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The Banach Fixed-Point Theorem and the Iteration Method 19 where Uo E M. Each solution of (13) is called a fixed point of the operator A. A (The fixed-point theorem of Banach). , by definition, IIAu - Avll :s: kllu - vii and fixed k, 0 for all u, v E M, (15) :s: k < 1. Then, the following hold true: (i) Existence and uniqueness. , the operator A has exactly one fixed point U on the set M. (ii) Convergence of the iteration method. For each given Uo E M, the sequence (un) constructed by (14) converges to the unique solution U of equation (13).

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