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2019 | Buch

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Fixed Point Theorems and Applications

verfasst von: Prof. Vittorino Pata

Verlag: Springer International Publishing

Buchreihe : UNITEXT

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book addresses fixed point theory, a fascinating and far-reaching field with applications in several areas of mathematics. The content is divided into two main parts. The first, which is more theoretical, develops the main abstract theorems on the existence and uniqueness of fixed points of maps. In turn, the second part focuses on applications, covering a large variety of significant results ranging from ordinary differential equations in Banach spaces, to partial differential equations, operator theory, functional analysis, measure theory, and game theory. A final section containing 50 problems, many of which include helpful hints, rounds out the coverage. Intended for Master’s and PhD students in Mathematics or, more generally, mathematically oriented subjects, the book is designed to be largely self-contained, although some mathematical background is needed: readers should be familiar with measure theory, Banach and Hilbert spaces, locally convex topological vector spaces and, in general, with linear functional analysis.

Inhaltsverzeichnis

Frontmatter

Fixed Point Theorems

Frontmatter

Chapter 1. The Banach Contraction Principle

Abstract

In this case, it is readily seen that there exists the smallest value $\lambda $ for which the inequality holds, called the Lipschitz constant of f.

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Chapter 2. The Boyd-Wong Theorem

Abstract

The following result is proved in Boyd and Wong (Proc Amer Math Soc 20:458–464, 1969 [8]).

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Chapter 3. Further Extensions of the Contraction Principle

Abstract

Further generalizations of the BCP Theorem 1.1 can be found in the literature (see, e.g., [30, 36]). We report here a couple of results in that direction.

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Chapter 4. Weak Contractions

Abstract

We now dwell on the case of maps on a metric space which are contractive without being contractions.

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Chapter 5. Contractions of -Type

Abstract

Let (X, d) be a complete metric space. Selecting an arbitrary $x_0\in X$, that we may call the zero of X, we denote.

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Chapter 6. Sequences of Maps and Fixed Points

Abstract

Let (X, d) be a complete metric space. We consider the problem of convergence of fixed points for a sequence of maps $f_n:X\rightarrow X$. Corollary 1.3 will be implicitly used in the statements of the next two theorems.

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Chapter 7. Fixed Points of Non-expansive Maps

Abstract

Let X be a Banach space, $C\subset X$ nonempty, closed, bounded and convex, and let $f:C\rightarrow C$ be a non-expansive map, namely, such that

$$d(f(x),f(y))\le d(x,y),\qquad \forall x, y\in C.$$

The problem is whether f admits a fixed point in C. The answer, in general, is false.

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Chapter 8. The Riesz Mean Ergodic Theorem

Abstract

If T is a non-expansive linear map of a uniformly convex Banach space, then all the fixed points of T are recovered by means of a limit procedure.

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Chapter 9. The Brouwer Fixed Point Theorem

Abstract

For every integer $n\ge 1$, we denote the n-dimensional unit disk.

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Chapter 10. The Schauder-Tychonoff Fixed Point Theorem

Abstract

In order to prove the main result of this chapter, the Schauder-Tychonoff fixed point theorem, we first need a definition and a lemma.

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Chapter 11. Further Consequences of the Schauder-Tychonoff Theorem

Abstract

In concrete applications it is somehow easier to work with functions defined on the whole space X, and rather ask more restrictive conditions, such as compactness, on the maps.

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Chapter 12. The Markov-Kakutani Theorem

Abstract

This result is concerned with common fixed points of a family of linear maps.

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Chapter 13. The Kakutani-Ky Fan Theorem

Abstract

Along this chapter, let X be a locally convex space.

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Some Applications

Frontmatter

Chapter 14. The Implicit Function Theorem

Abstract

Let X, Y be Banach spaces, let $x_0\in U\subset X$ with U open, and let $f:U\rightarrow Y$.

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Chapter 15. Location of Zeros

Abstract

Let X, Y be Banach spaces, and $f: B_X(x_0,r)\rightarrow Y$ be a Fréchet differentiable map. In order to find a zero for f, the idea is to apply an iterative method constructing a sequence $x_n$ starting from $x_0$ so that $x_{n+1}$ is the zero of the tangent $\tau $ of f at the point $x_{n}$ (see Definition 14.2). Assuming then $[f'(x)]^{-1}\in L(Y, X)$ on $B_X(x_0,r)$, one has

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Chapter 16. Ordinary Differential Equations in Banach Spaces

Abstract

Let X be a Banach space, and let [a, b] be a closed bounded interval of the real line. The notion of Riemann integral and the related properties can be extended with no differences from the case of real-valued functions to X-valued functions on [a, b].

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Chapter 17. The Lax-Milgram Lemma

Abstract

Let V be a real Hilbert space with scalar product $\langle \cdot ,\cdot \rangle _V$ and norm $\Vert \cdot \Vert _V$. We denote the action of an element $v^*$ of the dual space $V^*$ on $v\in V$ by $\langle v^*, v\rangle $.

