Convergence and Best Proximity Points for Generalized Contraction Pairs

Abstract

This paper is devoted to studying the existence of best proximity points and convergence for a class of generalized contraction pairs by using the concept of proximally-complete pairs and proximally-complete semi-sharp proximinal pairs. The obtained results are generalizations of the result of Sadiq Basha (Basha, S., Best proximity points: global optimal approximate solutions, J. Glob. Optim. 2011, 49, 15–21) As an application, we give a result for nonexpansive mappings in normed vector spaces.

1. Introduction and Preliminaries

Let $(X,d)$ be a metric space. Consider two nonempty subsets P and Q of X. Given a non-self mapping $f:P\to Q$, then if $P\cap f(P)=\varnothing ,$ the fixed point equation $f(t)=t$ has no solution, that is $d(t,f(t))>0$ for all t in P. The object of best proximity theory is to locate $z\in P$ such that $d(z,f(z))$ is minimum and so as to ensure the existence of a point $a\in X$ verifying $d(a,f(a))=d(P,Q)$, where $d(P,Q)=\inf \{d(\xi ,\vartheta ):\xi \in P,\vartheta \in Q\}$. In this case, a is called a best proximal point of f. Best proximity point theorems furnish sufficient conditions yielding the existence of approximate solutions, which are optimal, as well. The investigation of best proximity points is an attractive topic for optimization theory; see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]. Consider:

$$P_0=\{\xi \in P:d(\xi ,\vartheta )=d(P,Q),\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{m}\mathrm{e}\vartheta \in Q\}$$

(1)

and:

$$Q_0=\{\vartheta \in Q:d(\xi ,\vartheta )=d(P,Q),\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{m}\mathrm{e}\xi \in P\}.$$

(2)

In the case that $P\cap Q\ne \varnothing$, the subsets $P_0$ and $Q_0$ are nonempty. Moreover, if $P_0$ or $Q_0$ is nonempty, then again, $P_0$ and $Q_0$ are nonempty. In the same direction, the following lemma gives some sufficient conditions in the case of reflexive Banach spaces.

Lemma 1

([18]). Let P be a nonempty, bounded, closed, and convex subset of a reflexive Banach space $X.$ Then, $P_0$ and $Q_0$ are nonempty.

Let ${\mathbb{N}}^{*}$ (resp. $\mathbb{N}$) be the set of positive (resp. nonnegative) integers. In [3], Sadiq Basha proved the following result.

Theorem 1

([3]). Let P and Q be two nonempty compact subsets of a metric space $(X,d)$. Suppose that $f:P\to Q$ and $g:Q\to P$ are two mappings satisfying the following conditions:

(i) 

f and g are contractive;

(ii) 

$d(f\xi ,g\vartheta )<d(\xi ,\vartheta )$ whenever $d(P,Q)<d(\xi ,\vartheta )$ for $(\xi ,\vartheta )\in P\times Q.$

Then, there exist $z\in P$ and $w\in Q$ such that:

$$d(z,fz)=d(w,gw)=d(z,w)=d(P,Q).$$

Further, for an arbitrary element ${\xi }_0\in P_0$, let ${\xi }_{2n+1}=f{\xi }_{2n}$ and ${\xi }_{2n}=g{\xi }_{2n-1}$ for $n\ge 1$. Then, $({\xi }_{2n})$ converges to z, and $({\xi }_{2n+1})$ converges to $w.$

The concept of proximally complete pairs was first initiated by Espínola et al. [9] and was used to study the existence and convergence to best proximity points for cyclic contraction mappings.

Definition 1

([14]). Let P and Q be nonempty subsets of a metric space $(X,d)$. Let $({\xi }_n)$ be a sequence in $P\cup Q$ such that $({\xi }_{2n})$ in P and $({\xi }_{2n+1})$ in Q for $n\ge 0$. If for each $ϵ>0$, there exists an integer $n_0$ such that for all even integers $p\ge n_0$ and odd integers $q\ge n_0$, $d({\xi }_p,{\xi }_q)<d(P,Q)+ϵ$, then $({\xi }_n)$ is called a cyclical Cauchy sequence.

Lemma 2

([9]). (i)  The sequence $({\xi }_n)$ in $P\cup Q$ such that $({\xi }_{2n})$ in P and $({\xi }_{2n+1})$ in Q for $n\ge 0$ is cyclical Cauchy if:

$$\underset{n,m\to \infty }{\lim }d({\xi }_{2n},{\xi }_{2m+1})=d(P,Q).$$

(ii)  Any cyclical Cauchy sequence can have more than one accumulation point.

Example 1.

We endow on $X={\mathbb{R}}^2$ the metric:

$$d(({\xi }_1,{\vartheta }_1),({\xi }_2,{\vartheta }_2))=|{\xi }_1-{\xi }_2|+|{\vartheta }_1-{\vartheta }_2|.$$

Let $P=\{(1,u):-2\le u\le 2\}$ and $Q=\{(0,u):-1\le u\le 1\}$. Consider the sequence ${({\theta }_n)}_{n\ge 0}$ defined by ${\theta }_n=(\frac{1+{(-1)}^n}{2},1+\frac{{(-1)}^n}{n+1})$. Then, ${\theta }_{2n}=(1,1+\frac{1}{2n+1})$ and ${\theta }_{2n+1}=(0,1-\frac{1}{2n+1})$, so $({\theta }_{2n})$ is in P and $({\theta }_{2n+1})$ is in Q. Furthermore, ${\displaystyle \underset{n\to \infty }{\lim }{\theta }_{2n}=(1,1)\in P}$ and ${\displaystyle \underset{n\to \infty }{\lim }{\theta }_{2n+1}=(0,1)\in Q}$. Then, $({\theta }_n)$ does not converge. Moreover,

$$\underset{n,m\to \infty }{\lim }d({\theta }_{2n},{\theta }_{2m+1})=\underset{n,m\to \infty }{\lim }(1+|\frac{1}{2n+1}+\frac{1}{2m+2}|)=1=d(P,Q).$$

Thus, the sequence $({\theta }_n)$ is cyclical Cauchy.

