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Solvability of integrodifferential problems via fixed point theory in bmetric spaces
Fixed Point Theory and Applications volume 2015, Article number: 70 (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 bmetric spaces. Also, we present a data dependence theorem for the solutions set of fixed point problems.
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 nonempty, 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 Fcontraction. Consequently, Wardowski proved fixed point theorems generalizing the BanachCaccioppoli 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 bmetric 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 bmetric spaces. Clearly, the presented theorems extend wellknown results in the literature to bmetric spaces.
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
Let X be a nonempty set and let \(s\geq1\) be a given real number. A function \(d:X\times X\to\mathbb{R}^{+}\) is said to be a bmetric if and only if for all \(x,y,z\in X\) the following conditions are satisfied:

(1)
\(d(x,y)=0\) if and only if \(x=y\);

(2)
\(d(x,y)=d(y,x)\);

(3)
\(d(x,z)\leq s[d(x,y)+d(y,z)]\).
Then the triplet \((X, d, s)\) is called a bmetric space.
Clearly, a (standard) metric space is also a bmetric 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 \(d(x, y) = xy^{2}\) for all \(x, y \in X\). Clearly, \((X,d,2)\) is a bmetric space that is not a metric space.
Again, let \((X, d, s)\) be a bmetric space. The following notions are natural deductions from the corresponding metric versions:

(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 bmetric space \((X, d,s)\) is said to be complete if and only if each Cauchy sequence converges to some \(x \in X\).
Next, we give two significant examples of bmetric spaces from the literature.
Example 2.3
([19])
Let \(p \in(0,1)\). Then the set \(l^{p}(\mathbb{R}):=\{\{x_{n}\} \subset \mathbb{R} : \sum_{n=1}^{\infty} x_{n}^{p} < \infty\}\) endowed with the functional \(d: l^{p}(\mathbb{R}) \times l^{p}(\mathbb{R}) \to \mathbb{R}\) given by
for all \(\{x_{n}\},\{y_{n}\} \in l^{p}(\mathbb{R})\) is a bmetric space with \(s=2^{1/p}\).
Example 2.4
([19])
Let \(p \in(0,1)\). Then the space \(L^{p}([0,1])\) of all real functions \(f:[0,1] \to\mathbb{R}\) such that \(\int^{1}_{0}f(t)^{p}\,dt < \infty\) endowed with the functional \(d: L^{p}([0,1]) \times L^{p}([0,1]) \to\mathbb {R}\) given by
for all \(f,g \in L^{p}([0,1])\) is a bmetric space with \(s=2^{1/p}\).
Now, we give a brief background for multivalued mappings defined in a bmetric space \((X, d,s)\). For \(A, B \in CB(X)\), define the function \(H:CB(X)\times CB(X)\to\mathbb{R}^{+}\) by
where
with
Note that H is called the Hausdorff bmetric induced by the bmetric d.
We recall the following properties from [17, 24, 27].
Lemma 2.5
Let \((X,d,s)\) be a bmetric space. For any \(A, B,C \in CB(X)\) and any \(x,y\in X\), we have the following:

(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 bmetric 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
Let \((X,d,s)\) be a bmetric space. For \(A\in CB(X)\) and \(x\in X\), we have
where \(\overline{A}\) denotes the closure of the set A.
We conclude this section with two useful lemmas.
Lemma 2.8
Let \((X,d,s)\) be a bmetric 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
Let \((X,d,s)\) be a bmetric space and let \(\{x_{n}\}\) be a sequence in X such that
for some \(\lambda\in(0, s^{1})\) and each \(n\in{\mathbb{N}}\). Then \(\{x_{n}\}\) is a Cauchy sequence in X.
Fixed point theory in bmetric spaces
Wardowski type theorem
We study the existence of fixed points for multivalued mappings by adapting the ideas in [14] to the bmetric 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
Let \(s \geq1\) be a real number. We denote by \(\mathcal {F}_{s}\) the family of all functions \(F: \mathbb{R}^{+} \to\mathbb{R}\) with the following properties:

