System of multivariate pseudo-contractive operator equations and the existence of solutions

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,… ,T_N in a Banach space: {[ T_1(x_1,x_2,… ,x_N)=x_1,; T_2(x_1,x_2,… ,x_N)=x_2,; T_3(x_1,x_2,… ,x_N)=x_3,; ⋯; T_N(x_1,x_2,… ,x_N)=x_N.; ]. 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.