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Fixed Points for Maps with Weakly Sequentially Closed Graph

  • Afif Ben Amar
  • Donal O’Regan
Chapter

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 IS 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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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Afif Ben Amar
    • 1
  • Donal O’Regan
    • 2
  1. 1.Department of MathematicsUniversity of Sfax, Faculty of SciencesSfaxTunisia
  2. 2.School of MathematicsNational University of Ireland, GalwayGalwayIreland

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