Generalizations of the Intermediate Value Theorem for Approximating Fixed Points and Zeros of Continuous Functions

Generalizations of the traditional intermediate value theorem are presented. The obtained generalized theorems are particular useful for the existence of solutions of systems of nonlinear equations in several variables as well as for the existence of fixed points of continuous functions. Based on the corresponding criteria for the existence of a solution emanated by the intermediate value theorems, generalized bisection methods for approximating fixed points and zeros of continuous functions are given. These bisection methods require only algebraic signs of the function values and are of major importance for tackling problems with imprecise (not exactly known) information.

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