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Fixed points of dynamic processes of setvalued Fcontractions and application to functional equations
Fixed Point Theory and Applications volume 2015, Article number: 22 (2015)
Abstract
The article is a continuation of the investigations concerning Fcontractions which have been recently introduced in [Wardowski in Fixed Point Theory Appl. 2012:94,2012]. The authors extend the concept of Fcontractive mappings to the case of nonlinear Fcontractions and prove a fixed point theorem via the dynamic processes. The paper includes a nontrivial example which shows the motivation for such investigations. The work is summarized by the application of the introduced nonlinear Fcontractions to functional equations.
Introduction
Metric fixed point theory is a branch of mathematics, which is widely used not only in other mathematical theories, but also in many practical problems of natural sciences and engineering. One of the most common applications of fixed points of contractive mappings defined for different types of spaces is the verification of the existence and uniqueness of solutions of differential, integral or functional equations. The multiplicity of these nonlinear problems entails the search for more and better tools, which is nowadays very noticeable in the literature. One of such tools was recently delivered by Wardowski [1], where the author introduced a new type of contractive mapping called Fcontraction, i.e., the mapping \(T\colon X \to X\) with a domain in a metric space \((X,d)\) satisfying for all \(x, y\in X\) such that \(d(Tx,Ty)>0\) the contractive condition of the form
for some \(\tau> 0\) and \(F \colon\mathbb{R}_{+} \to\mathbb{R}\) satisfying the following conditions:

(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 \(\lim_{\alpha\to0^{+}} \alpha^{k} F(\alpha)=0\).
Substituting in (1) the appropriate mapping F satisfying (F1)(F3), we obtain different types of nonequivalent contractions. See the following examples of Fcontractions for concrete mappings F:
From (F1) and (1) it is easy to conclude that every Fcontraction T is a contractive mapping, i.e., \(d(Tx,Ty) < d(x,y) \) for all \(x,y \in X\), \(Tx \neq Ty\). Consequently, every Fcontraction 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 Fcontraction. 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 Fcontraction 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 Fcontractions; e.g., in [2] the authors investigated Fcontractions of HardyRogers type. Some of the articles present the application of Fcontractions to integral and functional equations [3], iterated function systems [4], Volterratype integral equations [5] and integral inequalities [6].
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.
Let \(T \colon X \to N(X)\) be a setvalued mapping, and let \(x_{0} \in X\) be arbitrary and fixed. Define
Each element of \(D(T,x_{0})\) is called a dynamic process of T starting at \(x_{0}\). The dynamic process \((x_{n})_{n \in\mathbb{N} \cup\{ 0\}}\) will be written simply \((x_{n})\). If T is a singlevalued mapping, then the dynamic process is uniquely defined and is of the form \((T^{n}x_{0})\) (see [7] and [8]).
Example 2.1
Consider a Banach space \(X=C([0,1])\) with a norm \(\x\=\sup_{t\in [0,1]}x(t)\), \(x \in X\). Let \(T\colon X\to2^{X}\) be such that for any \(x \in X\), Tx is a family of the functions \(t \mapsto c\int_{0}^{t} x(s)\,ds\), where \(c \in[0,1]\), i.e.,
and let \(x_{0}(t)=t\), \(t\in[0,1]\). Then the sequence \((\frac {1}{n!(n+1)!}t^{n+1} )\) is a dynamic process of the operator T starting at \(x_{0}\).
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 \(f(u) \leq\liminf_{n \to\infty} f(x_{n})\).
The results
In the following we will consider only the dynamic processes \((x_{n}) \in D(T,x_{0})\) satisfying the following condition:
Remark 3.1
Observe that if the investigated process does not satisfy property (D), then there exists \(n_{0} \in\mathbb{N}\) such that \(d(x_{n_{0}},x_{n_{0}+1})>0\) and \(d(x_{n_{0}1},x_{n_{0}})=0\). Then we get
which implies the existence of a fixed point. Therefore considering the dynamic processes satisfying condition (D) does not reduce a generality of our research.
We will use the notion of setvalued Fcontraction in the following sense.
Definition 3.1
Let \(x_{0} \in X\) and let \(F\colon\mathbb{R}_{+} \to\mathbb{R}\) satisfy (F1)(F3). The mapping \(T\colon X\to C(X)\) is called a setvalued Fcontraction with respect to a dynamic process \((x_{n}) \in D(T, x_{0})\) if there exists a function \(\tau\colon\mathbb{R}_{+} \to\mathbb{R}_{+}\) such that
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
Let \((X,d)\) be a complete metric space, \(T\colon X\to C(X)\), \(x_{0} \in X\) and let \(F\colon\mathbb{R}_{+} \to\mathbb{R}\) satisfy (F1)(F3). Assume that:

