Common endpoints of generalized weak contractive mappings via separation theorem with applications
- Sirous Moradi^{1},
- Ali P Farajzadeh^{2},
- Yeol Je Cho^{3, 4}Email author and
- Somyot Plubtieng^{5}
https://doi.org/10.1186/s13663-015-0449-4
© Moradi et al. 2015
Received: 30 April 2015
Accepted: 27 October 2015
Published: 4 November 2015
Abstract
In this paper, first, we give the separation theorem which is an extension of the separation theorem due to Jachymski and Jóźwik (J. Math. Anal. Appl. 300:147-159, 2004). Then, by using this and the related results, we prove that two generalized weak contraction multi-valued mappings have a unique common endpoint if and only if either they have the usual approximate endpoint property or they have the common approximate strict fixed point property. This result is an extension and correct version of the main result given by Khojasteh and Rakočević (Appl. Math. Lett. 25:289-293, 2012).
Keywords
MSC
1 Introduction
Let \((X,d)\) be a metric space and \(P_{cl,bd}(X)\) be the class of nonempty closed and bounded subsets of X. A point \(x\in X\) is called a fixed point of a multi-valued mapping \(T:X\longrightarrow P_{cl,bd}(X)\) if \(x \in Tx\). We denote \(\operatorname{Fix}(T)\) the set of fixed points of the mapping T, that is, \(\operatorname{Fix}(T)=\{x\in X:x\in Tx\}\).
An element \(x\in X\) is said to be an endpoint of a multi-valued mapping T if \(Tx=\{x\}\). We denote the set of all endpoints of T by \(\operatorname{End}(T)\).
Obviously, \(\operatorname{End}(T) \subseteq\operatorname{Fix}(T)\). The investigations on the existence of the endpoints for multi-valued mappings have been studied in recent years by many authors; see for example [1–9] and the references therein.
In 2010, Amini-Harandi [1] proved that, under sufficient conditions, the weak contractive mapping T has a unique endpoint if and only if T has the approximate endpoint property. After that, in 2011, Moradi and Khojasteh [6] could improve the result by replacing the weak contraction by a general form of weak contractive and, subsequently, this result was extended by Khojasteh and Rakočević [5] by introducing the concept of the approximate and common approximate K-boundary strict fixed point property. By an example, however, we show that their result is not correct and so we give the correct form of it applying a new method for its proof, by establishing a separation theorem
The paper is organized as follows.
In Section 2, we give some basic definitions and results which will be needed in the sequel.
In Section 4, we prove that the common approximate strict fixed point property and the usual approximate endpoint property are equivalent for generalized weak contraction multi-valued mappings \(T,S:X\longrightarrow P_{cl,bd}(X)\).
In Section 5, we give the main part in this paper. By using the separation theorem obtained in Section 3 and the results in Section 4, we prove that two generalized weak contraction multi-valued mappings have a unique common endpoint if and only if either they have the usual approximate endpoint property or they have the common approximate strict fixed point property.
Finally, in Section 6, we give some applications to integral equations by using the main result, Theorem 5.9.
The main results of this paper extend the recent results given by Zhang and Song [11], Moradi and Khojasteh [6], Daffer and Kaneko [12], Rouhani and Moradi [13], Ćirić’s theorems [14] and others.
2 Preliminaries
In this section, we give some definitions which are used in the sequel.
Definition 2.1
([14])
Definition 2.2
Definition 2.3
Definition 2.4
It is clear that, if S, T have the common approximate strict fixed point property, then they have the usual approximate endpoint property.
Definition 2.5
The concept of the approximate and common approximate K-boundary strict fixed point property were defined by Khojasteh and Rakočević [5].
We note that, if T and S have the common approximate strict fixed point property, then have the usual approximate endpoint property. But the converse is not true.