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Chapter 18. An Abstract Elliptic Problem

Abstract

Let $(V,\langle \cdot ,\cdot \rangle _V,\Vert \cdot \Vert _V)$ be a real Hilbert space. Again, we denote the action of an element $v^*$ of the dual space $V^*$ on $v\in V$ by $\langle v^*, v\rangle $.

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Chapter 19. Semilinear Evolution Equations

Abstract

Let X be a Banach space. Assume we are given an evolution equation in X of the form.

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Chapter 20. An Abstract Parabolic Problem

Abstract

Let $(V,H, V^*)$ be a Hilbert triple, with V separable and $V\Subset H$ (cf. Chap. 17 ff.), and let $A:V\rightarrow V^*$ be an elliptic operator associated to a continuous and coercive symmetric bilinear form on V (which can be assumed to be equal to the scalar product of V).

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Chapter 21. The Invariant Subspace Problem

Abstract

The invariant subspace problem is probably the problem of operator theory. The question, that attracted the attention of a great number of mathematicians, is quite simple to state.

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Chapter 22. Measure Preserving Maps on Compact Hausdorff Spaces

Abstract

Let X be a compact Hausdorff space, and let P(X) be the set of all Borel probability measures on X. By means of the Riesz representation theorem, the dual space of ${\mathcal C}(X)$ can be identified with the space M(X) of complex regular Borel measures on X. Recall that the norm $\Vert \mu \Vert $ of an element $\mu \in M(X)$ is given by the total variation of $\mu $. It is straightforward to check that P(X) is convex and closed in the weak$^*$ topology of M(X).

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Chapter 23. Invariant Means on Semigroups

Abstract

Let S be a semigroup, that is, a set endowed with an associative product

$$(s, t)\mapsto st.$$

We consider the (real) Banach space of all real-valued bounded functions on S, namely,

$$\ell ^\infty (S)=\Big \{f:S\rightarrow {\mathbb R}\,\,\,\text { such that}\,\,\,\Vert f\Vert \doteq \sup _{s\in S}|f(s)|<\infty \Big \}. $$

An element $f\in \ell ^\infty (S)$ is called positive if $f(s)\ge 0$ for every $s\in S$. A linear functional ${\Lambda }:\ell ^\infty (S)\rightarrow {\mathbb R}$ is called positive if ${\Lambda }f\ge 0$ for every positive element $f\in \ell ^\infty (S)$. We agree to denote a constant function on S by the value of the constant.

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Chapter 24. Haar Measures

Abstract

A topological group is a group G endowed with a Hausdorff topology that makes the group operations continuous; namely, the map $(x, y)\mapsto xy$ is continuous for every $x, y\in G$, and the map $x\mapsto x^{-1}$ is continuous for every $x\in G$.

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Chapter 25. Game Theory

Abstract

We consider a game with $n\ge 2$ players, under the assumption that the players do not cooperate among themselves. Each player pursues a strategy, in dependence of the strategies of the other players.

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Some Problems

Frontmatter

Chapter 26. Problems

Abstract

Let X be a metric space (not necessarily complete). A map $f:X\rightarrow X$ is said to be closed if, whenever $x_n\rightarrow x$ and $f(x_n)\rightarrow y$, it follows that $y=f(x)$.

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Backmatter

Titel: Fixed Point Theorems and Applications
verfasst von: Prof. Vittorino Pata
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-19670-7
Print ISBN: 978-3-030-19669-1
DOI: https://doi.org/10.1007/978-3-030-19670-7

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Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Fixed Point Theorems

Frontmatter

Chapter 1. The Banach Contraction Principle

Chapter 2. The Boyd-Wong Theorem

Chapter 3. Further Extensions of the Contraction Principle

Chapter 4. Weak Contractions

Chapter 5. Contractions of -Type

Chapter 6. Sequences of Maps and Fixed Points

Chapter 7. Fixed Points of Non-expansive Maps

Chapter 8. The Riesz Mean Ergodic Theorem

Chapter 9. The Brouwer Fixed Point Theorem

Chapter 10. The Schauder-Tychonoff Fixed Point Theorem

Chapter 11. Further Consequences of the Schauder-Tychonoff Theorem

Chapter 12. The Markov-Kakutani Theorem

Chapter 13. The Kakutani-Ky Fan Theorem

Some Applications

Frontmatter

Chapter 14. The Implicit Function Theorem

Chapter 15. Location of Zeros

Chapter 16. Ordinary Differential Equations in Banach Spaces

Chapter 17. The Lax-Milgram Lemma

Chapter 18. An Abstract Elliptic Problem

Chapter 19. Semilinear Evolution Equations

Chapter 20. An Abstract Parabolic Problem

Chapter 21. The Invariant Subspace Problem

Chapter 22. Measure Preserving Maps on Compact Hausdorff Spaces

Chapter 23. Invariant Means on Semigroups

Chapter 24. Haar Measures

Chapter 25. Game Theory

Some Problems

Frontmatter

Chapter 26. Problems

Backmatter

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