# On the best proximity point for the proximal contractive and nonexpansive mappings on the starshaped sets

- Meifang Guo
^{1}, - Xia Li
^{1}and - Yongfu Su
^{2}Email authorView ORCID ID profile

**Received: **13 November 2015

**Accepted: **6 March 2016

**Published: **15 March 2016

## Abstract

The purpose of this paper is to the best proximity point theorems for the proximal nonexpansive mapping on the starshaped sets by using a clever and simple method. The results improve and extend the recent results of Chen et al. (Fixed Point Theory Appl 2015:19, 2015). It should be noted that, the complex method is used by Jianren Chen et al. can be replaced by the clever and simple method presented in this paper.

## Keywords

## Introduction and preliminaries

The best proximity point problems and best proximity point theorems are basic part of nonlinear analysis and applications. In the recent years, many authors are studying the best proximity point problems. a lot of the best proximity point theorems and relatively results have been obtained in the metric spaces or normed spaces (see Fan 1969; Reich 1978; Prolla 1983; Sehgal and Singh 1988, 1989; Vetrivel et al. 1992; Basha 2000, 2011a, b; Kirk et al. 2003; Veeramani et al. 2005; Eldred and Veeramani 2006; Gabeleh 2013a, b, 2014; Sankar Raj 2011; Abkar and Gabeleh 2013a, b; Kosuru and Veeramani 2011; Lovaglia 1955; Opial 1967; Al-Thagafi and Shahzad 2009; Zhang et al. 2013; Chen et al. 2015; Hussain and Hezarjaribi 2016; Shayanpour et al. 2016; Yongfu and Yao 2015; AlNemer et al. 2016; Samet 2015; Yongfu et al. 2015; Kiran et al. 2015; Binayak 2015; Kong et al. 2015; Yongfu and Zhang 2014; Sun et al. 2014).

*A*,

*B*are two nonempty subsets of a metric space (

*X*,

*d*). Note that if \(A \cap B = \emptyset \), the equation \(Tx = x\) might have no solution. Under this circumstance it is meaningful to find a point \(x \in A\) such that

*d*(

*x*,

*Tx*) is minimum. Essentially, if \(d(x, Tx) = dist(A,B) = \inf \{d(x, y) {:} \,x \in A, y \in B\}\),

*d*(

*x*,

*Tx*) is the global minimum value

*dist*(

*A*,

*B*) and hence

*x*is an approximate solution of the equation \(Tx = x\) with the least possible error. Such a solution is known as a best proximity point of the mapping

*T*. A point \(x \in A\) is called the best proximity point of

*T*if

*T*. We can find an early classical work in Fan (1969), and afterward, there have been many interesting results such as in Reich (1978), Prolla (1983), Sehgal and Singh (1988, 1989), Vetrivel et al. (1992), Basha (2000, 2011a, b), Kirk et al. (2003), Veeramani et al. (2005) and Eldred and Veeramani (2006), and many others ( for example Gabeleh 2013a, b, 2014; Sankar Raj 2011; Abkar and Gabeleh 2013a, b; Kosuru and Veeramani 2011; Lovaglia 1955; Opial 1967; Al-Thagafi and Shahzad 2009; Zhang et al. 2013; Chen et al. 2015; Hussain and Hezarjaribi 2016; Shayanpour et al. 2016; Yongfu and Yao 2015; AlNemer et al. 2016; Samet 2015; Yongfu et al. 2015; Kiran et al. 2015; Binayak 2015; Kong et al. 2015; Yongfu and Zhang 2014; Sun et al. 2014).

Recently, Gabeleh introduced a new notion which is called the proximal nonexpansive mapping in Gabeleh (2013).

###
**Definition 1**

*A*,

*B*) be a pair of nonempty subsets of a metric space (

*X*,

*d*). A mapping \(T{:} A\rightarrow B\) is said to be proximal nonexpansive if

###
**Definition 2**

*A*,

*B*) be a pair of nonempty subsets of a metric space (

*X*,

*d*). A mapping \(T{:} \,A\rightarrow B\) is said to be proximal contraction if there exists a constant \(0<\alpha <1\) such that

###
**Definition 3**

(Kosuru and Veeramani 2011) Let (*A*, *B*) be a pair of nonempty subsets of a metric space (*X*, *d*). The pair (*A*, *B*) is said to be a semi-sharp proximinal pair if for each \(x \in A\) (respectively, in *B*) there exists at most one \(x^*\in B\) (respectively, in *A*) such that \(d(x, x^*) = dist(A,B)\).

