A Characterization of Weak Proximal Normal Structure and Best Proximity Pairs

Let $A,B$ be two non-empty subsets of a Banach space and $T$ be a cyclic mapping on $A\cup B$ ($T(A)\subseteq B,T(B)\subseteq A$). A pair $(x,y)\in A\times B$ is said to be a best proximity pair for $T$ if $\|x-Tx\|=\|y-Ty\|=d(A,B)=\inf\{\|x-y\|:x\in A,y\in B\}.$ The geometry of Banach spaces plays a crucial role for the existence of best proximity pairs. The analysis of proximal normal structure and weak or semi-normal structure, the property UC, the projectional property due to Eldred et al.[2], Moosa [4], Suzuki et al. [9], G. S. Raju et al. [7] etc., respectively are widely used to prove the existence of a best proximity pair for cyclic maps. We denote $\sup\{\|x-y\|:y\in B\}$ for $x\in A$ by $\delta(x,B).$ We shall say that the pair $(A,B)$ is proximinal pair if for every $x$ in $A$ (resp. in $B$), there exists $y$ in $B$ (resp. in $A$) such that $\|x-y\|=d(A,B).$ Further, if such a $y$ is unique, then $(A,B)$ is said to be a sharp proximinal pair [7]. In this case we denote $y$ by $x^{\prime}.$ Also, $(A,B)$ is said to be a proximinal parallel pair if $(A,B)$ is sharp proximinal and $B=A+h$ for some $h\in X$ [3]. It is shown in [3] that if $X$ is strictly convex and $A,B$ are weakly compact convex subsets of $X,$ then every $(A_{0},B_{0})$ is a non-empty proximinal parallel pair. Here $A_{0}=\{x\in A:\mbox{there exists}~{}y\in B~{}\mbox{such that}~{}\|x-y\|=d(A,B)\}$ and $B_{0}=\{x\in B:\mbox{there exists}~{}y\in A~{}\mbox{such that}~{}\|x-y\|=d(A,B)\}.$ Also, in [7], the authors have given example(s) of sharp proximinal pair which are not parallel. In [2], the author introduced a geometrical notion called proximal normal structure to prove the existence of a best proximity pair of a relatively nonexpansive mapping ($\|Tx-Ty\|\leq\|x-y\|$ for all $x\in A,y\in B.$) We say $(A,B)$ has proximal normal structure ([2]) [respectively weak proximal normal structure ([5])] if $(A,B)$ is convex and for any closed bounded [respectively weakly compact] convex proximinal pair $(H_{1},H_{2})$ of subsets of $(A,B)$ for which $d(H_{1},H_{2})=d(A,B)$ and $\delta(H_{1},H_{2})>d(H_{1},H_{2}),$ there exists $(x,y)\in(H_{1},H_{2})$ such that $\delta(x,H_{2})<\delta(H_{1},H_{2})$ and $\delta(y,H_{1})<\delta(H_{1},H_{2}).$ It is well known that every non-empty closed bounded convex pair $(A,B)$ of a uniformly convex Banach space has proximal normal structure. In fact, every non-empty compact convex pair $(A,B)$ of a Banach space has proximal normal structure. It is proved (Proposition 3.2 in [5]) that a bounded convex pair has proximal normal structure if and only if it doesn’t contain any proximal diametral sequence. A pair $\displaystyle\left(\{x_{n}\},\{y_{n}\}\right)$ of sequences in $(A,B)$ with $\|x_{n}-y_{n}\|=d(A,B),n\geq 1$ is said to be a proximal diametral sequence ([5]) if $d(A,B)<\delta(\{x_{n}\},\{y_{n}\})$ and $\displaystyle\max\{\lim_{n\to\infty}d\left(x_{n+1},\mbox{co}\left(\{y_{1},y_{2},...,y_{n}\}\right)\right),\lim_{n\to\infty}d\left(y_{n+1},\mbox{co}\left(\{x_{1},x_{2},...,x_{n}\}\right)\right)\}=\delta(\{x_{n}\},\{y_{n}\}).$ It is easy to see that proximal normal structure coincides with weak proximal normal structure in reflexive Banach spaces [5]. Moreover, therein the author proved the existence of a best proximity pair in the settings of a reflexive Banach space. Recently, in [6], the authors posed an open problem for the existence of a best proximity pair for a more general class of mappings, called relatively orbital nonexpansive mappings. Also therein the authors indicated that an affirmative answer may provide a characterization of proximal normal structure. Motivated by this, we aim to give a partial affirmative answer for the same. We also provide a characterization of weak proximal normal structure by using the existence of a best proximity pair for relatively orbital nonexpansive mappings. Finally, we introduce the notion of pointwise cyclic contraction wrt orbits and prove the minimal invariant subsets of such a map have nondiametral points. This guarantees the existence of a best proximity pair for such a class in the setting of a reflexive Banach space. Finally, we prove the existence of a best proximity pair for the class of pointwise cyclic contraction wrt orbits.

