Necessary and sufficient conditions of Lipschitz reversibility of a nonlinear difference operator in the space of almost periodic number sequences (Q2761526)

Let \(B\) be the Banach space of all almost periodic bounded sequences \(x:Z\to R\) with the norm \(\|x\|_{B}=\sup_{n\in Z}|x(n)|\). Consider the difference operator \(D:B\to B,\;(Dx)(n)=x(n+1)-f(x(n)),\;n\in Z\), where \(f:R\to R\) is a continuous function. The author proves that operator \(D\) has the inverse Lipschitz operator \(D^{-1}\) if and only if either \(\sup_{t,s\in R,t\neq s}|{f(t)-f(s)\over t-s}|<1\), or \(\inf_{t,s\in R,t\neq s}|{f(t)-f(s)\over t-s}|>1\). Some estimates for \(D^{-1}\) are obtained.