Theorems on the representation of nonlinear mapping families and implicit function theorems (Q1581459)

This article deals with a nonlinear mapping \(F:\Sigma\times X\to Y\) (\(\Sigma\) is a topological space, \(X\), \(Y\) are Banach spaces) twice strictly differentiable at points \((\sigma,x_*)\). If \(F\) is such a mapping and \(F_x'(\sigma, x_*)= A\), \(F''_{x^2}(\sigma, x_*)= B\) (do not depend on \(\sigma!\)), \[ \sup\{\|(A+ PB(h))^{-1}\|: h\in K, \|h\|\}< \infty \] (\(P= I-R\), \(R\) is a projection on the range of \(A\), \(K\) a cone in \(X\)) then, under natural conditions there exists a mapping \(\varphi:\Sigma\times X\) (\(\varphi(\sigma,0)=x_*\) such that \(F(\sigma,\varphi(\sigma, x))= F(\sigma,x_*)+ Ax+{1\over 2} PBx^2\) in a neighbourhood \((\sigma_0,x_*)\) and \[ \begin{multlined} \|\varsigma(\sigma,x)- (x_*+ x)\|\leq\\ C(R(F(\sigma, x_*+ x)- F(\sigma, x_*)- Ax)\|+\|x\|^{-1}\|P(F(\sigma, x_*+ x)- F(\sigma, x_*)-\textstyle{{1\over 2}} Bx^2)\|).\end{multlined} \] On the basis of this result (in the case of \(F(\sigma_0, x_*)= 0\)) the existence of an implicit function \(\chi(\sigma)\) satisfying \(F(\sigma, \chi(\sigma))= 0\) and the inequality \[ \|\chi(\sigma)- x_*\|\leq C(\|RF(\sigma, x_*)\|+\|PF(\sigma, x_*)\|^{1/2}) \] is stated. Another application gives a representation theorem for \(F\) as a superposition of ``rectifying'' mapping and quadratic one.