Mohamed A Khamsi

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• Ph D
• Professor (Full) at Khalifa University

We prove a new minimization theorem in weighted graphs endowed with a quasi-metric distance, which improves the graphical version of the Ekeland variational principle discovered recently (Alfuraidan and Khamsi in Proc Am Math Soc 147:5313–5321, 2019). As a powerful application in behavioral sciences, we consider how to improve the quality of life i...

Because of its many diverse applications, fixed point theory has been a flourishing area of mathematical research for decades. Banach’s formulation of the contraction mapping principle in the early twentieth century signaled the advent of an intense interest in the metric related aspects of the theory. The metric fixed point theory in modular funct...

In this paper, we reexamine the concept of firmly nonexpansiveness in the modular sense in the variable exponent sequence spaces ℓp(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \beg...

In this article, we introduced two iterative processes consisting of an inertial term, forward-backward algorithm and generalized contraction for approximating the solution of monotone variational inclusion problem. The motivation for this work is to prove the strong convergence of inertial-type algorithms under some relaxed conditions because many...

In this work, we introduce a variant form of uniform convexity in partially ordered Banach spaces. This uniform convexity property is more adequate than norm uniform convexity when studying the fixed point problem for monotone nonexpansive mappings.

Kannan maps have inspired a branch of metric fixed point theory devoted to the extension of the classical Banach contraction principle. The study of these maps in modular vector spaces was attempted timidly and was not successful. In this work, we look at this problem in the variable exponent sequence spaces lp(·). We prove the modular version of m...

In this work , we investigate the modular version of the Ekeland variational principle (EVP) in the context of variable exponent sequence spaces ℓ p ( · ) . The core obstacle in the development of a modular version of the EVP is the failure of the triangle inequality for the module. It is the lack of this inequality, which is indispensable in the e...

Kannan maps have inspired a branch of metric fixed point theory devoted to the extension of the classical Banach contraction principle. The study of these maps in modular vector spaces was attempted timidly and was not successful. In this work, we look at this problem in the variable exponent sequence spaces ℓ p ( · ) . We prove the modular version...

We present some new coincidence fixed point theorems for generalized multi-valued weak Γ-contraction mappings. Our outcomes extend several recent results in the framework of complete metric spaces endowed with a graph. Two illustrative examples are included and some consequences are derived.

We extend the notion of compact normal structure to binary relational systems. The notion was introduced by J.P. Penot for metric spaces. We prove that for involutive and reflexive binary relational systems, every commuting family of relational homomorphisms has a common fixed point. The proof is based upon the clever argument that J.B. Baillon dis...

In this paper, we consider the recently introduced C A T p ( 0 ) , where the comparison triangles belong to ℓ p , for p ≥ 2 . We first establish an inequality in these nonlinear metric spaces. Then, we use it to prove the existence of fixed points of asymptotically nonexpansive mappings defined in C A T p ( 0 ) . Moreover, we discuss the behavior o...

In this work, we initiate the study of the geometry of the variable exponent sequence space when . In 1931 Orlicz introduced the variable exponent sequence spaces while studying lacunary Fourier series. Since then, much progress has been made in the understanding of these spaces and of their continuous counterpart. In particular, it is well known t...

In this work, we give a graphical version of the Ekeland variational principle which enables us to discover a new version of the Caristi fixed point theorem in weighted digraphs not necessarily generated by a partial order. Then we show that both graphical versions of the Ekeland variational principle and Caristi’s fixed point theorem are equivalen...

In this paper, we introduce the concept of monotone Gregus-Ćirić-contraction mappings in weighted digraphs. Then we establish a fixed point theorem for monotone Gregus-Ćirić-contraction mappings defined in convex weighted digraphs.

Let Φ be the class of all real functions φ:[0,∞[×[0,∞[→[0,∞[ that satisfy the following condition: there exists α∈]0,1[such thatφ((1-α)r,αr)<r,for allr>0. In this paper, we show that if X is a nonempty compact convex subset of a real normed vector space, any two closed set-valued mappings T,S:X⇉X, with nonempty and convex values, have a common fixe...

