Topological Fixed Point Theory for Singlevalued and Multivalued Mappings and Applications pp 85-101 | Cite as

# Fixed Points for Maps with Weakly Sequentially Closed Graph

## Abstract

In this chapter, we discuss Sadovskii, Krasnoselskii, Leray–Schauder, and Furi–Pera type fixed point theorems for a class of multivalued mappings with weakly sequentially closed graph. We first discuss a Sadovskii type result for weakly condensing and 1-set weakly contractive multivalued maps with weakly sequentially closed graph. Next we discuss multivalued analogues of Krasnoselskii fixed point theorems for the sum *S* + *T* on nonempty closed convex of a Banach space where *T* is weakly completely continuous and *S* is weakly condensing (resp. 1-set weakly contractive). In particular we consider Krasnoselskii type fixed point theorems and Leray–Schauder alternatives for the sum of two weakly sequentially continuous mappings, *S* and *T* by looking at the multivalued mapping \((I - S)^{-1}T\), where *I* − *S* may not be injective. We note that the domains of all of the multivalued maps discussed here are not assumed to be bounded.

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