Best proximity points for tricyclic contractions in a (S) convex metric spaces

Abstract

In this paper, we introduce the notion of (S) convex structure, thereby, we acquire a best proximity point theorem for tricyclic contractions in the framework of convex metric spaces.Mathematics Subject Classification: 47H09, 47H10, 54H25

A New Best Approximation Result in (S) Convex Metric Spaces

“…Theorem 2 (see [10]). Let A, B, and C be nonempty closed, bounded, and convex subsets of a (S) convex metric space (X, d, W) which has the (C) property; suppose A, B, and C are disjoint subsets of…”

Best Proximity Point for Generalized and S -Geraphty Contractions

“…Just like cyclic mappings, nonself mappings may not have fixed points and a best proximity point for a nonself mapping T : A ⟶ B is a point x ∈ A such that dðx, TxÞ = distðA, BÞ: Later on, the current authors ( [10]) introduced the notion of tricyclic mappings and the best proximity point thereof. Let A, B, and C be nonempty subsets of a metric space ðX, dÞ: A mapping T : A ∪ B ∪ C ⟶ A ∪ B ∪ C is said to be tricyclic provided that TðAÞ ⊆ B,TðBÞ ⊆ C, and TðCÞ ⊆ A.…”

“…ð4Þ Some results about the best proximity points of tricyclic mappings can be found in [10][11][12][13][14][15].…”

A New Result on Tricyclic Mappings

“…A mapping T: A ∪ B ∪ C ⟶ A ∪ B ∪ C is said to be tricyclic provided that T(A) ⊆ B, T(B) ⊆ C, and T(C) ⊆ A, where A, B, and C are nonempty subsets of a metric space (X, d), and a best proximity point of T is a point x ∈ A ∪ B ∪ C such that D(x, Tx, T 2 x) � D(A, B, C) where the mapping D: X × X × X ⟶ [0, +∞) is defined by D(x, y, z): � d(x, y) + d(y, z) + d(z, x) and D (A, B, C): � inf D(x, y, z): x ∈ A, y ∈ B and z ∈ C 􏼈 􏼉. Moreover, the mapping T is said to be a tricyclic contraction if it is tricyclic and verifies D(Tx, Ty, Tz) ≤ kD(x, y, z) + (1 − k)δ (A, B, C), for some k ∈ (0, 1) and for all (x, y, z) ∈ A × B × C. For detailed information, we refer to [1,2].…”