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Nonconvex proximal pair and relatively nonexpansive maps with respect to orbits
Journal of Inequalities and Applications volume 2021, Article number: 124 (2021)
Abstract
Every nonconvex pair \((C, D)\) may not have proximal normal structure even in a Hilbert space. In this article, we use cyclic relatively nonexpansive maps with respect to orbits to show the presence of best proximity points in \(C\cup D\), where \(C\cup D\) is a cyclic Tregular set and \((C, D)\) is a nonempty, nonconvex proximal pair in a real Hilbert space. Moreover, we show the presence of best proximity points and fixed points for noncyclic relatively nonexpansive maps with respect to orbits defined on \(C\cup D\), where C and D are Tregular sets in a uniformly convex Banach space satisfying \(T(C)\subseteq C\), \(T(D)\subseteq D\) wherein the convergence of Kranoselskii’s iteration process is also discussed.
Introduction and preliminaries
Let \((X,\\cdot\)\) be a normed linear space and let C and D be nonempty subsets of X. A map \(T:C\cup D\to X\) with \(T(C)\subseteq D\), \(T(D)\subseteq C\) (or \(T(C)\subseteq C\), \(T(D)\subseteq D\)) and \(\TuTv\\leq \uv\\), for \(u\in C\), \(v\in D\) is known as a relatively nonexpansive map (see [1]). A relatively nonexpansive map may not be continuous (see for an example [2]). If \(C\cap D\neq \phi \), then \(T:C\cap D\to C\cap D\) is a nonexpansive map.
If \(Tw\neq w\), then it is endeavoured to get a point \(w_{0}\in C\) and \(d(w_{0}, Tw_{0})=\operatorname{dist}(C, D)\), where \(\operatorname{dist}(C, D):=\inf \{d(w, z):w\in C, z\in D\}\). A point \(w_{0}\in C\cup D\) is known as a best proximity point for T when \(d(w_{0}, Tw_{0})=\operatorname{dist}(C, D)\) holds true.
Eldred et al. [1] introduced proximal normal structure for a nonempty, convex pair \((C, D)\) of X and proved two interesting theorems (See Theorem 2.1 and Theorem 2.2 of [1]). Every nonempty, convex pair of subsets \((C, D)\) in a uniformly convex Banach space has proximal normal structure (see [1, 3]). Every nonempty, nonconvex pair of subsets \((C, D)\), even in a Hilbert space, a proximal normal structure may or may not exist (see [4]).
The notion of cyclic Tregular set was introduced by Rajesh et al. [4], which was an extension of Tregular set introduced by Veeramani [5]. The notion of cyclic Tregular set and Tregular set for relatively nonexpansive maps affirms the presence of best proximity points and fixed points on a nonempty nonconvex pair (see [4–6]). For any pair of subsets \((C, D)\) of X, let
Definition 1
A nonempty pair \((K_{1}, K_{2})\) of subsets in a normed linear space, X is known as a proximal pair [1] if for every \(w_{1}\in K_{1}\), \(z_{1}\in K_{2}\), there exist \(w_{2}\in K_{1}\), \(z_{2}\in K_{2}\) so that \(\w_{1}z_{2}\=\operatorname{dist}(K_{1},K_{2})=\w_{2}z_{1}\\) and proximal parallel pair [7] if

(i)
for any \((w_{1}, z_{1})\in K_{1}\times K_{2}\), there is unique \((w_{2}, z_{2})\in K_{1}\times K_{2}\) so that \(\w_{1}z_{2}\=\operatorname{dist}(K_{1},K_{2})=\w_{2}z_{1}\\) and

(ii)
\(K_{2}=K_{1}+h\), where \(h\in X\).
The proximal parallel pair \((K_{1}, K_{2})\) is said to have the rectangle property [8] if and only if \(\k_{1}+hk'_{1}\=\k'_{1}+hk_{1}\\), for \(k_{1}, k'_{1}\in K_{1}\), where \(K_{2}=K_{1}+h\), \(h\in X\).
Proposition 1
Let X be a strictly convex Banach space and \((K_{1}, K_{2})\) be a nonempty, nonconvex weakly compact proximal pair with \(\operatorname{dist}(K_{1}, K_{2})=\operatorname{dist}(\overline{\operatorname{conv}}(K_{1}), \overline{\operatorname{conv}}(K_{2}))\). Then the pairs \((K_{1}, K_{2})\) and \((\overline{\operatorname{conv}}(K_{1}), \overline{\operatorname{conv}}(K_{2}))\) are proximal parallel pair in X.
