Common best proximity points theorem for four mappings in metric-type spaces

Abstract

In this article, we first give an existence and uniqueness common best proximity points theorem for four mappings in a metric-type space (X, D, K) such that D is not necessarily continuous. An example is also given to support our main result. We also discuss the unique common fixed point existence result of four mappings defined on such a metric space.

Existence of common best proximity point for single and multivalued non-self mappings

“…Suppose we have two non-self mappings f, g : A → B, the equations f x = x and gx = x are likely to have no common solution, known as common fixed point of the mappings f and g. In this situation, one wants to find approximate solution x such that the errors d(x, f x) and d(x, gx) are minimum for these two fixed point equations, called as common best proximity point of the mappings f and g. For detailed analysis on common best proximity point, we direct the reader to see [5,11,13,14,15,17,19].…”

Best Proximity Point Results for Generalized Θ-Contractions and Application to Matrix Equations

“…Thus, a best proximity pair theorem furnishes sufficient conditions for the existence of an optimal approximate solution x, known as a best proximity point of the mapping F, satisfying the condition that d(x, Fx) = d(A, B). Many authors established the existence and convergence of fixed and best proximity points under certain contractive conditions in different metric spaces (see e.g., [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] and references therein).…”