Global optimality results for multivalued nonself mappings in bmetric spaces
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Abstract
In this paper, we introduce a new class of multivalued contractions with respect to bgeneralized pseudodistances and prove a best proximity point theorem for this class of nonself mappings. In this way, we improve and extend several existing results in the literature. Examples are given to support our main results. This work is a continuation of studies on the use of a new type of distances in the fixed point theory. The pioneering effort in direction of defining distance is inter alia paper of O. Kada, T. Suzuki and W. Takahashi.
Keywords
Best proximity point Multivalued contraction of Suzuki type bGeneralized pseudodistancesMathematics Subject Classification
47H10 47H09 46B201 Introduction
In 2008, Suzuki [1] presented a weaker notion of contractions in order to characterize the completeness of metric spaces and established the following interesting theorem.
Theorem 1.1
 (i)
X is complete.
 (ii)There exists \(r\in [0,1)\) such that every mapping T on X satisfying the following :has a fixed point.$$\begin{aligned} \theta (r)d(x,Tx)\le d(x,y)\quad \text {implies}\quad d(Tx,Ty)\le rd(x,y),\ \forall x,y\in X, \end{aligned}$$
After that the multivalued version of Theorem 1.1, which is an extension of Nadler’s fixed point theorem, was presented as below.
Theorem 1.2
Now, let (A, B) be a nonempty pair of subsets of a metric space (X, d) and let \(T:A\rightarrow 2^B\) be a multivalued nonself mapping. Then for each \(x\in A\) we have \({\mathcal {D}}(x,Tx)\ge dist(A,B)\), where \(dist(A,B):=\inf \{d(x,y) : (x,y)\in A\times B\}\) and \({\mathcal {D}}(x,Tx):=dist(\{x\},Tx)\). So, it is quite natural to seek an approximate solution \(x\in A\) that is optimal in the sense that the distance \({\mathcal {D}}(x,Tx)\) with respect to \({\mathcal {D}}\) is minimum. As the minimality of the value \({\mathcal {D}}(x,Tx)\) connotes the highest closeness between the elements x and Tx, one attempts to determine an element x for which \({\mathcal {D}}(x,Tx)\) assumes the least possible value dist(A, B). Such an optimal solution x for which \(\mathcal {D}(x,Tx)=dist(A,B)\), is called a best proximity point of the multivalued nonself mapping T. Existence of best proximity points for multivalued nonself mappings was first studied in [3] for multivalued nonexpansive nonself mappings in hyperconvex metric spaces and in Hilbert spaces (see also [4, 5, 6, 7, 8, 9, 10] for different approaches to the same problem).
The aim of this article is to elicit a best proximity point theorem for a new class of multivalued nonself mappings with respect to bgeneralized pseudodistances. Our results improve and extend some recent results in the previous works.
2 Preliminaries
Definition 2.1
The following notion is weaker than the notion of Pproperty which was first introduced in [12].
Definition 2.2
Example 2.1
[11] Let (A, B) be a nonempty, closed and convex pair of subsets of a Hilbert space \(\mathbb {H}\). Then (A, B) satisfies the Pproperty.
Example 2.2
Let (A, B) be a nonempty pair of subsets of a metric space (X, d) such that \(A_0\ne \emptyset \) and \(dist(A,B)=0\). Then (A, B) has the Pproperty.
Example 2.3
[13] Let (A, B) be a nonempty bounded, closed and convex pair of subsets of a uniformly convex Banach space X. Then (A, B) has the Pproperty.
Example 2.4
Here, we state the next best proximity point theorem which is a main result of [14].
Theorem 2.3
The notion of bmetric space was introduced by Czerwik [15] as below.
Definition 2.4
 \((d_{1})\)

\(d(x,y)=0\Leftrightarrow x=y\);
 \((d_{2})\)

\(d(x,y)=d(y,x)\);
 \((d_{3})\)

