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

: 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 ∆ -property to determine the existence of common best proximity point for such a pair of maps.


Preliminaries
Let ( , ρ) be a complete metric space and let CB( ) denote the class of all nonempty closed and bounded subsets of the nonempty set . θ ∈ is said to be a best proximity point (BPP, in short) of the multivalued map Γ : → CB( ) if ∆(θ, Γθ) = ρ(A, B). υ ∈ is called a fixed point of the multivalued map Γ : → CB( ) if υ ∈ Γυ.

2.
For two closed sets A, B, when A ∩ B = φ, we have ρ(A, B) = 0. In that case, a fixed point and a BPP are identical.
The following lemmas are significant in the present context. 2]). Let ( , ρ) be a metric space and A, B ∈ CB( ). Then In general, we may not obtain a point ξ ∈ B such that ρ(θ, ξ) ≤ H(A, 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 ( , ρ) be a metric space and A, B be two non-empty subsets of such that A B = φ. The pair (A, B) satisfies the P-property if and only if ρ(θ 1 ,

Definition 2 ([5]
). Let ( , ρ) be a metric space and A, B be two non-empty subsets of such that A B = φ. The pair (A, B) satisfies the weak P-property if and only if ρ(θ 1 , The following well known lemma will be used in the sequel. Lemma 3. If {θ n } is a sequence in a complete metric space ( , ρ) such that ρ(θ n+1 , θ n ) ≤ λρ(θ n , θ n−1 ) for all n ∈ N, where λ ∈ (0, 1), then {θ n } is a Cauchy sequence.
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 ∆-property and a multivalued Kannan-type contractive pair (MVKCP, in short) and prove an existence of CBPP result for that pair.

Common Best Proximity Point for MVBCP
In this section, first we define a MVBCP. The corresponding CBPP result follows. Definition 3. Let ( , ρ) be a metric space and A, B be two non-empty subsets of . The pair of mappings Ψ, Ω : for all θ, ξ ∈ . Theorem 1. Let ( , ρ) be a complete metric space and A, B be two non-empty closed subsets of such that A B = φ and that the pair (A, B) satisfies the weak P-property. Let the pair of mappings Ψ, Ω : A → CB(B) be a MVBCP such that Ψθ and Ωθ are compact for each θ ∈ A, and further Ψθ ⊆ B A and Ωθ ⊆ B A for all θ ∈ A B . Then Ψ and Ω have a CBPP.

Common Best Proximity Point for MVKCP
In this section, we define the concepts of weak ∆-property and a MVKCP. Combining these two concepts, we establish a CBPP result.
for any θ, ξ ∈ A B and λ ∈ [0, 1). It means that the composition map ΓoΩ : A B → A B is a Kannan map from A B to itself, which is a complete metric space.

Conclusions
The concepts of MVBCP, MVKCP and weak ∆-property have been introduced in this paper. Using weak P-property, a CBPP result has been proved for a MVBCP and using the weak ∆-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.