Some fixed points results for extended nonlinear contractions defined on $d-CS$ spaces

Abstract

In this paper we shall introduced the class of $d-CS$ spaces and so we shall obtained topological approach to large Kasahara spaces. This class include complete symmetric spaces, complete quasi $b$-metric spaces and complete $b$-spaces, but not include $d^*$-complete topological spaces and $d$-complete topological spaces. Further will be presented fixed theorem for nonlinear extended-contraction defined on this spaces, which generalize earlier results obtaned by Browder, Matkowski, Jachymski, Matkowski and \'Swiatkowski, Aran{\dj}elovi\'{c} and Ke\v{c}ki\'{c} and Alshehri, Aran{\dj}elovi\'c and Shahzad. Also, we obtain the fixed point theorem for nonlinear extended-contraction defined on quasi-metric spaces which extended recent result's of Pasicki, obtained for pseudo-metric spaces.