Global Optimization and Common Best Proximity Points for Some Multivalued Contractive Pairs of Mappings

Abstract

In this paper, we study a problem of global optimization using common best proximity point of a pair of multivalued mappings. First, we introduce a multivalued Banach-type contractive pair of mappings and establish criteria for the existence of their common best proximity point. Next, we put forward the concept of multivalued Kannan-type contractive pair and also the concept of weak $\Delta$-property to determine the existence of common best proximity point for such a pair of maps.

1. Preliminaries

Let $(\Im ,\rho )$ be a complete metric space and let $CB(\Im )$ denote the class of all nonempty closed and bounded subsets of the nonempty set ℑ. For $\mathcal{A},\mathcal{B}\in CB(\Im )$, the function $\mathcal{H}:CB(\Im )\times CB(\Im )\to [0,+\infty )$ defined by

$$\mathcal{H}(\mathcal{A},\mathcal{B})=max\{\underset{\xi \in \mathcal{B}}{sup}\Delta (\xi ,\mathcal{A}),\underset{\delta \in \mathcal{A}}{sup}\Delta (\delta ,\mathcal{B})\},$$

where $\Delta (\delta ,\mathcal{B})={inf}_{\xi \in \mathcal{B}}\rho (\delta ,\xi )$, is a metric on $CB(\Im )$.

For any two non-empty subsets $\mathcal{A},\mathcal{B}$ of the metric space $(\Im ,\rho )$, we shall use the following notations:

$${\mathcal{A}}_{\mathcal{B}}=\{\theta \in \mathcal{A}:\rho (\theta ,\xi )=\rho (\mathcal{A},\mathcal{B})\mathrm{for}\mathrm{some}\xi \in \mathcal{B}\},$$

$${\mathcal{B}}_{\mathcal{A}}=\{\xi \in \mathcal{B}:\rho (\theta ,\xi )=\rho (\mathcal{A},\mathcal{B})\mathrm{for}\mathrm{some}\theta \in \mathcal{A}\},$$

where $\rho (\mathcal{A},\mathcal{B})=inf\left\{\rho \right(\theta ,\xi ):\theta \in \mathcal{A},\xi \in \mathcal{B}\}$.

For $\mathcal{A},\mathcal{B}\in CB(\Im )$, we have

$$\rho (\mathcal{A},\mathcal{B})\le H(\mathcal{A},\mathcal{B}).$$

$\theta \in \Im$ is said to be a best proximity point (BPP, in short) of the multivalued map $\mathsf{\Gamma }:\Im \to CB(\Im )$ if $\Delta (\theta ,\mathsf{\Gamma }\theta )=\rho (\mathcal{A},\mathcal{B})$. $\upsilon \in \Im$ is called a fixed point of the multivalued map $\mathsf{\Gamma }:\Im \to CB(\Im )$ if $\upsilon \in \mathsf{\Gamma }\upsilon$.

Let $\mathsf{\Psi },\mathsf{\Omega }:\mathcal{A}\to CB\left(\mathcal{B}\right)$ be two multivalued maps. An element ${\theta }^{*}\in \mathcal{A}$ is said to be a common best proximity point (CBPP, in short) of $\mathsf{\Psi }$ and $\mathsf{\Omega }$ if and only if

$$\Delta ({\theta }^{*},\mathsf{\Psi }{\theta }^{*})=\rho (\mathcal{A},\mathcal{B})=\Delta ({\theta }^{*},\mathsf{\Omega }{\theta }^{*}).$$

Remark 1.

1. 

In the metric space $\left(CB\right(\Im ),\mathcal{H})$, $\theta \in \Im$ is a fixed point of Γ if and only if $\Delta (\theta ,\mathsf{\Gamma }\theta )=0$. In general, $\theta \in \mathsf{\Gamma }\xi$ if and only if $\Delta (\theta ,\mathsf{\Gamma }\xi )=0$ for any $\theta ,\xi \in \Im$.

2. 

For two closed sets $\mathcal{A},\mathcal{B}$, when $\mathcal{A}\cap \mathcal{B}\ne \varphi$, we have $\rho (\mathcal{A},\mathcal{B})=0$. In that case, a fixed point and a BPP are identical.

3. 

