Common best proximity points results for new proximal C-contraction mappings
- Parvaneh Lo’lo’^{1},
- Seiyed Mansour Vaezpour^{2} and
- Reza Saadati^{3}Email author
https://doi.org/10.1186/s13663-016-0545-0
© Lo’lo’ et al. 2016
Received: 23 September 2015
Accepted: 14 April 2016
Published: 30 April 2016
Abstract
We define a new version of proximal C-contraction and prove the existence and uniqueness of a common best proximity point for a pair of non-self functions. Then we apply our main results to get some fixed point theorems and we give an example to illustrate our results.
Keywords
MSC
1 Introduction and preliminaries
Consider a pair \((A,B)\) of nonempty subsets of a metric space \((X, d)\). Assume that f is a function from A into B. An \(w \in A\) is said to be a best proximity point whenever \(d(w, fw) = d(A,B)\), where \(d(A,B) = \inf\{d(s, t): s \in A, t\in B\}\).
Best proximity point theory of non-self functions was initiated by Fan [1] and Kirk et al. [2]; see also [3–13]. In this paper, we generalize some results of Kumam et al. [14] to obtain some new common best proximity point theorems. Next, by an example and some fixed point results, we support our main results and show some applications of them.
Definition 1.1
Definition 1.2
([14])
Let \((X,d)\) be a metric space and \(\emptyset\neq A,B\subset X\). We say the pair \((A,B)\) has the V-property if for every sequence \(\{ t_{n}\}\) of B satisfying \(d(s,t_{n})\rightarrow d(s,B)\) for some \(s\in A\), there exists a \(t \in B\) such that \(d(s,t)= d(s,B)\).
2 Main results
We denote by Ψ the family of all continuous functions from \([0, +\infty) \times[0, +\infty)\) to \([0, +\infty)\) such that \(\psi (u,v)=0 \) if and only if \(u=v=0\) where \(\psi\in\Psi\).
Definition 2.1
Definition 2.2
Definition 2.3
Theorem 2.4
- (i)
f, g are continuous,
- (ii)
\(f(A_{0})\subset B_{0}\) and \(g(A_{0}) \subset B_{0}\),
- (iii)
\((f,g)\) is a generalized proximal C-contraction pair,
- (iv)
there exist \(s_{0}, s_{1} \in A_{0}\) such that \(d(s_{1},fs_{0})=d(A,B)\).
Proof
We divide our further derivation into four steps.
Theorem 2.5
- (i)
f, g are continuous,
- (ii)
\(f(A_{0})\subset B_{0}\) and \(g(A_{0}) \subset B_{0}\),
- (iii)
\((f,g)\) is an α-proximal C1-contraction pair or an α-proximal C2-contraction pair,
- (iv)
\((f,g)\) is a triangular α-proximal admissible pair,
- (iv)
there exist \(s_{0}, s_{1} \in A_{0}\) such that \(d(s_{1},fs_{0})=d(A,B)\), \(\alpha(s_{1},s_{0}) \ge1\).
Proof
Definition 2.6
- (i)
\(p,q \in X\), \(\alpha(p,q) \ge1 \Longrightarrow \alpha(fp,gq)\ge 1\) or \(\alpha(gp,fq)\ge1\),
- (ii)
\(p, q, r \in X\), \(\bigl \{\scriptsize{ \begin{array}{l} \alpha(p,r)\ge1, \\ \alpha(r,q)\ge1, \end{array}} \bigr.\Longrightarrow\alpha(p,q) \ge1\).
The following corollary is a consequence of the last theorem.
Corollary 2.7
- (i)
f and g are continuous,
- (ii)
there exists \(s_{0} \in X\) such that \(\alpha(s_{0},fs_{0})\ge1\),
- (iii)
\((f,g)\) is a triangular α-admissible pair,
- (iv)
for all \(p, q \in X\), \(\alpha(p,q) d(fp,gq) \leq\frac{1}{2}(d(p,gq) + d(q,fp))-\psi (d(p,gq),d(q,fp))\) (or \((\alpha(p,q)+l)^{d(fp,gq)} \leq(l+1)^{\frac{1}{2}(d(p,gq) + d(q,fp))-\psi(d(p,gq),d(q,fp))} \)).
