The \((\alpha, \beta)\)-generalized convex contractive condition with approximate fixed point results and some consequence
- Abdul Latif^{1},
- Aphinat Ninsri^{2}Email author and
- Wutiphol Sintunavarat^{2}Email author
https://doi.org/10.1186/s13663-016-0546-z
© Latif et al. 2016
Received: 9 October 2015
Accepted: 22 April 2016
Published: 4 May 2016
Abstract
The aim of this work is to introduce some new notions of generalized convex contraction mappings and establish some approximate fixed point theorems for such mappings in the setting of complete metric spaces. Examples and application to approximate fixed point results for cyclic mappings are also given in order to illustrate the effectiveness of the obtained results.
Keywords
MSC
1 Introduction
It is well known that fixed point theory is one of the important tools for solving various problems in nonlinear analysis and various fields of applied mathematical analysis. The Banach contraction mapping principle presented by Banach [1] in his thesis is one of the cornerstones in the development of fixed point theory. This principle has been used to solve several problems such as the existence and uniqueness problems for a solution of nonlinear integral equations and nonlinear differential equations. Furthermore, it can be applied to the convergence theorem for solving some problems in computational mathematics. Hence, a large number of researchers have focused on the development of this topic. For instant, one of an interesting directions is the extension of fixed point results to approximate fixed point results. Indeed, in various practical situations, some conditions in the fixed point results are too strong and thus the existence of a fixed point is not guaranteed. In this case, one can consider points close to fixed points, which we call approximate fixed points.
On the other hand, the concept of convex contractions was introduced by Istratescu [2] in 1982. He also proved that each convex contraction self mapping on a complete metric space has a unique fixed point. Recently, Miandaragh et al. [3, 4] introduced two more general concepts of convex contractions, which are called generalized convex contractions and generalized convex contractions of order 2 and they also discussed some approximate fixed point results for such mappings.
The purpose of this paper is to formulate the concepts of generalized convex contractions and generalized convex contractions of order 2 in general terms and prove the existence results of approximate fixed points for these mappings on a complete metric space by using the idea of cyclic \((\alpha, \beta)\)-admissible mappings due to Alizadeh et al. [5]. We furnish an illustrative example to demonstrate the validity of the hypotheses and the degree of utility of our results. Our result extends, unifies, and generalizes various well-known fixed point and approximate fixed point results such as the Banach contraction mapping principle [1], Kannan’s fixed point results [6], fixed point and approximate fixed point results for convex contraction mappings due to Istratescu [2], and many results in the literature. As a consequence of the presented results, the approximate fixed point results for cyclic mappings are also given in order to illustrate the effectiveness of the obtained results.
2 Preliminaries
In this section, we give some definitions, examples, and remarks which are useful for the main results in this paper. Throughout this paper, \(\mathbb{Z}^{+}\) denotes the set of positive integers and \(\mathbb{R}\) denotes the set of real numbers.
Definition 2.1
([7])
Remark 2.2
We observe that a fixed point is an ε-fixed point, where ε is an arbitrary positive real number. However, the converse is not true.
Definition 2.3
([8])
Example 2.4
Example 2.5
In 1996, Browder and Petryshyn [9] defined the following notions.
Definition 2.6
([9])
It is not hard to prove the following results.
Lemma 2.7
Let \((X,d)\) be a metric space and \(T:X \rightarrow X\) be an asymptotically regular at some point \(z\in X\). Then T has the approximate fixed point property.
In 2014, Alizadeh et al. [5] introduced the notions of cyclic \((\alpha, \beta)\)-admissible mappings as follows.
Definition 2.8
- (i)
\(\alpha(x) \geq1\) for some \(x \in X\) implies \(\beta(Tx) \geq1\);
- (ii)
\(\beta(y) \geq1\) for some \(y \in X\) implies \(\alpha(Ty) \geq1\).