The function Δ is continuous in the sense that if ${\theta }_n\to \theta$ as $n\to +\infty$, then $\Delta ({\theta }_n,\mathcal{A})\to \Delta (\theta ,\mathcal{A})$ as $n\to +\infty$ for any $\mathcal{A}\subseteq \Im$.

4. 

A CBPP is an element at which the functions $\theta \to \Delta (\theta ,\mathsf{\Psi }\theta )$ and $\theta \to \Delta (\theta ,\mathsf{\Omega }\theta )$ achieve a global minimum, for $\Delta (\theta ,\mathsf{\Psi }\theta )\ge \rho (\mathcal{A},\mathcal{B})$ and $\Delta (\theta ,\mathsf{\Omega }\theta )\ge \rho (\mathcal{A},\mathcal{B})$ for all $\theta \in \mathcal{A}$.

The following lemmas are significant in the present context.

Lemma 1

([1,2]). Let $(\Im ,\rho )$ be a metric space and $\mathcal{A},\mathcal{B}\in CB(\Im )$. Then

1. 

$\Delta (\theta ,\mathcal{B})\le \rho (\theta ,\gamma )$ for any $\gamma \in \mathcal{B}$ and $\theta \in \Im$;

2. 

$\Delta (\theta ,\mathcal{B})\le \mathcal{H}(\mathcal{A},\mathcal{B})$ for any $\theta \in \mathcal{A}$.

Lemma 2

([3]). Let $\mathcal{A},\mathcal{B}\in CB(\Im )$ and let $\theta \in \mathcal{A}$. If $p>0$, then there exists $\xi \in \mathcal{B}$ such that

$$\rho (\theta ,\xi )\le \mathcal{H}(\mathcal{A},\mathcal{B})+p.$$

In general, we may not obtain a point $\xi \in \mathcal{B}$ such that

$$\rho (\theta ,\xi )\le \mathcal{H}(\mathcal{A},\mathcal{B}).$$

But when $\mathcal{B}$ is compact, then such a point ξ exists, i.e., $\rho (\theta ,\xi )\le \mathcal{H}(\mathcal{A},\mathcal{B}).$

The notion of P-property was introduced by Sankar Raj [4]. Further, the idea of weak P property was put forward by Zhang et al. [5] to improve the results of Caballero et al. [6] on Geraghty-contractions.

Definition 1

([4]). Let $(\Im ,\rho )$ be a metric space and $\mathcal{A},\mathcal{B}$ be two non-empty subsets of ℑ such that ${\mathcal{A}}_{\mathcal{B}}\ne \varphi$. The pair $(\mathcal{A},\mathcal{B})$ satisfies the P-property if and only if $\rho ({\theta }_1,{\xi }_1)=\rho (\mathcal{A},\mathcal{B})=\rho ({\theta }_2,{\xi }_2)$ implies $\rho ({\theta }_1,{\theta }_2)=\rho ({\xi }_1,{\xi }_2)$, where ${\theta }_1,{\theta }_2\in {\mathcal{A}}_{\mathcal{B}}$ and ${\xi }_1,{\xi }_2\in {\mathcal{B}}_{\mathcal{A}}$.

Definition 2

([5]). Let $(\Im ,\rho )$ be a metric space and $\mathcal{A},\mathcal{B}$ be two non-empty subsets of ℑ such that ${\mathcal{A}}_{\mathcal{B}}\ne \varphi$. The pair $(\mathcal{A},\mathcal{B})$ satisfies the weak P-property if and only if $\rho ({\theta }_1,{\xi }_1)=\rho (\mathcal{A},\mathcal{B})=\rho ({\theta }_2,{\xi }_2)$ implies $\rho ({\theta }_1,{\theta }_2)\le \rho ({\xi }_1,{\xi }_2)$, where ${\theta }_1,{\theta }_2\in \mathcal{A}$ and ${\xi }_1,{\xi }_2\in \mathcal{B}$.

The following well known lemma will be used in the sequel.

Lemma 3.

If $\left\{{\theta }_n\right\}$ is a sequence in a complete metric space $(\Im ,\rho )$ such that $\rho ({\theta }_{n+1},{\theta }_n)\le \lambda \rho ({\theta }_n,{\theta }_{n-1})$ for all $n\in \mathbb{N}$, where $\lambda \in (0,1)$, then $\left\{{\theta }_n\right\}$ is a Cauchy sequence.

