A Newton-type method and its application (Q885591)

This article deals with the following Newton-like approximations \[ x_{n+1} = x_n - (F'(x_n)^{-1}[F(x_n) + G(x_n)] \] to the solution \(x^*\) of the equation \(F(x) + G(x) = 0\). It is assumed that \(F\) is Gâteaux differentiable, the derivative \(F'\) is piecewise hemicontinuous, \(G\) is Lipschitzian, and the corresponding constants are sufficiently small. Under these assumptions, it is proved the existence and uniqueness of the solution \(x^*\) for the equation \(F(x) + G(x) = 0\), and the geometrical convergence of \(x_n\) to \(x^*\). As application, the Urysohn integral equation \[ x(t) + \int_\Omega K(t,s,x(s)) \, ds + g(t,x(t)) = 0, \quad x(t) \in C(\Omega), \] is considered. An illustrative numerical example is also presented.