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Integral inclusions of upper semi-continuous or lower semi-continuous type. (English) Zbl 0860.45007
The author establishes existence results for Volterra inclusions of the form \[ y(t) \in \int^t_0 k(t,s)F \bigl(s,y(s)\bigr) ds+g(t), \quad t \in[0,T], \] and for Hammerstein inclusions of the form \[ y(t)\in \int^T_0k(t,s) F\bigl(s,y(s) \bigr)ds + g(t), \quad t \in[0,T]. \] Here \(F\) is a multivalued function with nonempty compact values and \(F\) satisfies certain other conditions which are shown to imply either lower or upper semi-continuity for certain related mappings. The kernel \(k\) is assumed to be essentially bounded. The proofs rely on fixed-point theorems. Some of the results for superlinear inclusions are new even for integral equations.

45G10 Other nonlinear integral equations
Full Text: DOI
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