Example 2.6
Let \(X=\Bbb{R}\) with the Euclidian metric. If two mappings \(T,S:X \longrightarrow P_{cl,bd}(X)\) defined by \(Tx=\{ x \}\) and \(Sx=[x+1,x+2]\) (the closed interval between \(x+1\) and \(x+2\)), respectively, then T and S have the usual approximate endpoint property, while they do not have the common approximate strict fixed point property.
3 The separation theorem
In this section, we establish a separation theorem. In order to prove it, we need the following lemma.
Lemma 3.1
Let \(\psi\in\Psi\). Then, for any closed interval \([a,b] \subset(0,+\infty)\), there exists \(\alpha\in(0,1)\) such that \(\psi(t) < \alpha t\) for all \(t \in[a,b]\).
Proof
Suppose that the conclusion is not true. Assume that there exists \([a,b] \subset (0,+\infty)\) such that, for all \(\alpha\in(0,1)\), there exists \(t \in [a,b]\) such that \(\psi(t) \geq\alpha t\). Let \(\{ \alpha_{n} \}\) be a sequence in \((0,1)\) with \(\lim_{n \rightarrow\infty} \alpha _{n}=1\). Then, for all \(n \in\Bbb{N}\), there exists \(t_{n} \in[a,b]\) such that \(\psi(t_{n}) \geq\alpha_{n} t_{n}\). Since \(\{ t_{n} \} \subset[a,b]\) and \([a,b]\) is compact, there exist a subsequence \(\{t_{n(k)} \}\) of \(\{ t_{n} \}\) and \(t \in [a,b]\) such that \(\lim_{k \rightarrow\infty} t_{n(k)}=t\). Hence we have \(\lim_{k \rightarrow\infty}\alpha _{n(k)} t_{n(k)}=t\). Also, from \(\alpha_{n(k)} t_{n(k)} \leq\psi(t_{n(k)}) < t_{n(k)}\), it follows that \(\lim_{k \rightarrow\infty}\psi(t_{n(k)})=t\). Since ψ is upper semi-continuous, it follows that \(\lim_{k \rightarrow\infty}\psi(t_{n(k)}) \leq\psi(t)\) and so \(t \leq\psi(t)\), which is a contradiction (note \(\psi\in\Psi\)). This completes the proof. □
Theorem 3.2
Let \(\psi\in\Psi\). Then there exists \(\varphi\in\Phi\) such that \(\psi(t)< \varphi(t)\) for all \(t > 0\).
Proof
Let Ω be the class of all the functions \(\psi: [0,+\infty) \longrightarrow[0,+\infty)\) such that, for some \(\varphi\in \Phi\), \(\psi(t) < \varphi(t)\) for all \(t >0\). Obviously, \(\Phi \subset\Psi\subset\Omega\).
The following example shows that \(\Phi\subsetneq\Psi\subsetneq \Omega\).
Example 3.3
4 The approximate endpoint property
In this section, we prove that the common approximate strict fixed point property and the usual approximate endpoint property are equivalent for generalized weak contraction mappings \(T,S:X\longrightarrow P_{cl,bd}(X)\).
The following result plays an important role reaching the main goal of this section.
Lemma 4.1
Let \(\varphi\in\Phi\). Then the condition \(\lim_{n \rightarrow \infty}(t_{n} -\varphi(t_{n}))=0\) implies that \(\lim_{n \rightarrow\infty}t_{n}=0\).
Proof
Since \(\varphi\in\Phi\), \(\{t_{n} \}\) is a bounded sequence. If \(\lim_{n \rightarrow\infty}t_{n} \neq0\), then there exist \(t > 0\) and a subsequence \(\{t_{n(k)}\}\) such that \(\lim_{k \rightarrow\infty}t_{n(k)}=t\). Using \(\lim_{k \rightarrow\infty}(t_{n(k)} -\varphi(t_{n(k)}))=0\) and the continuity of φ, we get \(\varphi(t)=t\), which is a contradiction. □
Theorem 4.2
Proof
It is clear that, if T and S have the common approximate strict fixed point property, then they have the usual approximate endpoint property.
