Local existence of Lipschitz-continuous solutions of systems of nonlinear functional equations with iterated deviations
deviating argument, successive approximations
The following functional equation is studied\newline NEWLINE\[NEWLINE(E):\quad X(t)=F[t,\big(X(\alpha_j t +\varphi_j(t,X(\beta_j t+\psi_j(t,X(t)))))\big)_{j=1}^{k}],NEWLINE\]NEWLINE where \(\alpha_j\) and \(\beta_j\) are given real constants, \(1\leq j\leq k\), \(\varphi_j\) and \(\psi_j\) are real continuous functions defined in certain domain of \(\mathbb{R}^{n+1}\) containing the origin \((0,\mathbf{0})\in\mathbb{R}\times\mathbb{R}^n\), and \(F\) is a real vector continuous function defined in an appropriate neighbourhood of \((0,\mathbf{0},\ldots,\mathbf{0})\in\mathbb{R}\times(\mathbb{R}^n)^k\). Here the unknown \(X\) is a real vector function defined in an open interval \(J\) centered at the origin with values in a subset of \(\mathbb{R}^n\). A similar study was done by \textit{G. P. Pelyukh} [Ukrainian Math. J. 54, 75--90 (2001; Zbl 0985.39021)] in the case for which \(\beta_j=1\) and \(\psi_j\equiv 0,\) \(j=1,\ldots,k.\)NEWLINENEWLINEThe author imposes some sufficient conditions on the functions \(F, \varphi_j\) and \(\psi_j\) and on the coefficients \(\alpha_j,\beta_j\) in order to guarantee that Equation~\((E)\) has a unique Lipschitz continuous solution \(X(t)\) defined in a suitable neighborhood of the origin. It is interesting to notice that the conditions imposed to the real functions are of Lipschitzian type and that for constructing the solution the author employs an iterative method analogous to that of the successive approximations considered in the classical proof of the first order differential equation \(Z'=f(t,Z)\) being \(f\) a Lipschitz continuous map.
scientific article; zbMATH DE number 6081851
scientific article