Lemma 3

([9]). Let $(X,d)$ be a metric space. Given P and Q two nonempty subsets of X, then:

(i) 

Every cyclical Cauchy sequence is bounded.

(ii) 

If $d(P,Q)=0,$ then every cyclical Cauchy sequence $({\xi }_n)\subseteq P\cup Q$ is a Cauchy sequence.

Definition 2

([9]). Let P and Q be nonempty subsets of a metric space $(X,d)$. The pair $(P,Q)$ is called proximally complete if, for every cyclically Cauchy sequence $({\xi }_n)\subseteq P\cup Q$, $({\xi }_{2n})$ and $({\xi }_{2n+1})$ have convergent subsequences in P and Q, respectively.

In the following, we give cases where the pair $(P,Q)$ is proximally complete.

Theorem 2

([9]). Let $(X,d)$ be a metric space. Let P and Q be nonempty subsets of X. We have:

(i) 

If $(P,Q)$ is a boundedly-compact pair, then it is proximally complete.

(ii) 

If $(P,Q)$ is a closed pair such that $d(P,Q)=0$ and $(X,d)$ is complete, then $(P,Q)$ is proximally complete.

Theorem 3

([9]). Consider a uniformly-convex Banach space $(X,||.||)$. Then, any nonempty, closed and convex pair $(P,Q)$ of X is proximally complete.

Theorem 4

([9]). If $(P,Q)$ is a proximally-complete pair of a metric space X, then the subsets $P_0$ and $Q_0$ are closed in X.

Definition 3

([9]). Let P and Q be nonempty subsets of a metric space $(X,d)$. The pair $(P,Q)$ is called semi-sharp proximinal if, for all $\xi \in P$ and $\vartheta \in Q$, there exist at most ${\xi }^{\star }\in Q$ and at most ${\vartheta }^{\star }\in P$ such that $d(\xi ,{\xi }^{\star })=d({\vartheta }^{\star },\vartheta )=d(P,Q).$

Example 2

([19]). Let $(X,||.||)$ be a strictly Banach convex space. Then, every closed and convex pair $(P,Q)$ of X is semi-sharp proximinal.

Example 3.

Consider $X={\mathbb{R}}^2$ endowed with the metric defined by:

$$d(({\xi }_1,{\vartheta }_1),({\xi }_2,{\vartheta }_2))=|{\xi }_1-{\xi }_2|+|{\vartheta }_1-{\vartheta }_2|.$$

Let $P=\{(1,2),(2,2)\}$ and $Q=\{(2,1),(1,1)\}$. We have $d(P,Q)=1$. Furthermore,

$$d((1,2),(1,1))=d((2,2),(2,1))=1,d((1,2),(2,1))=d((2,2),(1,1))=2.$$

Then, $(P,Q)$ is semi-sharp proximinal.

Definition 4

([34]). A nonnegative function φ defined on $[0,\infty )$ is said to be a $(c)$-comparison function if:

$({\phi }_1)$ φ is non-decreasing;

$({\phi }_2)$ there are $p_0\in \mathbb{N}$ and $r\in (0,1)$ so that for $p\ge p_0$ and $s>0$,

$${\phi }^{p+1}(s)\le r{\phi }^p(s)+u_p,$$

(3)

where the series ${\sum }_{p=1}^{\infty }{u}_p$ is convergent and $u_p\ge 0$. ${\phi }^p$ is the $p^{\mathrm{th}}$ iterate of φ.

Lemma 4

([34]). Let $\phi :[0,\infty )\to [0,\infty )$ be a $(c)$-comparison function. Then,

$(i)$${({\phi }^n(s))}_{n\in \mathbb{N}}$ converges to zero as $n\to \infty$, for each $s>0$;

$(ii)$$\phi (s)<s$ for each $s>0$;

$(iii)$ φ is continuous at zero, and $\phi (0)=0$;

$(iv)$ the series ${\sum }_{n=0}^{\infty }{\phi }^n(s)<\infty$ for each $t>0$.

In the paper of Sadiq Bacha [3], the two considered mappings are supposed to be contractive. While in this paper (Theorem 5), the contractivity of mappings and Condition (b) in Theorem 2.1 of [3] are omitted. We just take weaker hypotheses, and we get the same result by considering proximally-complete pairs or proximally-complete semi-sharp proximinal pairs. We give conditions ensuring the existence of best proximity points via contraction pairs. We also provide a result for nonexpansive mappings in normed vector spaces. The obtained results are supported by some examples.

2. Main Results

The first theorem is:

Theorem 5.

Let $(P,Q)$ be a proximally-complete pair in a metric space $(X,d)$. Let $f:P\to Q$ and $g:Q\to P$ be non-self mappings such that for all $(x,y)\in P\times Q$,

$$D(fx,gy)\le \phi (D(x,y)),$$

(4)

where φ is a c-comparison function and $D(x,y):=d(x,y)-d(P,Q).$

Then, there exist ${\xi }_{\star }\in P$ and ${\vartheta }_{\star }\in Q$ such that:

$$d({\xi }_{\star },f{\xi }_{\star })=d({\vartheta }_{\star },g{\vartheta }_{\star })=d({\xi }_{\star },{\vartheta }_{\star })=d(P,Q).$$

Proof. 