The following notions are presented in Chen et al. (2015).

###
**Definition 4**

(Chen et al. 2015) A nonempty subset *A* of a linear space *X* is called a *p*-starshaped set if there exists a point \(p \in A\) such that \(\alpha p + (1-\alpha )x \in A\) , for all \(x \in A\), \(\alpha \in [0, 1]\), and *p* is called the center of *A*.

*A*,

*B*be two nonempty subsets of a metric space (

*X*,

*d*). We denote by \(A_0\) and \(B_0\) the following sets:

###
*Remark 5*

(Chen et al. 2015) It is easy to see that in a normed space \((X, \Vert \cdot \Vert )\), if *A* is a *p*-starshaped set and *B* is a *q*-qstarshaped set and \(\Vert p-q\Vert = dist(A,B)\), \(A_0\) is a *p*-starshaped set, and \(B_0\) is a *q*-starshaped set, respectively. If both of *A* and *B* are closed and \(A_0\) is nonempty, \(A_0\) is closed.

The purpose of this paper is to the best proximity point theorems for the proximal nonexpansive mapping on a starshaped sets by using a clever and simple method. The results improve and extend the recent results of Chen et al. (2015). It should be noted that, the complex method is used by Jianren Chen et al. can be replaced by the clever and simple method presented in this paper.

## A clever and simple method of proof

In Chen et al. (2015), authors proved the following conclusion, which plays an important role in the proof of their main results. However, the method used in the proof is relatively complicated. In this section, we will give a clever and simple method of proof.

###
**Conclusion 1**

*Let*(

*A*,

*B*)

*be a pair of nonempty closed subsets of a complete metric space*(

*X*,

*d*)

*and*\(A_0\)

*be closed and nonempty. Assume that*\(T{:} A\rightarrow B\)

*satisfies the following conditions*:

- (a)
*T**is a proximal contraction*; - (b)
\(T(A_0) \subset B_0\).

*Then there exists a unique*\(x^* \in A\)

*such that*\(d(x^*,Tx^*)=dist (A,B).\)

###
*Simple proof*

*T*is a proximal contraction, if \(d(u_1, Tx)=dist (A,B)\) and \(d(u_2, Tx)=dist (A,B)\), then \(u_1=u_2\), hence there exists a unique \(u \in A_0\) such that \(d(u, Tx)=dist (A,B)\), we denote by \(u=PTx\) this relation. That is, we define a mapping

*P*from \(T(A_0)\) into \(A_0\). Hence

*PT*is a mapping from complete metric subspace \(A_0\) into it-self. From (a) we know that,

The following conclusion is a main result of Jianren Chen et al. The method used in the proof is also complicated. In this section, we also give a very simple method of proof.

###
**Conclusion 2**

*Let*(

*A*,

*B*)

*be a pair of nonempty, closed subsets of a Banach space*

*X*

*such that*

*A*

*is a*

*p*-

*starshaped set*,

*B*

*is a*

*q*-

*starshaped set, and*\(\Vert p-q\Vert = dist(A,B)\).

*Suppose*

*A*

*is compact, and*(

*A*,

*B*)

*is a semi*-

*sharp proximinal pair. Assume that*\(T{:} A \rightarrow B\)

*satisfies the following conditions*:

- (a)
*T**is a proximal nonexpansive*; - (b)
\(T(A_0)\subset B_0\).