In this work, we extend the fundamental results of Schu to the class of monotone asymptotically nonexpansive mappings in modular function spaces. In particular, we study the behavior of the Fibonacci–Mann iteration process, introduced recently by Alfuraidan and Khamsi, defined by

In this work, we extend the fundamental results of Schu to the class of monotone asymptotically nonexpansive mappings in modular function spaces. In particular, we study the behavior of the Fibonacci-Mann iteration process defined by $$x_{n+1} = t_n T^{\phi(n)}(x_n) + (1-t_n)x_n,$$ for $n \in \mathbb{N}$, when $T$ is a monotone asymptotically nonex...

In this paper, we introduce the monotone Caristi inward mappings. As an example, we show that monotone inward contraction mappings are monotone Caristi inward mappings. A general fixed point theorem for such mappings is given. A mutlivalued version of these results is also introduced.

The purpose of this note is to discuss the recent paper of Espínola and Wiśnicki about the fixed point theory of monotone nonexpansive mappings. In their work, it is claimed that most of the fixed point results of this class of mappings boil down to the classical Knaster-Tarski fixed point theorem. We will show that their approach is very restricti...

In this paper, we establish several common fixed point theorems for families of set-valued mappings defined in hyperconvex metric spaces. Then we give several applications of our results.

In this paper, we introduce the concept of monotone Gregus-\'Ciri\'c-contraction mappings in weighted digraphs. Then we establish a fixed point theorem for monotone Gregus-\'Ciri\'c-contraction mappings defined in convex weighted digraphs.

In this work, we define the concept of mixed $G$-monotone mappings defined on a metric space endowed with a graph. Then we obtain sufficient conditions for the existence of coupled fixed points for such mappings when a weak contractivity type condition is satisfied.

In this work, we define the concept of mixed $G$-monotone mappings defined on a metric space endowed with a graph. Then we obtain sufficient conditions for the existence of coupled fixed points for such mappings when a weak contractivity type condition is satisfied.

In this work, we will discuss the recent work of Gornicki in the context of weighted graphs. This extension is valuable since it relaxes any order structure defined on a metric space. This approach finds its origin in the work of Jachymski. To be more specific, we prove that continuous Ciric-Jachymski-Matkowski contraction mappings monotone in the...

In this manuscript, we discuss the latest fixed point results of monotone mappings. The fixed point theory of such mappings has seen a tremendous interest in the last decade since the publication of Ran and Reurings paper in 2004. Fixed point theory for monotone mappings is useful and has many applications. For example when one is looking for a pos...

We extend the Gromov geometric definition of CAT(0) spaces to the case where the comparison triangles are not in the Euclidean plane but belong to a general Banach space. In particular, we study the case where the Banach space is ℓp, for p < 2.

We extend the results of Schu [‘Iterative construction of fixed points of asymptotically nonexpansive mappings’, J. Math. Anal. Appl.158 (1991), 407–413] to monotone asymptotically nonexpansive mappings by means of the Fibonacci–Mann iteration process $$\begin{eqnarray}x_{n+1}=t_{n}T^{f(n)}(x_{n})+(1-t_{n})x_{n},\quad n\in \mathbb{N},\end{eqnarray}...

Let C be a nonempty, ρ-bounded, ρ-closed, and convex subset of a modular function space Lρ and T: C → C be a monotone asymptotically ρ-nonexpansive mapping. In this paper, we investigate the existence of fixed points of T. In particular, we establish a modular monotone analogue to the original Goebel and Kirk's fixed point theorem for asymptoticall...

In this work, we investigate the existence of fixed points for multivalued G-monotoné Ciri´cCiri´c quasi-contraction and Reich contraction mappings in a metric space endowed with a graph G.

In this work, we extend the fixed point result of Kirk and Xu for asymptotic pointwise nonexpansive mappings in a uniformly convex Banach space to monotone mappings defined in a hyperbolic uniformly convex metric space endowed with a partial order.

We give a fixed point theorem for uniformly Lipschitzian mappings defined in modular vector spaces which have the uniform normal structure property in the modular sense. We also discuss this result in the variable exponent space p(.) = (x n) ∈ R N ; ∞ n=0 |λ x n | p(n) < ∞ for some λ > 0 .

We introduce the concept of a multivalued asymptotically nonexpansive mapping and establish Goebel and Kirk fixed point theorem for these mappings in uniformly hyperbolic metric spaces. We also define a modified Mann iteration process for this class of mappings and obtain an extension of some well-known results for singlevalued mappings defined on...