Let ${\xi }_0\in P$. Define the sequence $({\xi }_n)$ in $P\cup Q$ as follows:

$${\xi }_{2n+1}=f{\xi }_{2n}\mathrm{and}{\xi }_{2n}=g{\xi }_{2n-1},n=1,2,\cdots$$

By (4), we have:

$$D({\xi }_1,{\xi }_2)=D(f{\xi }_0,g{\xi }_1)\le \phi (D({\xi }_0,{\xi }_1)).$$

Again:

$$D({\xi }_2,{\xi }_3)=D(f{\xi }_2,g{\xi }_1)\le \phi (D({\xi }_2,{\xi }_1))\le {\phi }^2(D({\xi }_0,{\xi }_1)).$$

Continuing in this way, we find that:

$$D({\xi }_n,{\xi }_{n+1})\le {\phi }^n(D({\xi }_0,{\xi }_1))\forall n\ge 0.$$

(5)

If $d(P,Q)=0$, it is easy to show that there exists $x\in P\cap Q$ such that $d(x,fx)=d(x,gx)=0=d(P,Q).$ Moreover, the sequence $({\xi }_n)$ converges to a common fixed point of f and g.

From now on, suppose that $d(P,Q)>0.$ If $D({\xi }_{2n},{\xi }_{2n+1})=0$ for some n, then ${\xi }_{2n}$ is a best proximity point of f. From (4), we have:

$$D({\xi }_{2n+1},g{\xi }_{2n+1})=D(f{\xi }_{2n},g{\xi }_{2n+1})\le \phi (D({\xi }_{2n},{\xi }_{2n+1}))=\phi (0)=0$$

and so, ${\xi }_{2n+1}$ is a best proximity point of g. Similarly, if $D({\xi }_{2n+1},{\xi }_{2n+2})=0$ for some n, then ${\xi }_{2n+1}$ is a best proximity point of g, and ${\xi }_{2n+2}$ is a best proximity point of f.

Suppose now that $D({\xi }_n,{\xi }_{n+1})>0$ for all $n\ge 0$. Passing to the limit in Inequality (5), we get ${\displaystyle \underset{n\to \infty }{\lim }D({\xi }_n,{\xi }_{n+1})=0.}$ Hence,

$$\underset{n\to \infty }{\lim }d({\xi }_n,{\xi }_{n+1})=d(P,Q).$$

(6)

We claim that $({\xi }_n)$ is bounded. In view of (6), it suffices to prove that $({\xi }_{2n+1})$ is bounded. We argue by contradiction. Then, there exists $N\in {\mathbb{N}}^{*}$ such that:

$$d({\xi }_2,{\xi }_{2N+1})>M\mathrm{and}d({\xi }_2,{\xi }_{2N-1})\le M,$$

(7)

where the real $M>0$ is chosen in order that:

$$M-d(P,Q)>{\phi }^2\left(d({\xi }_0,{\xi }_1)+\phi [d({\xi }_0,{\xi }_1)-d(P,Q)]+M\right).$$

(8)

Using (4), we have:

$$\begin{array}{cc}\hfill d({\xi }_2,{\xi }_{2N+1})-d(P,Q)& =d(f{\xi }_{2N},g{\xi }_1)-d(P,Q)\hfill \\ & \le \phi \left(d({\xi }_{2N},{\xi }_1)-d(P,Q)\right)\hfill \\ & =\phi \left(d(f{\xi }_0,g{\xi }_{2N-1})-d(P,Q)\right)\hfill \\ & \le {\phi }^2\left(d({\xi }_0,{\xi }_{2N-1})-d(P,Q)\right).\hfill \end{array}$$

From (7), we get:

$$\begin{array}{cc}\hfill M-d(P,Q)& <{\phi }^2\left(d({\xi }_0,{\xi }_{2N-1})-d(P,Q)\right)\hfill \\ & \le {\phi }^2\left(d({\xi }_0,{\xi }_1)+d({\xi }_1,{\xi }_2)+d({\xi }_2,{\xi }_{2N-1})-d(P,Q)\right)\hfill \\ & ={\phi }^2\left(d({\xi }_0,{\xi }_1)+d(f{\xi }_0,g{\xi }_1)+d({\xi }_2,{\xi }_{2N-1})-d(P,Q)\right)\hfill \\ & \le {\phi }^2\left(d({\xi }_0,{\xi }_1)+\phi [d({\xi }_0,{\xi }_1)-d(P,Q)]+d({\xi }_2,{\xi }_{2N-1})\right)\hfill \\ & \le {\phi }^2\left(d({\xi }_0,{\xi }_1)+\phi [d({\xi }_0,{\xi }_1)-d(P,Q)]+M\right).\hfill \end{array}$$

We deduce that:

$$M-d(P,Q)<{\phi }^2\left(d({\xi }_0,{\xi }_1)+\phi [d({\xi }_0,{\xi }_1)-d(P,Q)]+M\right),$$

which is a contradiction with respect to (8). Hence, $({\xi }_n)$ is bounded.

