Differentiability with respect to parameters in the presence of a priori estimates (Q1178086)

Let \(X\), \(Y\) and \(Z\) be topological vector spaces, \(U\) and \(W\) be subsets (may be not open) of \(X\) and \(Z\) respectively, \(F\) be an injective map from \(U\times W\) into \(Y\times Z\), \(A_1\) be an injective linear map from \(X\) onto \(Y\), \(A_2\) be a linear map from \(Z\) into \(Y\) and \((x_0,z_0)\in U\times W\). Put \(A(x,z)=A_1x+A_2 z\) and \(\bar A(y,z)=A_1^{-1} y+A_1^{-1}A_2 z\). Using definitions of small maps in [\textit{V. I. Averbukh} and \textit{O. G. Smolyanov}, Russ. Math. Surv. 23, No. 4, 67--113 (1968); translation from Usp. Mat. Nauk 23, No. 4(142), 67-116 (1968; Zbl 0179.19103)], the author defines an equivalent relation between \(F\) and \(A\) at \((x_ 0,z_ 0)\) in \(U\times W\), and obtains results about the following question: Assume that \(F\) and \(A\) are equivalent at \((x_0,z_0)\) in \(U\times W\), then when is \(F^{-1}\) equivalent to \(\bar A\) at \(F(x_0,z_0)\) in \(F(U\times W)\)? These results are generalized versions of the implicit function theorem.