Example 2.9
3 Main results
In this section, we introduce the concepts of \((\alpha, \beta )\)-generalized convex contraction and \((\alpha, \beta)\)-generalized convex contraction of order 2 and prove the approximate fixed point theorems for such mappings.
Definition 3.1
Now, we establish a new approximate fixed point theorem for \((\alpha, \beta)\)-generalized convex contraction mappings in complete metric spaces.
Theorem 3.2
Let \((X,d)\) be a metric space and \(T : X \rightarrow X\) be an \((\alpha , \beta)\)-generalized convex contraction mapping. Assume that T is a cyclic \((\alpha,\beta)\)-admissible mapping and there exists \(x_{0}\in X\) such that \(\alpha(x_{0}) \geq1\) and \(\beta(x_{0}) \geq1\). Then T has the approximate fixed point property. Moreover, if T is continuous and \((X,d)\) is a complete metric space, then T has a fixed point.
Proof
Next, we will show that T has a fixed point provided that \((X,d)\) is a complete metric space and T is continuous. Now, we will prove that \(\{x_{n}\}\) is a Cauchy sequence in X. Without loss of generality, we may assume that \(m,n \in\mathbb{Z}^{+}\) such that \(n>m>1\). We distinguish two cases as follows.
Next we give an example to illustrate the usability of Theorem 3.2.
Example 3.3
Now, we show that Theorem 3.2 can guarantee the existence of fixed point of T. First of all, we will show that T is an \((\alpha, \beta )\)-generalized convex contraction mapping with \(a=\frac{2}{3}\) and \(b=\frac{1}{9}\).
Corollary 3.4
Proof
Corollary 3.5
Proof
Corollary 3.6
Proof
We observe that the Banach contractive condition due to Banach [1] implies the contractive condition (3.1) whenever \(\alpha,\beta:X \rightarrow[0,\infty)\) defined by \(\alpha(x) = \beta(x) = 1\) for all \(x\in X\). From the previous observation, we get the following result.
Corollary 3.7
Next, we introduce the concept of an \((\alpha, \beta)\)-generalized convex contraction of order 2 and also establish a new approximate fixed point theorem for such mappings in complete metric spaces.
Definition 3.8
Theorem 3.9
Let \((X,d)\) be a metric space and \(T : X \rightarrow X\) be an \((\alpha , \beta)\)-generalized convex contraction mapping of order 2. Assume that T is a cyclic \((\alpha,\beta)\)-admissible mapping and there exists \(x_{0}\in X\) such that \(\alpha(x_{0}) \geq1\) and \(\beta(x_{0}) \geq1\). Then T has the approximate fixed point property. Moreover, if T is continuous and \((X,d)\) is a complete metric space, then T has a fixed point.
Proof
Next, we show that T has a fixed point provided that \((X,d)\) is a complete metric space and T is continuous. First, we will claim that \(\{x_{n}\}\) is a Cauchy sequence in X. Let \(m,n \in\mathbb{Z}^{+}\) such that \(n>m>1\). Now, we distinguish the following cases.
Corollary 3.10
Corollary 3.11
Corollary 3.12
The following result is a special case of Theorem 3.9 because the Kannan contractive condition due to Kannan [6] implies the contractive condition (3.7) if \(\alpha,\beta:X \rightarrow[0,\infty)\) defined by \(\alpha(x) = \beta (x) = 1\) for all \(x\in X\).
Corollary 3.13
Corollaries 3.7 and 3.13, are interesting for defining the concepts of other classes of nonlinear mappings which are generalizations of several well-known mappings due to Chatterjea [10], Ćirić [11], Geraghty [12], Meir and Keeler [13], Mizoguchi and Takahashi [14], Suzuki [15], etc.
4 Some cyclic contractions via cyclic \((\alpha,\beta )\)-admissible mappings
In this section, we introduce the concept of cyclic generalized convex contraction mappings and prove some approximate fixed point results for such mappings in complete metric spaces.
