Some New Existence Theorems Of Coupled Quasi-solutions For A Class Of Operator Equations

In this paper, the semi-order method is used to study the existence of coupledquasi-solutions for a class of nonlinear operator equations Nx = A(x,x) in Banachspaces. Some new existence theorems are obtained. Main results are brie?y as follows:In Theorem 3.1, we discuss the existence of coupled quasi-solutions for operatorequation Nx = A(x,x) in Banach spaces;In Theorem 3.2, we further study the existence of coupled minimal-maximalquasi-solutions for equation Nx = A(x,x) in Banach spaces;Let N be the identical operator in Deduction 3.3. We obtain a special case ofTheorem 3.1. Suppose that A(x,x) = Ax in Deduction 3.4 and do not assume otherproperties of cones, then the result in Deduction 3.4 improves some results presentedin the paper of Zhang and Xie, Zhou and Yu, Liu and Wu. In Deduction 3.5, supposethat A(x,x) = Bx + Cx, where Cx is regarded as a disturbance of Bx. Hence, ourresults could be applied in disturbance equation problems.Theorems 3.1 and 3.2 generalize and improve the results of Theorems 2.1 and 2.2in the paper of Duan and Li, respectively. In discussing the boundary value problem ofmany ordinary differential equations, partial differential equations, integral equationsand abstract differential equations, we can consider the form of Nx = A(x,x) inproper function spaces, so that our results could be applied extensively.