\(d(x,z)\le s[d(x,y)+d(y,z)]\).
If d is a bmetric on X (with constant \(s\ge 1\)), then the pair (X, d) is called a bmetric space. Note that every metric space is a bmetric space. Throughout this paper, we assume that the bmetric \(d:X\times X\rightarrow [0,\infty )\) is continuous on \(X^{2}\).
Here, we mention the following fixed point theorem which is the main result of [15].
Theorem 2.5
Definition 2.6
 (J1)

\(J(x,z)\le s[J(x,y)+J(y,z)]\) for any \(x, y, z\in X\); and
 (J2)
 For any sequences \((x_{m}:m\in {\mathbb {N}})\) and \((y_{m}:m\in {\mathbb {N}})\) in X such thatand$$\begin{aligned} \lim _{n\rightarrow \infty }\sup _{m>n}J(x_{n},x_{m})=0, \end{aligned}$$(2.2)we have$$\begin{aligned} \lim _{m\rightarrow \infty }J(x_{m},y_{m})=0, \end{aligned}$$(2.3)$$\begin{aligned} \lim _{m\rightarrow \infty }d(x_{m},y_{m})=0\text {.} \end{aligned}$$(2.4)
Remark 2.7
If (X, d) is a bmetric space (with \(s\ge 1\)), then the bmetric \( d:X\times X\rightarrow [0,\infty )\) is a bgeneralized pseudodistance on X. However, there exists a bgeneralized pseudodistance on X which is not a bmetric (for details see Example 4.1 of [16]).
Remark 2.8
From (J1) and (J2) it follows that for any \(x, y\in X\) we have \(J(x,y)>0\) or \(J(y,x)>0\).
By using the notion of bgeneralized pseudodistance on a bmetric space X, we can define the \({\mathcal {H}}^{J}\) Hausdorff distance as below.
Definition 2.9
Similarly, the following definitions and notations can be constructed in b metric spaces equipped with a bgeneralized pseudodistance.
Definition 2.10
 (I)The pair (A, B) is said to have the \({WP}^{J}\)property if and only ifwhere \(x_{1},x_{2}\in A_{0}\) and \(y_{1},y_{2}\in B_{0}\).$$\begin{aligned} {\left\{ \begin{array}{ll} J(x_{1},y_{1}) =dist(A,B), \\ J(x_{2},y_{2})=dist(A,B) \end{array}\right. } \Rightarrow J(x_{1},x_{2})\le J(y_{1},y_{2}), \end{aligned}$$
 (II)We say that the bgeneralized pseudodistance J is associated with the pair (A, B) if for any sequences \((x_{m}:m\in {\mathbb {N}} )\) and \((y_{m}:m\in {\mathbb {N}})\) in X such that \(\lim _{m\rightarrow \infty }x_{m}=x\); \(\lim _{m\rightarrow \infty }y_{m}=y\); andwe have \(d(x,y)=dist(A,B)\).$$\begin{aligned} J(x_{m},y_{m1})=dist(A,B), \quad \forall m\in {\mathbb {N}}, \end{aligned}$$
We mention that for a bmetric space (X, d) if we put \(J=d\), then the map d is associated with each pair (A, B), where (A, B) is a nonempty pair in X because of the continuity of d.
Definition 2.11
The following lemma will be used in the sequel.
Lemma 2.12
3 Main results
We begin our main result of this section with the following notion.
Definition 3.1
It is clear that the class of multivalued nonself mappings which are contraction of Suzuki type with respect to bgeneralized pseudodistances contains the class of multivalued nonself mappings considered in Theorem 2.3. This can be seen by taking \(s=1\) and \(J=d\).
We now prove the main result of this article.
Theorem 3.2
Let X be a complete bmetric space (with \(s\ge 1)\) and let the map \( J:X\times X\rightarrow [0,\infty )\) be a bgeneralized pseudodistance on X. Let (A, B) be a pair of nonempty closed subsets of X with \(A_{0}\ne \emptyset \) and such that (A, B) has the \( WP^{J}\)property and J is associated with (A, B). Let \(T:A\rightarrow 2^{B}\) be a closed contraction multivalued nonself mapping of Suzuki type. If \(T(x)\in {\mathcal {C}}{\mathcal {B}}(X)\) for all \(x\in A,\) and \(T(x)\subset B_{0}\) for each \(x\in A_{0},\) then T has a best proximity point in A.
Proof