In this chapter, we present some of the known results about the concept of approximate fixed points of a mapping. In particular, we discuss some new results on approximating fixed points of monotone mappings. Then we conclude this chapter with an application of these results to the case of a nonlinear semigroup of mappings. It is worth mentioning t...

Let X be a Banach space or a complete hyperbolic metric space. Let C be a nonempty, bounded, closed, and convex subset of X and T : C → C be a monotone nonexpansive mapping. In this paper, we show that if X is a Banach space which is uniformly convex in every direction or a uniformly convex hyperbolic metric space, then T has a fixed point. This is...

Almost contraction mappings were introduced as an extension to the contraction mappings for which the conclusion of the Banach contraction principle (BCP in short) holds. In this paper, the concept of monotone almost contractions defined on a weighted graph is introduced. Then a fixed point theorem for such mappings is given. c 2016 all rights rese...

We prove the existence of fixed points of monotone ρ-nonexpansive mappings in ρ-uniformly convex modular function spaces. This is the modular version of Browder and Göhde fixed point theorems for monotone mappings. We also discuss the validity of this result in modular function spaces where the modular is uniformly convex in every direction. This p...

In this note, we discuss the definition of the multivalued weak contraction mappings defined in a metric space endowed with a graph as introduced by Hanjing and Suantai[A. Hanjing, S. Suantai, Fixed Point Theory Appl., 2015 (2015), 10 pages]. In particular, we show that this definition is not correct and give the correct definition of the multivalu...

Let C be a nonempty, bounded, closed, and convex subset of a Banach space X and $T : C \rightarrow C$ be a monotone asymptotic nonexpansive mapping. In this paper, we investigate the existence of fi?xed points of T. In particular, we establish an analogue to the original Goebel and Kirk's fixed point theorem for asymptotic nonexpansive mappings.

In this paper, we discuss the definition of the Reich multivalued monotone contraction mappings defined in a metric space endowed with a graph. In our investigation, we prove the existence of fixed point results for these mappings. We also introduce a vector valued Bernstein operator on the space C([0, 1], X), where X is a Banach space endowed with...

Fixed Point Theory and Graph Theory provides an intersection between the theories of fixed point theorems that give the conditions under which maps (single or multivalued) have solutions and graph theory which uses mathematical structures to illustrate the relationship between ordered pairs of objects in terms of their vertices and directed edges....

The purpose of this note is to give a natural approach to the extensions of the Banach contraction principle in metric spaces endowed with a partial order, a directed graph or a binary relation in terms of extended quasi-metric. This novel approach is new and may open the door to other new fixed point theorems. The case of multivalued mappings is a...

In this work we discuss the properties of the common fixed points set of a commuting family of monotone nonexpansive mappings. In particular, we show that under suitable assumptions, this set is a monotone nonexpansive retract.

We prove the existence of fixed points of monotone quasi-contraction mappings in metric and modular metric spaces. This is the extension of Ran and Reurings fixed point theorem for monotone contraction mappings in partially ordered metric spaces to the case of quasicontraction mappings introduced by ´ Ciri´c. The proofs are based on Lemmas 2.1 and...

In this work, we discuss the existence of solutions to the Fredholm integral equation x(t) =g(t) + ∫0¹f(t,s,x(s))ds, in the spaces [inline-equation], LP(I) (1 ≤ p < +∞) and L∞(I). The results obtained seem to be new and improve on known similar results.

Let the set $C \subset L_{1}([0,1])$ C ⊂ L 1 ( [ 0 , 1 ] ) be nonempty, convex and compact for the convergence almost everywhere and $T: C \rightarrow C$ T : C → C be a monotone nonexpansive mapping. In this paper, we study the behavior of the Krasnoselskii-Ishikawa iteration sequence $\{f_{n}\}$ { f n } defined by $f_{n+1} = \lambda f_{n} + (1-\la...

In this work, we discuss the existence of fixed points of monotone nonexpansive mappings defined on partially ordered Banach spaces. This work is a continuity of the previous works of Ran and Reurings, Nieto et al., and Jachimsky done for contraction mappings. As an application, we discuss the existence of solutions to an integral equations.

In this work we define the new concept of monotone pointwise contraction mappings in Banach and metric spaces. Then we prove the existence of fixed points of such mappings. MSC: 47H09, 46B20, 47H10, 47E10.