*Then there exists an element*\(x^* \in A\)

*such that*\(\Vert x^*-Tx^*\Vert =dist (A,B).\)

###
*Simple proof*

*T*is a proximal nonexpansive, there exists a unique \(u \in A_0\) such that \(d(u, Tx)=dist (A,B)\), we denote by \(u=PTx\) this relation. Hence

*PT*is a mapping from complete metric subspace \(A_0\) into it-self. From (a) we know that,

*PT*is a nonexpansive mapping from \(A_0\) into it-self. We define a mapping \(S: A_0\rightarrow A_0\) by

*S*is a contraction. From Remark 5, we know that \(A_0\) is closed, by using Banach contraction mapping principle, there exists a unique element \(x_\lambda \in A_0\) such that

*A*is compact. On the other hand, there exists a subsequence \(\{x_{n_k}\}\) to converge a element \(x^* \in A\). Therefore \(x^*=PTx^*\), this is equivalent to \(d(x^*,Tx^*)=dist (A,B)\). The proof is completes. \(\square \)

## Further generalized results

By using the same method as in the Conclusion 2, we can get the following further generalized results without assume the (*A*, *B*) is a semi-sharp proximinal pair, and the *A* is compact.

###
**Theorem 1**

*Let*(

*A*,

*B*)

*be a pair of nonempty, closed subsets of a Banach space*

*X*

*such that*

*A*

*is a*

*p*-

*starshaped set*,

*B*

*is a*

*q*-

*starshaped set, and*\(\Vert p-q\Vert = dist(A,B)\).

*Suppose*

*A*

*is weakly compact. Assume that*\(T{:} A \rightarrow B\)

*satisfies the following conditions*:

- (a)
*T**is a proximal nonexpansive*; - (b)
\(T(A_0)\subset B_0\).

*Then there exists an element*\(x^* \in A\)

*such that*\(\Vert x^*-Tx^*\Vert =dist (A,B).\)

###
*Proof*

Since *A* is weak compact, there exists a subsequence \(\{x_{n_k}\}\) to converge weakly a element \(x^* \in A\). Noting that, the nonexpansive mapping *PE* is demi-closed, then \(x^*=PTx^*\), this is equivalent to \(d(x^*,Tx^*)=dist (A,B)\). The proof is completes. \(\square \)

###
**Theorem 2**

*Let*

*X*

*be a uniformly convex Banach space with the Opail’s condition. Let*(

*A*,

*B*)

*be a pair of nonempty, closed subsets of*

*X*

*such that*

*A*

*is a*

*p*-

*starshaped set*,

*B*

*is a*

*q*-

*starshaped set, and*\(\Vert p-q\Vert = dist(A,B)\).

*Suppose*

*A*

*is bounded. Assume that*\(T{:}\, A \rightarrow B\)

*satisfies the following conditions*:

- (a)
*T**is a proximal nonexpansive*; - (b)
\(T(A_0)\subset B_0\).

*Then there exists an element*\(x^* \in A\)

*such that*\(\Vert x^*-Tx^*\Vert =dist (A,B).\)

###
*Proof*

Since *A* is bounded so it is weak compact, there exists a subsequence \(\{x_{n_k}\}\) to converge weakly a element \(x^* \in A\). Noting that, since *X* satisfies Opail’s condition, the nonexpansive mapping *PE* must be demi-closed (see Lemma 2 in Opial 1967), then \(x^*=PTx^*\), this is equivalent to \(d(x^*,Tx^*)=dist (A,B)\). The proof is completes. \(\square \)

Let \(g{:} A_0\rightarrow A_0\) be an isometry, we replace by \(S=g^{-1}PT\) the \(S=PT\), respectively in Theorems 1 and 2, we can get the following results which are further generalized forms to the result of Chen et al. (2015, Theorem 3.4).

###
**Theorem 3**

*Let*(

*A*,

*B*)

*be a pair of nonempty, closed subsets of a Banach space*

*X*

*such that*

*A*

*is a*

*p*-

*starshaped set*,

*B*

*is a*

*q*-

*starshaped set, and*\(\Vert p-q\Vert = dist(A,B)\).

*Suppose*

*A*

*is weakly compact. Assume that*\(T{:} \,A \rightarrow B , g: A_0\rightarrow A_0\)

*satisfy the following conditions*:

- (a)
*T**is a proximal nonexpansive*; - (b)
\(T(A_0)\subset B_0\).

- (c)
*g**is an isometry*.