Let $A,B$ be two closed convex subsets of a Banach space $X.$ Let $T:A\cup B\to A\cup B$ be a cyclic map. If $T$ admits a best proximity pair, then $A_{0}\neq\emptyset\neq B_{0}.$ Also, if $T$ is relatively nonexpansive, then $A_{0}\cup B_{0}$ is cyclically invariant under $T$ ($TA_{0}\subseteq B_{0},TB_{0}\subseteq A_{0}$). The following theorem is due to Eldred $\it{et~{}al.}$ [2]

The main tool to prove the same is to use the geometrical notion called “proximal normal structure” on $A_{0}\cup B_{0}$. Later many authors established the existence of a best proximity pair for relatively nonexpansive mappings in different settings using variants of geometry ([3],[4],[8],[9]). In [5], Moosa introduced pointwise relatively nonexpansive mappings involving orbits and therein proved the existence of a best proximity pair for such a class of mappings. Recently, in 2018, Kirk and Shahzad discussed the existence of a best proximity pair for relatively nonexpansive mappings and therein they raised the question “can the assumption that $T$ is relatively nonexpansive in Theorem 2.1 be replaced by the assumption that $T$ is relatively nonexpansive wrt orbits?” $T$ is said to be relatively nonexpansively mappings wrt orbits if $\|Tx-Ty\|\leq r_{x}\left(\mathcal{O}(y)\right)=\delta(x,\{y,Ty,T^{2}y,...\}).$ Using the following example, we can conclude that the answer is negative for the above open problem.

It is to be observed that a cyclic map $T$ on $A\cup B$ that satisfies $\|Tx-Ty\|\leq r_{x}\left(\mathcal{O}(y)\right)$ does not guarantee $A_{0}\cup B_{0}$ is cyclically invariant under $T.$ Hence, it is not reasonable to expect the existence of a best proximity pair for such a map $T.$ To overcome this, we redefine the relatively orbital nonexpansive mappings. For $x\in A\cup B,$ we denote $\{T^{2n}x:n\in\mathbb{N}\cup\{0\}\}$ by $\mathcal{O}^{2}(x)$.

It is worth mentioning that relatively orbital nonexpanive mapping is not necessarily relatively nonexpansive.

Let $(A,B)$ be a non-empty sharp proximinal pair in Banach space and $T$ be a relatively orbital nonexpansive mapping on $A\cup B.$ Then it is easy to see that $(A_{0},B_{0})$ is cyclically invariant under $T$ and $Tx^{\prime}=(Tx)^{\prime}.$ We say that $(A,B)$ is said to satisfy the weak best proximity pair property (WBPP) if every relatively orbital nonexpansive mapping on $A\cup B$ has a best proximity pair. The following theorem ensures that every non-empty weakly compact convex pair of subsets of a strictly convex Banach space satisfying the WBPP. The following theorem is in a way different than Theorem 2.6 of [5]. For the sake the completeness, we prove the same here.