(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 \(\tau+F(s\alpha_{n})\leq F(\alpha_{n1})\) for all \(n \in\mathbb{N}\) and some \(\tau\in\mathbb{R}^{+}\), then \(\tau +F(s^{n}\alpha_{n})\leq F(s^{n1}\alpha_{n1})\) 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).
Assume that, for all \(n \in\mathbb{N}\) and some \(\tau\in\mathbb{R}^{+}\), we have \(\tau+s \alpha_{n} + \ln(s\alpha_{n}) \leq\alpha_{n1} + \ln \alpha_{n1}\). Since \(x+\ln x\) is an increasing function, then \(s\alpha_{n} < \alpha_{n1}\). Thus
implies that
and hence (F4) holds true. Note that also \(F: \mathbb{R}^{+} \to \mathbb{R}\) defined by \(F(x)=\ln x\) satisfies (F1)(F4).
Definition 3.3
Let \((X,d,s)\) be a bmetric space. A multivalued mapping \(T:X \to CB(X)\) is called an Fcontraction of Nadler type if there exist \(F \in \mathcal{F}_{s}\) and \(\tau\in\mathbb{R}^{+}\) such that
for all \(x,y\in X\) with \(Tx \neq Ty\).
Now, we are ready to state and prove our first main theorem.
Theorem 3.4
Let \((X, d,s)\) be a complete bmetric space and let \(T: X \to CB(X)\). Assume that there exists a continuous from the right function \(F \in\mathcal{F}_{s}\) and \(\tau\in \mathbb{R}^{+}\) such that
for all \(x,y \in X\), \(Tx\neq Ty\). Then T has a fixed point.
Proof
Let \(x_{0} \in X\) be an arbitrary point of X and choose \(x_{1}\in Tx_{0}\). Clearly, if \(x_{1}\in Tx_{1}\), we deduce that \(x_{1}\) is a fixed point of T and so we can conclude the proof. Now, we assume that \(x_{1}\notin Tx_{1}\) and hence \(Tx_{0} \neq Tx_{1}\). Since \(F \in\mathcal{F}_{s}\) is continuous from the right, there exists a real number \(h>1\) such that
Next, from \(d(x_{1}, Tx_{1}) < h H(Tx_{0},Tx_{1})\), we deduce that there exists \(x_{2}\in Tx_{1}\) (obviously, \(x_{2} \neq x_{1}\)) such that \(d(x_{1}, x_{2}) \leq h H(Tx_{0},Tx_{1})\). Therefore, we can write
which implies
Consequently, we get
Iterating this procedure, we construct a sequence \(\{x_{n}\} \subset X\) such that \(x_{n} \notin Tx_{n}\), \(x_{n+1} \in Tx_{n}\) and
In order to simplify the reading of calculations, let \(d_{n}=d(x_{n},x_{n+1})>0\) for all \(n\in\mathbb{N}\cup \{0\}\). It follows by (3.2) and property (F4) that
Thus, by (3.3), we have
and so
which in view of property (F2) gives
Now, by property (F3) there exists \(k \in(0,1)\) such that
By (3.4), for all \(n \in \mathbb{N}\), we get
Passing to limit as \(n \to+ \infty\) in (3.5), we obtain