BPPs under different types of contractive conditions have been studied in [7,8,9,10,11,12,13,14,15]. Moreover, BPPs for different kinds of multivalued mappings have been studied in [16,17,18,19]. Some more relevant works may be found in [20,21,22,23,24].

In this paper, we put forward the idea of multivalued Banach-type contractive pair (MVBCP, in short) and with the help of weak P property, establish conditions under which such a pair admits a CBPP. Next, we define the notion of weak $\Delta$-property and a multivalued Kannan-type contractive pair (MVKCP, in short) and prove an existence of CBPP result for that pair.

2. Common Best Proximity Point for MVBCP

In this section, first we define a MVBCP. The corresponding CBPP result follows.

Definition 3.

Let $(\Im ,\rho )$ be a metric space and $\mathcal{A},\mathcal{B}$ be two non-empty subsets of ℑ. The pair of mappings $\mathsf{\Psi },\mathsf{\Omega }:\mathcal{A}\to CB\left(\mathcal{B}\right)$ is said to be a MVBCP if there exists $\tau \in [0,1)$ such that

$$\mathcal{H}(\mathsf{\Omega }\theta ,\mathsf{\Psi }\xi )\le \tau \rho (\theta ,\xi )$$

for all $\theta ,\xi \in \Im$.

Theorem 1.

Let $(\Im ,\rho )$ be a complete metric space and $\mathcal{A},\mathcal{B}$ be two non-empty closed subsets of ℑ such that ${\mathcal{A}}_{\mathcal{B}}\ne \varphi$ and that the pair $(\mathcal{A},\mathcal{B})$ satisfies the weak P-property. Let the pair of mappings $\mathsf{\Psi },\mathsf{\Omega }:\mathcal{A}\to CB\left(\mathcal{B}\right)$ be a MVBCP such that $\mathsf{\Psi }\theta$ and $\mathsf{\Omega }\theta$ are compact for each $\theta \in \mathcal{A}$, and further $\mathsf{\Psi }\theta \subseteq {\mathcal{B}}_{\mathcal{A}}$ and $\mathsf{\Omega }\theta \subseteq {\mathcal{B}}_{\mathcal{A}}$ for all $\theta \in {\mathcal{A}}_{\mathcal{B}}$. Then Ψ and Ω have a CBPP.

Proof. 

Fix ${\theta }_0\in {\mathcal{A}}_{\mathcal{B}}$ and choose ${\xi }_0\in \mathsf{\Omega }{\theta }_0\subseteq {\mathcal{B}}_{\mathcal{A}}$. By the definition of ${\mathcal{B}}_{\mathcal{A}}$, we choose ${\theta }_1\in {\mathcal{A}}_{\mathcal{B}}$ such that

$$\rho ({\theta }_1,{\xi }_0)=\rho (\mathcal{A},\mathcal{B}).$$

(1)

If ${\xi }_0\in \mathsf{\Omega }{\theta }_1\cap \mathsf{\Psi }{\theta }_1$, then we have

$$\rho (\mathcal{A},\mathcal{B})\le \Delta ({\theta }_1,\mathsf{\Psi }{\theta }_1)\le \rho ({\theta }_1,{\xi }_0)=\rho (\mathcal{A},\mathcal{B}),\mathrm{since}{\xi }_0\in \mathsf{\Psi }{\theta }_1,$$

and

$$\rho (\mathcal{A},\mathcal{B})\le \Delta ({\theta }_1,\mathsf{\Omega }{\theta }_1)\le \rho ({\theta }_1,{\xi }_0)=\rho (\mathcal{A},\mathcal{B}),\mathrm{since}{\xi }_0\in \mathsf{\Omega }{\theta }_1.$$

Thus $\rho (\mathcal{A},\mathcal{B})=\Delta ({\theta }_1,\mathsf{\Psi }{\theta }_1)=\Delta ({\theta }_1,\mathsf{\Omega }{\theta }_1)$, i.e., ${\theta }_1$ is a CBPP of $\mathsf{\Psi }$ and $\mathsf{\Omega }$. Therefore, assume that ${\xi }_0\notin \mathsf{\Omega }{\theta }_1\cap \mathsf{\Psi }{\theta }_1$. Consider the case ${\xi }_0\notin \mathsf{\Psi }{\theta }_1$.

