Solvability of integrodifferential problems via fixed point theory in b-metric spaces
- Monica Cosentino^{1},
- Mohamed Jleli^{2},
- Bessem Samet^{2}Email author and
- Calogero Vetro^{1}
https://doi.org/10.1186/s13663-015-0317-2
© Cosentino et al.; licensee Springer. 2015
Received: 3 January 2015
Accepted: 22 April 2015
Published: 14 May 2015
Abstract
The purpose of this paper is to study the existence of solutions set of integrodifferential problems in Banach spaces. We obtain our results by using fixed point theorems for multivalued mappings, under new contractive conditions, in the setting of complete b-metric spaces. Also, we present a data dependence theorem for the solutions set of fixed point problems.
Keywords
b-metric space differential inclusion fixed point multivalued mappingMSC
47H10 34A601 Introduction
Measure theory is a classical topic in mathematical analysis which is usually studied in the setting of real and complex numbers and functions. Indeed, measures have applications in the foundations of integration, probability and ergodic theories. On the other hand, theory of multivalued mappings has an important role in various branches of mathematics because of its applications in optimal control problems involving integrodifferential inclusions. As a matter of fact, the theory of integrodifferential equations and inclusions has undergone rapid development over the last decades (see, for instance, [1] and the references therein). Indeed, this theory has increased its significance in modern applied mathematical models of real processes arising in many engineering and scientific disciplines such as physics, biology, economics, signal processing and data fitting.
Notice that in the literature there are many papers focusing on the solution of differential problems approached via fixed point theory (see, for example, [2–4] and the references therein). On the other hand, it is well known that metric spaces and their generalizations furnish an useful tool for the study of multivalued mappings. In this regard, Nadler [5] was the first author who combined the ideas of contractions and multivalued mappings by providing a fixed point existence result.
Theorem 1.1
([5])
Let \((X, d)\) be a complete metric space and let \(T : X \to CB(X)\) be a multivalued mapping satisfying \(H(Tx, Ty) \leq k d(x, y)\) for all \(x,y \in X\), where k is a constant such that \(k \in(0, 1)\), and \(CB(X)\) denotes the family of non-empty, closed and bounded subsets of X. Then T has a fixed point, that is, there exists a point \(u \in X\) such that \(u\in Tu\).
Later on, many generalizations, extensions and applications of this theorem have appeared in the literature (see, for instance, [6–13]). In this literature review, we start from looking at the paper of Wardowski [14] who introduced a new concept of contraction, called F-contraction. Consequently, Wardowski proved fixed point theorems generalizing the Banach-Caccioppoli fixed point theorem (of which Theorem 1.1 is the multivalued version) in a new way than in the previous known theorems of the same class. Subsequently, Sgroi and Vetro [15] extended Wardowski’s ideas to the case of multivalued mappings and studied the solution of certain functional and integral equations under a suitable set of hypotheses. Yet another inspiration for our work comes from Feng and Liu’s paper [16] providing useful tools to establish both global and local fixed point theorems. Finally, we recall that the concept of metric space has been generalized in many directions to include measurements in a much more general sense. Here, we focus our attention on the notion of b-metric spaces, which are metric spaces satisfying a relaxed form of triangle inequality, see Czerwik [17] and Bakhtin [18]. Several researchers followed the idea of Czerwik and proved interesting results [19–25].
In this paper, we study the existence of solutions for certain integral problems of Fredholm type in Banach spaces. Also, we present a data dependence theorem for the solutions set of fixed point problems. We obtain our results by using fixed point theorems for multivalued mappings, under new contractive conditions, in the setting of complete b-metric spaces. Clearly, the presented theorems extend well-known results in the literature to b-metric spaces.
2 Preliminaries
In this section, we collect some basic definitions, lemmas and notations which will be used throughout the paper (see [17, 18, 26, 27] and the references therein). Let \(\mathbb{R}^{+}\) denote the set of all nonnegative real numbers and \(\mathbb{N}\) denote the set of positive integers.
Definition 2.1
- (1)
\(d(x,y)=0\) if and only if \(x=y\);
- (2)
\(d(x,y)=d(y,x)\);
- (3)
\(d(x,z)\leq s[d(x,y)+d(y,z)]\).
Clearly, a (standard) metric space is also a b-metric space, but the converse is not always true.
