A rapidly convergent iteration method and Gâteaux differentiable operators (Q1080119)

Let \(\{F_ r\}_{0\leq r\leq p}\) be a family of Banach spaces satisfying, if \(0\leq r_ 1\leq r_ 2\leq p\), (i) \(F_{r_ 1}\supseteq F_{r_ 2}\); (ii) \(| f|_{r_ 1}\leq | f|_{r_ 2}\) \((f\in F_{r_ 1})\); and (iii) \(\phi (r)=\ln (| f|_ r)\) is a convex function. Let \(G_ 0\) be a Banach space and \({\mathcal F}\) be a Gâteaux differentiable mapping, and suppose that \({\mathcal F}'(x)(F_ p)\) is dense in \(G_ 0\). Under appropriate assumptions, the equation \({\mathcal F}(x)=0\) has a solution in \(F_ r\) for \(0\leq r\leq p\). The results extend the Inverse Function Theorem of J. Moser to the class of Gâteaux differentiable operators.