and hence \(\lim_{n \to+ \infty}n^{1/k} s^{n}d_{n} = 0\). Now, the last limit implies that the series \(\sum_{n=1}^{+\infty}s^{n}d_{n}\) is convergent and hence \(\{x_{n}\}\) is a Cauchy sequence. Since \((X,d,s)\) is a complete bmetric space, then there exists \(u \in X\) such that \(\lim_{n \to+\infty}x_{n} = u\). Finally, we prove that u is a fixed point of T, that is, \(u \in Tu\).
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 \(Tx_{n1} \neq Tu\) for all \(n \geq n_{0}\).
Now, using (3.1) with \(x=x_{n}\) and \(y=u\), we obtain
which implies
Since F is strictly increasing and \(\tau\in \mathbb{R}^{+}\), we obtain
Also, we have
and passing to limit as \(n \to+\infty\) in the previous inequality, we get
which implies \(d(u,Tu)=0\). Finally, since Tu is closed, we obtain that \(u \in Tu\), that is, u is a fixed point of T. □
As an application of Theorem 3.4 we get the following proof of Nadler’s fixed point theorem in bmetric spaces [17].
Theorem 3.5
Let \((X, d,s)\) be a complete bmetric space and let \(T: X \to CB(X)\). Assume that there exists \(k \in(0,1)\) such that
for all \(x,y \in X\). Then T has a fixed point.
Proof
Let \(\tau\in\mathbb{R}^{+}\) be such that \(k=e^{2 \tau}\). From (3.6), for all \(x,y \in X\) with \(Tx \neq Ty\), we get
that is,
where \(F(x)= \ln x\). Thus we can apply Theorem 3.4 to deduce that T has a fixed point. □
Remark 3.6
Let \(CL(X)\) be the family of nonempty and closed subsets of X. Notice that Theorems 1.1 and 3.5 hold also in the case of a multivalued mapping \(T: X \to CL(X)\).
FengLiu 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 bmetric 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.
Let \((X,d,s)\) be a bmetric space and let \(T :X \to CL(X)\) be a multivalued mapping. Let \(\operatorname{Fix}(T):=\{x \in X : x \in Tx \}\) denote the fixed point set of T. Also, define the function \(f_{T} :X \to \mathbb{R}\) as \(f_{T}x = d(x,Tx)\). Then, for a positive constant \(\alpha\in(0, 1)\) and each \(x \in X\), define the set
Definition 3.7
Let \((X,d,s)\) be a bmetric space and let \(T :X \to CL(X)\) be a multivalued mapping. A function \(f:X \to\mathbb{R}\) is called Tlower semicontinuous if, for each \(\{x_{n}\} \subset X\) with \(x_{n+1} \in Tx_{n}\) and \(\lim_{n \to+\infty}x_{n} = x \in X\), we have
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
Let \((X,d,s)\) be a complete bmetric space and let \(T:X\to CL(X)\) be a multivalued mapping. Suppose that there exists \(r \in(0,s^{1}\alpha)\) with \(\alpha\in(0, 1)\) such that for any \(x \in X\) there is \(y \in I^{x}_{\alpha}\) satisfying
Then T has a fixed point in X provided that one of the following conditions holds:

