Mappings of Generalized Condensing Type in Metric Spaces with Busemann Convex Structure
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Mappings of Generalized Condensing Type in Metric Spaces with Busemann Convex Structure Moosa Gabeleh1 · Hans-Peter A. Künzi2 Received: 5 September 2019 / Accepted: 30 November 2019 © Iranian Mathematical Society 2020
Abstract Related to earlier work on existence results of best proximity points (pairs) for cyclic (noncyclic) condensing operators of integral type in the setting of reflexive Busemann convex spaces, in this paper, we introduce another class of cyclic (noncyclic) condensing operators and study the existence of best proximity points (pairs) as well as coupled best proximity points (pairs) in such spaces. Then, we present an application of our main existence result to study the existence of an optimal solution for a system of differential equations. Keywords Coupled best proximity point (pair) · Cyclic (noncyclic) condensing operator · Optimum solution · Busemann convex space Mathematics Subject Classification 53C22 · 47H09 · 34A12
1 Introduction Let A be a nonempty subset of a metric space (X , d) and T : A → X be a mapping. In this case, the fixed point equation T x = x may not have a solution. One of the interesting problems is to study the existence of approximate solutions of the equation T x = x in the absence of fixed points of the mapping T . A point x ∈ A is said to be an approximate solution of the equation T x = x provided that x is close to T x.
Communicated by Ali Abkar.
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Hans-Peter A. Künzi [email protected] Moosa Gabeleh [email protected]; [email protected]
1
Department of Mathematics, Ayatollah Boroujerdi University, Boroujerd, Iran
2
Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa
123
Bulletin of the Iranian Mathematical Society
In 1969, Ky Fan proved the following interesting existence theorem. Theorem 1.1 [15] Let A be a nonempty, compact, and convex subset of a locally convex Hausdorff topological vector space X , and T : A → X be a continuous mapping. Then, there exists a point x ∈ A, such that d(x, T x) = dist({T x}, A), where d is the semi-metric induced by a continuous semi-norm defined on X . Now, let A and B be two nonempty disjoint subsets of a metric space (X , d) and T : A → B be a non-self-mapping. Then, the absolute optimal approximate solution of the equation T x = x is a point p ∈ A for which d( p, T p) = dist(A, B). The point p is said to be a best proximity point for the mapping T . Notice that if A ∩ B = ∅, then the point p is an exact solution of the equation T x = x. Existence of best proximity points for various classes of non-self-mappings is an interesting subject in nonlinear functional analysis which has recently attracted the attention of many authors; see, for instance, [1,3,6,10,11]. In the present paper, we mainly focus on the study of best proximity points for certain classes of mappings T : A ∪ B → A ∪ B for which T (A) ⊆ B and T (B) ⊆ A. Such mappings are called cyclic mappings. Likewise, if T (A) ⊆ A and T (B) ⊆ B, then T is said to be a noncyclic mapping. For the noncycl
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