Fixed points of dynamic processes of set-valued F-contractions and application to functional equations
- Dorota Klim^{1} and
- Dariusz Wardowski^{1}Email author
https://doi.org/10.1186/s13663-015-0272-y
© Klim and Wardowski; licensee Springer. 2015
Received: 12 November 2014
Accepted: 23 January 2015
Published: 11 February 2015
Abstract
The article is a continuation of the investigations concerning F-contractions which have been recently introduced in [Wardowski in Fixed Point Theory Appl. 2012:94,2012]. The authors extend the concept of F-contractive mappings to the case of nonlinear F-contractions and prove a fixed point theorem via the dynamic processes. The paper includes a non-trivial example which shows the motivation for such investigations. The work is summarized by the application of the introduced nonlinear F-contractions to functional equations.
Keywords
MSC
1 Introduction
- (F1)
F is strictly increasing, i.e., for all \(\alpha,\beta\in\mathbb{R_{+}}\) such that \(\alpha< \beta\), \(F(\alpha) < F(\beta)\);
- (F2)
For each sequence \((\alpha_{n})\) of positive numbers \(\lim_{n \to\infty} \alpha_{n} = 0\) if and only if \(\lim_{n \to \infty} F(\alpha_{n}) =-\infty\);
- (F3)
There exists \(k \in(0,1)\) such that \(\lim_{\alpha\to0^{+}} \alpha^{k} F(\alpha)=0\).
From (F1) and (1) it is easy to conclude that every F-contraction T is a contractive mapping, i.e., \(d(Tx,Ty) < d(x,y) \) for all \(x,y \in X\), \(Tx \neq Ty\). Consequently, every F-contraction is a continuous mapping. If \(F_{1}\), \(F_{2}\) are the mappings satisfying (F1)-(F3), \(F_{1}(\alpha)\leq F_{2}(\alpha )\) for all \(\alpha>0\) and a mapping \(G = F_{2} - F_{1}\) is nondecreasing, then every \(F_{1}\)-contraction T is \(F_{2}\)-contraction; for details, see [1]. For the aforementioned mapping T, Wardowski proved the following theorem.
Theorem 1.1
(Wardowski [1])
Let \((X,d)\) be a complete metric space, and let \(T \colon X \to X\) be an F-contraction. Then T has a unique fixed point \(x^{*} \in X\) and for every \(x_{0} \in X\) a sequence \((T^{n}x_{0})\) is convergent to \(x^{*}\).
Theorem 1.1 is a generalization of the Banach contraction principle, since for the mapping F of the form \(F(t)=\ln t\), an F-contraction mapping becomes the Banach contraction condition (2). In [1] the author provided also the example showing the essentiality of this generalization.
In the literature one can find some interesting papers concerning F-contractions; e.g., in [2] the authors investigated F-contractions of Hardy-Rogers type. Some of the articles present the application of F-contractions to integral and functional equations [3], iterated function systems [4], Volterra-type integral equations [5] and integral inequalities [6].
2 Preliminaries
Throughout the article denoted by ℝ is the set of all real numbers, by \(\mathbb{R}_{+}\) is the set of all positive real numbers and by ℕ is the set of all natural numbers. \((X, d)\) (X for short) is a metric space with a metric d.
Denote by \(N(X)\) a collection of all nonempty subsets of X, by \(C(X)\) a collection of all nonempty closed subsets of X.
Example 2.1
An element \(x^{*} \in X\) is called a fixed point of T if \(x^{*} \in Tx^{*}\). A mapping \(f \colon X \to\mathbb{R}\) is called \(D(T,x_{0})\) -dynamic lower semicontinuous at \(u \in X\) if, for any dynamic process \((x_{n}) \in D(T,x_{0})\) and for any subsequence \((x_{n_{i}})\) of \((x_{n})\) convergent to u, we have \(f(u) \leq\liminf_{i \to\infty} f(x_{n_{i}})\). We say that f is \(D(T,x_{0})\) -dynamic lower semicontinuous if f is \(D(T,x_{0})\)-dynamic lower semicontinuous at each \(u \in X\). The mapping f is called lower semicontinuous if for any sequence \((x_{n}) \subset X\) and \(u \in X\) such that \(x_{n} \to u\), we have \(f(u) \leq\liminf_{n \to\infty} f(x_{n})\).