We claim that $({\xi }_n)$ is a cyclical Cauchy sequence. Letting $m\ge n,$ we have by (4),

$$\begin{array}{cc}\hfill D({\xi }_{2n},{\xi }_{2m+1})=D(f{\xi }_{2m},g{\xi }_{2n-1})& \le \phi (D({\xi }_{2m},{\xi }_{2n-1}))\hfill \\ & \le {\phi }^2(D({\xi }_{2m-1},{\xi }_{2n-2}))\hfill \\ & ⋮\hfill \\ & \le {\phi }^{2n}(D({\xi }_0,{\xi }_{2(m-n)+1})).\hfill \end{array}$$

Since $({\xi }_n)$ is bounded and $\phi$ is non-decreasing, by passing to the limit in the above inequality, we get ${\displaystyle \underset{m,n\to \infty }{\lim }D({\xi }_{2n},{\xi }_{2m+1})=0,}$ which implies that:

$$\underset{n,m\to \infty }{\lim }d({\xi }_{2n},{\xi }_{2m+1})=d(P,Q).$$

(9)

Then, $({\xi }_n)$ is a cyclical Cauchy sequence. Since $(P,Q)$ is a proximally-complete pair, the sequence $({\xi }_n)$ has a subsequence $({\xi }_{2n_k})$ converging to some element ${\xi }_{\star }\in P$. Again, $({\xi }_n)$ has a convergent subsequence $({\xi }_{2m_k+1})$ to some ${\vartheta }_{\star }\in Q.$

We claim that ${\xi }_{\star }$ is a best proximity of f. We have:

$$\begin{array}{c}\hfill D({\xi }_{\star },{\xi }_{2n_k-1})=d({\xi }_{\star },{\xi }_{2n_k-1})-d(P,Q)\le d({\xi }_{\star },{\xi }_{2n_k})+d({\xi }_{2n_k},{\xi }_{2n_k-1})-d(P,Q).\end{array}$$

Using (6), we obtain:

$$\underset{k\to \infty }{\lim }D({\xi }_{\star },{\xi }_{2n_k-1})=0.$$

(10)

By (4),

$$D({\xi }_{2n_k},f{\xi }_{\star })=D(f{\xi }_{\star },g{\xi }_{2n_k-1})\le \phi (D({\xi }_{\star },{\xi }_{2n_k-1})).$$

Taking $k\to \infty$ and using (10) together with the fact that $\phi$ is continuous at zero, we obtain that:

$$D({\xi }_{\star },f{\xi }_{\star })=\underset{k\to \infty }{\lim }D({\xi }_{2n_k},f{\xi }_{\star })=\phi (0)=0,$$

which implies that $D({\xi }_{\star },f{\xi }_{\star })=0$, and so, $d({\xi }_{\star },f{\xi }_{\star })=d(P,Q).$ Similarly, ${\vartheta }_{\star }$ is a best proximity of $g,$ i.e., $d({\vartheta }_{\star },g{\vartheta }_{\star })=d(P,Q).$ From (9), we have $d({\xi }_{\star },{\vartheta }_{\star })=d(P,Q).$ □

The following illustrates Theorem 5.

Example 4.

Consider $X={\mathbb{R}}^2$ with the metric defined as $d(({\xi }_1,{\vartheta }_1),({\xi }_2,{\vartheta }_2))=|{\xi }_1-{\xi }_2|+|{\vartheta }_1-{\vartheta }_2|.$ Let $P=\left\{1\right\}\times [0,1]$ and $Q=\left\{0\right\}\times [0,1]$. Note that $d(P,Q)=1$ and $(P,Q)$ is a proximally-complete pair. For $x\in [0,1]$, define $f:P\to Q$ and $g:Q\to P$ as follows:

$$f(1,x)=(0,\frac{x^2+1}{4})andg(0,x)=(1,\frac{x^2+1}{4}).$$

Taking $x,y\in [0,1],$ one writes:

$$\begin{array}{cc}\hfill D(f(1,x),g(0,y))=d(f(1,x),g(0,y))-1& =\frac{1}{4}|x^2-y^2|=\frac{1}{4}(x+y)|x-y|\hfill \\ & \le \frac{1}{2}|x-y|=\frac{1}{2}(d((1,x),(0,y))-1)\hfill \\ & =\frac{1}{2}D((1,x),(0,y)).\hfill \end{array}$$

Then, the condition contraction (4) is verified with $\phi (t)=\frac{1}{2}t$. Hence, f has a best proximity in P, and g has a best proximity in $Q.$ Here, $(1,2-\sqrt{3})$ is the unique best proximity of f and $(0,2-\sqrt{3})$ is the unique best proximity of g. Furthermore, $d((1,2-\sqrt{3}),(0,2-\sqrt{3}))=1=d(P,Q)$.

The following results are simple consequences of Theorem 5. We omit their proofs.

Corollary 1.

Let $(P,Q)$ be a proximally-complete pair in a metric space $(X,d)$. Let $f:P\to Q$ and $g:Q\to P$ be non-self maps such that for all $(x,y)\in P\times Q,$

$$d(fx,gy)\le \lambda d(x,y)+(1-\lambda )d(P,Q),$$

where $\lambda \in [0,1).$ Then, there are ${\xi }_{\star }\in P$ and ${\vartheta }_{\star }\in Q$ so that:

$$d({\xi }_{\star },f{\xi }_{\star })=d({\vartheta }_{\star },g{\vartheta }_{\star })=d({\xi }_{\star },{\vartheta }_{\star })=d(P,Q).$$

Corollary 2.