Moreover, if \((K_{1}, K_{2})\) is convex and X is a real Hilbert space, then, for \(x,y\in K_{1}\), \(\langle xy,h\rangle =0\), where \(h\in X\) and \(K_{2}=K_{1}+h\).
The notion of cyclic Tregular set was introduced by Rajesh et al. [4].
Definition 2
([4])
Let \((K_{1},K_{2})\) be a nonempty, nonconvex proximal pair in a normed linear space X. Let \(T:K_{1}\cup K_{2} \to X\) be a map with \(T(K_{1})\subseteq K_{2}\) and \(T(K_{2})\subseteq K_{1}\). The set \(K_{1}\cup K_{1}\) is known as a cyclic Tregular set if

(i)
\(\frac{u+Tu'}{2}\in K_{1}\), for \(u\in K_{1}\), \(u'\in K_{2}\) so that \(\uu'\=\operatorname{dist}(K_{1},K_{2})\) and

(ii)
\(\frac{v+Tv'}{2}\in K_{1}\), for \(v\in K_{2}\), \(v'\in K_{1}\) so that \(\vv'\=\operatorname{dist}(K_{1},K_{2})\).
In the above definition, if \(K_{1}=K_{2}\), then it reduces to being Tregular as defined by Veeramani [5].
Definition 3
([5])
Let X be a normed linear space, \(K\subseteq X\), and \(T: K\to K\). The set K is said to be a Tregular set if \(\frac{u+Tu}{2}\in K\), for \(u\in K\).
Let L and M be nonempty subsets and let \(T: L\cup M\to X\) with \(T(L)\subseteq M\), \(T(M)\subseteq L\) (or \(T(L)\subseteq L\), \(T(M)\subseteq M\)). Let \(a_{0}\in L \) (or M). (i) If \(T(L)\subseteq M\), \(T(M)\subseteq L\), then \(O(a_{0}):=\{a_{0}, Ta_{0}, \dots , T^{n}a_{0},\dots \}\), \(T^{2n}a_{0}\in L\) (or M) and \(T^{2n+1}a_{0}\in M\) (or L), \(n=0,1,2,\dots \); (ii) If \(T(L)\subseteq L\), \(T(M)\subseteq M\), then \(O(a_{0}):=\{a_{0}, Ta_{0},\dots , T^{n}a_{0},\dots \}\), \(O(a_{0})\subseteq L\) (or M), \(n=0,1,2,\dots \).
Definition 4
([9])
Let X be a Banach space and let L and M be nonempty subsets of X. A map \(T: L\cup M \to X\) with \(T(L)\subseteq L\), \(T(M)\subseteq M\) is said to be a noncyclic relatively nonexpansive map with respect to orbits provided that for every \(a\in L\), \(b\in M\) if \(\ab\=dist(L, M)\) then \(\TaTb\=dist(L, M)\), otherwise \(\TaTb\\leq R(a, O(b))\) and \(\TaTb\\leq R(b, O(a))\).
If \(L=M\), then it reduces to being nonexpansive with respect to orbits given by Harandi et al. [10]. Motivating by the definitions of Gabeleh et al. [9] and Harandi et al. [10], Shanjit et al. [11] introduced the following definition.
Definition 5
([11])
Let X be a Banach space and let L and M be nonempty subsets of X. A map \(T: L\cup M \to X\) with \(T(L)\subseteq M\) and \(T(M)\subseteq L\) is said to be a cyclic relatively nonexpansive map with respect to orbits provided that for every \(a\in L\), \(b\in M\) if \(\ab\=\operatorname{dist}(L, M)\), then \(\TaTb\=\operatorname{dist}(L, M)\), otherwise \(\TaTb\\leq R(a, O(b))\), \(\TbTa\\leq R(b, O(a))\).
Remark 1
Let \((L, M)\) be a nonempty, convex proximal pair in a Banach space X and \(T: L\cup M\to L\cup M\) be a relatively nonexpansive map.

(i)
If \(T(L)\subseteq M\) and \(T(M)\subseteq L\), then T is a relatively nonexpansive map with respect to orbits and \(L\cup M\) is a cyclic Tregular set.

(ii)
If \(T(L)\subseteq L\) and \(T(M)\subseteq M\), then T is a relatively nonexpansive map with respect to orbits and L and M are Tregular sets.