\(\ x_m\in A_{0} \ \text {and} \ y_m\in B_{0}\) for all \(m\in \{0\}\cup {\mathbb {N}} \text {,}\)

\(\ y_m\in Tx_m\) for all \(m\in \{0\}\cup {\mathbb {N}}\),

\(\ J(x_m,y_{m1}=dist(A,B)\) for all \(m\in {\mathbb {N}}\),

\(\ J(y_{m1},y_{m})\le \frac{r_1}{s} J(x_{m1},x_{m})\) for all \(m\in {\mathbb {N}}\).
Next corollaries are obtained from Theorem 3.2.
Corollary 3.3
Let X be a complete bmetric space (with \(s\ge 1).\) Let (A, B) be a pair of nonempty closed subsets of X with \(A_{0}\ne \emptyset \) and such that (A, B) has the WPproperty. Let \(T:A\rightarrow 2^{B}\) be a closed contraction multivalued nonself mapping of Suzuki type. If \(T(x)\in {\mathcal {C}}{\mathcal {B}}(X)\) for all \(x\in A,\) and \(T(x)\subset B_{0}\) for each \(x\in A_{0},\) then T has a best proximity point in A.
Corollary 3.4
(Compare with Theorem 2.3) Let (A, B) be a pair of nonempty closed subsets of a complete metric space (X, d) such that \(A_{0}\ne \emptyset \) and (A, B) satisfies the WPproperty. Let \(T:A\rightarrow 2^{B}\) be a closed contraction multivalued nonself mapping of Suzuki type. If \(T(x)\in {\mathcal {C}}{\mathcal {B}}(X)\) for all \(x\in A,\) and \(T(x_{0})\) \(\subset B_{0}\) for each \(x_{0}\in A_{0},\) then T has a best proximity point in A.
Corollary 3.5
Let X be a complete bmetric space (with \(s\ge 1)\) and let the map \( J:X\times X\rightarrow [0,\infty )\) be a bgeneralized pseudodistance on X. Let (A, B) be a pair of nonempty closed subsets of X with \(A_{0}\ne \emptyset \) and such that (A, B) has the \( WP^{J}\)property and J is associated with (A, B). Let \(T:A\rightarrow B\) be a continuous and contraction singlevalued nonself mapping of Suzuki type. If \(T(A_0)\subseteq B_0,\) then T has a best proximity point in A.
Corollary 3.6
Let X be a complete bmetric space (with \(s\ge 1).\) Let (A, B) be a pair of nonempty closed subsets of X with \(A_{0}\ne \emptyset \) and such that (A, B) has the WPproperty. Let \(T:A\rightarrow B\) be a continuous and contraction singlevalued nonself mapping of Suzuki type. If \( T(A_0)\subseteq B_0,\) then T has a best proximity point in A.
4 Examples illustrating Theorem 3.2 and some comparisons
In this section, we will present some examples illustrating the concepts having been introduced so far. We will show a fundamental difference between Theorems 3.2 and 2.3. The examples will show that Theorem 3.2 is an essential generalization of Theorem 2.3. First, we present an example of generalized pseudodistance in metric spaces and bmetric spaces, respectively.
Example 4.1
Next, we present an example which illustrate Theorem 3.2. To compare our results with some wellknown best proximity point theorems in the literature, we start by giving an example where X is a metric space.
Example 4.2
 I.We show that pair (A, B) has the \(WP^{J}\)property. Indeed, we observe that \(dist(A,B)=1/2\) andHence, the pair (A, B) has the \(WP^{J}\)property.$$\begin{aligned}&A_{0}=\{x\in A: J(x,y)=dist(A,B) \ \text {for some} \ y\in B\}=\{1\},\\&B_{0}=\{y\in B: J(x,y)=dist(A,B)\ \text {for some} \ x\in A\}=\{3/2\}\text {.} \end{aligned}$$
 II.
We see that A is complete and by (4.2) we have \(T(A_{0})=T(\{1\})=3/2\in B_{0}\).
 III.We see that T is contraction of Banach type (i.e. \(J(Tx,Ty)\le rJ(x,y)\) for some \(r\in [0,1)\) and for all \(x,y\in A\)). Indeed, let \(r=1/2\) and let \(x,y\in A\) be arbitrary and fixed. We see that by (4.2)We consider the following cases: Case 1 If \(x,y\in [0,1]\), then by (4.2), \(Tx=Ty=3/2\). Now, by (4.1) we have:$$\begin{aligned} Tx\in E, \quad \forall x\in A . \end{aligned}$$(4.3)Case 2 If \(\{x,y\}\cap [3,4]\ne \emptyset \), then by (4.3), \(\{Tx,Ty\}\cap E=\{Tx,Ty\}\) which, by (4.1) gives \(J(Tx,Ty)=d(Tx,Ty)\) . Moreover, since \(\{x,y\}\cap E\ne \{x,y\}\), by (4.1), we obtain \(J(x,y)=4\). Hence$$\begin{aligned} J(Tx,Ty)=J(3/2,3/2)=0\le rd(x,y)=rJ(x,y). \end{aligned}$$(4.4)In consequence (4.4) and (4.5) implies that T is contraction of Banach type.$$\begin{aligned} J(Tx,Ty)=d(Tx,Ty)\le 1<2=r\cdot 4=rJ(x,y). \end{aligned}$$(4.5)