We prove the existence of fixed points of monotone-contraction mappings in modular function spaces. This is the modular version of the Ran and Reurings fixed point theorem. We also discus the extension of these results to the case of pointwise monotone-contraction mappings in modular function spaces. MSC: 47H09, 47H10.

In this paper, we investigate the common fixed points set of nonexpansive semigroups of nonlinear mappings { T t } t ≥ 0 , i.e., a family such that T 0 ( x ) = x , T s + t = T s ( T t ( x ) ) , where the domain is a metric space ( M , d ) . In particular we prove that under suitable conditions, the common fixed points set is the same as the common...

Let ( X , ∥ ⋅ ∥ ) be a Banach space. Let C be a nonempty, bounded, closed, and convex subset of X and T : C → C be a monotone nonexpansive mapping. In this paper, it is shown that a technique of Mann which is defined by x n + 1 = t n T ( x n ) + ( 1 − t n ) x n , n = 1 , 2 , … , is fruitful in finding a fixed point of monotone nonexpansive mappings...

In this work, we discuss the concept of Banach operator pairs in modular vector spaces. We prove the existence of common fixed points for these type of operators which satisfy a modular continuity in modular compact sets. On the basis of our result, we are able to give an analog of DeMarr’s common fixed point theorem for a family of symmetric Banac...

Banach’s contraction mapping principle is remarkable in its simplicity, yet it is perhaps the most widely applied fixed point theorem in all of analysis with special applications to the theory of differential and integral equations. Because the underlined space of this theorem is a metric space, the theory that developed following its publication i...

In this paper, we investigate the common approximate fixed points of monotone nonexpansive semigroups of nonlinear mappings { T ( t ) } t ≥ 0 , i.e., a family such that T ( 0 ) x = x , T ( s + t ) x = T ( s ) ∘ T ( t ) x , where the domain is a Banach space. In particular we prove that under suitable conditions, the common approximate fixed points...

In this work, we define the concept of G-monotone nonexpansive multivalued mappings defined on a metric space with a graph G. Then we obtain sufficient conditions for the existence of fixed points for such mappings in hyperbolic metric spaces. This is the first kind of such results in this direction. MSC: 47H09, 46B20, 47H10, 47E10.

In this work, we define the concept of G-monotone nonexpansive multivalued mappings defined on a metric space with a graph G. Then we obtain sufficient conditions for the existence of fixed points for such mappings in hyperbolic metric spaces. This is the first kind of such results in this direction.

Assume \(\rho \in \Re\) is \((UUC1)\). Let C be a ρ-closed ρ-bounded convex nonempty subset of \(L_{\rho}\). Let \(T: C\rightarrow C\) be a pointwise asymptotically nonexpansive mapping. According to Theorem 5.7 the mapping T has a fixed point. The proof of this important theorem is of the existential nature and does not describe any algorithm for...

This chapter presents a series of fixed point existence theorems for nonlinear mappings acting in modular functions spaces. We cover a range of different types of mappings including ρ-contractions and their pointwise asymptotic versions and ρ-nonexpansive mappings and pointwise asymptotic ρ-nonexpansive mappings, under various assumptions on the fu...

This chapter introduces the general notions related to modular function spaces. The results discussed in this chapter is used throughout the rest of the book. This chapter also presents the reader with an exhaustive list of examples that will frequently reoccur in later parts of the book.

The concept of a metric space is closely related to our intuitive understanding of space is the 3-dimensional Euclidean space. In fact, the notion of metric is a generalization of the Euclidean metric arising from the basic long known properties of the Euclidean distance. Maurice Fréchet1is credited as the mathematician who introduced the abstract...

Let us recall that a family \(\{T_t\}_{t \geq 0}\) of mappings forms a semigroup if \(T_0(x)=x\), and \(T_{s+t}=T_s(T_t(x))\). Such a situation is quite typical in mathematics and applications. For instance, in the theory of dynamical systems, the modular function space \(L_{\rho}\) would define the state space and the mapping \((t,x)\rightarrow T_...

This chapter introduces general notions related to the geometry of modular function spaces. We define the modular version of uniform convexity and property (R) which will equip us with powerful tools for proving the fixed point theorems in modular function spaces. The geometrical theory also provides a set of powerful techniques for proving existen...

