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The purpose of this paper is to study the following system of nonlinear operator equations for N-variable pseudo-contractive mappings $$T_1, T_2,\ldots ,T_N$$ in a Banach space: $$\begin{aligned} {\left\{ \begin{array}{ll} T_1(x_{1},x_{2},\ldots ,x_{N})=x_1,\\ T_2(x_{1},x_{2},\ldots ,x_{N})=x_2,\\ T_3(x_{1},x_{2},\ldots ,x_{N})=x_3,\\ \ \ \ \ \ \ \ \cdots \\ T_N(x_{1},x_{2},\ldots ,x_{N})=x_N.\\ \end{array}\right. } \end{aligned}$$ The existence theorems of solutions are proved using a comprehensive analytical method. To get the expected results, the reflexivity of the product of Banach space and Opial’s condition of the product of Banach spaces are discussed. Meanwhile, the weak topology of the product of Banach spaces and normal structure of the product of Banach spaces are also discussed. The results of this article improve and extend the previous classical results given in the literature.

We prove some existence theorems for solutions of a certain system of multivariate nonexpansive operator equations and calculate the solutions by using the generalized Mann and Halpern iterative algorithms in uniformly convex and uniformly smooth Banach spaces. The results of this paper improve and extend the previously known ones in the literature.

Let K be a nonempty closed convex and bounded subset of a reflexive Banach space X. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A_{1}, A_{2},\ldots,A_{N}$\end{document}A1,A2,…,AN be N-variables monotone demi-continuous mappings from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$K^{N}$\end{document}KN into X. Then: (1) the system of multivariate variational inequalities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textstyle\begin{cases} \langle A_{1}(x_{1},x_{2},\ldots,x_{N}), y_{1}-x_{1} \rangle\geq0, &\forall y_{1} \in K,\\ \langle A_{2}(x_{1},x_{2},\ldots,x_{N}), y_{2}-x_{2} \rangle\geq0, &\forall y_{2} \in K,\\ \cdots\\ \langle A_{N}(x_{1},x_{2},\ldots,x_{N}), y_{N}-x_{N} \rangle\geq0, &\forall y_{N} \in K,\\ \end{cases} $$\end{document}{〈A1(x1,x2,…,xN),y1−x1〉≥0,∀y1∈K,〈A2(x1,x2,…,xN),y2−x2〉≥0,∀y2∈K,⋯〈AN(x1,x2,…,xN),yN−xN〉≥0,∀yN∈K, has a solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(x_{1}^{*},x_{2}^{*},\ldots,x_{N}^{*}) \in K^{N}$\end{document}(x1∗,x2∗,…,xN∗)∈KN; (2) the set of solutions of this system of multivariate variational inequalities is closed convex in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$K^{N}$\end{document}KN; (3) if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A_{1}, A_{2},\ldots,A_{N}$\end{document}A1,A2,…,AN are also strictly monotone, this system of multivariate variational inequalities has a unique solution.

The purpose of this paper is to present the definition of $$(\phi ,\varphi )$$ -contractive mapping and to discuss the relation of $$\psi$$ -contractive mappings and $$(\phi ,\varphi )$$ -contractive mappings. Furthermore, the generalized $$(\phi ,\varphi )$$ -contraction mapping principle has been proved without the uniqueness condition. Meanwhile, the generalized $$\psi$$ -contraction mapping principle has been obtained by using an ingenious method.