Best Proximity Points for Alternative Maps

Abstract

Let ( X , d ) be a metric space and Ω i , i = 1 , 2 , … , m , be a nonempty subset of ( X , d ) . An operator T : ∪ 1 ≤ i ≤ m Ω i → ∪ 1 ≤ i ≤ m Ω i is called an alternative map if T ( Ω j ) ⊆ ∪ i ≠ j Ω i , j = 1 , 2 , … , m . In addition, if for any x, y ∈ ∪ 1 ≤ i ≤ m Ω i , there exists a constant α ∈ [ 0 , 1 ) such that d ( T x , T y ) ≤ α d ( x , y ) + ( 1 − α ) d ( Ω j , Ω k ) for some Ω j and Ω k ∈ { Ω i } i = 1 m with x ∈ Ω j and y ∈ Ω k , then we call T an alternative contraction. Moreover, if ( X , d ) has an alternative UC property and T is an alternative contraction, then the best proximity point of T exists.