A discussion on best proximity point and coupled best proximity point in partially ordered metric spaces

Abstract

In this paper we establish some best proximity point results using generalized weak contractions with discontinuous control functions. The theorems are established in metric spaces with a partial order. We view the main problem in the paper as a problem of finding an optimal approximate solution of a fixed point equation. We also discuss several corollaries and give an illustrative example. We apply our result to obtain some coupled best proximity point results.MSC: 47H10; 54H10; 54H25

Relation-Theoretic Fixed Point Theorems for Generalized Weakly Contractive Mappings

“…We have already noted that the main result is an actual generalization of a previous result in [4]. The methodology here is comparable with several other similar results dealing with contractive inequalities noted in [3][4][5][6][7][8][9][10]. There are possibilities of applying relation-theoretic methods to other weakly contractive mappings.…”

“…Afterwards, the idea was generalized and extended in several ways by which a large amount of literature on the fixed point theory of weakly contractive mappings was produced. Some instance of these works are [3][4][5][6][7][8][9][10][11][12][13].…”

Prešić Type Nonself Operators and Related Best Proximity Results

“…In the literature, we have seen that the existence of the best proximity points has been investigated by several researchers by using different techniques, for example: Jleli and Samet [11] used α-ψ-proximal contraction to studied the best proximity points of single-valued mappings; Abkar and Gbeleh [12] used asymptotic cyclic contraction in their results; Abkar and Gbeleh [13] also proved the existence of best proximity points for multivalued nonself mappings satisfying contraction and nonexpansive condition along with P-property; Alghamdi et al [14] studied the best proximity point theorems in geodesic metric spaces; Choudhury et al [15] used the structure of partially-ordered metric spaces to discuss best proximity and couple best proximity points; Bari et al [16] used cyclic Meir-Keeler contraction in their discussion; Eldred and Veeramani [17] used cyclic proximal contraction to discuss the existence of best proximity point in metric space, and they further provided an algorithm to calculate a best proximity point over the structure of a uniformly-convex Banach space; Jacob et al [18] gave hybrid algorithms for nonself nonexpansive mappings and provided an iterative sequence of the algorithm, which converges to the proximity point of the mapping; Markin and Shahzad [19] studied the best proximity points of relatively u-continuous mappings; Sadiq Basha et al [20] discussed the existence of best proximity points of two mappings satisfying the min-max condition; Shatanawi and Pitea [21] used the notions of P-property and weak P-property in their best proximity theorems; Vetro [22] gave the existence and convergence theorems for best proximity points of the mappings satisfying the p-cyclic φ-contraction.…”

Best proximity points of roximal contractive mappings in partially ordered complete metric spaces

“…Therefore, it is concluded that best proximity point theorems generalize fixed point theorems in a natural way. For more works on the existence of best proximity points, we refer [1,3,8,14,17] and references therein.…”