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Journal of Inequalities and Applications volume 2022, Article number: 132 (2022)
Abstract
Very recently Jain et al. (J. Inequal. Appl. 2022:73, 2022) introduced the concept of multivalued Fcontraction with altering distance function and investigated the existence of best proximity points for this class of mappings. In this article we prove that the existence of best proximity points for multivalued Fcontraction nonself mappings can be obtained from the corresponding fixed point result for multivalued Fcontraction self mappings, and so the main conclusion due to Jain et al. is not a real generalization of fixed point theory.
1 Introduction
In metric fixed point theory, sufficient conditions are derived to ensure the existence of solutions of the equation \(U(x)=x\), where U is a self mapping defined on a metric space \((X,d)\). The Banach contraction principle [2] for standard metric spaces is one of the important results in metric fixed point theory, and it has a lot of applications in differential equations and integral equations for the existence of solutions. Let A and B be two nonempty subsets of a metric space \((X,d)\) and \(Q:A\rightarrow B\) be a nonself mapping. A necessary condition to guarantee the existence of solutions of the equation \(Qx=x\) is \(Q(A)\cap A\neq \emptyset \). If \(Q(A)\cap A= \emptyset \), then the mapping Q has no fixed points. In this case, one seeks for an element in the domain space whose distance from its image is minimum, i.e., one interesting problem is to minimize \(d(x,Qx)\) such that \(x\in A\). Since \(d(x,Qx)\geq \sigma (A,B):=\inf \{d(x,y):x\in A, y\in B\}\), one searches for an element \(x\in A\) for which \(d(x,Qx)= \sigma (A,B)\). Best proximity point problems deal with this situation. Authors usually discover best proximity point theorems to generalize the corresponding fixed point theorems.
Just recently, Jain et al. [1] introduced the concept of multivalued Fcontraction with altering distance function and investigated the existence of best proximity point for this class of nonself mappings. After proving their main result [1, Theorem 2.1], they stated the corresponding fixed point theorem [1, Corollary 2.2] for self mappings. Then the authors furnished nontrivial examples to validate their claim and also provided applications to fractional calculus and on equation of motion modelling to differential equations.
In this note, we show that the main result of [1], which is related to the existence of a best proximity point for multivalued Fcontraction nonself mappings, is a straightforward consequence of the corresponding fixed point result.
2 Preliminaries
We first recall some notations and definitions from [1], which will be needed throughout this paper.
Let W be a nonempty set and \((W, \sigma )\) be a metric space. By \(CB(W)\) we denote the collection of all closed and bounded subsets of W.
Suppose that \(L~\text{and}~M\) are nonempty subsets of W and \(u\in W\). We shall adopt the following notations:
Definition 2.1
([3])
Let \((L,M)\) be a nonempty pair of subsets of a metric space \((W, \sigma )\). The pair \((L,M)\) with \(L_{0}\neq \emptyset \) is said to have the Pproperty if for every \(u_{1}, u_{2} \in L_{0}\) and \(v_{1}, v_{2} \in M_{0}\) we have
Definition 2.2
([4])
Let \((L,M)\) be a nonempty pair of subsets of a metric space \((W, \sigma )\) and \(T:L\rightarrow 2^{M}\) be a multivalued mapping. A point \(u\in L\) is called a best proximity point of the mapping T if \(D(u,Tu)=\sigma (L,M)\).
Definition 2.3
A mapping \(\phi :(0,\infty )\rightarrow (0,\infty )\) is called an altering distance function if ϕ is continuous, monotonically increasing and \(\phi (u)>0\) for all \(u>0\).
In the year 2012, Wardowski introduced the following class of functions in order to present an interesting extension of the Banach contraction principle (see also [6, 7] for more information).
Definition 2.4
([5])
A function \(F:(0,+\infty )\to \mathbb{R}\) is said to be a Wardowski function provided that
 \((F_{1})\):

F is strictly increasing;
 \((F_{2})\):

For each sequence \(\{\gamma _{n}\}\) of positive numbers, \(\lim_{n\to \infty}\gamma _{n}=0\Leftrightarrow \lim_{n\to \infty}F( \gamma _{n})=\infty \);
 \((F_{3})\):

There exists \(c\in (0,1)\) such that \(\lim_{\gamma \to 0}\gamma ^{c}F(\gamma )=0\).
