The article deals with the nonlinear inclusion problem \[ F(x) \in C, \] where \(F:\;D (\subset {\mathbb X}) \to {\mathbb Y}\) is a continuously Fréchet differentiable function between a reflexive Banach space \({\mathbb X}\) and an arbitrary Banach space \({\mathbb Y}\), \(D\) an open set, \(C \subset {\mathbb Y}\) a nonempty closed convex cone. Approximate solutions \(x_k\) to this inclusion are defined by the equations \[ x_{k+1} \in x_k + \text{argmin}\, \{\|d\|:\;F(x_k) + F'(x_k)d \in C\}, \quad k = 0,1,2,\dots, \] with an initial \(x_0 = \overline{x}\); it is assumed that \(F\) satisfies Robinson’s condition at \(\overline{x}\). The authors construct some scalar majorant equations for the inclusion under consideration and prove estimates \[ \|x_{k+1} - x_k\| \leq t_{k+1} - t_k, \quad \|x_{k+1} - x_k\| \leq \frac{t_{k+1} - t_k}{(t_k - t_{k-1})^2} \, \|x_k - x_{k-1}\|^2, \qquad k = 0,1,2,\dots, \] where \(t_k\), \(k = 0,1,2,\dots\), are Newton approximations for majorant equations. In the end of the article the authors present a numerical example.
There are some vague places in the article; in particular, the Robinson condition is not defined in a correct way.