In this paper, we prove the existence of best proximity points of cyclic contractions and generalised proximal rational contractions of first and second kind in the setting of complete b-metric space. Some results are also given in the form of corollaries.
In this article, we introduce cyclic relatively nonexpansive mappings with respect to orbits and prove that every cyclic relatively nonexpansive mapping with respect to orbits T satisfying T A ⊆ B , T B ⊆ A has a best proximity point. We also prove that Mann’s iteration process for a noncyclic relatively nonexpansive mapping with respect to orbits converges to a fixed point. These relatively nonexpansive mappings with respect to orbits need not be continuous. Some illustrations are given in support of our results.
In this paper, we prove a sufficient condition that every nonempty closed convex bounded pair M , N in a reflexive Banach space B satisfying Opial’s condition has proximal normal structure. We analyze the relatively nonexpansive self-mapping T on M ∪ N satisfying T M ⊆ M and T N ⊆ N , to show that Ishikawa’s and Halpern’s iteration converges to the best proximity point. Also, we prove that under relatively isometry self-mapping T on M ∪ N satisfying T N ⊆ N and T M ⊆ M , Ishikawa’s iteration converges to the best proximity point in the collection of all Chebyshev centers of N relative to M . Some illustrative examples are provided to support our results.
Every non-convex pair $(C, D)$ ( C , D ) may not have proximal normal structure even in a Hilbert space. In this article, we use cyclic relatively nonexpansive maps with respect to orbits to show the presence of best proximity points in $C\cup D$ C ∪ D , where $C\cup D$ C ∪ D is a cyclic T-regular set and $(C, D)$ ( C , D ) is a non-empty, non-convex proximal pair in a real Hilbert space. Moreover, we show the presence of best proximity points and fixed points for non-cyclic relatively nonexpansive maps with respect to orbits defined on $C\cup D$ C ∪ D , where C and D are T-regular sets in a uniformly convex Banach space satisfying $T(C)\subseteq C$ T ( C ) ⊆ C , $T(D)\subseteq D$ T ( D ) ⊆ D wherein the convergence of Kranoselskii’s iteration process is also discussed.
In this paper, we introduce the concept of coupled fixed point in partially ordered multiplicative metric space by proving some theorems for the existence and uniqueness of coupled fixed point. Also, we discuss the application to the existence and uniqueness of solution for a periodic boundary value problem.
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