Let C be a ρ-bounded, ρ-closed, convex subset of a modular function space Lρ. We investigate the problem of constructing common fixed points for asymptotic pointwise nonexpansive semigroups of mappings Tt:C→C, i.e. a family such that T0(f)=f, Ts+t(f)=Ts∘Tt(f), and ρ(T(f)−T(g))≤αt(f)ρ(f−g), where lim supt→∞αt(f)≤1, for every f∈C. MSC: 47H09, 46B20,...

We introduce a general viscosity iterative method for a finite family of generalized asymptotically quasi-nonexpansive mappings in a convex metric space. The new iterative method contains several well-known iterative methods as its special case including multistep iterative method of Khan et al. [Common fixed points Noor iteration for a finite fami...

The purpose of this paper is to study the existence of fixed points for contractive-type multivalued maps in the setting of modular metric spaces. The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. In this paper we...

In this paper we develop a fixed point theorem in the partially ordered vector metric space C([−τ,0],Rn) by using vectorial norm. Then we use it to prove the existence of periodic solutions to nonlinear delay differential equations. MSC: 06F30, 46B20, 47E10, 34K13, 34K05.

In 1983 A. Quilliot published his original work on graphs and ordered sets viewed as metric spaces. His approach was revolutionary. It was the first time that metric ideas and concepts could be defined in discrete sets. In particular one can show that graphs or order preserving maps are exactly the class of nonexpansive mappings defined on metric s...

The theory of fixed points is one of the most powerful tools of modern mathematics” quoted by Felix Browder, gave a new impetus to the modern fixed point theory via the development of nonlinear functional analysis as an active and vital branch of mathematics. The flourishing field of fixed point theory started in the early days of topology (the wor...

We establish strong convergence and Δ-convergence theorems of an iteration scheme associated to a pair of nonexpansive mappings on a nonlinear domain. In particular we prove that such a scheme converges to a common fixed point of both mappings. Our results are a generalization of well-known similar results in the linear setting. In particular, we a...

The flourishing field of fixed point theory started in the early days of topology with seminal contributions by Poincare, Lefschetz-Hopf, and Leray-Schauder at the turn of the 19th and early 20th centuries. The theory vigorously developed into a dense and multifaceted body of principles, results, and methods from topology and analysis to algebra an...

We discuss Caristi's fixed point theorem for mappings defined on a metric space endowed with a graph. This work should be seen as a generalization of the classical Caristi's fixed point theorem. It extends some recent works on the extension of Banach contraction principle to metric spaces with graph.

We establish convergence in the modular sense of an iteration scheme associated with a pair of mappings on a nonlinear domain in modular function spaces. In particular, we prove that such a scheme converges to a common fixed point of the mappings. Our results are generalization of known similar results in the non-modular setting. In particular, we...

The purpose of this contributed volume is to provide a primary resource for anyone interested in fixed point theory with a metric flavor. The book presents information for those wishing to find results that might apply to their own work and for those wishing to obtain a deeper understanding of the theory. The book should be of interest to a wide ra...

In this paper, we investigate the common approximate fixed point sequences of nonexpansive semigroups of nonlinear mappings {T-t}(t >= 0), i.e., a family such that T-0(x) = x, Ts+t = T-s(T-t(x)) where the domain is a metric space (M, d). In particular, we prove that under suitable conditions the common approximate fixed point sequences set is the s...

Let C be a bounded, closed, convex subset of a uniformly convex metric space (M,d). In this paper, we introduce the concept of asymptotic pointwise nonexpansive semigroups of nonlinear mappings T t :C→C, i.e., a family such that T 0 (x)=x, T s+t =T s (T t (x)), and d(T t (x),T t (y))≤α t (x)d(x,y), where limsup t→∞ α t (x)≤1 for every x∈C. Then we...

Let C be rho-bounded, rho-closed, convex subset of a modular function space L-rho. We investigate the existence of common fixed points for asymptotic pointwise nonexpansive semigroups of nonlinear mappings T-t : C -> C, i.e. a family such that T-o(f) = f, Ts+t(f) = T-s o T-t(f) and rho(T(f) -T(g)) <= alpha(t)(f)rho(f - g), where lim sup(t ->infinit...