Now, we remove the continuity hypothesis of f and g, and get the following theorem.
Theorem 2.8
- (i)
\(f(A_{0})\subset B_{0}\) and \(g(A_{0}) \subset B_{0}\),
- (ii)
\((f,g)\) is a generalized proximal C-contraction pair,
- (iii)
there are \(s_{0}, s_{1} \in A_{0}\) such that \(d(s_{1},fs_{0})=d(A,B)\).
Proof
Similarly, it is easy to prove that z is a best proximity point of g. Then z is a common best proximity point of the functions f and g. By the proof of Theorem 2.4 we conclude that f and g have unique common best proximity point. □
Theorem 2.9
- (i)
\(f(A_{0})\subset B_{0}\) and \(g(A_{0}) \subset B_{0}\),
- (ii)
\((f,g)\) is an α-proximal C1-contraction pair or an α-proximal C2-contraction pair,
- (iii)
\((f,g)\) is a triangular α-proximal admissible pair,
- (iv)
there exist \(s_{0}, s_{1} \in A_{0}\) such that \(d(s_{1},fs_{0})=d(A,B)\), \(\alpha(s_{1},s_{0}) \ge1\),
- (v)
if \(\{s_{n}\}\) is a sequence in A such that \(\alpha (s_{n},s_{n+1})\ge1\) and \(s_{n} \rightarrow s_{0}\) as \(n \rightarrow\infty \), then \(\alpha(s_{n},s_{0})\ge1\) for all \(n\in\mathbb{N} \cup\{0\}\).
Proof
We can derive from the proof of Theorem 2.5 that there exist a sequence \(\{s_{n}\}\) and z in A such that \(s_{n} \rightarrow z\) and \(\alpha(s_{n},s_{n+1}) \ge1\). Also, by (v), \(\alpha(s_{n},z)\ge1\) for every \(n\in\mathbb{N} \cup\{ 0\}\). Let \(s=q\), \(t=s_{2n+2}\), \(p=z\), \(q=s_{2n+1}\). If \((f,g)\) is an α-proximal C1-contraction pair or α-proximal C2-contraction pair, then \((f,g)\) is a generalized proximal C-contraction pair. Then by the proof of the last theorem, z is a common best proximity of f and g. □
The following corollary is an immediate consequence of the main theorem of this section.
Corollary 2.10
- (i)
\((f,g)\) is a triangular α-admissible pair,
- (ii)
there exists an \(s_{0} \in X\) such that \(\alpha(s_{0},fs_{0})\ge1\),
- (iii)
if \(\{s_{n}\}\) is a sequence in A such that \(\alpha (s_{n},s_{n+1})\ge1\) and \(s_{n} \rightarrow s_{0} \in A\) as \(n \rightarrow \infty\), then \(\alpha(s_{n},s_{0})\ge1\) for all \(n \in\mathbb{N}\cup\{ 0\}\),
- (iv)
for all \(x,y \in X\), \(\alpha(p,q) d(fp,gq) \leq\frac{1}{2}(d(p,gq) + d(q,fp))-\psi (d(p,gq),d(q,fp))\) (or \((\alpha(p,q)+l)^{d(fp,gq)} \leq(l+1)^{\frac{1}{2}(d(p,gq) + d(q,fp))-\psi(d(p,gq),d(q,fp))} \)).
In order to illustrate our results, we present the following example.
Example 2.11
Example 2.12
The following example shows that the triangular α-proximal admissible condition for \((f,g)\) cannot be relaxed from Theorem 2.9.
Example 2.13
Declarations
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Authors’ Affiliations
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