On the midpoint method for solving equations
The article deals with the problem of approximating a solution \(x^*\) to the generalized equation \[ 0 \in f(x) + g(x) + F(x), \] where \(f: \;X \to Y\) is a Fréchet-differentiable operator in a neighborhood \(U\) of \(x^*\), \(g: \;X \to Y\) is continuous at \(x^*\) and \(F: \;X \to 2^Y\) is a set-valued map (\(X\), \(Y\) are Banach spaces). The corresponding approximations \(\{x_n\}|_{n=0}^\infty\) are defined with the following recursive formulas \[ 0\in f(x_n) + g(x_N) + (f'(x_n) + [x_{n-1},x_n;g])(x_{n+1} - x_n) + F(x_{n+1}) \] where \([x,y;g]\) is the first order divided difference of \(g\) at the points \(x,y\), i.e. the linear operator from \(L(X,Y)\) satisfying the equation \([x,y;g](y - x) = g(y) - g(x)\) for \(x, y \in U\), \(x \neq y\). The authors formulate some natural conditions \(A_1\)--\(A_6\) for the existence of the sequence \(\{x_n\}\) and for the validity of the following estimates \[ \|x_{n+1} - x^*\| \leq c\|x_n - x^*\| \;\max \;\{\|x_n - x^*\|,\|x_{n-1} - x^*\|\} \quad (n = 1,2,\ldots). \] Reviewer's remark: However, the last condition ``the set valued map \[ G(x)^{-1} = [f(x^*) + f'(x^*)(x - x^*) + g(x) + F(x)]^{-1} \] is \(M\)-pseudo-Lipschitz around \((0,x^*)\) seems to be vague and cumbersome.
scientific article; zbMATH DE number 5736352