Fixed Point in Topological Vector Space-Valued Cone Metric Spaces

Abstract

We obtain common fixed points of a pair of mappings satisfying a generalized contractive type condition in TVS-valued cone metric spaces. Our results generalize some well-known recent results in the literature.

Common fixed point theorems for four mappings on cone b-metric spaces over Banach algebras

“…Wherein, some authors extend cone metric spaces into several more general cases. The most famous ones of them are three cases as follows: tvs-cone metric spaces or vector spaces valued cone metric spaces (see [3,4,24]), cone b-metric spaces or cone metric type spaces (see [6,16,20]), and tvs-cone b-metric spaces (see [23]). We have the following diagram for these four classes of abstract metric spaces including cone metric spaces: cone metric space − −−− → tvs-cone metric space     cone b-metric space − −−− → tvs-cone b-metric space…”

Mizoguchi-Takahashi-type theorems in tvs-cone metric spaces

“…Du in [13] introduced the concept of topological vector space (tvs)-cone metric and tvs-cone metric space to improve and extend the concept of cone metric space in the sense of Huang and Zhang [1]. In [7,9,13,14] the authors tried to generalize this approach using cones in tvs instead of Banach spaces. However, it should be noted that an old result shows that if the underlying cone of an ordered tvs is solid and normal, then such tvs must be an ordered normed space.…”

“…Definition 1.1. [7,13,14] Let X be a nonempty set and (E, P ) be an ordered tvs. A vector-valued function d : X × X E is said to be a tvs-cone metric, if the following conditions hold:…”

“…Clearly, a cone metric space in the sense of Huang and Zhang is a special case of tvs-cone metric spaces when (X, d) is tvs-cone metric space with respect to a normal cone P. Definition 1.3. [7,13,14] Let (X, d) be a tvs-cone metric space, x X and let {x n } be a sequence in X. Then (i) {x n } tvs-cone converges to x whenever for every c E with θ ≪ c there is a natural number n 0 such that d(x n , x) ≪ c, for all n ≥ n 0 .…”

On set-valued contractions of Nadler type in tυs-G-cone metric spaces

“…Later, in the articles [16][17][18][19], the authors tried to generalize this approach by using cones in topological vector spaces tυs instead of Banach spaces. However, it should be noted that an old result shows that if the underlying cone of an ordered tυs is solid and normal, then such tυs must be an ordered normed space.…”