(i)
\(f_{T}\) is Tlower semicontinuous,

(ii)
T is closed.
Proof
Since Tx is a nonempty closed set for any \(x \in X\), \(I^{x}_{\alpha}\) is nonempty for any constant \(\alpha\in(0,1)\). Now, for a fixed point \(x_{0} \in X\), there exists \(x_{1} \in I^{x_{0}}_{\alpha}\) such that
If \(x_{1}\) is not a fixed point of T, we choose \(x_{2}\in I^{x_{1}}_{\alpha}\) such that
Again, if \(x_{2}\) is not a fixed point of T (and so on), by iterating this procedure, we can get an iterative sequence \(\{x_{n}\}\), where \(x_{n+1} \in I^{x_{n}}_{\alpha}\) and
On the other hand, \(x_{n+1} \in I^{x_{n}}_{\alpha}\) implies
The next step of the proof is to show that the sequence \(\{x_{n}\}\) is a Cauchy sequence. Using (3.8) and (3.9), we get
Since \(r/\alpha< s^{1}\), by Lemma 2.9 we deduce that \(\{x_{n}\}\) is a Cauchy sequence and so, by completeness of the bmetric space \((X,d,s)\), \(\{x_{n}\}\) converges to some \(u\in X\). Now we claim that x is a fixed point of T. Therefore, we distinguish two cases.
Case 1: Suppose that (i) holds true. Again, by (3.8) and (3.9), we get
which implies
Consequently,
Since \(x_{n+1} \in Tx_{n}\), \(f_{T}\) is Tlower semicontinuous and \(\lim_{n \to+\infty}x_{n} = u\), we have
Since Tu is closed, we get that \(u\in Tu\), that is, u is a fixed point of T.
Case 2: If (ii) holds true, then from \(x_{n+1} \in Tx_{n}\) for all \(n \in\mathbb{N} \cup \{0\}\) and
we get that \((u,u) \in Gr(T)\) and hence \(u \in Tu\). Thus u is a fixed point of T.
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 bmetric spaces.
Theorem 3.10
Let \((X,d,s)\) be a complete bmetric 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
We have to show that all the hypotheses of Theorem 3.9 hold true. Firstly, we prove that T satisfies condition (3.7) of Theorem 3.9. Indeed, for all \(x \in X\) and \(y \in T x\), we write
and hence the assertion holds trivially for each \(x \in X\) and \(y \in I^{x}_{\alpha}\) with \(\alpha\in(0,1)\) such that \(r <\alpha s^{1}\). It would remain to show that \(f_{T}: X \to\mathbb{R}\) defined by \(f_{T}x=d(x,Tx)\) is Tlower semicontinuous. Indeed, let \(\{x_{n}\} \subset X\) be a sequence with \(x_{n+1} \in Tx_{n}\) and \(\lim_{n \to+\infty}x_{n} = x \in X\). Clearly, we have
and hence, passing to limit as \(n \to+ \infty\), we get \(f_{T}x=0\). This implies that
This completes the proof. □
Finally, we give a local version of Theorem 3.9.
Theorem 3.11
Let \((X,d,s)\) be a complete bmetric space, \(x_{0} \in X\), \(R>0\) and let \(T:X\to CL(X)\) be a multivalued mapping. Suppose that there exists \(r \in(0,s^{1}\alpha)\) with \(\alpha \in(0,1)\) such that for any \(x \in\overline{B}(x_{0},R)\) there is \(y \in I^{x}_{\alpha}\) satisfying
If \(d(x_{0}, Tx_{0}) \leq\frac{\alpha}{s} (1 \frac{sr}{\alpha} )R\), then T has a fixed point in \(\overline{B}(x_{0},R)\) provided one of the following conditions holds:

(i)
\(f_{T}\) is Tlower semicontinuous,

(ii)
T is closed.
Proof
Proceeding as in the proof of Theorem 3.9, we construct an iterative sequence \(\{x_{n}\}\) with initial point \(x_{0}\), with \(x_{n+1} \in I^{x_{n}}_{\alpha}\) and satisfying the conditions (3.8)(3.10) for all \(n \in\mathbb{N} \cup \{0\}\). From (3.10) and \(d(x_{0}, Tx_{0}) \leq\frac{\alpha}{s} (1 \frac{sr}{\alpha} )R\), we obtain
for all \(n \in\mathbb{N} \cup\{0\}\).
This implies \(x_{n} \in\overline{B}(x_{0},R)\). Indeed, we have
and so \(x_{n} \in \overline{B}(x_{0},R)\). Thus, following the proof of Theorem 3.9, it is easy to show that T has a fixed point in \(\overline {B}(x_{0},R)\). □
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.
Now, we consider the following integral inclusion of Fredholm type:
where \(G:[0,1]\times[0,1] \times\mathbb{R} \to K_{cv}(\mathbb{R})\) is a multivalued operator, where \(K_{cv}\) denotes the family of nonempty compact and convex subsets of \(\mathbb{R}\).
Let \(I = [0,1]\) and let \(C(I,\mathbb{R})\) be the space of all continuous functions \(f:I \to\mathbb{R}\). It is well known that such a space with the metric given by
is a complete bmetric space with \(s=2\).
Adapting an idea in [4], we prove the following theorem.
Theorem 4.1
Suppose that the following conditions hold:

