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Fixed Points for Maps with Weakly Sequentially Closed Graph

In: Topological Fixed Point Theory for Singlevalued and Multivalued Mappings and Applications

Author

Listed:
  • Afif Ben Amar

    (University of Sfax, Faculty of Sciences, Department of Mathematics)

  • Donal O’Regan

    (National University of Ireland, Galway, School of Mathematics)

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 $$(I - S)^{-1}T$$ , 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.

Suggested Citation

  • Afif Ben Amar & Donal O’Regan, 2016. "Fixed Points for Maps with Weakly Sequentially Closed Graph," Springer Books, in: Topological Fixed Point Theory for Singlevalued and Multivalued Mappings and Applications, chapter 0, pages 85-101, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-31948-3_4
    DOI: 10.1007/978-3-319-31948-3_4
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