Necessary and sufficient conditions of existence of periodic solutions of non-linear equation \(x(n+1)-f(x(n))=h(n),\;n\in Z\) (Q2768800)

Let us denote by \(l_{\infty}\) the Banach space of all bounded maps \(x:\mathbb{Z}\to \mathbb{R}\) with the norm \(\|x\|_{l_{\infty}}=\sup_{n\in \mathbb{Z}}|x(n)|\), and let us denote by \(T^{m}\), \(m\in \mathbb{Z}\) the shift operator \((T^{m}x)(n)=x(n+m)\), \(n\in \mathbb{Z}\), \(x\in L_{\infty}\). An element \(x\in l_{\infty}\) is called periodic if \(T^{m}x=x\) for some \(m\in \mathbb{Z}\setminus \{0\}\). Let \(Pl_{\infty}\) be a linear space of all periodic elements in \(l_{\infty}\), and \(\text{per}(x)\buildrel \text{def}\over=\min\{m\in \mathbb{N}\); \(T^{m}x=x\}\). This paper deals with the difference equation \(x(n+1)-f(x(n))=h(n)\), \(n\in \mathbb{Z}\), where \(f:\mathbb{R}\to \mathbb{R}\) is a continuous function, \(h\in l_{\infty}\). Let a function \(\lambda_{c}:\mathbb{R}\to \mathbb{R}\) be such that \(\lambda_{c}(x)=c\) for all \(x\in \mathbb{R}\), let \(I:\mathbb{R}\to \mathbb{R}\) be the identity mapping, \((g_2\circ g_1)(x)=g_2(g_1(x))\), \(\forall x\in \mathbb{R}\), \(g_{i}:\mathbb{R}\to \mathbb{R}\), \(i=1,2\). The author obtains the necessary and sufficient conditions of existence of periodic solutions of the considered equation. In particular, the following result is proved. A solution \(x\in Pl_{\infty},\text{per}(x)=\text{per}(h)\) of the considered equation exists for any \(h\in Pl_{\infty}\) if and only if for arbitrary \(m\in \mathbb{N}\) and \(\delta_{k}\in \mathbb{R}\), \(k\in[1,m]\cap \mathbb{Z}\) the following equality holds true: \(\mathbb{R}((f-\lambda_{\delta_{m}})\circ (f-\lambda_{\delta_{m-1}})\circ\ldots\circ (f-\lambda_{\delta_{1}})-I)=\mathbb{R}\).