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 setvalued Fcontraction 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.
Then there exists a fixed point of T.
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 \(n\in\mathbb{N}\).
From (H1) we get
Thus the sequence \((d(x_{n},x_{n+1}))\) is decreasing and hence convergent.
We show that \(\lim_{n\to\infty}d(x_{n},x_{n+1})=0\). From (H2) there exists \(b>0\) and \(n_{0} \in\mathbb{N}\) such that \(\tau (d(x_{n},x_{n+1}))>b\) for all \(n > n_{0}\). Thus, we obtain for all \(n > n_{0}\) the following inequalities:
From the above we have \(\lim_{n\to\infty} F(d(x_{n},x_{n+1}))=\infty\) and, by (F2), \(\lim_{n \to\infty} d(x_{n},x_{n+1})=0\).
From (F3) there exists \(k \in(0,1)\) such that
By (5) the following holds for all \(n > n_{0}\):
Letting \(n \to\infty\) in the above and using (6), we obtain
Now, let us observe that from (7) there exists \(n_{1} \in\mathbb {N}\) such that \(n {d(x_{n},x_{n+1})}^{k} \leq1\) for all \(n \geq n_{1}\). Consequently, we have
In order to show that \((x_{n})\) is a Cauchy sequence, consider \(m,n \in\mathbb{N}\) such that \(m>n\geq n_{1}\). From the definition of the metric and from (8), we get
From the above and from the convergence of the series \(\sum_{i=1}^{\infty} 1/i^{\frac{1}{k}}\), we receive that \((x_{n})\) is a Cauchy sequence.
From the completeness of X there exists \(x^{*} \in X\) such that \(\lim_{n \to\infty}x_{n}=x^{*}\). Denote \(s_{n} = d(x_{n},x_{n+1})\), \(n\in\mathbb {N}\). Obviously, \(s_{n} \to0\), \(n \to\infty\). Thus, by \(D(T,x_{0})\)dynamic lower semicontinuity of the mapping \(X \ni x \mapsto d(x,Tx)\), we have
The closedness of \(Tx^{*}\) implies \(x^{*} \in Tx^{*}\). □
The direct consequence of Theorem 3.1 for singlevalued maps is the following.
Corollary 3.1
Let \((X,d)\) be a complete metric space, \(T\colon X\to X\), \(x_{0} \in X\), and let \(F\colon\mathbb{R}_{+} \to\mathbb{R}\) satisfy (F1)(F3). Assume that:

1.
There exists a function \(\tau\colon\mathbb{R}_{+} \to\mathbb {R}_{+}\) such that

(P1)
\(\forall_{n \in\mathbb{N}} \ [d(T^{n}x_{0},T^{n+1}x_{0})>0 \Rightarrow \tau(d(T^{n1}x_{0},T^{n}x_{0}))+F(d(T^{n}x_{0},T^{n+1}x_{0}))\leq F(d(T^{n1}x_{0},T^{n}x_{0})) ]\);

(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.
Then there exists a fixed point of T.
Corollary 3.2
Let \((X,d)\) be a complete metric space, \(T\colon X\to X\), and let \(F\colon\mathbb{R}_{+} \to\mathbb{R}\) satisfy (F1)(F3). Assume that:

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.
Then there exists a fixed point of T.
Example
In this section we present an example which shows the motivation for investigating nonlinear Fcontractions. 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 nonconstant function significantly extends the applicability of Theorem 1.1 and Corollaries 3.1 and 3.2.
Example 4.1
First, we recursively define a sequence \((x_{n})\) as follows:
One can verify that
Hence, we can put
Let \(X = \{x_{n}\colon n \in\mathbb{N}\} \cup\{x_{\infty}\}\). Together with the standard metric, X constitutes a complete metric space. Consider a mapping \(T\colon X\to X\) by the formulae
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.
Now, from the definition of the sequence \((x_{n})\) and (9), we obtain for all \(n\in\mathbb{N}\)
where \(\tau\colon\mathbb{R}_{+} \to\mathbb{R}_{+}\) is of the form
Finally, observe that taking \(F(t)= \ln(t^{2} +t)\), \(t>0\) (see (4)) we get that T satisfies the following contractive condition:
Therefore, the assumptions of Corollary 3.1 are fulfilled and \(x_{\infty}\) is a fixed point of T.
Application
Now, we apply our results in order to prove the existence of a solution of the following functional equation:
where \(f \colon W \times D \to\mathbb{R}\) and \(G \colon W \times D \times\mathbb{R}\to\mathbb{R}\) are bounded, \(\eta\colon W \times D \to W\), W and D are Banach spaces. Equations of the type (10) find their application in mathematical optimization, computer programming and in dynamic programming, which gives tools for solving boundary value problems arising in engineering and physical sciences (see, e.g., [9]).
Let \(B(W)\) denote the set of all bounded realvalued functions on W. The pair \((B(W), \ \cdot\)\), where
is a Banach space.
In order to show the existence of a solution of equation (10), we consider the operator \(T \colon B(W) \to B(W)\) of the form
for all \(h \in B(W)\) and \(x \in W\). Obviously, T is well defined, since f and G are bounded.
We will prove the following theorem.
Theorem 5.1
Let \(T \colon B(W) \to B(W)\) be an operator defined by (11) and assume that the following conditions are satisfied:

1.
\(X \ni x \mapsto\xTx\\) 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)
\(\forall_{h,k \in B(W)} \ \forall_{x \in W}\ \forall_{y \in D} \ G(x,y,h(x))G(x,y,k(x)) \leq\hk\  C(\hk\)\).

(a)
Then the functional equation (10) has a bounded solution.
Proof
Let \(\lambda>0\) be arbitrary, \(x \in W\) and \(h \in B(W)\). Without loss of generality, we can assume that \(Th \neq h\). There exist \(y_{1}, y_{2} \in D\) such that
Then we get
and
for all \(\lambda>0\). Hence
and
Therefore
and
In consequence, we get
Using simple calculations, we obtain
which is equivalent to
for \(F(t) = \ln t + t\), \(t >0\) (see (3)) and \(\tau= C\). Corollary 3.2 implies the existence of a bounded solution of equation (11). □
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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.
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MSC
 47H09
 47H10
 46N10
 54E50
Keywords
 setvalued Fcontraction
 dynamic process
 fixed point
 functional equation