Since $\mathsf{\Psi }{\theta }_1$ is compact, by Lemma 2 and the definition of MVBCP, there exist ${\xi }_1\in \mathsf{\Psi }{\theta }_1\subseteq {\mathcal{B}}_{\mathcal{A}}$ and $\tau \in [0,1)$ such that

$$0<\Delta ({\xi }_0,\mathsf{\Psi }{\theta }_1)<\rho ({\xi }_0,{\xi }_1)\le \mathcal{H}(\mathsf{\Omega }{\theta }_0,\mathsf{\Psi }{\theta }_1)\le \tau \rho ({\theta }_0,{\theta }_1).$$

(2)

Since ${\xi }_1\in {\mathcal{B}}_{\mathcal{A}}$, there exists ${\theta }_2\in {\mathcal{A}}_{\mathcal{B}}$ such that

$$\rho ({\theta }_2,{\xi }_1)=\rho (\mathcal{A},\mathcal{B}).$$

(3)

From (1), (3) and weak P-property, we have that

$$\rho ({\theta }_1,{\theta }_2)\le \rho ({\xi }_0,{\xi }_1).$$

(4)

From (2) and (4), we have that

$$\rho ({\theta }_1,{\theta }_2)\le \rho ({\xi }_0,{\xi }_1)\le \tau \rho ({\theta }_0,{\theta }_1).$$

(5)

If ${\xi }_1\in \mathsf{\Omega }{\theta }_2\cap \mathsf{\Psi }{\theta }_2$, then like earlier we can show that ${\theta }_2$ is a CBPP of $\mathsf{\Omega }$ and $\mathsf{\Psi }$. Thus assume that ${\xi }_1\notin \mathsf{\Omega }{\theta }_2\cap \mathsf{\Psi }{\theta }_2$. Consider the case ${\xi }_1\notin \mathsf{\Omega }{\theta }_2$. Since $\mathsf{\Omega }{\theta }_2$ is compact, there exists ${\xi }_2\in \mathsf{\Omega }{\theta }_2$ such that

$$\begin{array}{cc}\hfill 0<\Delta ({\xi }_1,\mathsf{\Omega }{\theta }_2)<\rho ({\xi }_1,{\xi }_2)& \le \mathcal{H}(\mathsf{\Omega }{\theta }_2,\mathsf{\Psi }{\theta }_1)\hfill \\ \hfill & \le \tau \rho ({\theta }_1,{\theta }_2).\hfill \end{array}$$

(6)

Since ${\xi }_2\in \mathsf{\Omega }{\theta }_2\subseteq {\mathcal{B}}_{\mathcal{A}}$, there exists ${\theta }_3\in {\mathcal{A}}_{\mathcal{B}}$ such that

$$\rho ({\theta }_3,{\xi }_2)=\rho (\mathcal{A},\mathcal{B}).$$

(7)

From (3), (7) and weak P-property, we have that

$$\rho ({\theta }_2,{\theta }_3)\le \rho ({\xi }_1,{\xi }_2).$$

(8)

Also, from (5) and (6),

$$\rho ({\xi }_1,{\xi }_2)\le \tau \rho ({\xi }_0,{\xi }_1).$$

(9)

Continuing in this way, we obtain two sequences $\left\{{\theta }_n\right\}$ and $\left\{{\xi }_n\right\}$ in ${\mathcal{A}}_{\mathcal{B}}$ and ${\mathcal{B}}_{\mathcal{A}}$ respectively, satisfying

(B1)${\xi }_{2n}\in \mathsf{\Omega }{\theta }_{2n}\subseteq {\mathcal{B}}_{\mathcal{A}}$ and ${\xi }_{2n+1}\in \mathsf{\Psi }{\theta }_{2n+1}\subseteq {\mathcal{B}}_{\mathcal{A}}$,

(B2)$\rho ({\theta }_{n+1},{\xi }_n)=\rho (\mathcal{A},\mathcal{B})$,

(B3)$\rho ({\theta }_n,{\theta }_{n+1})\le \tau \rho ({\theta }_{n-1},{\theta }_n)$ and $\rho ({\xi }_n,{\xi }_{n+1})\le \tau \rho ({\xi }_{n-1},{\xi }_n)$,

for each $n=0,1,2,\dots$.