The class of all Wardowski functions will be denoted by \(\mathcal{F}\).
Using the class of Wardowski functions, the following family of multivalued contractions was introduced in [1].
Definition 2.5
Let \((W, \sigma , \leq )\) be a partially ordered complete metric space and \((L,M)\) be a nonempty pair of subsets of W such that \(L_{0}\) is nonempty. A mapping \(T:L\rightarrow CB(M)\) is said to be a multivalued nonself Fcontraction depending on an altering distance function ϕ if \(Tu_{0}\subseteq M_{0}\) for any \(u_{0}\in L_{0}\), and there exist \(\tau >0\) and \(F\in \mathcal{F}\) such that
for all \(u,v \in L\) with \(u\leq v\), where \(N(u,v)=\max \{\sigma (u,v), D(u,Tu), D(v,Tv), \frac{D(u,Tv)+D(v,Tu)}{2}\}\) and ϕ satisfies \(\phi (x+y)\leq \phi (x)+\phi (y)\) for all \(x,y>0\).
Notice that if in the above definition \(L=M\), then the mapping T is said to be a multivalued Fcontraction.
Here we state the main result of [1].
Theorem 2.6
([1, Theorem 2.1])
Let \((W, \sigma , \leq )\) be a partially ordered complete metric space and \((L,M)\) be a nonempty pair of closed subsets of W such that \(L_{0}\) is nonempty and \((L,M)\) satisfies the Pproperty. Let \(T:L\rightarrow CB(M)\) be a multivalued nonself Fcontraction depending on an altering distance function ϕ such that the following conditions are satisfied:
(i) There exist two elements \(u_{0},u_{1} \in L_{0}\) and \(v_{0}\in Tu_{0}\) such that \(\sigma (u_{1},v_{0})=\sigma (L,M)\) and \(u_{0}\leq u_{1}\);
(ii) For all \(u,v\in L_{0}\), \(u\leq v\) implies \(Tu\subseteq Tv\);
(iii) If \(\{u_{n}\}\) is a nondecreasing sequence in L such that \(u_{n}\rightarrow u\) as \(n\rightarrow \infty \), then \(u_{n}\leq u\) for all \(n\geq 1\).
Then T has a best proximity point.
It is worth noticing that if \(L=M\) in Theorem 2.6, then we obtain the following fixed point theorem.
Theorem 2.7
Let \((W, \sigma , \leq )\) be a partially ordered complete metric space and L be a nonempty closed subset of the metric space W, and let \(T:L\rightarrow CB(L)\) be a multivalued Fcontraction depending on an altering distance function ϕ such that the following conditions are satisfied:
(i) There exist two elements \(u_{0},u_{1} \in L\) and \(v_{0}\in Tu_{0}\) such that \(\sigma (u_{1},v_{0})=0\) and \(u_{0}\leq u_{1}=v_{0}\);
(ii) For all \(u,v\in L\), \(u\leq v\) implies \(Tu\subseteq Tv\);
(iii) If \(\{u_{n}\}\) is a nondecreasing sequence in L such that \(u_{n}\rightarrow u\) as \(n\rightarrow \infty \), then \(u_{n}\leq u\) for all \(n\geq 1\).
Then T has a fixed point.
3 Main results
The main motivation of the current work is to show that Theorem 2.6 is a special case of Theorem 2.7, and so the main conclusion of [1] is not a real generalization of fixed point theory.
To this end, we first present a brief proof of Theorem 2.7.