Let $(P,Q)$ be a proximally-complete pair in a metric space $(X,d)$. Let $f:P\cup Q\to P\cup Q$ be a non-self mapping such that $f(P)\subseteq Q$, $f(Q)\subseteq P$ and for all $(x,y)\in P\times Q$,

$$D(fx,fy)\le \phi (D(x,y)),$$

where φ is a c-comparison function and $D(x,y):=d(x,y)-d(P,Q).$ Then, there are ${\xi }_{\star }\in P$ and ${\vartheta }_{\star }\in Q$, so that:

$$d({\xi }_{\star },f{\xi }_{\star })=d({\vartheta }_{\star },f{\vartheta }_{\star })=d({\xi }_{\star },{\vartheta }_{\star })=d(P,Q).$$

Corollary 3.

Let $(P,Q)$ be a proximally-complete pair in a metric space $(X,d)$. Let $f:P\cup Q\to P\cup Q$ be a given non-self map such that $f(P)\subseteq Q,f(Q)\subseteq P$ and for all $(x,y)\in P\times Q,$

$$d(fx,fy)\le \lambda d(x,y)+(1-\lambda )d(P,Q),$$

where $\lambda \in [0,1).$ Then, there are ${\xi }_{\star }\in P$ and ${\vartheta }_{\star }\in Q$ so that:

$$d({\xi }_{\star },f{\xi }_{\star })=d({\vartheta }_{\star },f{\vartheta }_{\star })=d({\xi }_{\star },{\vartheta }_{\star })=d(P,Q).$$

Our second main result is:

Theorem 6.

Let $(P,Q)$ be a proximally-complete semi-sharp proximinal pair in a metric space $(X,d)$. Let $f:P\to Q$ and $g:Q\to P$ be non-self maps such that for all $(x,y)\in P\times Q,$

$$D(fx,gy)\le \phi (D(x,y)),$$

(11)

where φ is a c-comparison function and $D(x,y):=d(x,y)-d(P,Q).$ Then, the following hold:

(i) 

There is ${\xi }_{\star }\in P$ such that $d({\xi }_{\star },f{\xi }_{\star })=d(P,Q);$

(ii) 

${\xi }_{\star }$ is a fixed point of $gf$, i.e., $gf{\xi }_{\star }={\xi }_{\star }$, and $f{\xi }_{\star }$ is a fixed point of $fg$, i.e., $fg(f{\xi }_{\star })=f{\xi }_{\star }$;

(iii) 

For any ${\xi }_0\in P$, let ${\xi }_{2n+1}=f{\xi }_{2n}$ and ${\xi }_{2n}=g{\xi }_{2n-1}.$ Then, the sequence $({\xi }_{2n})$ converges to ${\xi }_{\star }$, and the sequence $({\xi }_{2n+1})$ converges to $f{\xi }_{\star }$.

Proof. 

Let ${\xi }_0\in P.$ Define the sequence $({\xi }_n)$ by ${\xi }_{2n+1}=f{\xi }_{2n}$ and ${\xi }_{2n}=g{\xi }_{2n-1}.$ By Theorem 5, there exists $({\xi }_{\star },{\vartheta }_{\star })\in P\times Q$ so that:

$$d({\xi }_{\star },f{\xi }_{\star })=d({\vartheta }_{\star },g{\vartheta }_{\star })=d({\xi }_{\star },{\vartheta }_{\star })=d(P,Q).$$

From (11),

$$D(f{\xi }_{\star },gf{\xi }_{\star })\le \phi (D({\xi }_{\star },f{\xi }_{\star }))=\phi (0)=0.$$

Then, $D(f{\xi }_{\star },gf{\xi }_{\star })=0$, and so, $d(f{\xi }_{\star },gf{\xi }_{\star })=d(f{\xi }_{\star },{\xi }_{\star })=d({\xi }_{\star },{\vartheta }_{\star })=d(P,Q).$ Since $(P,Q)$ is semi-sharp proximinal, then ${\vartheta }_{\star }=f{\xi }_{\star }$ and $gf{\xi }_{\star }={\xi }_{\star }.$ It follows that $fg(f{\xi }_{\star })=f(gf{\xi }_{\star })=f{\xi }_{\star }.$ By Theorem 5, the sequence $({\xi }_n)$ is cyclical Cauchy in $P\cup Q$. Furthermore, the sequence $({\xi }_{2n})$ has a convergent subsequence $({\xi }_{2n_k})$ to ${\xi }_{\star }$, and the sequence $({\xi }_{2n+1})$ has a convergent subsequence $({\xi }_{2n_k+1})$ to ${\vartheta }_{\star }=f{\xi }_{\star }$. Following Theorem 3.3 of [9] and since $(P,Q)$ is a semi-sharp proximinal pair, the sequence $({\xi }_{2n})$ is Cauchy. Furthermore, $({\xi }_{2n})$ has a convergent subsequence to ${\xi }_{\star }$. Then, $({\xi }_{2n})$ converges to ${\xi }_{\star }$. Similarly, we show that $({\xi }_{2n+1})$ converges to $f{\xi }_{\star }$.  □

The following examples support Theorem 6.

Example 5.

Consider $X={\mathbb{R}}^2$ with the metric defined as $d(({\xi }_1,{\vartheta }_1),({\xi }_2,{\vartheta }_2))=|{\xi }_1-{\xi }_2|+|{\vartheta }_1-{\vartheta }_2|.$ Let $P=\{(1,1),(1,2)\}$ and $Q=\{(2,1),(2,2)\}$. We have $d(P,Q)=1$, and $(P,Q)$ is a proximally-complete semi-sharp pair. Define $f:P\to Q$ and $g:Q\to P$ as follows:

$$f(1,1)=f(1,2)=(2,2)andg(2,2)=g(2,1)=(1,2).$$

The condition (11) is verified for each c-comparison function φ. Here, $(1,2)$ is the unique best proximity of $f.$ Furthermore, $(1,2)$ is the unique fixed point of $gf$, and $f(1,2)=(2,2)$ is the unique fixed point of $fg.$ Again, if ${\theta }_0=(1,1)$ with ${\theta }_{2n+1}=f{\theta }_{2n}$ and ${\theta }_{2n}=g{\theta }_{2n-1},$ then ${\theta }_{2n}=(1,2)$ for all $n\ge 1$ and ${\theta }_{2n+1}=(2,2)$ for all $n\ge 0.$

Example 6.