From (B3) and Lemma 3, we observe that $\left\{{\theta }_n\right\}$ and $\left\{{\xi }_n\right\}$ both are Cauchy sequences. Since $\mathcal{A}$ and $\mathcal{B}$ are closed subsets of a complete metric space, we conclude that $\mathcal{A}$ and $\mathcal{B}$ both are complete subspaces.

Hence, there exists $\theta \in \mathcal{A}$ and $\xi \in \mathcal{B}$ such that ${\theta }_n\to \theta$ and ${\xi }_n\to \xi$ as $n\to +\infty$.

We claim that $\mathsf{\Omega }{\theta }_n$ converges to $\mathsf{\Omega }\theta$. Indeed, if $m>n$, then

$$\begin{array}{cc}\hfill \mathcal{H}(\mathsf{\Omega }{\theta }_n,\mathsf{\Omega }\theta )& \le \mathcal{H}(\mathsf{\Omega }{\theta }_n,\mathsf{\Psi }{\theta }_m)+\mathcal{H}(\mathsf{\Psi }{\theta }_m,\mathsf{\Omega }\theta )\hfill \\ \hfill & \le \tau [\rho ({\theta }_n,{\theta }_m)+\rho ({\theta }_m,\theta )]\hfill \\ \hfill & \to 0\mathrm{as}n\to +\infty .\hfill \end{array}$$

Similarly, we can show that $\mathsf{\Psi }{\theta }_n$ converges to $\mathsf{\Psi }\theta$.

From (B2) we have that

$$\rho ({\theta }_{n+1},{\xi }_n)=\rho (\mathcal{A},\mathcal{B})$$

for each $n=0,1,2,\dots$.

This implies

$$\begin{array}{c}\hfill \underset{n\to +\infty }{lim}\rho ({\theta }_{n+1},{\xi }_n)=\rho (\theta ,\xi )=\rho (\mathcal{A},\mathcal{B}).\end{array}$$

(10)

Again, we claim that $\xi \in \mathsf{\Omega }\theta \cap \mathsf{\Psi }\theta$. Since ${\xi }_{2n}\in \mathsf{\Omega }{\theta }_{2n}$, we have

$$\begin{array}{cc}\hfill & \underset{n\to +\infty }{lim}\Delta ({\xi }_{2n},\mathsf{\Omega }\theta )\le \underset{n\to +\infty }{lim}\mathcal{H}(\mathsf{\Omega }{\theta }_{2n},\mathsf{\Omega }\theta )=0,\left(\mathrm{since}\mathsf{\Omega }{\theta }_n\mathrm{converges}\mathrm{to}\mathsf{\Omega }\theta \right)\hfill \\ \hfill \Rightarrow & \Delta (\xi ,\mathsf{\Omega }\theta )=0.\hfill \end{array}$$

Hence $\xi \in \mathsf{\Omega }\theta$.

Also since ${\xi }_{2n+1}\in \mathsf{\Psi }{\theta }_{2n+1}$, we have

$$\begin{array}{cc}\hfill & \underset{n\to +\infty }{lim}\Delta ({\xi }_{2n+1},\mathsf{\Psi }\theta )\le \underset{n\to +\infty }{lim}\mathcal{H}(\mathsf{\Psi }{\theta }_{2n+1},\mathsf{\Psi }\theta )=0,\left(\mathrm{since}\mathsf{\Psi }{\theta }_n\mathrm{converges}\mathrm{to}\mathsf{\Psi }\theta \right)\hfill \\ \hfill \Rightarrow & \Delta (\xi ,\mathsf{\Psi }\theta )=0.\hfill \end{array}$$

Hence $\xi \in \mathsf{\Psi }\theta$. Therefore,

$$\begin{array}{c}\hfill \xi \in \mathsf{\Omega }\theta \cap \mathsf{\Psi }\theta .\end{array}$$

(11)

Finally, using (10) and (11) we have that

$$\begin{array}{cc}\hfill & \rho (\mathcal{A},\mathcal{B})\le \Delta (\theta ,\mathsf{\Psi }\theta )\le \rho (\theta ,\xi )=\rho (\mathcal{A},\mathcal{B})\hfill \\ \hfill \Rightarrow & \Delta (\theta ,\mathsf{\Psi }\theta )=\rho (\mathcal{A},\mathcal{B}),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \rho (\mathcal{A},\mathcal{B})\le \Delta (\theta ,\mathsf{\Omega }\theta )\le \rho (\theta ,\xi )=\rho (\mathcal{A},\mathcal{B})\hfill \\ \hfill \Rightarrow & \Delta (\theta ,\mathsf{\Omega }\theta )=\rho (\mathcal{A},\mathcal{B}),\hfill \end{array}$$

Hence $\theta$ is a CBPP of $\mathsf{\Omega }$ and $\mathsf{\Psi }$. □

Next, we present an example in which the pair $(\mathcal{A},\mathcal{B})$ satisfies only the weak P-property but not the P-property.

