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This is a monograph covering topological fixed point theory for several classes of single and multivalued maps. The authors begin by presenting basic notions in locally convex topological vector spaces. Special attention is then devoted to weak compactness, in particular to the theorems of Eberlein–Šmulian, Grothendick and Dunford–Pettis. Leray–Schauder alternatives and eigenvalue problems for decomposable single-valued nonlinear weakly compact operators in Dunford–Pettis spaces are considered, in addition to some variants of Schauder, Krasnoselskii, Sadovskii, and Leray–Schauder type fixed point theorems for different classes of weakly sequentially continuous operators on general Banach spaces. The authors then proceed with an examination of Sadovskii, Furi–Pera, and Krasnoselskii fixed point theorems and nonlinear Leray–Schauder alternatives in the framework of weak topologies and involving multivalued mappings with weakly sequentially closed graph. These results are formulated in terms of axiomatic measures of weak noncompactness.
The authors continue to present some fixed point theorems in a nonempty closed convex of any Banach algebras or Banach algebras satisfying a sequential condition (P) for the sum and the product of nonlinear weakly sequentially continuous operators, and illustrate the theory by considering functional integral and partial differential equations. The existence of fixed points, nonlinear Leray–Schauder alternatives for different classes of nonlinear (ws)-compact operators (weakly condensing, 1-set weakly contractive, strictly quasi-bounded) defined on an unbounded closed convex subset of a Banach space are also discussed. The authors also examine the existence of nonlinear eigenvalues and eigenvectors, as well as the surjectivity of quasibounded operators. Finally, some approximate fixed point theorems for multivalued mappings defined on Banach spaces. Weak and strong topologies play a role here and both bounded and unbounded regions are considered. The authors explicate a method developed to indicate how to use approximate fixed point theorems to prove the existence of approximate Nash equilibria for non-cooperative games.
Fixed point theory is a powerful and fruitful tool in modern mathematics and may be considered as a core subject in nonlinear analysis. In the last 50 years, fixed point theory has been a flourishing area of research. As such, the monograph begins with an overview of these developments before gravitating towards topics selected to reflect the particular interests of the authors.



Chapter 1. Basic Concepts

In this chapter we discuss some concepts needed for the results presented in this book.

Afif Ben Amar, Donal O’Regan

Chapter 2. Nonlinear Eigenvalue Problems in Dunford–Pettis Spaces

In this chapter, we present some variants of Leray–Schauder type fixed point theorems and eigenvalue results for decomposable single-valued nonlinear weakly compact operators in Dunford–Pettis spaces.

Afif Ben Amar, Donal O’Regan

Chapter 3. Fixed Point Theory in Locally Convex Spaces

In this section we discuss the existence of fixed points for weakly sequentially continuous mappings on domains of Banach spaces. We first present some applicable Leray–Schauder type theorems for weakly condensing and 1-set weakly contractive operators. The main condition is formulated in terms of De Blasi’s measure of weak noncompactness β (see Sect. 1.12).

Afif Ben Amar, Donal O’Regan

Chapter 4. Fixed Points for Maps with Weakly Sequentially Closed Graph

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)−1T $$(I - S)^{-1}T$$ , 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.

Afif Ben Amar, Donal O’Regan

Chapter 5. Fixed Point Theory in Banach Algebras

In this chapter we discuss x=AxBx+Cx $$\displaystyle{ x = AxBx + Cx }$$ in suitable Banach algebras. We present some fixed point theory in Banach spaces under a weak topology setting. One difficulty that arises is that in a Banach algebra equipped with its weak topology the product of two weakly convergent sequences is not necessarily weakly convergent.

Afif Ben Amar, Donal O’Regan

Chapter 6. Fixed Point Theory for (ws)-Compact Operators

In this chapter we present fixed point theory and study eigenvalues and eigenvectors of nonlinear (ws)-compact operators.

Afif Ben Amar, Donal O’Regan

Chapter 7. Approximate Fixed Point Theorems in Banach Spaces

Let Ω $$\Omega $$ be a nonempty convex subset of a topological vector space X. An approximate fixed point sequence for a map F:Ω→Ω¯ $$F: \Omega \longrightarrow \overline{\Omega }$$ is a sequence {xn}n∈Ω $$\{x_{n}\}_{n} \in \Omega $$ so that xn−F(xn)→θ $$x_{n} - F(x_{n})\longrightarrow \theta$$ . Similarly, we can define approximate fixed point nets for F. Let us mention that F has an approximate fixed point net if and only if θ∈{x−F(x):x∈Ω}¯. $$\displaystyle{\theta \in \overline{\{x - F(x): x \in \Omega \}}.}$$

Afif Ben Amar, Donal O’Regan


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