Main results
We prove the following proposition.
Proposition 2
Let \((L, M)\) be a nonempty, nonconvex weakly compact proximal pair in a real Hilbert space satisfying \(\operatorname{dist}(\overline{\operatorname{conv}}(L), \overline{\operatorname{conv}}(M))=\operatorname{dist}(L, M)\). Then \((L, M)\) has the rectangle property.
Proof
From Proposition 1, the pairs \((\overline{\operatorname{conv}}(L), \overline{\operatorname{conv}}(M))\) and \((L, M)\) are proximal parallel pair in X. Let \(s_{1}, s_{2}\in L\). Then we have \(s_{1}+h, s_{2}+h\in M\), where \(h\in X\). Now,
Since \((s_{1}, s_{1}+h), (s_{2}, s_{2}+h)\in (\overline{\operatorname{conv}}(L), \overline{\operatorname{conv}}(M))\), from Proposition 1, \(s_{1}s_{2}\) is orthogonal to h that is, \(\langle s_{1}s_{2},h\rangle =0\). Hence, \(\s_{1}+hs_{2}\=\s_{2}+hs_{1}\\) for every \(s_{1}, s_{2}\in L\). This shows that the pair \((L, M)\) has the rectangle property. □
Lemma 1
Let X be a strictly convex Banach space and let \((L, M)\) be a nonempty, nonconvex weakly compact proximal pair satisfying
Let \(T: L\cup M\to X\) be a cyclic relatively nonexpansive map with respect to orbits so that \(L\cup M\) is a cyclic Tregular set.
Additionally, it is assumed that \((L, M)\) is a minimal proximal pair. Then \(L \subseteq \overline{\operatorname{conv}}(T(M))\) and \(M\subseteq \overline{\operatorname{conv}}(T(L))\).
Proof
Let \(E=\overline{\operatorname{conv}}(T(M))\cap L\) and \(F=\overline{\operatorname{conv}}(T(L))\cap M\). Then \(E\subseteq L\) and \(F\subseteq M\) are nonconvex weakly compact subsets of X. Suppose \((u, v)\in (L, M)\) so that \(\uv\=\operatorname{dist}(L, M)\). Then \((Tv, Tu)\in T(M)\times T(L)\), which implies \((Tv, Tu)\in (E, F)\). Since \(\uv\=\operatorname{dist}(L, M)\), it follows that \(\TvTu\=\operatorname{dist}(L, M)\). Hence \(\operatorname{dist}(E, F)=\operatorname{dist}(L, M)\). To claim that the pair \((E, F)\) is a proximal, it suffices to prove that, for every \(u\in E\), we have \(v\in F\) so that
Let \(u\in E=\overline{\operatorname{conv}}(T(M))\cap L\). Then \(u=\sum^{\infty }_{i=1}\alpha _{i}Tv_{i}\), where \(v_{i}\in M\), \(\alpha _{i}\geq 0\) and \(\sum^{\infty }_{i=1}\alpha _{i}=1\). Since \((L, M)\) is a proximal pair, we have \(v'_{i}\in L\) so that
Then \(u'=\sum^{\infty }_{i=1}\alpha _{i}Tv'_{i}\in \overline{\operatorname{conv}}(T(L))\) so that \(\uu'\=\operatorname{dist}(L, M)\) and \(u'\in F\). Hence, \((E, F)\) is a proximal parallel pair (and hence proximal parallel pair). Let \((u_{1},v_{1})\in E\times F\). Then we have \((v'_{1},u'_{1})\in E\times F\) so that
As \(u_{1}\in \overline{\operatorname{conv}}(T(M))\) and \(Tu'_{1}\in \overline{\operatorname{conv}}(T(M))\), which implies \(\frac{u_{1}+Tu'_{1}}{2}\in \overline{\operatorname{conv}}(T(M))\). Again, \(\frac{u_{1}+Tu'_{1}}{2}\in L\). This shows that \(\frac{u_{1}+Tu'_{1}}{2}\in E\), where \(\u_{1}u'_{1}\=dist(L,M)\). Similarly, \(\frac{v_{1}+Tv'_{1}}{2}\in F\), where \(\v_{1}v'_{1}\=\operatorname{dist}(L,M)\). This shows that \(E\cup F\) is a cyclic Tregular set and \((L, M):=(E, F)\). Hence, \(L \subseteq \overline{\operatorname{conv}}(T(M))\) and \(M\subseteq \overline{\operatorname{conv}}(T(L))\). □
Lemma 2
Let X be a strictly convex Banach space and let \((L, M)\) be a nonempty, nonconvex weakly compact proximal pair in X with
Let \(T: L\cup M\to X\) be a relatively nonexpansive map with respect to orbits with \(T(L)\subseteq L\), \(T(M)\subseteq M\) and let L and M be cyclic Tregular sets.