Example 1.

Consider $\Im ={\mathbb{R}}^2$ with the Euclidean metric ρ. Let $\mathcal{A}=\left\{\right(-5,0),(0,1),(5,0\left)\right\}$ and $\mathcal{B}=\{(\theta ,\xi ):\xi =2+\sqrt{2-{\theta }^2},\theta \in [-\sqrt{2},\sqrt{2}]\}$. Then $\rho (\mathcal{A},\mathcal{B})=\sqrt{3}$ and ${\mathcal{A}}_{\mathcal{B}}=\left\{(0,1)\right\}$, ${\mathcal{B}}_{\mathcal{A}}=\{(\sqrt{2},2),(-\sqrt{2},2)\}$.

Define a pair of multivalued maps $\mathsf{\Omega },\mathsf{\Psi }:\mathcal{A}\to CB\left(\mathcal{B}\right)$ in the following manner:

$$\mathsf{\Omega }(-5,0)=\left\{(0,2+\sqrt{2})\right\},\mathsf{\Omega }(0,1)=\{(-\sqrt{2},2),(0,2+\sqrt{2})\},\mathsf{\Omega }(5,0)=\{(-1,3),(1,3)\},$$

and

$$\mathsf{\Psi }(-5,0)=\{(-\sqrt{2},2),(-1,3)\},\mathsf{\Psi }(0,1)=\left\{(\sqrt{2},2)\right\},\mathsf{\Psi }(5,0)=\{(\sqrt{2},2),(1,3)\}.$$

By routine calculations, it is easy to check that the condition

$$\mathcal{H}(\mathsf{\Omega }\theta ,\mathsf{\Psi }\xi )\le \tau \rho (\theta ,\xi )$$

is satisfied for all $\theta ,\xi \in \Im$ and for $\tau =\frac{19}{20}\in [0,1)$.

Thus the pair $\mathsf{\Psi },\mathsf{\Omega }$ is a MVBCP.

Finally, we observe that

$$\rho ((0,1),(\sqrt{2},2))=\rho ((0,1),(-\sqrt{2},2))=\sqrt{3}=\rho (\mathcal{A},\mathcal{B}),$$

but

$$\rho ((0,1),(0,1))=0<\rho ((\sqrt{2},2),(-\sqrt{2},2))=2\sqrt{2}.$$

Thus, $(\mathcal{A},\mathcal{B})$ satisfies weak P-property, but not the P-property. Therefore, all conditions of Theorem 1 are satisfied and since $\Delta ((0,1),\mathsf{\Psi }(0,1))=\Delta ((0,1),\mathsf{\Omega }(0,1))=\sqrt{3}=\rho (\mathcal{A},\mathcal{B})$, we conclude that $(0,1)$ is a CBPP of Ψ and Ω.

3. Common Best Proximity Point for MVKCP

In this section, we define the concepts of weak $\Delta$-property and a MVKCP. Combining these two concepts, we establish a CBPP result.

Definition 4.

Consider the metric space $\left(CB\right(\Im ),\mathcal{H})$ and let $\mathcal{A},\mathcal{B}$ be two non-empty subsets in $CB(\Im )$ such that ${\mathcal{A}}_{\mathcal{B}}\ne \varphi$. The pair $(\mathcal{A},\mathcal{B})$ is said to have the weak Δ-property if and only if $\Delta (\theta ,\mathcal{U})=\rho (\mathcal{A},\mathcal{B})=\Delta (\xi ,\mathcal{V}))$ implies $\rho (\theta ,\xi )\le \mathcal{H}(\mathcal{U},\mathcal{V})$, for all $\theta ,\xi \in {\mathcal{A}}_{\mathcal{B}}$ and $\mathcal{U},\mathcal{V}\subseteq {\mathcal{B}}_{\mathcal{A}}$.

