Fixed point theorems for generalized quasi-contractions in cone $b$-metric spaces over Banach algebras without the assumption of normality with applications

Xu, Shaoyuan; Cheng, Suyu; Aleksic, Suzana

doi:10.22075/ijnaa.2017.1857.1483

Fixed point theorems for generalized quasi-contractions in cone $b$-metric spaces over Banach algebras without the assumption of normality with applications

Document Type : Research Paper

Authors

Shaoyuan Xu ¹

Suyu Cheng ²

Suzana Aleksic ³

¹ School of Mathematics and Statistics, Hanshan Normal University, Chaozhou, 521041, China

² Library, Hanshan Normal University, Chaozhou, 521041, China

³ Department of Mathematics and Informatics, Faculty of Science, University of Kragujevac, Radoja Domanovi'ca 12, 34000 Kragujevac, Serbia

10.22075/ijnaa.2017.1857.1483

Abstract

In this paper, we introduce the concept of generalized quasi-contractions in the setting of cone $b$-metric spaces over Banach algebras. By omitting the assumption of normality we establish common fixed point theorems for the generalized quasi-contractions with the spectral radius $r(\lambda)$ of the quasi-contractive constant vector $\lambda$ satisfying $r(\lambda)\in [0,\frac{1}{s})$ in the setting of cone $b$-metric spaces over Banach algebras, where the coefficient $s$ satisfies $s\ge 1$. As consequences, we obtain common fixed point theorems for the generalized $g$-quasi-contractions in the setting of such spaces. The main results generalize, extend and unify several well-known comparable results in the literature. Moreover, we apply our main results to some nonlinear equations, which shows that these results are more general than corresponding ones in the setting of $b$-metric or metric spaces.

Keywords

cone $b$-metric spaces over Banach algebras

non-normal cones

$c$-sequences

generalized quasi-contractions

Fixed point theorem