A semilocal convergence of a secant-type method for solving generalized equations
Let \(X,Y\) be two Banach spaces and let \(f:X\rightarrow Y\) be continuous and \(G:X\rightarrow \mathcal{P}(Y)\) be a set-valued map with closed graph. In order to solve the inclusion \[ 0\in f(x)+G(x), \] the authors consider the iterative method defined by \(x_0,x_1\in X\) and \[ y_k=\alpha x_k+(1-\alpha)x_{k-1},\;0\in f(x_k)+[y_k,x_k;f](x_{k+1}-x_k)+G(x_{k+1}),\;k\geq 1, \] where \(\alpha\) is fixed in \([0,1)\) and \([y_k,x_k;f]\) denotes a first order divided difference of \(f\) on the points \(y_k\) and \(x_k\). The main result of the paper (Theorem~3.1) is a semilocal convergence theorem for the iterative process considered.
scientific article; zbMATH DE number 5125397