Definition 5.

Let $(\Im ,\rho )$ be a metric space and $\mathcal{A},\mathcal{B}$ be two non-empty subsets of ℑ. The pair of mappings $\mathsf{\Psi },\mathsf{\Omega }:\mathcal{A}\to CB\left(\mathcal{B}\right)$ (Ψ and Ω may be identical) is said to be a multivalued Kannan-type contractive pair (MVKCP, in short) if there exists $\lambda \in [0,1)$ such that

$$\mathcal{H}(\mathsf{\Omega }\theta ,\mathsf{\Psi }\xi )\le \frac{\lambda }{2}[\Delta (\theta ,\mathsf{\Omega }\theta )+\Delta (\xi ,\mathsf{\Psi }\xi )-2\rho (\mathcal{A},\mathcal{B})]$$

(12)

for all $\theta ,\xi \in \Im$.

Remark 2.

If $\mathsf{\Psi },\mathsf{\Omega }$ is an MVKCP, the condition (12) is satisfied when $\mathsf{\Psi }=\mathsf{\Omega }$ as well.

Definition 6

([25]). Let $(\Im ,\rho )$ be a metric space and R be a self-map on ℑ. R is said to be a Kannan mapping if there exists $0\le \lambda <\frac{1}{2}$ such that

$$\rho (R\theta ,R\xi )\le \lambda \left\{\rho \right(\theta ,R\theta )+\rho (\xi ,R\xi \left)\right\},$$

for all $\theta ,\xi \in \Im$.

Remark 3.

If $(\Im ,\rho )$ is a complete metric space, then a Kannan mapping on ℑ possesses a unique fixed point.

Now we present the main result of this section.

Theorem 2.

Let $(\Im ,\rho )$ be a complete metric space and $\mathcal{A},\mathcal{B}$ be two non-empty closed subsets of ℑ such that ${\mathcal{A}}_{\mathcal{B}}\ne \varphi$ and that the pair $(\mathcal{A},\mathcal{B})$ satisfies the weak Δ-property. Let the pair of mappings $\mathsf{\Psi },\mathsf{\Omega }:\mathcal{A}\to CB\left(\mathcal{B}\right)$ be a MVKCP such that $\mathsf{\Psi }\theta \subseteq {\mathcal{B}}_{\mathcal{A}}$ and $\mathsf{\Omega }\theta \subseteq {\mathcal{B}}_{\mathcal{A}}$ for all $\theta \in {\overline{\mathcal{A}}}_B$. Then Ψ and Ω have a CBPP.

Proof. 

Define the map $\mathsf{\Gamma }:\mathsf{\Omega }\left({\overline{\mathcal{A}}}_B\right)\to {\mathcal{A}}_{\mathcal{B}}$ by

$$\mathsf{\Gamma }\left(S\right)=\{\theta \in {\mathcal{A}}_{\mathcal{B}}:\Delta (\theta ,S)=\rho (\mathcal{A},\mathcal{B})\},$$

(13)

for all $S\in {\overline{\mathcal{A}}}_B$. The map $\mathsf{\Gamma }$ is well defined, for if $\mathsf{\Gamma }\left(S\right)={\theta }_1$ and $\mathsf{\Gamma }\left(S\right)={\theta }_2$, then $\Delta ({\theta }_1,S)=\rho (\mathcal{A},\mathcal{B})$ and $\Delta ({\theta }_2,S)=\rho (\mathcal{A},\mathcal{B})$. By weak $\Delta$-property, we have $\rho ({\theta }_1,{\theta }_2)\le \mathcal{H}(S,S)=0$, i.e., ${\theta }_1={\theta }_2$.

From (13), we have $\Delta \left(\mathsf{\Gamma }\right(\mathsf{\Omega }\theta ),\mathsf{\Omega }\theta )=\rho (\mathcal{A},\mathcal{B})$ and $\Delta \left(\mathsf{\Gamma }\right(\mathsf{\Omega }\xi ),\mathsf{\Omega }\xi )=\rho (\mathcal{A},\mathcal{B})$ for any $\theta ,\xi \in {\overline{A}}_B$.

