Pythagorean Property and Best-Proximity Point Theorems

Global Optimization for Quasi-Noncyclic Relatively Nonexpansive Mappings with Application to Analytic Complex Functions

“…By Assumption (iii), (2) and 3, we have Ty = y . Therefore, we get (x , y ) ∈ Prox A×B (T), and hence, the minimization problem (1) has a solution.…”

“…The necessary condition to guarantee the existence of x in A is satisfying d(Tx, x) = d(Tx, A) := inf{d(Tx, y) : y ∈ A}, which is called the best proximity point. After that, many authors studied and developed Fan's theorem by using different assumptions on various kinds of mappings in many directions; one can refer to [2][3][4][5][6][7][8].…”

A note on “Convergence and best proximity points for Berinde's cyclic contraction with proximally complete property”

“…Definition Let A and B be nonempty subsets of a metric space ( X , d ). A sequence ( x n ) in , with x 2 n ∈ A and x 2 n +1 ∈ B , is said to be a cyclically Cauchy sequence if and only if for any ϵ >0 there exists such that d ( x n , x m )< d ( A , B )+ ϵ , when n is even, m is odd and n , m ≤ N .…”

“…Definition A pair ( A , B ) of nonempty subsets of a metric space ( X , d ) is called semi‐sharp proximally if, for each ( x , y )∈ A × B , there exists at most such that .…”

“…Definition A pair ( A , B ) of nonempty subsets of a metric space ( X , d ) is proximally complete if any cyclically Cauchy sequence ( x n ) in the sequences ( x 2 n ) and ( x 2 n +1 ) have convergent subsequences in A and B , respectively.…”