Consider the metric space $(X,d)$ given by Example 5. Consider the subsets $P=\{(s,0),s\in [0,1\left]\right\}$ and $Q=\{(t,1),t\in [0,1\left]\right\}$. Here, $d(P,Q)=1$. For all $x=(s,0)\in P$ and $y=(t,1)\in Q$, there exist a unique $u=(s,1)\in Q$ and a unique $v=(t,0)\in P$ such that $d(x,u)=d(y,v)=1=d(P,Q)$, so $(P,Q)$ is a proximally-complete semi-sharp pair. For $s,t\in [0,1]$, define $f:P\to Q$ by $f(s,0)=(\frac{s+1}{2},1)$ and $g:Q\to P$ $g(t,1)=(\frac{t+1}{2},0)$. Let $x=(s,0)\in P$ and $y=(t,1)\in Q$, then:

$$\begin{array}{cc}\hfill D(fx,gy)& =D(((\frac{s+1}{2},1),(\frac{t+1}{2},0))\hfill \\ & =|\frac{s-t}{2}|+1-d(P,Q)\hfill \\ & =|\frac{s-t}{2}|\hfill \\ & =\phi (D(x,y)|,\hfill \end{array}$$

where $\phi (t)=\frac{t}{2}$. There exists a unique point $x^{\star }=(1,0)\in P$ such that $d(x^{\star },f{x}^{\star })=1=d(P,Q)$ and $gf{x}^{\star }=g(1,1)=x^{\star }$. Here, $(1,2)$ is the unique fixed point of $gf$ and $f(1,2)=(2,2)$ is the unique fixed point of $fg.$ For any ${\theta }_0=(s,0)\in P$, let ${\theta }_{2n+1}=f{\theta }_{2n}$ and ${\theta }_{2n}=g{\theta }_{2n-1}.$ Then, the sequence $({\theta }_{2n})$ converges to $x^{\star }$, and the sequence $({\theta }_{2n+1})$ converges to $f{x}^{\star }$.

The following corollaries are consequences of Theorem 6.

Corollary 4.

Let $(P,Q)$ be a proximally-complete semi-sharp proximinal pair in a metric space $(X,d)$. Let $f:P\to Q$ and $g:Q\to P$ be non-self mappings such that for all $(x,y)\in P\times Q,$

$$d(fx,gy)\le \lambda d(x,y)+(1-\lambda )d(P,Q),$$

(12)

where $\lambda \in [0,1).$ Then, the following hold:

(i) 

There exists a point ${\xi }_{\star }\in P$ such that $d({\xi }_{\star },f{\xi }_{\star })=d(P,Q);$

(ii) 

${\xi }_{\star }$ is a fixed point of $gf$, i.e., $gf{\xi }_{\star }={\xi }_{\star }$, and $f{\xi }_{\star }$ is a fixed point of $fg$, i.e., $fg(f{\xi }_{\star })=f{\xi }_{\star }$;

(iii) 

For any ${\xi }_0\in P$, let ${\xi }_{2n+1}=f{\xi }_{2n}$ and ${\xi }_{2n}=g{\xi }_{2n-1}.$ Then, the sequence $({\xi }_{2n})$ converges to ${\xi }_{\star }$, and the sequence $({\xi }_{2n+1})$ converges to $f{\xi }_{\star }$.

Corollary 5.

Let $(P,Q)$ be a proximally-complete semi-sharp proximinal pair in a metric space $(X,d)$. Let $f:P\cup Q\to P\cup Q$ be a non-self mapping such that $f(P)\subseteq Q,f(Q)\subseteq P$ and for all $(x,y)\in P\times Q,$

$$D(fx,fy)\le \phi (D(x,y)),$$

where φ is a c-comparison function and $D(x,y):=d(x,y)-d(P,Q).$ Then, the following hold:

(i) 

There exists a point ${\xi }_{\star }\in P$ such that $d({\xi }_{\star },f{\xi }_{\star })=d(P,Q);$

(ii) 

${\xi }_{\star }$ is a fixed point of $f^2$ in P, and $f{\xi }_{\star }$ is a fixed point of $f^2$ in Q;

(iii) 

For any ${\xi }_0\in P$, let ${\xi }_{n+1}=f{\xi }_n.$ Then, the sequence $({\xi }_{2n})$ converges to ${\xi }_{\star }$, and the sequence $({\xi }_{2n+1})$ converges to $f{\xi }_{\star }$.

Corollary 6.