Example 2.2
Let \(X=[0, 1]\) and \(d:X \times X \to\mathbb{R}^{+}\) be defined by \(d(x, y) = |x-y|^{2}\) for all \(x, y \in X\). Clearly, \((X,d,2)\) is a b-metric space that is not a metric space.
- (i)
A sequence \(\{x_{n}\} \subseteq X\) converges to \(x \in X\) if \(\lim_{n \to+\infty} d(x_{n}, x) = 0\).
- (ii)
A sequence \(\{x_{n}\} \subseteq X\) is said to be a Cauchy sequence if, for every given \(\varepsilon > 0\), there exists a positive integer \(n(\varepsilon)\) such that \(d(x_{m}, x_{n}) < \varepsilon\) for all \(m, n \geq n(\varepsilon)\).
- (iii)
A b-metric space \((X, d,s)\) is said to be complete if and only if each Cauchy sequence converges to some \(x \in X\).
Example 2.3
([19])
Example 2.4
([19])
We recall the following properties from [17, 24, 27].
Lemma 2.5
- (i)
\(d(x,B)\leq d(x,b)\) for any \(b\in B\);
- (ii)
\(\delta(A,B)\leq H(A,B)\);
- (iii)
\(d(x,B)\leq H(A,B)\) for any \(x\in A\);
- (iv)
\(H(A,A)=0\);
- (v)
\(H(A,B)=H(B,A)\);
- (vi)
\(H(A,C)\leq s[H(A,B)+H(B,C)]\);
- (vii)
\(d(x,A)\leq s[d(x,y)+d(y,A)]\).
Lemma 2.6
Let \((X,d,s)\) be a b-metric space and \(A,B\in CB(X)\). Then, for each \(h>1\) and for each \(a\in A\), there exists \(b(a)\in B\) such that \(d(a,b(a))\leq h H(A,B)\).
Lemma 2.7
We conclude this section with two useful lemmas.
Lemma 2.8
Let \((X,d,s)\) be a b-metric space and let \(\{x_{n}\}\) be a sequence in X. If \(\lim_{n \to+\infty}x_{n} = y\) and \(\lim_{n \to +\infty}x_{n} = z\), then \(y=z\).
Lemma 2.9
3 Fixed point theory in b-metric spaces
3.1 Wardowski type theorem
We study the existence of fixed points for multivalued mappings by adapting the ideas in [14] to the b-metric setting. The motivation of this research is to solve certain classes of integrodifferential problems. First, inspired by Wardowski [14], we give the following definitions.
Definition 3.1
- (F1)
F is strictly increasing;
- (F2)
for each sequence \(\{\alpha_{n}\} \subset\mathbb{R}^{+}\) of positive numbers \(\lim_{n \to+ \infty}\alpha_{n} = 0\) if and only if \(\lim_{n \to+ \infty}F(\alpha_{n}) = -\infty\);
- (F3)
for each sequence \(\{\alpha_{n}\} \subset\mathbb{R}^{+}\) of positive numbers with \(\lim_{n \to+ \infty}\alpha_{n} = 0\), there exists \(k \in(0, 1)\) such that \(\lim_{n \to+\infty} (\alpha_{n})^{k} F(\alpha_{n}) = 0\);
- (F4)
for each sequence \(\{\alpha_{n}\} \subset\mathbb{R}^{+}\) of positive numbers such that \(\tau+F(s\alpha_{n})\leq F(\alpha_{n-1})\) for all \(n \in\mathbb{N}\) and some \(\tau\in\mathbb{R}^{+}\), then \(\tau +F(s^{n}\alpha_{n})\leq F(s^{n-1}\alpha_{n-1})\) for all \(n \in\mathbb{N}\).
Example 3.2
Let \(F: \mathbb{R}^{+} \to\mathbb{R}\) be defined by \(F(x)=x+ \ln x\). Clearly, F satisfies (F1)-(F4). Here we show only (F4).
Definition 3.3
Now, we are ready to state and prove our first main theorem.
Theorem 3.4
Proof
Firstly, we observe that if there exists an increasing sequence \(\{ n_{k} \} \subset \mathbb{N}\) such that \(x_{n_{k}} \in Tu\) for all \(k \in\mathbb{N}\), since Tu is closed and \(\lim_{k \to+ \infty}x_{n_{k}} = u\), we deduce \(u \in Tu\) and hence the proof is completed. Then we assume that there exists \(n_{0} \in \mathbb{N}\) such that \(x_{n} \notin Tu\) for all \(n \in\mathbb{N}\) with \(n \geq n_{0}\). It follows that \(Tx_{n-1} \neq Tu\) for all \(n \geq n_{0}\).