 IV.We see that T is contraction of Suzuki type (when \(s=1\), and T is single valued), i.e.for some \(r\in [0,1)\) and for all \(x,y\in A\). It is consequence of Step III.$$\begin{aligned} \frac{1}{1+r}J^{*}(x,Tx)\le J(x,y) \quad \text {implies} \quad J(Tx,Ty)\le rJ(x,y), \end{aligned}$$
 V.
We see that there exists a best proximity point of T. Indeed, for \(z=1\) we have \(d(z,T(z))=d(1,3/2)=1/2=dist(A,B)\).
Now, we will compare our results with two important results which existing in the literature. In 2013, Zhang et al. [18] proved the following theorem.
Theorem 4.1
 (i)
(A, B) has the WPproperty.
 (ii)
A is complete.
 (iii)
\(T(A_{0}) \subset B_{0}\).
In this same year, Suzuki [19] established the following interesting result.
Theorem 4.2
 (iv)There exists \(\alpha \in [0,1/2)\) such that$$\begin{aligned} d(Tx, Ty) \le \alpha (d(x, Tx)  dist(A,B))+\alpha (d(y,Ty)dist (A,B))), \end{aligned}$$
Remark 4.3
Proof
 II.We see that the mapping T is not contraction in the sense of Theorem 4.2. In this order, suppose the following condition holdsfor some \(\alpha \in [0,\frac{1}{2})\) and for all \(x,y\in A\). In particular, for \(x_{0}=\frac{1}{2}\) and \(y_{0}=\frac{7}{2}\) we have \( d(x_{0},Tx_{0})=d(\frac{1}{2},\frac{3}{2})=1\), \(d(y_{0},Ty_{0})=d(\frac{7}{2} ,\frac{5}{2})=1\) and \(d(Tx_{0},Ty_{0})=d(\frac{3}{2},\frac{5}{2})=1\). Hence, and by (4.7) we get$$\begin{aligned} d(Tx,Ty)\le \alpha [d(x,Tx)dist(A,B)]+\alpha [d(y,Ty)dist(A,B)], \end{aligned}$$(4.7)which is impossible.\(\square \)$$\begin{aligned} 1= & {} d(Tx_{0},Ty_{0})\le \alpha [d(x_{0},Tx_{0})dist(A,B)]+\alpha [d(y_{0},Ty_{0})dist(A,B)] \\= & {} \alpha [1\frac{1}{2}]+\alpha [1\frac{1}{2}]=\alpha <\frac{ 1}{2}, \end{aligned}$$
Remark 4.4
Remark 4.5
In 2013, Abkar and Gabeleh [20] proved that some recent results concerning the existence of best proximity points can be obtained from the same results in fixed point theory. The Authors used a bijective isometry \(g: A_{0} \rightarrow B_{0}\) such that \(d(x,g(x))=dist(A,B)\) (see Theorem 10 of [20]). It is worth noticing, that in our results such kind consideration is not true. Indeed, the fact \(J(x,y)=dist(A,B)\), \( J(x,y^{\prime })=dist(A,B)\) and \(P^{J}\)property or \(WP^{J}\)property does not imply that \(J(y,y^{\prime })=J(y^{\prime },y)=0\) and \(y=y^{\prime }\). That would be possible if \(max\{J(x,x),J(x,x)\}=0\). However, in general it does not hold. Moreover, in the literature there are no fixed point theorem for such kind contraction with respect to Jgeneralized pseudodistances. We obtain such kind result as immediate corollary from Theorem 3.2 (see Corollary 3.5).
Example 4.3
 I.We show that (A, B) has the \(WP^{J}\)property. Indeed, we observe that \(dist(A,B)=1\) andHence, the pair (A, B) has the \(WP^{J}\)property.$$\begin{aligned}&A_{0}=\{x\in A: J(x,y)=dist(A,B) \ \text {for some} \ y\in B\}=\{5\},\\&B_{0}=\{y\in B: J(x,y)=dist(A,B)\ \text {for some} \ x\in A\}=\{6\}. \end{aligned}$$
 II.
We see that A (4.9) we have \(T(A_{0})=\{6 \} \subseteq B_{0}\).
 III.We see that T is contraction of Suzuki type with \(s=2\), i.e. there exists \(r\in [0,1)\) so thatfor all \(x,y\in A\). Indeed, let \(r=\frac{1}{4}\) and let \(x,y \in A\) be arbitrary and fixed. Then by (4.9), we may consider the following cases:$$\begin{aligned} \frac{1}{1+r}J^{*}(x,Tx)\le J(x,y) \quad \text {implies} \quad {\mathcal {H}}^{J}(Tx,Ty)\le rJ(x,y), \end{aligned}$$(4.10)In consequence, using the symmetry of J, we conclude that the map T is contraction of Suzuki type.

