Fixed Point Results in W-Distance Spaces

Vladimir Rakočević

• Publisher: Chapman and Hall/CRC
• Year: 2021
• ISBN:
  9781000489477
• Edition: 1
• Topic: Mathématiques
• Subtopic: Topologie

Fixed Point Results in W-Distance Spaces is a self-contained and comprehensive reference for advanced fixed-point theory and can serve as a useful guide for related research. The book can be used as a teaching resource for advanced courses on fixed-point theory, which is a modern and important field in mathematics. It would be especially valuable for graduate and postgraduate courses and seminars.

Features

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• Written in a concise and fluent style, covers a broad range of topics and includes related topics from research.

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• Suitable for researchers and postgraduates.

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• Contains brand new results not published elsewhere.

Chapter 1Introduction

DOI: 10.1201/9781003213444-1

Chapter 1 provides some basic definitions and notations used throughout the book necessary for understanding the main topics and ideas discussed in the remaining chapters. For this reason, the fundamental concepts and results concerning metric spaces are given. Furthermore, three famous fixed point theorems of Banach, Kannan and Ćirić are given with their proofs.

1.1 Metric spaces

Throughout this book, we assume that $ℝ=(-\infty ,+\infty )$, ${ℝ}^{+}=[0,+\infty )$, ${\mathbb{N}}_0=\left\{0\right\}\cup \mathbb{N}$, where ℕ denotes the set of all positive integers and ℂ denotes the set of all complex numbers. In this section, we revisit the essential notions of set theory and metric spaces. For details, [122] may be consulted.

We say that sets A and B are equivalent if there is a $1-1$, onto function $f:A\to B$. A finite set is one that is equal to the empty set denoted by ∅, or equivalent to $\{1,\dots ,n\}$ for some $n\in \mathbb{N}$; any set that is not finite is called infinite. Infinite sets equivalent to ℕ are called countable, all other infinite sets said to be uncountable. Often the term countable is applied to a set that is equivalent to any ...