(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 that
$$H\bigl(G(t,s,x),G(t,s,y)\bigr) \leq l(t,s)\biglx(s)y(s)\bigr $$for all \(t,s \in I\) and for all \(x,y \in\mathbb{R}\).
Then the integral inclusion (4.1) has at least one solution in \(C(I,\mathbb{R})\).
Proof
Let us define the multivalued operator \(T : C(I,\mathbb{R}) \to CL(C(I,\mathbb{R}))\) by
for each \(x \in C(I,\mathbb{R})\). Let \(x \in C(I,\mathbb{R})\) and denote \(G_{x}(t,s) :=G(t,s,x(s))\), \(t,s \in I\). For the multivalued operator \(G_{x} : I \times I \to K_{cv}(\mathbb{R})\), by Michael’s selection theorem, we get that there exists a continuous operator \(g_{x} : I \times I \to\mathbb{R}\) such that \(g_{x}(t, s) \in G_{x}(t, s)\) for all \(t,s \in I\). This implies that \(f(t)+ \int^{1}_{0} g_{x}(t, s)\,ds \in Tx\) and so Tx is a nonempty set. It is an easy matter to show that Tx is closed, and so details are omitted (see also [4]). This implies that Tx is closed in \((C(I,\mathbb{R}),d)\).
Next, we show that the multivalued operator T satisfies all the hypotheses of Theorem 3.5. To this aim, let \(x,y \in C(I,\mathbb{R})\) be such that \(v \in Tx\). Then there exists \(g_{x}(t, s) \in G_{x}(t, s)\) with \(t,s \in I\) such that \(v(t) =f(t)+ \int^{1}_{0} g_{x}(t, s)\,ds\), \(t \in I\). On the other hand, by hypothesis (iii), we get
for all \(t,s \in I\) and for all \(x,y \in\mathbb{R}\). Consequently, there exists \(w(t, s) \in G_{y} (t, s)\) such that
for all \(t,s \in I\). Now, we can consider the multivalued operator S defined by
for all \(t,s \in I\). Taking into account the fact that the multivalued operator G is lower semicontinuous, it follows that there exists a continuous operator \(g_{y} : I \times I \to\mathbb{R}\) such that \(g_{y} (t, s) \in S(t, s)\) for all \(t,s \in I\). We have
and
for all \(t \in I\). Thus, \(d(v,z) \leq\frac{k}{2} d(x,y)\). Interchanging the roles of x and y, we obtain that
for all \(x,y \in C(I,\mathbb{R})\). Thus, all the conditions of Theorem 3.5 are satisfied and hence we deduce the existence of a solution of (4.1). □
Remark 4.2
Consider the following differential inclusion:
where \(x \in C(I,\mathbb{R})\) and \(G:I \times I \times\mathbb{R} \to K_{cv}(\mathbb{R})\) is a multivalued operator. Notice that (4.2) is equivalent to (4.1) with \(f(t)=0\). Consequently, if the operator G satisfies the hypotheses of Theorem 4.1, then the existence of a solution of (4.2) follows by an application of Theorem 4.1.
Example 4.3
For all \(x \in C(I,\mathbb{R})\) and \(t,s \in I\), let \(G(t,s,x(s))=\{\nu \in\mathbb{R}: g_{1}(t,s,x(s)) \leq\nu\leq g_{2}(t,s,x(s))\}\), where \(g_{1}(t,s,x(s))\) is upper semicontinuous in \(I \times I\) and \(g_{2}(t,s,x(s))\) is lower semicontinuous in \(I \times I\). Consider the following differential inclusion:
and assume that there exists \(l(t,\cdot) \in L^{1}(I)\), for each \(t \in I\) and \(\sup_{t \in I}\int^{1}_{0} l(t,s)\,ds \leq\frac{\sqrt{2}}{2}\), such that
for all \(t,s \in I\) and for all \(x,y \in\mathbb{R}\).
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 twopoint boundary value problem has at least one solution.
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
Let \((X,d,s)\) be a complete bmetric space and let \(S,T:X\to CL(X)\) be two multivalued mappings such that \(\sup_{x \in X}H(Sx,Tx) < +\infty\). Suppose that there exists \(r \in(0,s^{1}\alpha)\) with \(\alpha\in(0,1)\) such that for any \(x \in X\) there is \(y \in I^{x}_{\alpha}\) satisfying
Then we have
provided that one of the following conditions holds:

