Abstract
In this manuscript, p-cyclic orbital ϕ-contraction map over closed, nonempty, convex subsets of a uniformly convex Banach space X possesses a unique best proximity point if the auxiliary function ϕ is strictly increasing. The given result unifies and extend some existing results in the related literature. We provide an illustrative example to indicate the validity of the observed result.
Highlights
Fixed point theory appeared first in the solution of the certain differential equations, see, e.g., Liouville [15] and Picard [18]
We introduce a notion called p-cyclic orbital φ-contraction, which is defined as follows
(ii) Let d(x, y) be the metric induced by the norm x − y, x, y ∈ X
Summary
Fixed point theory appeared first in the solution of the certain differential equations, see, e.g., Liouville [15] and Picard [18]. (See [1, Def. 1].) Let A and B be nonempty subsets of a metric space X and φ : [0, ∞) → [0, ∞) be a strictly increasing map. Let T be a p-cyclic orbital φ-contraction map satisfying (1) for some x ∈ Ai (1 i p). Ap (p ∈ N, p 2) be nonempty subsets of a metric space a p-cyclic orbital φ-contraction map satisfying (1) for some x ∈ Ai (1 i p), the following hold:. A p-cyclic orbital φ-contraction map satisfying for some x ∈ Ai (1 i p), the following hold:. Orbital φ-contraction map of type one satisfying there exists a fixed point of T , say, ξ ∈. This implies that ξ = T ξ, and ξ is a fixed point in Ai. Since T is p-cyclic, ξ∈
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