To Favard's theory for functional equations (Q2360273)

Let \(E\) be a Banach space over \(\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}\) with the norm \(\|\cdot\|_E\), let \({\mathcal K}\) denote the set of all nonempty compact subsets of \(E\), and let \(C^0\) be the Banach space of all bounded continuous functions \(x: \mathbb{R} \to E\) with the norm \[ \| x \|_{C^{0}} := \sup_{t \in \mathbb{R}} \|x(t)\|_E. \] The following notations are needed \[ R(x) := \{x(t): t \in \mathbb{R} \}, \;x \in C^0, \] \[ {\mathcal D}_K := \{x \in C^0: R(x) \subset K \}, \;K \in {\mathcal K}, \] and \[ \left(S_h x \right)(t) := x(t + h), \;t, h \in \mathbb{R},\, x \in C^0. \] A function \(y \in C^0\) is said to be almost periodic if the closure of the set \(\{S_h y : h \in \mathbb{R} \}\) in \(C^0\) is compact. The set of all almost periodic functions \(y \in C^0\) is denoted by \(B^0\). An operator \(H: C^0 \to C^0\) is called almost periodic if for each \(K \in {\mathcal K}\) and a sequence \(\left( h_k \right)_{k \geq 1}\) of reals there exists a subsequence \(\left( h_{k_l} \right)_{l \geq 1}\) such that \[ \lim_{l_1,l_2 \to \infty} \sup_{x \in {\mathcal D}_K} \|S_{h_{l_1}}HS_{-h_{l_1}}x - S_{h_{l_2}}HS_{-h_{l_2}}x\|_{C^0} = 0. \] The main result is the following. Theorem 1. Assume that \(K \in {\mathcal K}\), \(y \in B^0\) and operator \({\mathcal F}: C^0 \to C^0\) is almost periodic. If \(x^* \in C^0\) is a solution of the equation \({\mathcal F}x^* = y\) such that \(R(x^*) \subset K\), \(\text{diam }R(x^*) > 0\) and \[ \inf \{ \|{\mathcal F}z - {\mathcal F}{x^*} \|: \;z \in C^0, \;R(z) \subset K, \;\|z - x^* \|_{C^0} \geq \varepsilon \} > 0 \] whenever \(0 < \varepsilon < r(x^*,K) := \sup \{\|u - v \|_E: \;u \in R(x^*), \;v \in K \}\), then \(x^*\) is almost periodic.