Additionally, it is assumed that \((L, M)\) is a minimal proximal pair. Then \(L \subseteq \overline{\operatorname{conv}}(T(L))\) and \(M\in \overline{\operatorname{conv}}(T(M))\).
Proof
Let \(E=\overline{\operatorname{conv}}(T(L))\cap L\) and \(F=\overline{\operatorname{conv}}(T(M))\cap M\). Then \(E\subseteq L\) and \(F\subseteq M\) are nonempty, nonconvex weakly compact subsets. Suppose \(u\in L\) and \(v\in M\) so that \(\uv\=\operatorname{dist}(L, M)\). Then \((Tu, Tv)\in T(L)\times T(M)\), which implies \((Tu, Tv)\in E\times F\). Since \(\uv\=\operatorname{dist}(L, M)\), it follows that \(\TvTu\=\operatorname{dist}(L, M)\). Hence \(\operatorname{dist}(E, F)=\operatorname{dist}(L, M)\). Also, \((E, F)\) is a proximal parallel pair with \(T(E)\subseteq E\), \(T(F)\subseteq F\) and E and F are Tregular sets. This proves that \((L,M):=(E,F)\). Hence \(L \subseteq \overline{\operatorname{conv}}(T(L))\) and \(M\in \overline{\operatorname{conv}}(T(M))\). □
Theorem 1
Let X be a real Hilbert space and let \((C, D)\) be a nonempty, nonconvex weakly compact proximal pair of subsets with
Let \(T: C\cup D\to X\) be a cyclic relatively nonexpansive map with respect orbits. Suppose \(C\cup D\) is a cyclic Tregular set. Then we have \(u\in C\cup D\) so that \(\uTu\=\operatorname{dist}(C, D)\).
Proof
Let \(\mathcal{F}\) be the collection of all nonempty, nonconvex weakly closed proximal pair of subsets \((L, M)\) in \((C, D)\), with \(\operatorname{dist}(C, D)=\operatorname{dist}(L, M)\) and \(L\cup M\) is a cyclic Tregular set. \(\mathcal{F}\) is nonempty as \((C_{0}, D_{0})\in \mathcal{F}\).
By Zorn’s lemma, partially ordered set \(\mathcal{F}\) has a minimal pair under set inclusion order, say \((L, M)\). Therefore, from Lemma 1, we see that
If \(\delta (L, M)=\operatorname{dist}(C, D)\), we get our result and the theorem is complete. Suppose \(\delta (L, M)>\operatorname{dist}(C, D)\). Then \(Tu\neq u+h\) and \(T(u+h)\neq u\) for every \(u\in L\). Fix \(u_{0}\in L\). Since X is a real Hilbert space and \(L\cup M\) is a cyclic Tregular set, we have \(\beta \in ]0,1[\) so that \(R(w, M)\leq \beta \delta (L, M)\), \(R(w', L)\leq \beta \delta (L, M)\), where \(w=\frac{u_{0}+T(u_{0}+h)}{2}\in L\) and \(w'=\frac{u_{0}+h+Tu_{0}}{2}\in M\). Define
Then \((P, Q)\) is a nonempty, nonconvex weakly compact proximal parallel pair with \(\operatorname{dist}(P, Q)=\operatorname{dist}(C, D)\). From Proposition 2, the pair \((P, Q)\) has the rectangle property and, for \(u\in P\),
This shows that \(T(P)\subseteq Q\). Similarly, for \(v\in Q\),
This shows that \(T(Q)\subseteq P\). Since \((P, Q)\) is a proximal parallel pair, for every \((u, v)\in P\times Q\) we have \((v', u')\in P\times Q\) so that \(\uu'\=\vv'\=\operatorname{dist}(C, D)\) and \((Tu', Tv')\in P\times Q\). Clearly, \(\frac{u+Tu'}{2}\in L\) and \(\frac{v+Tv'}{2}\in M\). Now,
which means \(\frac{u+Tu'}{2}\in P\). Similarly, \(\frac{v+Tv'}{2}\in Q\). This shows that \(P\cup Q\) is a cyclic Tregular set. Therefore, \((P, Q)\in \mathcal{F}\). But \(\delta (L, M)=\sup_{u\in P}R(u, M)\leq \beta \delta (L, M)<\delta (L, M)\), which is a contradiction. Hence, L and M are singleton sets. Therefore, we have \(u\in C\cup D\) so that \(\uTu\=\operatorname{dist}(C, D)\). □
Example 1
Let \(X=(\mathcal{R}^{2},\\cdot\)\) be a Euclidean space. Let
where \(\mathcal{Q}:=\) the set of rational numbers. Then \((L, M)\) is a nonempty, nonconvex proximal parallel pair with \(\operatorname{dist}(L, M)=\operatorname{dist}(\overline{\operatorname{conv}}(L), \overline{\operatorname{conv}}(M))=2\) and \(M=L+h\), \(h=(2,0)\). Also, \((L, M)\) has the rectangle property.