The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, have been recently introduced. In this paper we investigate the existence of fixed points of modular nonexpansive mappings. We also discuss some compactness properties of the family of admis...

The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. In this paper we investigate the existence of fixed points of modular contractive mappings in modular metric spaces. These are related to the successive approximati...

In this work, we investigate the additivity of the Minkowski functionals associated to a cone in a linear vector space. As an application we discuss the equivalence of the classical Caristi fixed point theorem in metric spaces and its vectorial version in cone metric spaces.

Let (M,d)(M,d) be a complete 2-uniformly convex metric space. Let CC be a nonempty, bounded, closed, and convex subset of MM, and let T:C→CT:C→C be an asymptotic pointwise nonexpansive mapping. In this paper, we prove that the modified Mann iteration process defined by xn+1=tnTn(xn)⊕(1−tn)xnxn+1=tnTn(xn)⊕(1−tn)xn converges in a weaker sense to a fi...

In this work, we introduce a density property in ordered sets that is weaker than the order density. Then, we prove a strong version of a result proved by T. Büber and W. A. Kirk [in: World congress of nonlinear analysts ’92. Proceedings of the first world congress, Tampa, FL, USA, August 1992. Berlin: de Gruyter. 2115–2125 (1996; Zbl 0844.47031)],...

Let X be a metric space and {T 1, ..., T N } be a finite family of mappings defined on D ⊂ X. Let r : ℕ → {1,..., N} be a map that assumes every value infinitely often. The purpose of this article is to establish the convergence of the sequence (x n ) defined by In particular, we extend the study of Bauschke [1] from the linear case of Hilbert spa...

In this paper, we introduce and study the concept of one-local retract in modular function spaces. In particular, we prove that any commutative family of ρ-nonexpansive mappings defined on a nonempty, ρ-closed and ρ-bounded subset of a modular function space has a common fixed point provided its convexity structure of admissible subsets is compact...

In this paper, we first introduce the concept of NR-map and then use this concept to establish the existence of common fixed points for Banach operator pairs in the context of uniformly convex geodesic metric spaces. New proofs of main results (Theorems 2.1 and 3.5) of Chen and Li [J. Chen, Z. Li, Banach operator pair and common fixed points for no...

In this article, we introduce a new approach to common fixed point theory for a weak compatible pair. We first introduce the concepts of R-pair and NR-pair and establish some new common fixed point theorems for a weak compatible pair in hyperconvex metric spaces and uniformly convex metric spaces. We shall also establish the well-known De Marr's th...

Let X be a metric space and {T 1, ..., T N } be a finite family of mappings defined on D ⊂ X. Let r : ℕ → {1,..., N} be a map that assumes every value infinitely often. The purpose of this article is to establish the convergence of the sequence (x n ) defined by In particular we prove Amemiya and Ando's theorem in metric trees without compactness...

The purpose of this paper is to establish DeMarr’s well-known theorem for an arbitrary family of symmetric Banach operator pairs in hyperconvex metric spaces without the compactness assumption. We also give necessary and sufficient criteria for the existence of a common fixed point of a semigroup of isometric mappings. As an application, several re...

In this study, we introduce the concept of externally complete ordered sets. We discuss the properties of such sets and characterize them in ordered trees. We also prove some common fixed point results for order preserving mappings. In particular, we introduce for the first time the concept of Banach Operator pairs in partially ordered sets and pro...

In this article, we introduce the concept of a Banach operator pair in the setting of modular function spaces. We prove some common fixed point results for such type of operators satisfying a more general condition of nonexpansiveness. We also establish a version of the well-known De Marr's theorem for an arbitrary family of symmetric Banach operat...

Banach's Contraction PrincipleFurther extensions of Banach's PrincipleThe Caristi-Ekeland PrincipleEquivalents of the Caristi-Ekeland PrincipleSet-valued contractionsGeneralized contractions

A fixed point theoremStructure of the fixed point setUniform normal structureUniform relative normal structureQuasi-normal structureStability and normal structureUltrametric spacesFixed point set structure—separable case

IntroductionBrouwer's TheoremFurther comments on Brouwer's TheoremSchauder's TheoremStability of Schauder's TheoremBanach algebras: Stone Weierstrass TheoremLeray-Schauder degreeCondensing mappingsContinuous mappings in hyperconvex spaces