Order-convergence and iterative interval methods

In the first part of this paper we introduce order-convergence in partially ordered spaces having lattice properties. Lipschitz assumptions are made for an operator equation Tx = Θ, and additional operators are then derived from the Lipschitz operators. We show how to solve the operator equation by means of these operators, using iterative methods which produce interval sequences, and we state some theorems on the inclusion and the existence of a solution of the equation as well as on the convergence of the interval sequences. In the second part of the paper we show how these theorems can be used to find the solution of a real equation, a nonlinear system of equations in $\text{R}$ⁿ and an algebraic eigenvalue problem.