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

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