Fixed point results for nonlinear contractions of Perov type in abstract metric spaces with applications

In this paper, we present some common fixed point results for $g$-quasi-contractions of Perov type in cone $b$-metric spaces without the assumption of continuity. Besides, by constructing a non-expansive mapping from a real Banach algebra $\mathcal{A}$ to $\mathcal{B}(\mathcal{A})$, the space of all of its bounded linear operators, we explore the relationship between the results for the mappings of Perov type on cone metric (cone $b$-metric) spaces and that for the corresponding mappings on cone metric (cone $b$-metric) spaces over Banach algebras. As consequences, without the assumption of normality, we obtain common fixed point theorems for generalized $g$-quasi-contractions with the spectral radius $r(\lambda )$ of the $g$-quasi-contractive constant vector $\lambda$ satisfying $r(\lambda )\in [0,\frac{1}{s})$ (where $s\ge 1$) in the setting of cone $b$-metric spaces over Banach algebras. In addition, we also get some fixed point theorems for nonlinear contractions of Perov type in the setting of cone normed spaces. The main results generalize, extend and unify several well-known comparable results in the literature. Finally, we apply our main results to some nonlinear equations.