Best proximity point for the proximal nonexpansive mapping on the starshaped sets

Abstract

The existence of the best proximity point for the proximal nonexpansive mapping on starshaped sets is studied. Our results are established without the assumptions of continuity, affinity or the P-property. Finally, as applications of the theorems, analogs for the nonexpansive mappings are also given.

On the best proximity point for the proximal contractive and nonexpansive mappings on the starshaped sets

“… The purpose of this paper is to the best proximity point theorems for the proximal nonexpansive mapping on the starshaped sets by using a clever and simple method. The results improve and extend the recent results of Chen et al (Fixed Point Theory Appl 2015:19, 2015 ). It should be noted that, the complex method is used by Jianren Chen et al can be replaced by the clever and simple method presented in this paper.…”

“… 2013 ; Chen et al. 2015 ; Hussain and Hezarjaribi 2016 ; Shayanpour et al. 2016 ; Yongfu and Yao 2015 ; AlNemer et al.…”

“…(Chen et al. 2015 ) A nonempty subset A of a linear space X is called a p -starshaped set if there exists a point such that , for all , , and p is called the center of A .…”

Best proximity point for proximal Berinde nonexpansive mappings on starshaped sets

“…Chen [4] proved an interesting existence theorem of proximity points for proximal nonexpansive mappings under starshape sets A and B.…”

“…Next, let us show that x * is a best proximity point of T . Since T x * n ∈ B 0 for any n, there exist x n ∈ A 0 such that (4) x n − T x * n = D(A, B) . From D(A 0 , B 0 ) ≤ (1 − a n )x n + a n p − T n x * n = (1 − a n )x n + a n p − (1 − a n )T x * n − a n q ≤ (1 − a n ) x n − T x * n + a n p − q = D(A 0 , B 0 ) , which implies (5) (1 − a n )x n + a n p − T n x * n = D(A, B) .…”