Proof of Theorem 2.7
Let \(u_{0}\in L\). So, according to condition (i), there exists \(u_{1}\in Tu_{0}\) such that \(u_{0}\leq u_{1}\). Now, from condition (ii), \(Tu_{0}\subseteq Tu_{1}\). So, there exists \(u_{2}\in Tu_{1}\) such that \(u_{1}\leq u_{2}\). Continuing this process, we obtain a nondecreasing sequence \(\{u_{n}\}_{n\geq 0}\) in L such that \(u_{n+1}\in Tu_{n}\) for any \(n\geq 1\). If for some \(n_{0}\in \mathbb{N}\), \(u_{n_{0}}=u_{n_{0}+1}\), then
that is, \(u_{n_{0}}\in L\) is a fixed point of the mapping T, and we are finished. Assume that \(u_{n}\neq u_{n+1}\) for all \(n\geq 1\). Thus
where
which means that
Note that if \(\sigma (u_{n1},u_{n})<\sigma (u_{n},u_{n+1})\), then
which is a contradiction. Hence, \(\sigma (u_{n1},u_{n})\geq \sigma (u_{n},u_{n+1})\) for all \(n\geq 1\). Let \({\lim_{n\rightarrow \infty}}\sigma (u_{n}, u_{n+1}):=r \geq 0\). Now,
Continuing in this way, we obtain
This shows that \(r=0\). Corresponding to the function F, there exists \(k\in (0,1)\) such that
Therefore,
where \(B_{n}:=\phi (\sigma (u_{n}, u_{n+1}))\). Thus, for given \(\varepsilon >0\), there exists \(n_{1}\in \mathbb{N}\) such that
Let \(m>n\geq n_{1}\). Then
This implies that the sequence \(\{u_{n}\}\) is a Cauchy sequence. Since L is complete, there exists \(u\in L\) such that \(u_{n}\rightarrow u\) as \(n\rightarrow \infty \). It now follows from condition (iii) that \(u_{n}\leq u\) for all \(n\geq 1\). Suppose that \(u\notin Tu\). Then
Letting \(n\rightarrow \infty \) in the above relation, we obtain
which is a contradiction. Therefore, \(u\in Tu\), and this completes the proof. □
We are now ready to state the main result of this article.
Theorem 3.1
Theorem 2.6is a straightforward consequence of Theorem 2.7.
Proof
Let \(x_{0} \in L_{0}\). Then \(Tx_{0}\subseteq M_{0}\). As the pair \((L,M)\) has the Pproperty, for every \(y_{0}\in Tx_{0}\), there exists unique \(z_{0}\in L_{0}\) such that \(\sigma (z_{0},y_{0})=\sigma (L,M)\). Let
We show that \(D_{x_{0}}\in CB(L_{0}) \). Let \(p,q \in D_{x_{0}}\). Then there exist \(p',q'\in Tx_{0}\) such that
Since the pair \((L,M)\) has the Pproperty, \(\sigma (p,q)=\sigma (p',q')\leq {\mathrm{diam}}(Tx_{0})\). This shows that \(D_{x_{0}}\) is bounded. Now, let \(x\in \overline{D_{x_{0}}}\). Then there exists a sequence \(\{x_{n}\}\subset D_{x_{0}}\) such that \(x_{n} \rightarrow x\) as \(n\rightarrow \infty \). By definition of the set \(D_{x_{0}}\), for all \(n\geq 1\), there exists \(y_{n}\in Tx_{0}\) such that \(\sigma (x_{n},y_{n})=\sigma (L,M)\). Thus, for any \(m,n\geq 1\), we have \(\sigma (x_{n},y_{n})=\sigma (L,M)=\sigma (x_{m},y_{m})\). Again using the fact that \((L,M)\) has the Pproperty, we must have \(\sigma (y_{m},y_{n})=\sigma (x_{m},x_{n})\), which ensures that the sequence \(\{y_{n}\}\) is Cauchy. Since \(Tx_{0}\) is closed, there exists \(y\in Tx_{0}\) such that \(y_{n} \rightarrow y\) as \(n\rightarrow \infty \). Thereby, \(\sigma (x,y)=\sigma (L,M)\), which concludes that \(x\in D_{x_{0}}\), i.e., \(D_{x_{0}}\) is a closed subset of \(L_{0}\). Now, let us define a multivalued mapping \(S:L_{0}\rightarrow CB(L_{0})\) by \(Sx_{0}=D_{x_{0}}\). Since for every \(y_{0}\in Tx_{0}\) there exists unique \(x\in L_{0}\) with \(\sigma (x,y_{0})=\sigma (L,M)\), so this mapping is well defined.
