On the existence and uniqueness of solutions continuous and bounded on the real axis for nonlinear functional equations (Q2722236)

This paper deals with the functional equation NEWLINE\[NEWLINEx(t)=T[x](t):=f\bigl(t,x(\varphi_1(t,x(t)),\ldots,x(\varphi_m(t,x(t)))\bigl),NEWLINE\]NEWLINE where \(f:\mathbb R\times \mathbb R^m \to \mathbb R, \varphi_i:\mathbb R\times \mathbb R \to \mathbb R, i=1,\ldots ,m\). The functions \(f\) and \(\varphi_i\) are bounded and Lipschitzian. The author imposes conditions on Lipschitz constants of these functions which allow to apply contraction principle to the operator \(T:C^{0,L}(\mathbb R) \mapsto C^0(\mathbb R),\) where \(C^0(\mathbb R)\) is the space of continuous and bounded on \(\mathbb R\) functions and the subspace \(C^{0,L}(\mathbb R)\) consists of functions satisfying Lipschitz conditions with constant \(L\). In such a way the author proves the existence of a unique solution \(x(t)\in C^{0,L}(\mathbb R)\).