Abstract
AbstractWe 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.
Highlights
Ordered normed spaces and cones have applications in applied mathematics, for instance, in using Newton’s approximation method 1–4 and in optimization theory 5
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
Summary
Ordered normed spaces and cones have applications in applied mathematics, for instance, in using Newton’s approximation method 1–4 and in optimization theory 5. Huang and Zhang 7 reintroduced such spaces under the name of cone metric spaces but went further, defining convergent and Cauchy sequences in the terms of interior points of the underlying cone. These and other authors see, e.g., 8–22 proved some fixed point and common fixed point theorems for contractive-type mappings in cone metric spaces and cone uniform spaces. 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
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