Let $T$ be a cyclic map on $A\cup B$. We say that the pair $(A,B)$ has a proximinal nondiametral pair if there exists $(x,y)\in A\times B$ such that $\max\{\delta(x,B),\delta(y,A)\}<\delta(A,B)$ whenever $d(A,B)<\delta(A,B).$ A similar technique can be used to obtain the following:

Let $(A,B)$ be a bounded convex proximinal pair of a Banach space $X.$ A non-constant pair of sequences $\left(\{x_{n}\},\{y_{n}\}\right)$ of $(A,B)$ is said to be a proximinal diametral sequence if $\|x_{n}-y_{n}\|=d(A,B)$ for every $n\in\mathbb{N}$ and $\delta(\{x_{n}\},\{y_{n}\})=\displaystyle\lim_{n\to\infty}d\left(x_{n+1},\mbox{co}\left(\{y_{1},y_{2},...,y_{n}\}\right)\right)=\lim_{n\to\infty}d\left(y_{n+1},\mbox{co}\left(\{x_{1},x_{2},...,x_{n}\}\right)\right).$ It is to be observed that if $d(A,B)=0,$ then the proximinal diametral sequence turns out to be a diametral sequence in $A\cap B$ in the sense of Brodskii and Milman ([1]). Using a similar argument employed in the proof of Theorem 2.5 ([2]) one can obtain the following:

Let $(A,B)$ be a non-empty weakly compact convex sharp proximinal pair of subsets of a Banach space having WBPP. Suppose $(A,B)$ does not have proximal weak normal structure. Then by Theorem 3.1, $(A,B)$ has a proximinal diametral sequence, say, $\left(\{x_{n}\},\{y_{n}\}\right).$ Consequently, $\displaystyle\lim_{n\to\infty}d\left(x_{n+1},\mbox{co}\left(\{y_{1},y_{2},...,y_{n}\}\right)\right)=\delta(\{x_{n}\},\{y_{n}\})=\lim_{n\to\infty}d\left(y_{n+1},\mbox{co}\left(\{x_{1},x_{2},...,x_{n}\}\right)\right).$

Since, $(A,B)$ is weakly compact, there exists a subsequence $\left(\{x_{n_{k}}\},\{y_{n_{k}}\}\right)$ of $\left(\{x_{n}\},\{y_{n}\}\right)$ which is weakly convergent. It is easy to see that the sequence $\left(\{x_{n_{k}}\},\{y_{n_{k}}\}\right)$ is a proximinal diametral subsequence. Hence, without loss of any generality, we may assume that the sequence $\left(\{x_{n}\},\{y_{n}\}\right)$ is proximinal diametral and weakly convergent. Now, $H=\overline{\mbox{co}}\left(\{x_{1},x_{2},...\}\right),K=\overline{\mbox{co}}\left(\{y_{1},y_{2},...\}\right)$ are weakly compact convex subsets of $A,B$ respectively. Define $T:H\cup K\to H\cup K$ by

Clearly, $\delta(H,K)=\delta(\{x_{n}\},\{y_{n}\})$ and $\displaystyle\lim_{n\to\infty}\|x_{n}-z\|=\delta(H,K)=\lim_{n\to\infty}\|y_{n}-v\|$ for any $z\in K,v\in H.$ Hence, $r_{x}\left(\mathcal{O}^{2}(y)\right)=\delta(H,K)$ for each $x\in H,y\in K.$ Now,

Also, if $(x,y)\in H\times K$ with $\|x-y\|=d(H,K),$ then $\|Tx-Ty\|=d(H,K).$ Therefore $T$ is a relatively orbital nonexpansive mapping. As $(A,B)$ is a sharp proximinal pair, then so is $(H,K)$ and $T$ does not have any best proximity pair. Thus we have the following:

By Theorem 2.5 and Proposition 3.2 we have the following characterization:

Let $(A,B)$ be a pair of subsets of a normed linear space. A cyclic map $T$ on $A\cup B$ is said to be a proximal pointwise contraction if for any $x\in A,$ there exists $\alpha(x)\in[0,1)$ such that $\|Tx-Ty\|\leq\alpha(x)\|x-y\|$ ([10]). Later many authors obtained the existence of a best proximity pair for certain types of pointwise cyclic contractions ([8], [11], [12]). Now we introduce the notion of pointwise cyclic contraction wrt orbits and prove the existence of a best proximity pair for such a map. Our result is a generalization of the main results given in the aforementioned articles.

It is easy to see that every pointwise cyclic contraction mapping wrt orbits is relatively orbital nonexpansive.