By the given condition there exist \(u_{0}, u_{1}\in L_{0}\) and \(v_{0}\in Tu_{0}\) such that \(\sigma (u_{1},v_{0})=\sigma (L,M)\) and \(u_{0}\leq u_{1}\). By the definition of the mapping S, \(u_{1}\in Su_{0}\) and \(u_{0}\leq u_{1}\). Assume that \(u,v \in L_{0}\) with \(u\leq v\). We assert that \(Su\subseteq Sv\). Let \(y\in Su=D_{u}\). Then there exists an element \(w\in Tu\) for which \(\sigma (y,w)=\sigma (L,M)\). By the fact that \(Tu\subseteq Tv\), we conclude that \(w\in Tv\), and so
So, the first and second conditions of Theorem 2.7 are satisfied. Now let \(u,v\in L_{0}\) with \(u\leq v\). Since \(T:L\rightarrow CB(M)\) is a multivalued nonself Fcontraction with altering distance function ϕ,
where \(N(u,v)=\max \{\sigma (u,v), D(u,Tu), D(v,Tv), \frac{D(u,Tv)+D(v,Tu)}{2}\}\).
Let \(p_{0}\in Su~\text{and}~q_{0} \in Sv\). Then there exists \(p_{1}\in Tu\) such that \(\sigma (p_{0},p_{1})=\sigma (L,M)\). Similarly, there exists \(q_{1}\in Tv\) such that \(\sigma (q_{0},q_{1})=\sigma (L,M)\). Since the pair \((L,M)\) has the Pproperty, we obtain
Besides, if \(p\in Tu\), \(q\in Tv\), then there exists unique \((p',q') \in L_{0}\times L_{0}\) such that \(\sigma (p',p)=\sigma (L,M)\) and \(\sigma (q',q)=\sigma (L,M)\). Thus \((p',q')\in Su\times Sv\). Since \((L,M)\) has the Pproperty,
Thereby, \(\delta (Tu,Tv)= \delta (Su,Sv)\). Let \(p'\in Su\). Then there exists \(p\in Tu\) such that \(\sigma (p',p)=\sigma (L,M)\), which deduces that
Moreover, if \(h'\in Sv\), then there exists \(h\in Tv\) such that \(\sigma (h',h)=\sigma (L,M)\), which implies that
Similarly, if \(q'\in Sv\), then there exists \(q\in Tv\) such that \(\sigma (q',q)=\sigma (L,M)\), and so
Equivalently,
This shows that \(N(u,v)\leq N'(u,v)+\sigma (L,M)\), where
So, we have
This shows that the mapping \(S:L_{0}\rightarrow CB(L_{0})\) is a multivalued Fcontraction with altering distance function ϕ. Also, the third condition of Theorem 2.7 is satisfied. It now follows from Theorem 2.7 that there exists \(x^{*}\in L_{0}\) such that \(x^{*}\in Sx^{*}\), which deduces that there exists \(y^{*}\in Tx^{*}\) such that \(\sigma (x^{*},y^{*})=\sigma (L,M)\). Since
we conclude that \(D(x^{*},Tx^{*})=\sigma (L,M)\), that is, \(x^{*}\in L_{0}\) is a best proximity point of the mapping \(T:L\rightarrow CB(M)\), and we are finished. □
We now apply our technique to illustrate Example 2.1 of [1].
Example 3.2
(Example 2.1 of [1])
Let \(W=\mathbb{R}^{2}\) with the order ≤ defined with \((x,y)\leq (z,t)\Leftrightarrow x\leq z\), \(y\leq t\). Consider the metric σ on W as follows:
Then \((W,\sigma )\) is a complete partially ordered metric space. Assume that
Clearly, L and M are closed with \(\sigma (L,M)=5\) and \((L_{0},M_{0})=(L,M)\). Define the multivalued nonselfmapping \(T:L\rightarrow CB(M)\) as
By the used technique as in the proof of Theorem 3.1, the mapping \(S:L_{0}\rightarrow CB(L_{0})\) is defined by
Obviously, \((0,5)\in L_{0}\) is a fixed point of the mapping S which implies that it is a best proximity point of the mapping T.
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References
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The authors would like to thank the anonymous referees for the careful reading of the manuscript and their useful comments.
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Som, S., Gabeleh, M. Comments on the paper “Best proximity point results with their consequences and applications”. J Inequal Appl 2022, 132 (2022). https://doi.org/10.1186/s13660022028714
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DOI: https://doi.org/10.1186/s13660022028714