Let $(P,Q)$ be a proximally-complete semi-sharp proximinal pair in a metric space $(X,d)$. Let $f:P\cup Q\to P\cup Q$ be a non-self mapping such that $f(P)\subseteq Q,f(Q)\subseteq P$ and for all $(x,y)\in P\times Q,$

$$d(fx,fy)\le \lambda d(x,y)+(1-\lambda )d(P,Q),$$

where $\lambda \in [0,1).$ Then, the following hold:

(i) 

There exists a point ${\xi }_{\star }\in P$ such that $d({\xi }_{\star },f{\xi }_{\star })=d(P,Q);$

(ii) 

${\xi }_{\star }$ is a fixed point of $f^2$ in P and $f{\xi }_{\star }$ is a fixed point of $f^2$ in Q;

(iii) 

For any ${\xi }_0\in P$, let ${\xi }_{n+1}=f{\xi }_n.$ Then, the sequence $({\xi }_{2n})$ converges to ${\xi }_{\star }$, and the sequence $({\xi }_{2n+1})$ converges to $f{\xi }_{\star }$.

In the following, we give a result from Corollary 1 for nonexpansive mappings in normed vector spaces.

Theorem 7.

Let X be a normed vector space and $P,Q$ be two nonempty subsets of X. Given $f:P\to Q$ and $g:Q\to P$ are non-self mappings such that for all $(x,y)\in P_0\times Q_0$,

$$\left|\right|fx-gy\left|\right|\le \left|\right|x-y\left|\right|,$$

(13)

where $P_0$ and $Q_0$ are defined by (1) and (2), respectively. Suppose that:

(i) 

$\varnothing \ne P_0$ is convex and boundedly compact;

(ii) 

$Q_0$ is compact;

(iii) 

The functions $z\to \left|\right|z-fz\left|\right|$ and $z\to \left|\right|z-gz\left|\right|$ are lower semi-continuous in $P_0$ and $Q_0$, respectively.

Then, there exists $({\xi }_{\star },{\vartheta }_{\star })\in P_0\times Q_0$ such that:

$$\left|\right|{\xi }_{\star }-f{\xi }_{\star }\left|\right|=\left|\right|{\vartheta }_{\star }-g{\vartheta }_{\star }\left|\right|=d(P,Q).$$

Proof. 

Since $P_0\ne \varnothing ,$ there exists $({\xi }_0,{\vartheta }_0)\in P_0\times Q_0$ such that $\left|\right|{\xi }_0-{\vartheta }_0\left|\right|=d(P,Q).$ We claim that $f:P_0\to Q_0$ and $g:Q_0\to P_0.$ Let $x\in P_0,$ so there exists $y\in Q_0,$ such that $\left|\right|x-y\left|\right|=d(P,Q).$ From (13),

$$d(P,Q)\le \left|\right|fx-gy\left|\right|\le \left|\right|x-y\left|\right|=d(P,Q),$$

which implies that $\left|\right|fx-gy\left|\right|=d(P,Q)$, and so, $f(P_0)\subset Q_0.$ Similarly, we show that $g(Q_0)\subset P_0.$

For $n\ge 1,$ consider:

$$\left\{\begin{array}{c}{f}_{n}x=\frac{1}{n}{\vartheta }_0+(1-\frac{1}{n})fx\mathrm{f}\mathrm{o}\mathrm{r}x\in P_0,\hfill \\ g_{n}y=\frac{1}{n}{\xi }_0+(1-\frac{1}{n})gy\mathrm{f}\mathrm{o}\mathrm{r}y\in Q_0.\hfill \end{array}\right.$$

Since $P_0$ is convex, we have that $g_n:Q_0\to P_0.$ Again, for $x\in P_0,$, there exists $y\in Q_0$ such that $\left|\right|x-y\left|\right|=d(P,Q).$ From (13),

$$\begin{array}{cc}\hfill d(P,Q)\le \left|\right|f_{n}x-g_{n}y\left|\right|& \le \frac{1}{n}\left|\right|{\xi }_0-{\vartheta }_0\left|\right|+(1-\frac{1}{n})\left|\right|fx-gy\left|\right|\hfill \\ & \le \frac{1}{n}\left|\right|{\xi }_0-{\vartheta }_0\left|\right|+(1-\frac{1}{n})\left|\right|x-y\left|\right|=d(P,Q),\hfill \end{array}$$

which implies that $\left|\right|f_{n}x-g_{n}y\left|\right|=d(P,Q)$, and so, $f_{n}x\in Q_0$, that is $f_n:P_0\to Q_0.$

Let $(x,y)\in P_0\times Q_0$. Then:

$$\begin{array}{cc}\hfill \left|\right|f_{n}x-g_{n}y\left|\right|& \le \frac{1}{n}\left|\right|{\xi }_0-{\vartheta }_0\left|\right|+(1-\frac{1}{n})\left|\right|fx-gy\left|\right|\hfill \\ & \le (1-\frac{1}{n})\left|\right|x-y\left|\right|+\frac{1}{n}d(P,Q).\hfill \end{array}$$

Since $(P_0,Q_0)$ is proximally complete, by Corollary 1, there exists $({\xi }_n,{\vartheta }_n)\in P_0\times Q_0$ such that:

$$\left|\right|{\xi }_n-f_n{\xi }_n\left|\right|=\left|\right|{\vartheta }_n-g_n{\vartheta }_n\left|\right|=\left|\right|{\xi }_n-{\vartheta }_n\left|\right|=d(P,Q)\mathrm{f}\mathrm{o}\mathrm{r}n\in {\mathbb{N}}^{*}.$$

We have:

$$\left|\right|{\xi }_n-f{\xi }_n\left|\right|\le \left|\right|{\xi }_n-f_n{\xi }_n\left|\right|+\frac{1}{n}\left|\right|{\vartheta }_0-f{\xi }_n\left|\right|.$$

Since $f{\xi }_n\in Q_0$ and $Q_0$ is compact, we get:

$$\underset{n\to \in \infty }{\lim }\left|\right|{\xi }_n-f{\xi }_n\left|\right|=d(P,Q).$$

