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.

It is well known that sublinear operators and superlinear operators are two classes of important nonlinear operators in nonlinear analysis and dynamical systems. Since sublinear operators have only weak nonlinearity, this advantage makes it easy to deal with them. However, superlinear operators have strong nonlinearity, and there are only a few results about them. In this paper, the convergence of Picard iteration for the superlinear operator $A$ is obtained based on the conditions that the fixed point equation $Ax=x$ has a strong upper solution and a lower solution (or alternatively, an upper solution and a strong lower solution). Besides, the uniqueness of the fixed point of strongly increasing operators as well as the global attractivity of strongly monotone dynamical systems are also discussed. In addition, the main results are applied to monotone dynamics of superlinear operators and nonlinear integral equations. The method used in our work develops the traditional method of upper and lower solutions. Since a strong upper (upper) solution and a lower (strong lower) solution are easily checked, the obtained results are effective and practicable in the study of nonlinear equations and dynamical systems. The main novelty is that this paper provides new fixed point results for increasing superlinear operators and the obtained results are applied to strongly monotone systems to investigate their global attractivity.

In this paper, we introduced a new concept of generalized $\phi$-concave-convex operator and proved the existence and uniqueness of fixed points of such operators with mixed monotonicity. As consequences, several new fixed point results about mixed monotone operators with some concavity and convexity were gained. In addition, the main results were applied to nonlinear integral equations on unbounded regions. The research findings generalized and developed recent relevant results in the literature.