Definition 4.1
- (i)
\(T(A) \subseteq B\) and \(T(B) \subseteq A\),
- (ii)there exist \(a,b \in[0,\infty)\) with \(a+b<1\), satisfying the following condition:for all \(x \in A\), \(y \in B\).$$\begin{aligned} d \bigl(T^{2}x,T^{2}y \bigr)\leq ad(Tx,Ty)+bd(x,y) \end{aligned}$$(4.1)
Example 4.2
Now, we establish approximate fixed point theorems for cyclic generalized convex contraction mappings as follows.
Theorem 4.3
Let A and B be two nonempty closed subsets of a metric space \((X,d)\) such that \(A \cap B \neq\emptyset\) and \(T : A \cup B \rightarrow A \cup B\) be a cyclic generalized convex contraction mapping. Then T has the approximate fixed point property. Moreover, if T is continuous and \((X,d)\) is a complete metric space, then T has a fixed point in \(A \cap B\).
Proof
Now we will claim that T is a cyclic \((\alpha,\beta)\)-admissible mapping. Assume that \(\alpha(x) \geq1\) for some \(x \in X\) and then \(x \in A\). By condition (i) in Definition 4.1, we get \(Tx \in B\) and so \(\beta(Tx) \geq1\). On the other hand, we may assume that \(\beta(y) \geq1\) for some \(y \in X\) and so \(y \in B\). Again, by using the condition (i) in Definition 4.1, we get \(Tx \in A\) and then \(\alpha(Tx) \geq1\). Therefore T is a cyclic \((\alpha,\beta)\)-admissible mapping. Since \(A \cap B\) is nonempty, there exists \(x_{0} \in A \cap B\) such that \(\alpha(x_{0}) \geq1\) and \(\beta(x_{0}) \geq1\). From Theorem 3.2, we can conclude that T has the approximate fixed point property.
Finally, we will show that T has a fixed point provided that T is continuous and \((X,d)\) is a complete metric space. Since A and B are two closed subsets of complete metric space \((X,d)\), we see that \(A \cup B\) is also a complete metric space. From Theorem 3.2, we see that T has a fixed point in \(A \cup B\), say z. If \(z \in A\), then we have \(z=Tz \in B\). Also, if \(z \in B\), we have \(z=Tz \in A\). Therefore, \(z \in A \cap B\). This completes the proof. □
Next, we introduce the concept of cyclic generalized convex contraction mappings of order 2 and establish the approximate fixed point theorem for such mappings in complete metric spaces.
Definition 4.4
- (i)
\(T(A) \subseteq B\) and \(T(B) \subseteq A\),
- (ii)there exist \(a_{1},a_{2},b_{1},b_{2}\in[0,1)\) with \(a_{1}+a_{2}+b_{1}+b_{2}<1\), satisfying the following condition:for all \(x \in A\), \(y \in B\).$$\begin{aligned} d \bigl(T^{2}x,T^{2}y \bigr)\leq a_{1}d(x,Tx)+a_{2}d \bigl(Tx,T^{2}x \bigr)+b_{1}d(y,Ty)+b_{2}d \bigl(Ty,T^{2}y \bigr) \end{aligned}$$(4.2)
By a similar technique to the proof of Theorem 4.3, we get the following result.
Theorem 4.5
Let A and B be two nonempty closed subsets of a metric space \((X,d)\) such that \(A \cap B \neq\emptyset\) and \(T : A \cup B \rightarrow A \cup B\) be a cyclic generalized convex contraction mapping of order 2. Then T has the approximate fixed point property. Moreover, if T is continuous and \((X,d)\) is a complete metric space, then T has a fixed point in \(A \cap B\).
Declarations
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR for the technical and financial support. The third author would like to thank the Thailand Research Fund and Thammasat University under Grant No. TRG5780013 for financial support during the preparation of this manuscript.
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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