Case 1. If \(Tx=[6,\frac{13}{2}] \cup \{7\}\), \(Ty=\{7\} \cup [\frac{15}{2},8]\), then \(x=0\), \(y=1\) and \({\mathcal {H}}^{J}(Tx,Ty)=1\). Moreover, by (4.8), we calculate: \(J(x,Tx)=J(0,[6,\frac{13}{2}] \cup \{7\})=36\); \(J^{*}(x,Tx)=\frac{1}{2}J(x,Tx)dist(A,B)=17\). Hence, \(\frac{1 }{1+r}J^{*}(x,Tx)=\frac{68}{5}\ge 1= J(x,y)\), which gives that in this case the condition (4.10) holds.

Case 2. If \(Tx=\{7\} \cup [\frac{15}{2},8]\), \(Ty=[6, \frac{13}{2}] \cup \{7\}\), then \(x=1\), \(y=0\) and \(\mathcal {H}^{J}(Tx,Ty)=1\). Moreover, by (4.8), we calculate: \(J(x,Tx)=J(1,\{7\} \cup [\frac{15}{2 },8])=36\); \(J^{*}(x,Tx)=\frac{1}{2}J(x,Tx)dist(A,B)=17\). Hence, \(\frac{1 }{1+r}J^{*}(x,Tx)=\frac{68}{5}\ge 1= J(x,y)\), which gives that in this case the condition (4.10) holds.