5 The endpoint and fixed point results
The main motivation for this section is to present an exact version and correct proof for the following theorem.
Theorem 5.1
([5])
The following example shows that the aforementioned theorem is not correct.
Example 5.2
Let \(X= \Bbb{R}\) be endowed with the Euclidian metric, \(K=[-1,+1]\) and \(T,S:K \longrightarrow P_{bd,cl}(X)\) defined by \(Tx=Sx=\{ \frac{x}{2} \}\). Obviously, \(\partial K= \{-1,+1 \}\). We define the mapping \(\psi:[0,+\infty)\longrightarrow [0,+\infty)\) by \(\psi(t)=\frac{t}{2}\). One can show that all hypotheses in Theorem 5.1 hold. Also T, S have a unique common endpoint \(x=0\) in K. But T and S do not have the common approximate K-boundary strict fixed point property.
The following theorem is a modification and generalization form of the above theorem which is the most important consequence of this article.
Theorem 5.3
- (1)
\(\psi(0)=0\) and T and S have the usual approximate endpoint property.
- (2)
T and S have the common approximate strict fixed point property.
Proof
It is clear that, if T and S have a common endpoint, then they have the common approximate strict fixed point property.
Similarly, \(Tx=\{x\}\). Therefore, T and S have a common endpoint.
The uniqueness of the common endpoint follows from (5.3). This completes the proof. □
Corollary 5.4
Proof
If T has a unique endpoint, then T has the approximate endpoint property.
Conversely, let T have the approximate endpoint property. Define \(S=T\). Then T and S have the common approximate strict fixed point property. Hence, using Theorem 5.3, T has a unique endpoint. □
Corollary 5.5
Proof
Let \(\psi(t)=kt\) and apply Theorem 5.3. □
The following corollary extends the results given by Nadler [15], Daffer and Kaneko [12] and Rouhani and Moradi [13].
Corollary 5.6
Proof
Using Theorem 3.1 of Rouhani and Moradi [13], there exists \(x\in X\) such that \(x\in Tx\) and \(x \in Sx\). Also, from (5.11), we conclude that \(\operatorname{Fix}(T)=\operatorname{Fix}(S)\). If T and S have the usual approximate endpoint property, by Corollary 5.5, we conclude that T and S have a unique endpoint \(x_{0}\). So \(\operatorname{End}(T)=\operatorname{End}(S)=\{x_{0}\}\).
The following corollary is a direct result of Theorem 5.3.
Corollary 5.7
Proof
There exists \(\varphi\in\Phi\) such that \(\psi(t) < \psi(t)\) for all \(t>0\). It is clear that, if f and g have a unique fixed point, then f and g have the usual approximate endpoint property.
Conversely, let f and g have the usual approximate fixed point property. Hence there exists a sequence \(\{x_{n} \}\) such that \(\lim_{n \rightarrow\infty}d(x_{n},f(x_{n}))=0\) or \(\lim_{n \rightarrow\infty}d(x_{n},g(x_{n}))=0\). Suppose that \(\lim_{n \rightarrow\infty}d(x_{n},f(x_{n}))=0\).
Using Theorem 5.3, f and g have a unique common fixed point. This completes the proof. □
Theorem 5.8
Proof
As an application of Corollary 5.7 and Theorem 5.8, we obtain the following fixed point result, which extends the Ćirić theorem [14], Theorem 2.5.
Theorem 5.9
Using Theorem 5.9, we can conclude to the corresponding theorem given by Zhang and Song [11].
Theorem 5.10
Proof
Let \(\psi(t)=t-\varphi(t)\) and apply Theorem 5.9. □
Example 5.11
6 Applications to integral equations
Fixed point theorems in complete metric spaces are widely investigated and have found various applications in differential and integral equations. Motivated by [16], we study the existence of solutions for a system of nonlinear integral equations using the results proved in the previous section.
Theorem 6.1
Proof
Declarations
Acknowledgements
The third author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (2014R1A2A2A01002100).
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.
Authors’ Affiliations
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