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Written by a team of leading experts in the field, this volume presents a self-contained account of the theory, techniques and results in metric type spaces (in particular in G-metric spaces); that is, the text approaches this important area of fixed point analysis beginning from the basic ideas of metric space topology.
The text is structured so that it leads the reader from preliminaries and historical notes on metric spaces (in particular G-metric spaces) and on mappings, to Banach type contraction theorems in metric type spaces, fixed point theory in partially ordered G-metric spaces, fixed point theory for expansive mappings in metric type spaces, generalizations, present results and techniques in a very general abstract setting and framework.
Fixed point theory is one of the major research areas in nonlinear analysis. This is partly due to the fact that in many real world problems fixed point theory is the basic mathematical tool used to establish the existence of solutions to problems which arise naturally in applications. As a result, fixed point theory is an important area of study in pure and applied mathematics and it is a flourishing area of research.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction with a Brief Historical Survey

Abstract
In 1906, Fréchet [78] gave the formal definition of the distance by introducing a function d that assigns a nonnegative real number d(x, y) (the distance between x and y) to every pair (x, y) of elements (points) of a nonempty set X. It is assumed that this function satisfies the following conditions:
Ravi P. Agarwal, Erdal Karapınar, Donal O’Regan, Antonio Francisco Roldán-López-de-Hierro

Chapter 2. Preliminaries

Abstract
In this section we present fundamental definitions and elementary results (see Apostol [23], Bourbaki [51], and Schweizer and Sklar [186]).
Ravi P. Agarwal, Erdal Karapınar, Donal O’Regan, Antonio Francisco Roldán-López-de-Hierro

Chapter 3. G-Metric Spaces

Abstract
In this chapter we introduce the concept of G -metric on a set X, and we show some of its basic properties. We provide any G-metric space with a Hausdorff topology in which the notions of convergent and Cauchy sequences will be a key tool in almost all proofs. Later, we will study the close relationships between G-metrics and quasi-metrics.
Ravi P. Agarwal, Erdal Karapınar, Donal O’Regan, Antonio Francisco Roldán-López-de-Hierro

Chapter 4. Basic Fixed Point Results in the Setting of G-Metric Spaces

Abstract
The Banach contractive mapping principle is the most celebrated result in fixed point theory. The simplicity of its proof and the possibility of attaining the fixed point by using successive approximations makes it a useful tool in analysis and in applied mathematics. In this chapter, we present a variety of fixed (and coincidence) point results in the context of G-metric spaces.
Ravi P. Agarwal, Erdal Karapınar, Donal O’Regan, Antonio Francisco Roldán-López-de-Hierro

Chapter 5. Fixed Point Theorems in Partially Ordered G-Metric Spaces

Abstract
In [168], Ran and Reurings established a fixed point theorem that extends the Banach contraction principle to the setting of partially ordered metric spaces (see Theorem A.1.1). In their original version, Ran and Reurings used a continuous function. Nieto and Rodríguez-López established a similar result replacing the continuity of the nonlinear operator by a property on the partially ordered metric space (see Theorem A.1.2). In this chapter, we present some fixed point theorems in the setting of partially ordered G-metric spaces. In particular, we will use a binary relation weaker than a partial order.
Ravi P. Agarwal, Erdal Karapınar, Donal O’Regan, Antonio Francisco Roldán-López-de-Hierro

Chapter 6. Further Fixed Point Results on G-Metric Spaces

Abstract
In this chapter we present some fixed point theorems in the context of G-metric spaces.
Ravi P. Agarwal, Erdal Karapınar, Donal O’Regan, Antonio Francisco Roldán-López-de-Hierro

Chapter 7. Fixed Point Theorems via Admissible Mappings

Abstract
In this chapter we explain how to use functions in order to extend the notion of partial order or, more precisely, how non-decreasing mappings can be interpreted involving certain classes of admissible functions. The results we present are inspired by Samet et al. [183].
Ravi P. Agarwal, Erdal Karapınar, Donal O’Regan, Antonio Francisco Roldán-López-de-Hierro

Chapter 8. New Approaches to Fixed Point Results on G-Metric Spaces

Abstract
Recently, Samet et al. [184], and Jleli and Samet [97], observed that some fixed point theorems in the context of G-metric space in the literature can be concluded from existence results in the setting of quasi-metric spaces. In fact, if the contractivity condition of the fixed point theorem on a G-metric space can be reduced to two variables instead of there variables, then one can construct an equivalent fixed point theorem in the setup of usual metric spaces. More precisely, in [97, 184], the authors noticed that q(x, y) = G(x, y, y) forms a quasi-metric.
Ravi P. Agarwal, Erdal Karapınar, Donal O’Regan, Antonio Francisco Roldán-López-de-Hierro

Chapter 9. Expansive Mappings

Abstract
In this chapter we present some fixed point theorems for expansive mappings.
Ravi P. Agarwal, Erdal Karapınar, Donal O’Regan, Antonio Francisco Roldán-López-de-Hierro

Chapter 10. Reconstruction of G-Metrics: G ∗-Metrics

Abstract
The main aim of the present chapter is to prove new unidimensional and multidimensional fixed point results in the framework of G-metric spaces provided with a partial preorder (not necessarily a partial order). However, we need to overcome the well-known fact that the usual product of G-metrics is not necessarily a G-metric unless they come from classical metrics. Hence, we will omit one of the axioms that define a G-metric and we consider a new class of metrics, called G -metrics. Notice that our main results are valid in the context of G-metric spaces.
Ravi P. Agarwal, Erdal Karapınar, Donal O’Regan, Antonio Francisco Roldán-López-de-Hierro

Chapter 11. Multidimensional Fixed Point Theorems on G-Metric Spaces

Abstract
In this chapter we introduce several notions of multidimensional fixed points. To prove results, it is usual to consider a number of sequences equal to the dimension of the product space in which the main mapping is defined. Also, using the techniques described in Sect. 10.​3, we will show that most of multidimensional results can be deduced from the corresponding unidimensional result in G -metric spaces.
Ravi P. Agarwal, Erdal Karapınar, Donal O’Regan, Antonio Francisco Roldán-López-de-Hierro

Chapter 12. Recent Motivating Fixed Point Theory

Abstract
In this chapter, we present some recent fixed/coincidence point results. They show some current research, thoughts and directions on fixed point theory in metric type spaces. However, in order not to enlarge the present book we will not include their proofs. We give the references so that the interested reader can find the proofs.
Ravi P. Agarwal, Erdal Karapınar, Donal O’Regan, Antonio Francisco Roldán-López-de-Hierro

Backmatter

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