Again, using the weak $\Delta$-property, we have

$$\begin{array}{cc}\hfill \rho \left(\mathsf{\Gamma }\right(\mathsf{\Omega }\theta ),\mathsf{\Gamma }(\mathsf{\Omega }\xi \left)\right)& \le \mathcal{H}(\mathsf{\Omega }\theta ,\mathsf{\Omega }\xi )\hfill \\ \hfill & \le \frac{\lambda }{2}[\Delta (\theta ,\mathsf{\Omega }\theta )+\Delta (\xi ,\mathsf{\Omega }\xi )-2\rho (\mathcal{A},\mathcal{B})]\hfill \\ \hfill & \le \frac{\lambda }{2}[\rho (\theta ,\mathsf{\Gamma }\left(\mathsf{\Omega }\theta \right))+\Delta (\mathsf{\Gamma }\left(\mathsf{\Omega }\theta \right),\mathsf{\Omega }\theta )+\rho (\xi ,\mathsf{\Gamma }\left(\mathsf{\Omega }\xi \right))+\Delta (\mathsf{\Gamma }\left(\mathsf{\Omega }\xi \right),\mathsf{\Omega }\xi )-2\rho (\mathcal{A},\mathcal{B})]\hfill \\ \hfill & =\frac{\lambda }{2}[\rho (\theta ,\mathsf{\Gamma }\left(\mathsf{\Omega }\theta \right))+\rho (\xi ,\mathsf{\Gamma }\left(\mathsf{\Omega }\xi \right))-2\rho (\mathcal{A},\mathcal{B})],\hfill \end{array}$$

for any $\theta ,\xi \in {\overline{\mathcal{A}}}_B$ and $\lambda \in [0,1)$.

It means that the composition map $\mathsf{\Gamma }o\mathsf{\Omega }:{\overline{\mathcal{A}}}_B\to {\overline{\mathcal{A}}}_B$ is a Kannan map from ${\overline{\mathcal{A}}}_B$ to itself, which is a complete metric space.

Thus, $\mathsf{\Gamma }o\mathsf{\Omega }$ has a unique fixed point ${\theta }_1$, i.e., $\mathsf{\Gamma }o\mathsf{\Omega }\left({\theta }_1\right)={\theta }_1\in {\mathcal{A}}_{\mathcal{B}}$, which implies that $\Delta ({\theta }_1,\mathsf{\Omega }\left({\theta }_1\right))=\rho (\mathcal{A},\mathcal{B})$.

Similarly, we can define $\Pi :\mathsf{\Psi }\left({\overline{\mathcal{A}}}_B\right)\to {\mathcal{A}}_{\mathcal{B}}$ and obtain a unique fixed point ${\theta }_2$ of $\Pi o\mathsf{\Psi }$ and consequently $\Delta ({\theta }_2,\mathsf{\Psi }\left({\theta }_2\right))=\rho (\mathcal{A},\mathcal{B})$.

Using the weak $\Delta$-property, we have that

$$\begin{array}{cc}\hfill \rho ({\theta }_1,{\theta }_2)& \le \mathcal{H}(\mathsf{\Omega }{\theta }_1,\mathsf{\Psi }{\theta }_2)\hfill \\ \hfill & \le \frac{\lambda }{2}[\Delta ({\theta }_1,\mathsf{\Omega }{\theta }_1)+\Delta ({\theta }_2,\mathsf{\Psi }{\theta }_2)-2\rho (\mathcal{A},\mathcal{B})]\hfill \\ \hfill & =0,\hfill \end{array}$$

which implies that ${\theta }_1={\theta }_2=\theta$ (say).

Therefore, $\Delta (\theta ,\mathsf{\Omega }(\theta \left)\right)=\Delta (\theta ,\mathsf{\Psi }(\theta \left)\right)=\rho (\mathcal{A},\mathcal{B})$. Thus $\theta$ is a CBPP of $\mathsf{\Omega }$ and $\mathsf{\Psi }$. □

4. Conclusions

The concepts of MVBCP, MVKCP and weak $\Delta$-property have been introduced in this paper. Using weak P-property, a CBPP result has been proved for a MVBCP and using the weak $\Delta$-property, a similar result has been established for a MVKCP. The current study is interesting because the proof of our main theorem in Section 2 provides us with a scheme on how to find a CBPP for two multivalued maps. An application of the same has also been discussed in Example 1.