Let \(T: L\cup M\to L\cup M\) by
Clearly, \(L\cup M\) is a cyclic Tregular set. The map T is not a relatively nonexpansive map but a relatively nonexpansive map with respect to orbits. Then, from Theorem 1, we have \(((1,0),(1,0))\in L\times M\) so that \(\(1,0)T(1,0)\=\operatorname{dist}(L, M)=\(1,0)T(1,0)\\).
If the nonempty pair \((C, D)\) is convex, then from Theorem 1, we obtain the following corollary.
Corollary 1
([11])
Let X be a uniformly convex Banach space and let \((C, D)\) be a nonempty, convex weakly compact proximal pair of subsets in X having the rectangle property. Let \(T: C\cup D\to X\) be a cyclic relatively nonexpansive map with respect to orbits. Then we have \(u\in C\cup D\) so that \(\uTu\=\operatorname{dist}(C, D)\).
The following theorem proves that a relatively nonexpansive map with respect to orbits T defined on \(C\cup D\) has fixed points in C and D.
Theorem 2
Let X be a uniformly convex Banach space and let \((C, D)\) be a nonempty, nonconvex weakly compact proximal pair in X with
Let \(T: C\cup D\to X\) be a relatively nonexpansive map with respect to orbits with \(T(C)\subseteq C\), \(T(D)\subseteq D\). Suppose C and D are Tregular sets. Then we have \((Tu, Tv)=(u, v)\in C\times D\) so that \(\uv\=\operatorname{dist}(C, D)\).
Proof
Let \(\mathcal{F}\) be the collection of all nonempty, nonconvex weakly closed proximal pair of subsets \((L, M)\) in \((C, D)\), satisfying \(\operatorname{dist}(L, M)=\operatorname{dist}(C, D)\), \(T(L)\subseteq L\), \(T(M)\subseteq M\) and let L and M be Tregular sets. \(\mathcal{F}\) is nonempty as \((C_{0}, D_{0})\in \mathcal{F}\). By Zorn’s lemma, the partially ordered set \(\mathcal{F}\) has the minimal pair under set inclusion order, say \((L, M)\). Therefore, from Lemma 2, we see
If \(\delta (L, M)=\operatorname{dist}(C, D)\), we get our result and the theorem is complete. Suppose
Fix \(u_{0}\in L\). Since X is a uniformly convex space and L and M are Tregular sets, we have \(\beta \in ]0,1[\) so that \(R(w, M)\leq \beta \delta (L, M)\) and \(R(w', L)\leq \beta \delta (L, M)\), where \(w=\frac{u_{0}+Tu_{0}}{2}\in L\) and \(w'=w+h\). Define
Then \((P, Q)\) is a nonconvex weakly compact proximal pair (and hence proximal parallel pair). Since \(L \subseteq \overline{\operatorname{conv}}(T(L))\), \(M \subseteq \overline{\operatorname{conv}}(T(M))\) and, for \(u\in P\),
This shows that \(T(P)\subseteq P\). Similarly, for \(v\in Q\),
This shows that \(T(Q)\subseteq Q\). Let \(u\in P\), then \(Tu\in P\). Since L is a Tregular set, \(\frac{u+Tu}{2}\in L\). Now,
This shows that \(\frac{u+Tu}{2}\in P\). Similarly, \(\frac{v+Tv}{2}\in Q\), \(v\in Q\). Hence, P and Q are Tregular sets. Therefore, \((P, Q)\in \mathcal{F}\). This forces that \(\beta =1\). Thus, \(\delta (L, M)=\operatorname{dist}(L, M)\). Since \(M=L+h\), we have \(L=\{u\}\) and \(M=\{u+h\}\) for some \(u\in C\). Therefore, we have \((Tu, Tv)=(u, v)\in C\times D\) so that \(\uv\=\operatorname{dist}(C, D)\). □
If the nonempty pair \((C, D)\) is convex, then from Theorem 2, we obtain the following corollary.
