## 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  
**Lemma 1**. : _Let us consider the operators \(L\) and \(N\) give by the relations on (21) and (22) with \(X=Y\) defined in (20) and \(\Omega\) given by the set in (23). Then, the following assertions are satisfied:_\(L\) is a Fredholm operator of index zero and_  
\[\text{Ker }L\cong\mathbb{R}^{4},\quad\text{Im }L=\bigg{\{}\mathbf{y}\in Y:\int_{0}^{ \omega}\mathbf{y}(\tau)^{T}d\tau=0\bigg{\}}. \tag{24}\]  
\(N\) _is a continuous operator._  
_The operators \(P\) and \(Q\) defined by_  
\[P:X\to X,\quad Q:Y\to Y,\quad P\Big{(}\mathbf{x}^{T}\Big{)}=Q\Big{(} \mathbf{x}^{T}\Big{)}=\frac{1}{\omega}\int_{0}^{\omega}\mathbf{x}(\tau)^{T}d\tau \tag{25}\]  
_satisfy the relations (i)-(iii) given in Proposition 1._  
_If the assumptions of Theorem 1 are met, the operator \(N\) is \(L\)-compact on \(\overline{\Omega}\)._  
Proof. : Proof of _(a)_. Utilizing the definition of \(L\), we deduce the following:  
\[\text{Ker }(L) =\Big{\{}\mathbf{x}^{T}\in\text{Dom }(L)\ :\ L(\mathbf{x})=(0,0,0,0)^{T}\Big{\}}\] \[=\Big{\{}\mathbf{x}^{T}\in\text{Dom }(L)\ :\ \frac{dx_{i}}{dt}(t)=0,\ i= \overline{1,4}\Big{\}}\] \[=\Big{\{}\mathbf{x}^{T}\in\text{Dom }(L)\ :\ \mathbf{x}(t)=\mathbf{x}(0),\ t\in[0, \omega]\Big{\}}\cong\mathbb{R}^{4}.\]  
Consequently, \(\text{dim}(\text{Ker }(L))=4<\infty\). Moreover, employing a result from elementary algebra, it is established that there exists a unique isomorphism \(\varphi:X/\text{Ker }(L)\to\text{Im }(L)\), which ensures the commutativity visualized by the following scheme:  
Here, \(\pi:X\to X/\text{Ker }(L)\) represents the canonical projection, and \(i:\text{Im }(L)\to Y\) signifies the inclusion. This implies that \(X/\text{Ker }(L)\cong\text{Im }(L)\). Additionally, based on elementary algebra, we can identify the isomorphisms \(X\cong\text{Ker }(L)\bigoplus X/\text{Ker }(L)\) and \(Y\cong\text{Im }(L)\bigoplus X/\text{Im }(L)\). It follows that \(X\cong\text{Ker}(L)\bigoplus X/\text{ker }(L)\cong\text{Ker }(L)\bigoplus\text{Im }(L)\). Given that \(\text{Im }(L)\bigoplus X/\text{Im }(L)\cong Y=X\cong\text{Ker }(L)\bigoplus\text{Im }(L)\), this implies that \(\text{Ker }(L)\cong X/\text{Im }(L)\), and thus, \(\text{dim}(\text{Ker }(L))=\text{cod}(\text{Im }(L))=4<\infty\). Therefore, by Definition 1 we have that \(L\) is a Fredholm operator of index zero. We have deduced that \(\text{Ker }L\cong\mathbb{R}^{4}\). To characterize the set defining \(\text{Im }(L)\), we choose arbitrarily \(\mathbf{y}^{T}\in\text{Im }(L)\), and find \(\mathbf{x}\in\text{Dom }(L)\) such that \(L\mathbf{x}^{T}=\mathbf{y}^{T}\), i.e., \(\mathbf{x}^{\prime}(t)=\mathbf{y}(t)\). The periodicity of \(\mathbf{x}\), implies that \(\int_{0}^{\omega}\mathbf{x}^{\prime}(\tau)d\tau=0\). Consequently, \(\mathbf{y}^{T}\in\text{Im }(L)\) if and only if \(\int_{0}^{\omega}\mathbf{y}(\tau)d\tau=0\), and thus, \(\text{Im }(L)\) is characterized by the set defined in (24). Proof of _(b)_. Consider the sequence \(\{\mathbf{x}_{n}\}\subset X\) and \(\overline{\mathbf{x}}\in X\) such that \(\mathbf{x}_{n}\to\overline{\mathbf{x}}\) in the topology induced by the norm of \(X\). Based on the definition of the operator \(N\) as given in (22), and applying the inequality component-wise,  
\[|\exp(z_{2})-\exp(z_{1})|=\bigg{|}\int_{z_{2}}^{z_{1}}\exp(s)ds\bigg{|}\leq \max\Big{\{}\exp(z_{1}),\exp(z_{2})\Big{\}}|z_{2}-z_{1}|,\quad\forall z_{1},z_ {2}\in\mathbb{R},\]  
we establish the existence of a constant \(C>0\) depending only on \(h,g\), \(\alpha\), \(\gamma\), \(\mu\), \(c\), \(\epsilon\), \(\theta\), and \(K\), such that \(\|N(\mathbf{x}_{n})-N(\overline{\mathbf{x}})\|\leq C\|\mathbf{x}_{n}-\overline {\mathbf{x}}\|\). This leads to the conclusion that \(N(\mathbf{x}_{n})\to N(\overline{\mathbf{x}})\) in the norm sense, proving that \(N\) is continuous. Proof of (_c_). Our proof of Ker \(L=\text{Im }P\) is developed by double inclusion.