Best proximity point of generalized F-proximal non-self contractions

A note on the paper "Best proximity point of generalized $F$-proximal non-self contractions

“…Moreover, if a + e + L ≤ 1 then T has a unique fixed point. Now, we recall the definition of generalized F -proximal contraction of the first kind from [2] as follows.…”

“…Definition 2.5. [2] Let (X, d) be a metric space and A, B be two non-empty subsets of X. Then B is said to be approximatively compact with respect to A if for every sequence {y n } of B satisfying d(x, y n ) → d(x, B) as n → ∞ for some x ∈ A has a convergent subsequence where d(x, B) = inf {d(x, y) : y ∈ B}.…”

“…By using definition 2.4, Beg et al proved the following best proximity point result in [2] Theorem 2.8. [2] Let A, B be non-void, closed subsets of a complete metric space (X, d) such that B is approximatively compact with respect to A.…”

“…By using definition 2.4, Beg et al proved the following best proximity point result in [2] Theorem 2.8. [2] Let A, B be non-void, closed subsets of a complete metric space (X, d) such that B is approximatively compact with respect to A. Let T : A → B satisfies the following conditions: i) T (A 0 ) ⊆ B 0 and (A, B) satisfies the p-property; ii) T is a generalized F -proximal contraction of the first kind.…”

“…Then in the year 2014, Cosentino and Vetro [3] introduced the notion of F -contraction of Hardy-Rogerstype as a generalization of F -contraction mapping and showed the existence of fixed points for such class of mappings in a complete metric space. Then in the year 2021, Beg et al [2] introduced the notion of generalized F -proximal contraction of the first kind and generalized F -proximal contraction of the second kind respectively for non-self mappings and investigated about unique best proximity point for these two classes of non-self mappings to generalize the fixed point result of Cosentino and Vetro [3]. In this short note, we show that the existence of best proximity point for generalized F -proximal contraction of the first kind follows from the same conclusion in fixed point theory.…”