Some fixed point theorems and existence of weak solutions of Volterra integral equation under Henstock-Kurzweil-Pettis integrability

summary:In this paper we examine the set of weakly continuous solutions for a Volterra integral equation in Henstock-Kurzweil-Pettis integrability settings. Our result extends those obtained in several kinds of integrability settings. Besides, we prove some new fixed point theorems for function spaces relative to the weak topology which are basic in our considerations and comprise the theory of differential and integral equations in Banach spaces

Critical types of Krasnoselskii fixed point theorems in weak topologies

In this note, by means of the technique of measures of weak noncompactness, we establish a generalized form of fixed point theorem for the sum of T + S in weak topology setups of a metrizable locally convex space, where S is not weakly compact, I − T allows to be noninvertible, and T is not necessarily continuous. The obtained results unify and significantly extend a lot of previously known extensions of Krasnoselskii fixed-point theorems. The analysis presented here reveals the essential characteristics of the Krasnoselskii type fixed-point theorem in weak topology settings.Keywords: Fixed point, noncompact mapping, multi-valued mappin

Topological fixed point theory for singlevalued and multivalued mappings and applications

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.

Topological Fixed Point Theory for Singlevalued and Multivalued Mappings and Applications

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.

Measures of weak noncompactness and fixed point theory in banach algebras satisfying condition (<i>p</i>)

The aim of this paper is to prove some new fixed point theorems in a nonempty closed convex subset of a Banach algebra satisfying a sequential condition (P) in a weak topology setting

Measures of weak noncompactness and fixed point theory in banach algebras satisfying condition (<i>p</i>)

The aim of this paper is to prove some new fixed point theorems in a nonempty closed convex subset of a Banach algebra satisfying a sequential condition (P) in a weak topology setting

Measures of noncompactness in locally convex spaces and fixed point theory for the sum of two operators on unbounded convex sets

summary:In this paper we prove a collection of new fixed point theorems for operators of the form $T+S$ on an unbounded closed convex subset of a Hausdorff topological vector space $(E,\Gamma)$. We also introduce the concept of demi-$\tau$-compact operator and $\tau$-semi-closed operator at the origin. Moreover, a series of new fixed point theorems of Krasnosel'skii type is proved for the sum $T+S$ of two operators, where $T$ is $\tau$-sequentially continuous and $\tau$-compact while $S$ is $\tau$-sequentially continuous (and $\Phi_{\tau}$-condensing, $\Phi_{\tau}$-nonexpansive or nonlinear contraction or nonexpansive). The main condition in our results is formulated in terms of axiomatic $\tau$-measures of noncompactness. Apart from that we show the applicability of some our results to the theory of integral equations in the Lebesgue space

Measures of weak noncompactness, nonlinear Leray-Schauder alternatives in Banach algebras satisfying condition (P) and an application

In this paper, we establish some new nonlinear Leray-Schauder alternatives for the sum and the product of weakly sequentially continuous operators in Banach algebras satisfying certain sequential condition (P). The main condition in our results is formulated in terms of axiomatic measures of weak noncompactness. As an application, our results are used to prove the existence of solutions for a nonlinearintegral equation.Keywords: Measures of weak noncompactness, weakly condensing, weakly sequentially continuous, fixed point theorems, integral equatio

Fixed-Point Theory on a Frechet Topological Vector Space

We establish some versions of fixed-point theorem in a Frechet topological vector space E. The main result is that every map A=BC (where B is a continuous map and C is a continuous linear weakly compact operator) from a closed convex subset of a Frechet topological vector space having the Dunford-Pettis property into itself has fixed-point. Based on this result, we present two versions of the Krasnoselskii fixed-point theorem. Our first result extend the well-known Krasnoselskii's fixed-point theorem for U-contractions and weakly compact mappings, while the second one, by assuming that the family {T(⋅,y):y∈C(M) where M⊂E and C:M→E a compact operator} is nonlinear φ equicontractive, we give a fixed-point theorem for the operator of the form Ex:=T(x,C(x))