On the existence of local smooth solutions of systems of nonlinear functional equations with deviations depending on unknown functions (Q2768823)

This paper deals with the functional equation NEWLINE\[NEWLINEx(t)=f\bigl[t,x\bigl(\lambda_1t+\varphi_1(t,x(t)),\ldots,x(\lambda_pt+\varphi_p(t,x(t)) \bigr)\bigr)\bigr].\tag{1}NEWLINE\]NEWLINE Here \(0<\lambda_i<1\), the functions \(f:\mathbb R\times \mathbb R^p \to \mathbb R\), \(\varphi_i:\mathbb R\times \mathbb R \to \mathbb R\), \(i=1,\ldots,p\), satisfy Lipschitz conditions near the origins of the spaces \(\mathbb R\times \mathbb R^p\) and \(\mathbb R\times \mathbb R\) respectively, and, besides, \(f(0,{\mathbf 0})=0\), \(\varphi_i(0,0)=0\). First, under assumption that the Lipschitz constant of the function \(f(0,\lambda_1t,\ldots,\lambda_pt)\) is less than 1 and Lipschitz constants of \(\varphi_i(t,x)\) are sufficiently small, the author proves the existence of a unique local Lipschitzian solution to (1). NEWLINENEWLINENEWLINENext, sufficient conditions are found providing the existence of \(C^k\)-smooth local solution. Finally, in the case of analytic functions \(f\) and \(\varphi_i\) and under the assumption that \(\lambda_1,\ldots,\lambda_p\) satisfy certain non-resonant conditions, the author constructs a solution of the form \(x(t)=\sum_{i=1}^{\infty}c_ix^i\). The convergence of this series is established by means of a standard majorant method.