Case 3. If \(Tx=[6,\frac{13}{2}] \cup \{7\}\) and \(Ty=\{7\}\), then \(x=0\), \(y \in (0,1) \cup \{4\}\) and \(\mathcal {H}^{J}(Tx,Ty)=1\). Moreover, by (4.8), we calculate: \(J(x,y)=65\) if \(y\in (0,1)\) (since \( (0,1)\cap E=\emptyset \)) or \(J(x,y)=d(x,y)=16\) if \(y=4\); \(s\mathcal {H} ^{J}(Tx,Ty)=2 \le \frac{1}{4} J(x,y)\), which gives that in this case the condition (4.10) holds.

Case 4. If \(Tx=[6,\frac{13}{2}] \cup \{7\}\) and \(Ty=\{6\}\), then \(x=0\), \(y \in \{4,5\}\) and \(\mathcal {H}^{J}(Tx,Ty)=1\). Moreover, by (4.8), we calculate: \(J(x,y)=16\) (if \(y=4\)) or \(J(x,y)=25\) (if \(y=5\)); in both cases we get \(s\mathcal {H}^{J}(Tx,Ty)=2 \le \frac{1}{4} J(x,y)\), which gives that in this case the condition (4.10) holds.

Case 5. If \(Tx=\{7\}\cup [\frac{15}{2},8]\) and \(Ty=\{6\}\), then \(x=1\), \(y \in \{4,5\}\) and \(\mathcal {H}^{J}(Tx,Ty)=1\). Moreover, by (4.8), we calculate: \(J(x,y)=16\) (if \(y=4\)) or \(J(x,y)=25\) (if \(y=5\)); in both cases we get \(s\mathcal {H}^{J}(Tx,Ty)=2 \le \frac{1}{4} J(x,y)\), which gives that in this case the condition (4.10) holds.

Case 6. If \(Tx=\{7\}\cup [\frac{15}{2},8]\) and \(Ty=\{7\}\), then \(x=1\), \(y \in (0,1)\) and \(\mathcal {H}^{J}(Tx,Ty)=1\). Moreover, by (4.8), we calculate: \(J(x,y)=65\) (since \((0,1)\cap E=\emptyset \)); \(s\mathcal {H} ^{J}(Tx,Ty)=2 \le \frac{1}{4} J(x,y)\), which gives that in this case the condition (4.10) holds.

Case 7. If \(Tx=\{7\}, Ty=\{6\}\), then \(x \in (0,1)\), \(y \in \{4,5\}\) and \(\mathcal {H}^{J}(Tx,Ty)=1\). Moreover, by (4.8), we calculate: \(J(x,y)=65\) (since \((0,1)\cap E=\emptyset \)) and then \(s\mathcal {H}^{J}(Tx,Ty)=2 \le \frac{1}{4} J(x,y)\), which gives that in this case the condition (4.10) holds.

Case 8. If \(Tx=\{6\}, Ty=\{7\}\), then \(x\i \{4,5\}\), \(y \in (0,1) \) and \(\mathcal {H}^{J}(Tx,Ty)=1\). Moreover, by (4.8), we calculate: \(J(x,y)=65\) if \(y\in (0,1)\) (since \((0,1)\cap E=\emptyset \)) and then \(s\mathcal {H} ^{J}(Tx,Ty)=2 \le \frac{1}{4} J(x,y)\), which gives that in this case the condition (4.10) holds.

 IV.
We see that there exists a best proximity point of T. Indeed, for \(z=5\) we have \(d(z,Tz)=d(5,\{6\})=1=dist(A,B)\). Now, we will compare our result with another result for Jgeneralized pseudodistance [16].
Theorem 4.6
Remark 4.7
Let X, A, B, T, E and J be as in Example 4.3. We see that the map T is not contraction of Nadler type (in sense of Theorem 4.6.)
Proof
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