Inverse function theorem and extended differentiability in Banach spaces (Q1916565)

A version of the inverse function theorem is given for the situation, where the derivative is not invertible at the initial point. It is shown that under certain conditions locally ``extended'' solutions exist and the corresponding multi-valued mapping is ``extendedly'' differentiable in a sense defined by the same author in [``On differentiability of implicit function in Banach spaces'', Archives of Viniti, 2270 (1989)]. This is applied to a certain nonlinear elliptic equation without Gateaux differentiable solutions.