3 The results
Remark 3.1
We will use the notion of set-valued F-contraction in the following sense.
Definition 3.1
Observe that, due to Remark 3.1, the above contractive condition is well defined.
The main result of the paper is the following.
Theorem 3.1
- 1.There exist a function \(\tau\colon\mathbb{R}_{+} \to\mathbb{R}_{+}\) and a dynamic process \((x_{n})\in D(T,x_{0})\) such that
- (H1)
T is a set-valued F-contraction with respect to \((x_{n})\);
- (H2)
\(\forall_{t \geq0}\ \liminf_{s\to t^{+}} \tau(s)>0\).
- (H1)
- 2.
A mapping \(X \ni x \mapsto d(x,Tx)\) is \(D(T,x_{0})\)-dynamic lower semicontinuous.
Proof
Consider a dynamic process \((x_{n})\) of the mapping T starting at \(x_{0}\) satisfying condition (H1). Observe that if there exists \(n_{0} \in\mathbb{N}\) such that \(x_{n_{0}}=x_{n_{0}+1}\), then the existence of a fixed point is clear. Hence we can assume that \(d(x_{n},x_{n+1})>0\) for all \(n\in\mathbb{N}\).
The direct consequence of Theorem 3.1 for single-valued maps is the following.
Corollary 3.1
- 1.There exists a function \(\tau\colon\mathbb{R}_{+} \to\mathbb {R}_{+}\) such that
- (P1)
\(\forall_{n \in\mathbb{N}} \ [d(T^{n}x_{0},T^{n+1}x_{0})>0 \Rightarrow \tau(d(T^{n-1}x_{0},T^{n}x_{0}))+F(d(T^{n}x_{0},T^{n+1}x_{0}))\leq F(d(T^{n-1}x_{0},T^{n}x_{0})) ]\);
- (P2)
\(\forall_{t \geq0}\ \liminf_{s\to t^{+}} \tau(s)>0\);
- (P1)
- 2.
\(X \ni x \mapsto d(x,Tx)\) is \(D(T,x_{0})\)-dynamic lower semicontinuous.
Corollary 3.2
- 1.There exists a function \(\tau\colon\mathbb{R}_{+} \to\mathbb {R}_{+}\) such that
- (C1)
\(\forall_{x \in X} \ [d(Tx,T^{2}x)>0 \Rightarrow\tau(d(x,Tx)) + F(d(Tx,T^{2}x)) \leq F(d(x,Tx)) ]\);
- (C2)
\(\forall_{t \geq0}\ \liminf_{s\to t^{+}} \tau(s)>0\);
- (C1)
- 2.
\(X \ni x \mapsto d(x,Tx)\) is lower semicontinuous.
4 Example
In this section we present an example which shows the motivation for investigating nonlinear F-contractions. It is worthy to note that the presented example does not apply to the mapping F, which gives the contractive condition of the known type in the literature. The example also shows that considering τ as a non-constant function significantly extends the applicability of Theorem 1.1 and Corollaries 3.1 and 3.2.
Example 4.1
Take \(x_{0} =x_{1}\). Observe that \((x_{n})\) is a dynamic process of the mapping T starting at \(x_{1}\), which is convergent to \(x_{\infty}\). Moreover, note that \(X \ni x \mapsto d(x,Tx)\) is \(D(T,x_{0})\)-dynamic lower semicontinuous.
5 Application
We will prove the following theorem.
Theorem 5.1
- 1.
\(X \ni x \mapsto\|x-Tx\|\) is lower semicontinuous;
- 2.There exists a function \(C \colon\mathbb{R}_{+} \to\mathbb{R}_{+}\) such that
- (a)
\(\forall_{t \geq 0} \ \liminf_{s\to t^{+}} C(s) > 0\);
- (b)
\(\forall_{t > 0} \ C(t) < t\);
- (c)
\(\forall_{h,k \in B(W)} \ \forall_{x \in W}\ \forall_{y \in D} \ |G(x,y,h(x))-G(x,y,k(x))| \leq\|h-k\| - C(\|h-k\|)\).
- (a)
Proof
Declarations
Acknowledgements
The authors are very grateful to the reviewers and editors for their careful reading the manuscript and valuable comments. The second author was financially supported by University of Łódź as a part of donation for the research activities aimed in the development of young scientists.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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