*Then there exists an element*\(x^* \in A\)

*such that*\(\Vert gx^*-Tx^*\Vert =dist (A,B).\)

###
**Theorem 4**

*Let*

*X*

*be a uniformly convex Banach space with the Opail’s condition. Let*(

*A*,

*B*)

*be a pair of nonempty, closed subsets of*

*X*

*such that*

*A*

*is a*

*p*-

*starshaped set*,

*B*

*is a*

*q*-

*starshaped set, and*\(\Vert p-q\Vert = dist(A,B)\).

*Suppose*

*A*

*is bounded. Assume that*\(T {:} A \rightarrow B , g{:} A_0\rightarrow A_0\)

*satisfy the following conditions*:

- (a)
*T**is a proximal nonexpansive*; - (b)
\(T(A_0)\subset B_0\).

- (c)
*g*is an isometry.

*Then there exists an element*\(x^* \in A\)

*such that*\(\Vert gx^*-Tx^*\Vert =dist (A,B).\)

###
**Definition 6**

*A*,

*B*) be a pair of nonempty subsets of a metric space (

*X*,

*d*). The pair (

*A*,

*B*) is said to have the weak

*P*-property if

By using the well-known Schauder fixed point theorem, we can improve the result of (Theorem 4.1, Chen et al. 2015) to the following generalized result.

###
**Theorem 5**

*Let* (*A*, *B*) *be a pair of nonempty, closed, and convex subsets of a Banach space*
*X*
*such that*
*A*
*is compact. Suppose that*
\(A_0\)
*is nonempty and* (*A*, *B*) *has the weak*
*P*-*property. Let*
\(T{:}\,A \rightarrow B\)
*be a nonexpansive non-self-mapping such that*
\(T(A_0) \subset B_0\). *Then*
*T*
*has at least one best proximity point in*
*A*.

###
*Proof*

In this case, the mapping *PT* defined in Conclusion 2 is also a nonexpansive mapping from \(A_0\) into it-self. Since *A* is compact and convex, by using Schauder fixed point theorem, there exists a element \(x^* \in A\) such that \(x^*=PTx^*\), this is equivalent to \(\Vert x^*-Tx^*\Vert =dist (A,B).\) This completes the proof. \(\square \)