(i)
\(f_{S}\) and \(f_{T}\) are, respectively, Slower and Tlower semicontinuous,

(ii)
S and T are closed.
Proof
By Theorem 3.9 we deduce that \(\operatorname{Fix} (S)\) and \(\operatorname{Fix}(T)\) are nonempty sets. Also, notice that \(\operatorname{Fix} (S)\) and \(\operatorname{Fix}(T)\) are closed. Indeed, for instance, let \(\{u_{n}\}\subset \operatorname{Fix}(S)\) be a sequence such that \(\lim_{n \to+\infty }u_{n} = u\). Then, if S is closed, the conclusion follows easily. On the other hand, if \(f_{S}x := d(x,Sx)\) is Slower semicontinuous, then we have
It follows that \(u \in Su\), that is, \(u \in \operatorname{Fix}(S)\). The same holds for \(\operatorname{Fix}(T)\).
Let us consider \(x_{0} \in \operatorname{Fix}(S)\). Hence, there exists \(x_{1} \in I^{x_{0}}_{\alpha}\) with \(d(x_{1},Tx_{1})\leq rd(x_{0},x_{1})\). Now, since \(\alpha d(x_{0},x_{1}) \leq d(x_{0},Tx_{0})\), we obtain
Iterating this process, we can construct an iterative sequence \(\{x_{n}\}\) such that

(i)
\(x_{0} \in \operatorname{Fix}(S)\),

(ii)
\(d(x_{n},Tx_{n})\leq(r\alpha^{1})^{n}d(x_{0},Tx_{0})\),

(iii)
\(d(x_{n},x_{n+1})\leq\alpha^{1}(r\alpha^{1})^{n} d(x_{0},Tx_{0})\).
From (iii), we deduce that \(\{x_{n}\}\) is a Cauchy sequence and therefore it converges to an element \(u \in X\). From (ii), by following the same lines of proof as in Theorem 3.9, we get that \(u \in \operatorname{Fix}(T)\). Again, if \(m>n\) from
letting \(m \to+\infty\), we deduce
for each \(n \in\mathbb{N} \cup\{0\}\). Then, for \(n =0\), we get
In a similar way we can prove that, for each \(y_{0} \in \operatorname{Fix}(T)\), there exists \(v \in \operatorname{Fix}(S)\) such that
Thus, the proof is complete. □
Building on Theorem 5.1 and dealing with a sequence of multivalued mappings, we obtain the following result.
Theorem 5.2
Let \((X,d,s)\) be a complete bmetric space and let \(T_{i}:X\to CL(X)\) be a sequence of multivalued mappings. Suppose that there exists \(r \in(0,s^{1}\alpha)\) with \(\alpha\in(0, 1)\) such that for any \(x \in X\) there is \(y \in I^{x}_{\alpha}\) satisfying \(d(y,T_{i}y) \leq r d(x,y)\) and \(\lim_{i \to+\infty}H(T_{i}x,T_{0}x) =0\) uniformly, where \(i \in\mathbb{N}\cup\{0\}\). Then we have
Proof
Let \(\varepsilon>0\) be arbitrary and choose \(N \in\mathbb {N}\) such that
for \(i \geq N\) and for each \(x \in X\). Consequently, by Theorem 5.1 we get
for \(i \geq N\). Therefore, for the arbitrarity of ε, the proof is complete. □
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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. IRG1404.
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Calogero Vetro is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
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Cosentino, M., Jleli, M., Samet, B. et al. Solvability of integrodifferential problems via fixed point theory in bmetric spaces. Fixed Point Theory Appl 2015, 70 (2015). https://doi.org/10.1186/s1366301503172
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MSC
 47H10
 34A60
Keywords
 bmetric space
 differential inclusion
 fixed point
 multivalued mapping