On interval operators obtained by splitting the Lipschitz matrix (Q1114342)

The author presents a class of interval operators M with which he constructs iterative methods to enclose the set \(X^*=\{x\in D| \quad 0\in [\underline{g}(x),\overline{g}(x)]\}.\) Here each \(g\in \{\underline{g},\overline{g})\) is a function which maps \(D\subseteq {\mathbb{R}}^ n\) into \({\mathbb{R}}^ n\) and which satisfies an interval Lipschitz condition \(g(x_ 1)-g(x_ 2)\in A(x_ 1-x_ 2),\) \(x_ 1,x_ 2\in D\) with the same \(n\times n\) interval matrix A. The operators M are based on interval splittings B-C of A. Properties of M and of the corresponding classes of iterations are derived. Among these properties are the inclusion monotonicity of the sequence of iterates and its convergence to a limit which encloses \(X^*\). Stable fixed points are considered as well as special splittings (midpoint splitting, Gauss- Seidel splitting, Jacobi-splitting). Comparisons between these splittings and examples are given.