Esistenza, unicità e calcolo della soluzione di un sistema non lineare

Abstract

LetX be a point of the realn-dimensional Euclidean space ℝⁿ,G(X) a given vector withn real components defined in ℝ^u,U an unknown vector withs real components,K a known vector withs real components andA a given reals×n matrix of ranks. Assuming that, for every pair of pointsX ₁ , X₂of ℝⁿ,G(X) satisfies the conditions

$$(G(X_1 ) - G(X_2 ), X_1 - X_2 ) \geqslant o (X_1 - X_2 , X_1 - X_2 )$$

and

$$\left\| {(G(X_1 ) - G(X_2 )\left\| { \leqslant M} \right\|X_1 - X_2 )} \right\|$$

wherec andM are positive constants, we prove that a unique solution of the system

$$\left\{ \begin{gathered} G(X) + A ^T U = 0 \hfill \\ AX = K \hfill \\ \end{gathered} \right.$$

exists and we show a method for finding such a solution