Best Proximity Point Theorems for p-Cyclic Meir-Keeler Contractions

Abstract

We consider a contraction map T of the Meir-Keeler type on the union of p subsets A 1 , . . . , A p , p ≥ 2 , of a metric space X, d to itself. We give sufficient conditions for the existence and convergence of a best proximity point for such a map.

Best Proximity Points for Cyclical Contractive Mappings

“…The contractive conditions given in this paper behave differently from the ones used in [7] and [3], in the sense that the nonexpansive implication is nontrivial as we shall see in section 3.…”

“…In this paper we give analogous results to the above fixed point theorems using cyclical contractive conditions which does not force ∩ p i=1 A i = ∅ as in [7] and thereby we investigate the existence of best proximity point x ∈ A i satisfying d(x, T x) = d(A i , A i+1 ). The contractive conditions given in this paper behave differently from the ones used in [7] and [3], in the sense that the nonexpansive implication is nontrivial as we shall see in section 3.…”

Some results on best proximity points of cyclic alpha-psi contractions in Menger probabilistic metric spaces

“…The distances in-between adjacent subsets are assumed to be identical just to facilitate the exposition by simplifying the contractive condition to the form (2.1) so as to make less involved their associate calculations. Note that the distances inbetween adjacent subsets in non-expansive cyclic selfmappings are identical in uniformly convex Banach spaces [27].…”

“…It has also to be pointed out that the parallel background literature related to results on best proximity points and fixed points in cyclic mappings in metric and Banach spaces as well as topics related to common fixed points is exhaustive including studies of fixed point theory and applications in the fuzzy framework. See, for instance, [5,6,13,[17][18][19][20][21][22][23][24][25][26][27][31][32][33][34][35][36][37] as well as references therein.…”

Best proximity point results for Hardy–Rogers p-proximal cyclic contraction in uniform spaces

“…In [11], Eldred and Veeramani extended the cyclic contractive condition above to the case when A ∩ B is empty and proved the existence of best proximity point. For further results in this area, see [3,5,17,20,23,25,26,30].…”