## Declarations

### Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

**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

## References

- Abkar A, Gabeleh M (2013) A note on some best proximity point theorems proved under P-property. Abstr Appl Anal 2013:189567View ArticleGoogle Scholar
- Abkar A, Gabeleh M (2013) Best proximity points of non-self mappings. Top 21(2):287–295View ArticleGoogle Scholar
- AlNemer G, Markin J, Shahzad N (2016) On best proximity points of upper semicontinuous multivalued mappings. Fixed Point Theory Appl 2015:237View ArticleGoogle Scholar
- Al-Thagafi MA, Shahzad N (2009) Convergence and existence results for best proximity points. Nonlinear Anal Theory Methods Appl 70(10):3665–3671View ArticleGoogle Scholar
- Basha SS (2011a) Best proximity points: optimal solutions. J Optim Theory Appl 151(1):210–216View ArticleGoogle Scholar
- Basha SS (2011b) Best proximity point theorems. J Approx Theory 163(11):1772–1781View ArticleGoogle Scholar
- Basha SS, Veeramani P (2000) Best proximity pair theorems for multifunctions with open fibres. J Approx Theory 103(1):119–129View ArticleGoogle Scholar
- Chen J, Xiao S, Wang H, Deng S (2015) Best proximity point for the proximal nonexpansive mapping on the starshaped sets. Fixed Point Theory Appl 2015:19View ArticleGoogle Scholar
- Choudhury BS, Metiya N, Postolache M, Konar P (2015) A discussion on best proximity point and coupled best proximity point in partially ordered metric spaces. Fixed Point Theory Appl 2015:170View ArticleGoogle Scholar
- Eldred AA, Veeramani P (2006) Existence and convergence of best proximity points. J Math Anal Appl 323(2):1001–1006View ArticleGoogle Scholar
- Fan K (1969) Extensions of two fixed point theorems of F.E. Browder. Math Z 112(3):234–240View ArticleGoogle Scholar
- Gabeleh M (2013) Proximal weakly contractive and proximal nonexpansive non-self-mappings in metric and Banach spaces. J Optim Theory Appl 158(2):615–625View ArticleGoogle Scholar
- Gabeleh M (2013) Global optimal solutions of non-self mappings. Sci Bull Politeh Univ Buchar Ser A 75:67–74Google Scholar
- Gabeleh M (2014) Best proximity point theorems via proximal non-self mappings. J Optim Theory Appl 164:565–576View ArticleGoogle Scholar
- Hussain N, Hezarjaribi M, Kutbi MA, Salimi P (2016) Best proximity results for Suzuki and convex type contractions. Fixed Point Theory Appl 2016:14View ArticleGoogle Scholar
- Kiran Q, Ali M, Kamran T, Karapinar E (2015) Existence of best proximity points for controlled proximal contraction. Fixed Point Theory Appl 2015:207View ArticleGoogle Scholar
- Kirk WA, Reich S, Veeramani P (2003) Proximinal retracts and best proximity pair theorems. Numer Funct Anal Optim 24(7–8):851–862View ArticleGoogle Scholar
- Kong D, Liu L, Wu Y (2015) Best proximity point theorems for \(\alpha \)-nonexpansive mappings in Banach spaces. Fixed Point Theory Appl 2015:159View ArticleGoogle Scholar
- Kosuru GSR, Veeramani P (2011) A note on existence and convergence of best proximity points for pointwise cyclic contractions. Numer Funct Anal Optim 32(7):821–830View ArticleGoogle Scholar
- Lovaglia AR (1955) Locally uniformly convex Banach spaces. Trans Am Math Soc 78:225–238View ArticleGoogle Scholar
- Opial Z (1967) Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull Am Math Soc 73(4):591–597View ArticleGoogle Scholar
- Prolla JB (1983) Fixed-point theorems for set-valued mappings and existence of best approximants. Numer Funct Anal Optim 5(4):449–455View ArticleGoogle Scholar
- Reich S (1978) Approximate selections, best approximations, fixed points, and invariant sets. J Math Anal Appl 62(1):104–113View ArticleGoogle Scholar
- Samet B (2015) Best proximity point results in partially ordered metric spaces via simulation functions. Fixed Point Theory Appl 2015:232View ArticleGoogle Scholar
- Sankar Raj V (2011) A best proximity point theorem for weakly contractive non-self-mappings. Nonlinear Anal Theory Methods Appl 74(14):4804–4808View ArticleGoogle Scholar
- Sehgal VM, Singh SP (1988) A generalization to multifunctions of Fans best approximation theorem. Proc Am Math Soc 102(3):534–537Google Scholar
- Sehgal VM, Singh SP (1989) A theorem on best approximations. Numer Funct Anal Optim 10(1–2):181–184View ArticleGoogle Scholar
- Shayanpour H, Shams M, Nematizadeh A (2016) Some results on best proximity point on star-shaped sets in probabilistic Banach (Menger) spaces. Fixed Point Theory Appl 2016:13View ArticleGoogle Scholar
- Su Y, Gao W, Yao J-C (2015) Generalized contraction mapping principle and generalized best proximity point theorems in probabilistic metric spaces. Fixed Point Theory Appl 2015:76View ArticleGoogle Scholar
- Sun Y, Su Y, Zhang J (2014) A new method for the research of best proximity point theorems of nonlinear mappings. Fixed Point Theory Appl 2014:116View ArticleGoogle Scholar
- Su Y, Yao J-C (2015) Further generalized contraction mapping principle and best proximity theorem in metric spaces. Fixed Point Theory Appl 2015:120View ArticleGoogle Scholar
- Su Y, Zhang J (2014) Fixed point and best proximity point theorems for contractions in new class of probabilistic metric spaces. Fixed Point Theory Appl 2014:170View ArticleGoogle Scholar
- Veeramani P, Kirk WA, Eldred AA (2005) Proximal normal structure and relatively nonexpansive mappings. Stud Math 171(3):283–293View ArticleGoogle Scholar
- Vetrivel V, Veeramani P, Bhattacharyya P (1992) Some extensions of Fans best approximation theorem. Numer Funct Anal Optim 13(3–4):397–402View ArticleGoogle Scholar
- Zhang J, Su Y, Cheng Q (2013) A note on A best proximity point theorem for Geraghty-contractions. Fixed Point Theory Appl 2013:99View ArticleGoogle Scholar