Convergence analysis of a method for variational inclusions
Mohammed Harunor RASHİD, J.H. Wang, C. Li · Applicable Analysis · 2011
Consider the following variational inclusion problem: where f is differentiable in a neighbourhood of a solution and g is differentiable at , and F is a set-valued mapping, and the method introduced in Jean-Alexis and Pietrus [C. Jean-Alexis and A. Pietrus, On the convergence of some methods for variational inclusions, Rev. R. Acad. Cien. serie A. Mat. 102(2) (2008), pp. 355–361] for solving this problem: where ∇f(x) denotes the Fréchet derivative of f at x and [x, y; g] the first-order divided difference of g on the points x and y. Local converge analysis are provided for the method under the weaker conditions than Jean-Alexis and Pietrus (2008). Moreover, if ∇f and the first-order divided difference of g are p-Hölder continuous at a solution, then we show that this method converges superlinearly. In particular, our results extend the corresponding ones Jean-Alexis and Pietrus (2008), and fix a gap in the proof in (Jean-Alexis and Pietrus (2008), Theorem 1).