Corollary 2
([9])
Let X be a uniformly convex Banach space, and let \((C, D)\) be a nonempty, convex weakly compact proximal pair of subsets in X. Let \(T: C\cup D\to X\) be a relatively nonexpansive map with respect to orbits with \(T(C)\subseteq C\), \(T(D)\subseteq D\). Then we have \((Tu, Tv)=(u, v)\in C\times D\) so that \(\uv\=\operatorname{dist}(C, D)\).
In the year 2020, Kim et al. introduced a modified Kranoselskii–Mann interactive method and gave some interesting results (see [12]). Next, we show the convergence of Kranoselskii’s iteration process (see [1, 13]) for a nonconvex proximal pair.
Theorem 3
Let \((L, M)\) be a nonempty, nonconvex weakly compact proximal pair with \(\operatorname{dist}(\overline{\operatorname{conv}}(L), \overline{\operatorname{conv}}(M))=\operatorname{dist}(L, M)\) in a uniformly convex Banach space X. Let \(T: L\cup M\to X\) be a relatively nonexpansive map with respect to orbits satisfying \(T(L)\subseteq L\), \(T(M)\subseteq M\). Further, assume that L and M are Tregular sets. Let an initial point \(s_{0} \in L\) and define a sequence
Then \(\lim_{n\to +\infty }\s_{n}Ts_{n}\=0\). Moreover, if T is continuous and \(T(L)\) is contained in a compact set, then \(\lim_{n\to +\infty }s_{n}=s\) and \(Ts=s\).
Proof
Suppose \(\operatorname{dist}(L, M)>0\). Since \(\operatorname{dist}(\overline{\operatorname{conv}}(L), \overline{\operatorname{conv}}(M))=\operatorname{dist}(L, M)\), by Proposition 1, the pairs \((L, M)\) and \((\overline{\operatorname{conv}}(L), \overline{\operatorname{conv}}(M))\) are proximal parallel pairs in X. From Theorem 2, there exist \(s\in L\), \(t\in M\) so that \(Ts=s\), \(Tt=t\) and \(\st\=\operatorname{dist}(L, M)\). L and M being Tregular sets, the sequence \(\{s_{n}\}\subseteq L\). Now,
Hence, \(\{\s_{n}t\\}\) is nonincreasing and \(\lim_{n\to +\infty }\s_{n}t\=k\).
Suppose \(\lim_{n\to +\infty }\s_{n}Ts_{n}\\neq 0\). Then there exists a subsequence \(\{s_{n_{i}}\}\) of \(\{s_{n}\}\) such that \(\s_{n_{i}}Ts_{n_{i}}\\geq \varepsilon >0\) for \(i=1,2,\dots \). Choose \(\theta \in ]0,1[\) and \(\varepsilon _{1}\) so that \(\frac{\varepsilon }{\theta }>k\) and \(0<\varepsilon _{1}<\min \{ \frac{k\delta (\theta )}{1\delta (\theta )}, \frac{\varepsilon }{\theta }k \} \).
Since X is uniformly convex, \(\delta (\varepsilon _{1})>0\) for \(\varepsilon _{1}>0\) is a strictly increasing function. Hence, \(0<\delta (\theta )<\frac{\varepsilon }{k+\varepsilon _{1}}\). So, it is possible to choose \(\varepsilon _{1}>0\) so small that
As \(\lim_{n\to +\infty }\s_{n_{i}}t\=k\), choose i, so that \(\s_{n_{i}}t\\leq k+\varepsilon _{1}\). Since \(Tt=t\), we have \(\Ts_{n_{i}}Tt\\leq R(s_{n_{i}}, O(t))=\s_{n_{i}}t\\leq k+ \varepsilon _{1}\). Now,
By choosing \(\varepsilon _{1}>0\) so small, we get
This shows that \(\lim_{n\to +\infty }\s_{n}Ts_{n}\=0\).
