About nonlinear contractive conditions on cone metric spaces

TL;DR

This paper demonstrates that nonlinear contractive conditions in TVS-cone metric spaces can be translated into conditions in standard metric spaces, extending previous fixed point theory results.

Contribution

It provides a method to reduce nonlinear contractive conditions in TVS-cone metric spaces to those in usual metric spaces, broadening the applicability of fixed point results.

Findings

Reduction of TVS-cone metric space conditions to metric space conditions

Extension of fixed point results to broader classes of spaces

Unification of fixed point theories in different spaces

Abstract

In the present work we shall show that nonlinear contractive conditions on TVS-cone metric spaces can be reduced to nonlinear contractive conditions on usual metric spaces. Our results extend the results of Du (2010) [W.S. Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal. 72 (2010), 2259-2261] and results of Abdeljawad and Rezapour [T. Abdeljawad, Sh. Rezapour, Some fixed point results in TVS-cone metric spaces, Fixed Point Theory (to appear 2012)].