Topological Vector Space-Valued Cone Metric Spaces and Fixed Point Theorems

Abstract

We develop the theory of topological vector space valued cone metric spaces with nonnormal cones. We prove three general fixed point results in these spaces and deduce as corollaries several extensions of theorems about fixed points and common fixed points, known from the theory of normed-valued cone metric spaces. Examples are given to distinguish our results from the known ones.

Tripled coincidence points for mixed comparable mappings in partially ordered cone metric spaces over Banach algebras

“…The results given by V. Berinde and M. Borcut [1] generalized and extended the works of Bhaskar and Lakshmikantham and Sabetghadam. In 2007, Huang and Zhang [7] introduced the concept of cone metric spaces as a generalization of general metric spaces, in which the distance d(x, y) of x and y is defined to be a vector in an ordered Banach space E and proved that the Banach contraction principle remains true in the setting of cone metric spaces. Since then, many fixed point results of the mappings with certain contractive property on cone metric spaces have been proved on the basis of the work of Huang and Zhang [7] (see [2,3,4,5,8,9,10,11,12,13,14,15,16,17,18,20] and the references therein). Among those works, the results of [15] attract much attention since the authors of [15] introduced the concept of cone metric spaces over Banach algebras by replacing Banach spaces with Banach algebras in order to generalize the Banach contraction principle to a more general form.…”

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.…”

“…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. Thus, proper generalizations when passing from norm-valued cone metric spaces to tvs-valued cone metric spaces can be obtained only in the case of nonnormal cones (for more details see [14]). …”

“…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 .…”