Suppose \(T(L)\) is contained in a compact set. Then \(\{s_{n}\}\) has a subsequence \(\{s_{n_{i}}\}\) so that \(\lim_{i\to +\infty } s_{n_{i}}=s\in L\). Thus, we have \(z\in M\) so that \(\sz\=\operatorname{dist}(L,M)\). Now,
Since T is continuous, from Eq. (3), when \(i\to +\infty \), we have
Since \(\sz\=\operatorname{dist}(L, M)\), it follows that \(\TsTz\=\operatorname{dist}(L, M)\). Therefore, \(\sTz\\leq \operatorname{dist}(L, M)\), which implies \(\sTz\= \operatorname{dist}(L, M)\). By strict convexity of the norm, \(Tz=z\), which implies \(Ts=s\), because s is the unique point of L nearest to z. □
Example 2
Let \(X=(\mathcal{R}^{2}, \\cdot\)\) be a Euclidean space. Let
where \(\mathcal{Q}:=\) the set of rational numbers. Then \((L, M)\) is a nonempty, nonconvex proximal parallel pair with \(\operatorname{dist}(L, M)=\operatorname{dist}(\overline{\operatorname{conv}}(L), \overline{\operatorname{conv}}(M))=1\) and \(M=L+h\), \(h=(1,0)\).
Let \(T: L\to L\) by
and \(T: M\to M\) by
Clearly, \(T(L)\subseteq L\), \(T(M)\subseteq M\) and L and M are Tregular sets. The map T is not a relatively nonexpansive map but a relatively nonexpansive map with respect to orbits. Then, by Theorem 2, there exist \((0,0)\in L\), \((1,0)\in M\) so that \(\(0,0)(1,0)\=\operatorname{dist}(L, M)\).
Let \(s_{0}=(u_{0}, v_{0})\in L\) be an initial point. Then \(Ts_{0}=T(u_{0}, v_{0})=(0, \frac{v_{0}}{2})\). Now,
Similarly, \(s_{2}=(u_{2}, v_{2})= (0, \frac{v_{0}}{2^{4}} )\), \(s_{3}=(u_{3}, v_{3})= (0, \frac{v_{0}}{2^{6}} )\) and so on. In general, \(s_{n}=(u_{n}, v_{n})= (0, \frac{v_{0}}{2^{2n}} )\) and \(\lim_{n\to +\infty }(u_{n}, v_{n})=(0, 0)\) and \(T(0, 0)=(0, 0)\). In a similar way, if \(s'_{0}=(u'_{0}, v'_{0})\in M\) be an initial point, then \(\lim_{n\to +\infty }(u'_{n}, v'_{n})=(1, 0)\) and \(T(1, 0)=(1, 0)\).
From Theorem 3, if \(\operatorname{dist}(L, M)=0\), \(L\cap M\) is convex and T is a nonexpansive map, then we have the next result.
Corollary 3
([13])
Let L be a nonempty, bounded closed convex subset in a uniformly convex Banach space X and let \(T:L\to L\) be a nonexpansive map. Let an initial point \(s_{0}\in L\) and define a sequence
Then \(\lim_{n\to +\infty }\s_{n}Ts_{n}\=0\). Moreover, if \(T(L)\) is contained in a compact set, then \(\lim_{n\to +\infty }s_{n}=s\) and \(Ts=s\).
Conclusion
Relatively nonexpansive maps with respect to orbits, cyclic Tregular sets and Tregular sets are used to obtain our main results. The results, Theorem 1, Theorem 2 and Theorem 3, that are obtained in this article are more generalized than the results obtained in the literature. To converge Kranoselskii’s iteration process to a fixed point, the map T in Theorem 3 should be continuous.
Availability of data and materials
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Acknowledgements
The authors are thankful to the anonymous reviewer for their valuable comments and suggestions.
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The first author, Laishram Shanjit, is financially supported by the University Grant Commission, India, fellowship granting no. 420004.
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LS and YR contributed equally to the preparation of this manuscript. All authors read and approved the final manuscript.
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Shanjit, L., Rohen, Y. Nonconvex proximal pair and relatively nonexpansive maps with respect to orbits. J Inequal Appl 2021, 124 (2021). https://doi.org/10.1186/s13660021026605
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MSC
 47H09
 47H10
 54H25
Keywords
 Proximal parallel pair
 Cyclic Tregular set
 Kranoselskii’s iteration
 Fixed point
 Best proximity point