Principles of Transpose in the Fixed Point Theory for Cone Metric Spaces

Milan Tasković

Mathematica Moravica, Tome 15 (2011) no. 2.

Résumé

This paper presents new principles of transpose in the fixed point theory as for example: Let $X$ be a nonempty set and let $\mathfrak{C}$ be an arbitrary formula which contains terms $x,y \in X$, $\leq$, $+$, $\preccurlyeq$, $\oplus$, $T: X \to X$, and $\rho$. Then, as assertion of the form: For every $T$ and for every $\rho(x,y)\in \mathbb{R}_{+}^{0} := [0,+\infty)$ the following fact (A) $\qquad \mathfrak{C}(x,y\in X,\leq, +, T, \rho)$ implies $T$ has a fixed point is a theorem if and only if the assertion of the form: For every $T$ and for every $\rho(x,y)\in C$, where $C$ is a cone of the set $G$ of all cones, the following fact in the form (TA) $\qquad \mathfrak{C}(x,y\in X, \preccurlyeq, T, \rho)$ implies $T$ has a fixed point is a theorem. Applications of the principles of transpose in nonlinear functional analysis and fixed point theory are numerous.

Détail

Mots-clés : Coincidence points, common fixed points, cone metric spaces, Principles of Transpose, Banach’s contraction principle, numerical and nonnumerical distances, characterizations of contractive mappings, Banach’s mappings, nonnumerical transversals