As an application of Theorem 3.4 we get the following proof of Nadler’s fixed point theorem in b-metric spaces [17].
Theorem 3.5
Proof
3.2 Feng-Liu type theorems
Another very interesting approach to studying the existence of fixed points for multivalued mappings was proposed by Feng and Liu [16]. Here, we investigate the possibility to extend this approach to the b-metric setting. The main reason of this research is to obtain data dependence results for fixed points set. We recall some notions and fix notation as follows.
Definition 3.7
Definition 3.8
Let \(T :X \to CL(X)\) be a multivalued mapping. The graph of T is the subset \(\{(x,y) : x \in X, y \in Tx \}\) of \(X \times X\); we denote the graph of T by \(G(T)\). Then T is a closed multivalued mapping if the graph \(G(T)\) is a closed subset of \((X \times X,d^{*})\), where the metric \(d^{*}\) is given by \(d^{*}((x,y),(u,v))=d(x,u)+d(y,v)\) for all \((x,y),(u,v) \in X \times X\).
Now, we state and prove the following theorem.
Theorem 3.9
- (i)
\(f_{T}\) is T-lower semicontinuous,
- (ii)
T is closed.
Proof
This completes the proof. □
Now, we show that Theorem 3.9 is a generalization of the following version of Nadler’s fixed point theorem in b-metric spaces.
Theorem 3.10
Let \((X,d,s)\) be a complete b-metric space and let \(T:X\rightarrow CL(X) \) be a multivalued mapping such that for all \(x,y\in X\), we have \(H(Tx,Ty)\leq r d(x,y)\), where \(r\in(0,s^{-1})\), then T has a fixed point.
Proof
Finally, we give a local version of Theorem 3.9.
Theorem 3.11
- (i)
\(f_{T}\) is T-lower semicontinuous,
- (ii)
T is closed.
Proof
4 Existence of solution for integral inclusions of Fredholm type
In this section, we study the solvability of integral inclusions of Fredholm type. Precisely, we present an existence result of solution under general conditions on multivalued operators. For more on the solution of integral inclusions and related problems, the reader is referred to [1, 4, 28] and the references therein.
Adapting an idea in [4], we prove the following theorem.
Theorem 4.1
- (i)
for each \(x \in C(I,\mathbb{R})\), the multivalued operator \(G:I \times I \times\mathbb{R} \to K_{cv}(\mathbb{R})\) is such that \(G(t,s,x(s))\) is lower semicontinuous in \(I \times I\);
- (ii)
\(f \in C(I,\mathbb{R})\);
- (iii)there exists \(l(t,\cdot) \in L^{1}(I)\), for each \(t \in I\) and \(\sup_{t \in I}\int^{1}_{0} l(t,s)\,ds \leq\sqrt{\frac{k}{2}}\) with \(k \in(0,1)\), such thatfor all \(t,s \in I\) and for all \(x,y \in\mathbb{R}\).$$H\bigl(G(t,s,x),G(t,s,y)\bigr) \leq l(t,s)\bigl|x(s)-y(s)\bigr| $$
Proof
Remark 4.2
Example 4.3
Clearly, the multivalued operator G is compact and convex valued. Thus, all the hypotheses of Theorem 4.1 are satisfied with \(f(t)=0\), and hence the above two-point boundary value problem has at least one solution.
5 Stability of solutions set for fixed point problems
We study data dependence of solutions set for fixed point problems by using the technique presented in Section 3.2. Indeed, in view of Theorem 3.9, we prove a data dependence theorem of the fixed points set for two multivalued mappings.
Theorem 5.1
- (i)
\(f_{S}\) and \(f_{T}\) are, respectively, S-lower and T-lower semicontinuous,
- (ii)
S and T are closed.
Proof
- (i)
\(x_{0} \in \operatorname{Fix}(S)\),
- (ii)
\(d(x_{n},Tx_{n})\leq(r\alpha^{-1})^{n}d(x_{0},Tx_{0})\),
- (iii)
\(d(x_{n},x_{n+1})\leq\alpha^{-1}(r\alpha^{-1})^{n} d(x_{0},Tx_{0})\).
Building on Theorem 5.1 and dealing with a sequence of multivalued mappings, we obtain the following result.
Theorem 5.2
Proof
Declarations
Acknowledgements
The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the International Research Group Project No. IRG14-04.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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