## 3 The Banach Space of Continuous Periodic Functions and Particular Operators \(L\) and \(N\)
### \(L\) Is a Fredholm Operator of Index Zero Operator and \(N\) Is an \(L\)-Compact Operator
Firstly, we obtain that Ker \(L\subset\text{Im }P\) as follows: from the isomorphism Ker \(L\cong\mathbb{R}^{4}\) (see Lemma 1-_(a)_), we note that \(\mathbf{x}^{T}\in\text{Ker }L\) is equivalent to the fact that \(\mathbf{x}(t)\) is a constant function on \([t_{0},\infty]\), which implies that \(\mathbf{x}\in\text{Im }P\), since \(P(\mathbf{x}^{T})=\mathbf{x}^{T}\) for constant \(\mathbf{x}(t)\). Secondly, we show that \(\text{Im }P\subset\text{Ker }L\): for \(\mathbf{y}^{T}\subset E\text{Im }P\), there exists \(\mathbf{z}\in X\) such that \(P(\mathbf{z}^{T})=\mathbf{y}^{T}\), and from Equation (25), we have \(\omega^{-1}\int_{0}^{\omega}\mathbf{z}(\tau)^{T}d\tau=\mathbf{y}^{T}\), which implies, by differentiation, that \(L(\mathbf{y}^{T})=0\) or \(\mathbf{y}\in\text{Ker }L\). We prove that Ker \(Q=\text{Im }L\) by using the definition of the operator \(Q\) provided in (25). We observe the fact that \(\mathbf{y}^{T}\in\text{Ker }Q\) is rewritten as \(\int_{0}^{\omega}\mathbf{y}(\tau)^{T}d\tau=0\), which, according to (24), is equivalent to \(\mathbf{y}^{T}\in\text{Im }L\). The proof that \(\text{Im }(I-Q)=\text{Im }L\) is carried out through double inclusion. Given \(\mathbf{y}^{T}\in\text{Im }(I-Q)\), there exists \(\mathbf{z}\in X\) such that \((I-Q)\left(\mathbf{z}^{T}\right)=\mathbf{y}^{T}\), implying  
\[\int_{0}^{\omega}\mathbf{y}(\tau)^{T}d\tau=\int_{0}^{\omega}\left(\mathbf{z}( \tau)^{T}-\frac{1}{\omega}\int_{0}^{\omega}\mathbf{z}(m)^{T}dm\right)d\tau=(0,0,0,0),\]  
and, according to (24), we obtain that \(\mathbf{y}(\tau)^{T}\in\text{Im }L\). Hence, we deduce that \(\text{Im }(I-Q)\subset\text{Im }L\). Similarly, we establish the inclusion \(\text{Im }L\subset\text{Im }(I-Q)\). Let us prove the existence of the operators \(K_{P}\) and \(L_{P}\). The operator \(L_{P}\) is the restriction of \(L\) to \(\text{Dom }L\cap\text{Ker }P\), i.e., \(L_{P}:\text{Dom }L\cap\text{Ker }P\rightarrow\text{Im }L\) and \(L_{P}=L\) on \(\text{Dom }L\cap\text{Ker }P\). The operator \(K_{P}\) is the inverse operator of \(L_{P}\), defined by the relation  
\[K_{P}\left(\mathbf{x}^{T}\right)(t)=\int_{0}^{t}\mathbf{x}(\tau)^{T}d\tau- \frac{1}{\omega}\int_{0}^{\omega}\int_{0}^{\eta}\mathbf{x}(m)^{T}dmd\eta. \tag{26}\]  
Then, to prove that \(K_{P}=L_{P}^{-1}\), we apply the following identity:  
\[\int_{0}^{t}\frac{d}{ds}\mathbf{x}(s)ds-\frac{1}{\omega}\int_{0}^{\omega}\int_ {0}^{t}\frac{d}{dm}\mathbf{x}(m)dmdt=\mathbf{x}(t),\]  
which is satisfied only on \(\text{Dom }L\cap\text{Ker }P\). Proof of (_d_). The proof is facilitated by applying Definition 2. Considering the definition of the operator \(Q\), we note that \(QN(\mathbf{x}^{T})=\frac{1}{\omega}\int_{0}^{\omega}N(\tau)^{T}d\tau\). Then, for \(\mathbf{x}\in\overline{\Omega}\) we obtain that \(\|QN(\mathbf{x}^{T})\|\leq\frac{1}{\omega}\int_{0}^{\omega}\|N\|d\tau=\|N\|\), which implies that \(QN(\overline{\Omega})\) is bounded on \(\overline{\Omega}\). From the definitions of the operators \(K_{P},N\), and \(Q\), we have  
\[(K_{P}(I-Q)N)(\mathbf{x}^{T})(t)=\int_{0}^{t}N(\tau)^{T}d\tau+\left(\frac{1}{ 2}-\frac{t}{\omega}\right)\int_{0}^{\omega}N(\tau)^{T}d\tau-\frac{1}{\omega} \int_{0}^{\omega}\int_{0}^{\eta}N(m)^{T}dmd\eta.\]  
Consequently, we deduce the estimate \(\|K_{P}(I-Q)N\|\leq 2\omega\|N\|\), indicating that \((K_{P}(I-Q)N)(\overline{\Omega})\) is bounded on \(\overline{\Omega}\), as \(N\) is bounded on \(\overline{\Omega}\). Additionally, we clearly deduce that  
\[P:X\to X,\quad Q:Y\to Y,\quad P\Big{(}\mathbf{x}^{T}\Big{)}=Q\Big{(} \mathbf{x}^{T}\Big{)}=\frac{1}{\omega}\int_{0}^{\omega}\mathbf{x}(\tau)^{T}d\tau \tag{25}\]0  
or equivalently, \(K_{P}(I-Q)N\) is an equicontinuous operator. By applying the Arzela-Ascoli theorem, it follows that \(K_{P}(I-Q)N\) is a compact operator on \(\overline{\Omega}\). Hence, we have satisfied the conditions of Definition 2, thereby concluding that \(N\) is \(L\)-compact on \(\overline{\Omega}\).