(14)

Again,

$$\left|\right|{\xi }_n\left|\right|\le \left|\right|{\xi }_n-f_n{\xi }_n\left|\right|+\frac{1}{n}\left|\right|{\vartheta }_0\left|\right|+(1-\frac{1}{n})\left|\right|f{\xi }_n\left|\right|=d(P,Q)+\left|\right|{\vartheta }_0\left|\right|+\left|\right|f{\xi }_n\left|\right|,$$

which implies that $({\xi }_n)$ is bounded. Since $P_0$ is boundedly compact, there exist ${\xi }_{\star }\in P_0$ and $({\xi }_{n_k})$ a subsequence of $({\xi }_n)$ such that ${\displaystyle \underset{k\to \infty }{\lim }{\xi }_{n_k}={\xi }_{\star }.}$ From (14) and Assumption $(iii)$, we have:

$${\displaystyle \left|\right|{\xi }_{\star }-f{\xi }_{\star }\left|\right|\le \underset{k\to \infty }{lim\; inf}\left|\right|{\xi }_{n_k}-f{\xi }_{n_k}\left|\right|=\underset{k\to \infty }{\lim }\left|\right|{\xi }_{n_k}-f{\xi }_{n_k}\left|\right|=d(P,Q),}$$

which implies that $\left|\right|{\xi }_{\star }-f{\xi }_{\star }\left|\right|=d(P,Q).$

On the other hand, we have:

$$\begin{array}{cc}\hfill \left|\right|{\vartheta }_n-g{\vartheta }_n\left|\right|& \le \left|\right|{\vartheta }_n-g_n{\vartheta }_n\left|\right|+\frac{1}{n}(||{\xi }_0\left|\right|+\left|\right|g{\vartheta }_n\left|\right|)\hfill \\ & \le d(P,Q)+\frac{1}{n}(||{\xi }_0\left|\right|+\left|\right|f{\xi }_n-g{\vartheta }_n\left|\right|+\left|\right|f{\xi }_n\left|\right|)\hfill \\ & \le d(P,Q)+\frac{1}{n}(||{\xi }_0\left|\right|+\left|\right|{\xi }_n-{\vartheta }_n\left|\right|+\left|\right|f{\xi }_n\left|\right|)\hfill \\ & =d(P,Q)+\frac{1}{n}(||{\xi }_0\left|\right|+d(P,Q)+\left|\right|f{\xi }_n\left|\right|).\hfill \end{array}$$

This implies that:

$${\displaystyle \underset{n\to \infty }{\lim }\left|\right|{\vartheta }_n-g{\vartheta }_n\left|\right|=d(P,Q).}$$

(15)

Notice that $({\vartheta }_n)$ is bounded because $\left|\right|{\xi }_n-{\vartheta }_n\left|\right|=d(P,Q)$ and $({\xi }_n)$ is bounded. Since $Q_0$ is compact, there exist ${\vartheta }_{\star }\in Q_0$ and a subsequence $({\vartheta }_{n_k})$ of $({\vartheta }_n)$ such that ${\displaystyle \underset{k\to \infty }{\lim }{\vartheta }_{n_k}={\vartheta }_{\star }.}$ By assumption $(iii)$, we have $\left|\right|{\vartheta }_{\star }-g{\vartheta }_{\star }\left|\right|=d(P,Q).$ □

As particular cases from Theorem 7, we have:

Corollary 7.

Let X be a normed vector space and $P,Q$ be two nonempty subsets of X. Let $f:P\cup Q\to P\cup Q$ be a non-self map such that $f(P)\subseteq Q,f(Q)\subseteq P$ and for all $(x,y)\in P_0\times Q_0,$

$$\left|\right|fx-fy\left|\right|\le \left|\right|x-y\left|\right|.$$

Suppose that:

(i) 

$\varnothing \ne P_0$ is convex and boundedly compact;

(ii) 

$Q_0$ is compact;

(iii) 

The function $z\to \left|\right|z-fz\left|\right|$ is lower semi-continuous in $P_0$.

Then, there exists ${\xi }_{\star }\in P_0$ such that:

$$\left|\right|{\xi }_{\star }-f{\xi }_{\star }\left|\right|=d(P,Q).$$

Corollary 8.

Let X be a normed vector space and $P,Q$ be two nonempty subsets of X. Let $f:P\cup Q\to P\cup Q$ be a non-self map such that $f(P)\subseteq Q,f(Q)\subseteq P$ and for all $(x,y)\in P_0\times Q_0,$

$$\left|\right|fx-fy\left|\right|\le \left|\right|x-y\left|\right|.$$

Suppose that:

(i) 

$\varnothing \ne P_0$ is convex and boundedly compact;

(ii) 

$Q_0$ is compact;

(iii) 

The function $z\to \left|\right|z-fz\left|\right|$ is lower semi-continuous in $Q_0$.

Then, there is ${\vartheta }_{\star }\in Q_0$ such that:

$$\left|\right|{\vartheta }_{\star }-f{\vartheta }_{\star }\left|\right|=d(P,Q).$$

Remark 1.

Corollaries 2, 3, 5, 6, 7, and 8 remain true by replacing $f:P\cup Q\to P\cup B$ with $f:A\times Q\to P\times B$ (keeping other hypotheses).

3. Conclusions

In this paper, we considered proximally-complete pairs and proximally-complete semi-sharp proximinal pairs as weaker hypotheses with respect to [3] to get convergence and best proximity points. We applied Theorem 5 to provide a result for nonexpansive mappings in normed vector spaces.