally convex spaces. Diss. Math. 92 (1972; Zbl 0211.25903)] for multivalued operators defined on locally convex topological spaces.
45N05 Abstract integral equations, integral equations in abstract spaces
45G10 Other nonlinear integral equations
45J05 Integro-ordinary differential equations
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