## 3 The Banach Space of Continuous Periodic Functions and Particular Operators \(L\) and \(N\)
### The Items (C1)-(C3) of Theorem 2 Are Satisfied  
Let us consider the assumptions and notation of Lemma 2. Additionally, we define the radius \(\sigma\) for the set \(\Omega\) as follows:  
\[\sigma=\max\Biggl{\{}\sum_{i=1}^{4}\Big{(}\ln(\Lambda_{i})\Big{)}^{2},\sum_{ i=1}^{4}\Big{(}|\ln(\Psi_{i})+\zeta_{i}|\Big{)}^{2}\Biggr{\}}, \tag{60}\]  
where \(\mathbf{\Psi},\mathbf{\zeta}\), and \(\mathbf{\Lambda}\) are as defined in (28)-(32) and (54)-(57). Then, the items (C1)-(C3) of Theorem 2 are satisfied. Proof. : The proof of each item is developed basically by applying contradiction arguments. Proof of (C1). : Let us assume that there exists a pair \((\mathbf{x},\lambda)\in(\partial\Omega\cap\text{Dom}~{}(L)\times(0,1)\) such that \(L(\mathbf{x}^{T})=\lambda N(\mathbf{x}^{T})\). However, according to Lemma 2, this assumption implies that \(\mathbf{x}\in Int(\Omega)\), which is a direct contradiction to the initial assumption that \(\mathbf{x}\in\partial\Omega\). Therefore, such a pair \(\mathbf{x}\in Int(\Omega)\) cannot exist, and item (C1) is proved. Proof of (C2). : Consider \(\mathbf{x}\in\partial\Omega\cap\text{ker}(L)\) such that \(QN(\mathbf{x}^{T})=\mathbf{0}^{T}\). From Lemma 3, it follows that \(\mathbf{x}\in Int(\Omega)\). This again leads to a contradiction, as it conflicts with our initial assumption that \(\mathbf{x}\in\partial\Omega\). Proof of (C3). We define the application \(\Phi:\text{Dom }(L)\times[0,1]\to X\), defined as follows:  
\[\Phi(\mathbf{x}^{T},\varepsilon)=\left[ \begin{array}{c}\overline{\varphi}- \overline{\beta}\exp(x_{4})-\overline{h}\\ \overline{\psi}+\overline{\beta}\exp(x_{1}+x_{4}-x_{2})\\ \overline{\rho}-\overline{\gamma}\exp(x_{2})\\ \overline{\gamma}\exp(x_{2}+x_{3}-x_{4})-\overline{\eta}\end{array} \right]+ \varepsilon\left[ \begin{array}{c}\overline{\delta}\exp(x_{2}-x_{1})\\ -\overline{\alpha}+h+g\\ \overline{\rho}\exp(x_{3}+x_{4})-\overline{\eta}\end{array} \right].\]  
Our goal is to show that \(\Phi(\mathbf{x}^{T},\varepsilon)\
eq 0\), for \(\varepsilon\in[0,1]\) and \(\mathbf{x}^{T}\in\partial\Omega\cap\ker(L)\). Recall that \(\ker(L)\cong\mathbb{R}^{4}\), such that \(\mathbf{x}^{T}\) is a constant function, say \(\mathbf{x}^{T}(t)=(c_{1},c_{2},c_{3},c_{4})^{T}\). Assuming the existence of \(\varepsilon\in[0,1]\) and \((c_{1},c_{2},c_{3},c_{4})^{T}\in\partial\Omega\cap\ker(L)\), with \(\|(c_{1},c_{2},c_{3},c_{4})\|=\sigma\), such that \(\Phi((c_{1},c_{2},c_{3},c_{4})^{T},\varepsilon)=0\). It follows that  
\[0 = \overline{\varphi}-\overline{\beta}\exp(c_{4})-\overline{h}+ \varepsilon\overline{\delta}\exp(c_{2}-c_{4}),\] \[0 = \overline{\psi}+\overline{\beta}\exp(c_{1}+c_{4}-c_{2})- \varepsilon(\overline{\alpha}+h+g),\] \[0 = \overline{\rho}-\overline{\gamma}\exp(c_{2})+\varepsilon(\overline {\rho}\exp(c_{4}-c_{3})-\overline{\eta}),\] \[0 = \overline{\eta}\exp(c_{2}+c_{3})-\overline{\eta}.\]  
However, similar arguments to those used in the proof of Lemma 2, imply \(\|(c_{1},c_{2},c_{3},c_{4})\|<\sigma\), contradicting the initial assumption \(\|(c_{1},c_{2},c_{3},c_{4})\|=\sigma\). Applying the homotopy invariance theorem of topological degree, performing \(J=I:\text{Im }(Q)\to\ker(L)\) such that \(\mathbf{x}^{T}\mapsto\mathbf{x}^{T}\), using the fact that the system  
\[0 = \varphi-\beta\exp(x_{4})-h+g\exp(x_{2}-x_{4}),\] \[0 = \psi+\beta\exp(x_{1}+x_{4}-x_{2})-(\alpha+h+g),\] \[0 = \rho-\gamma\exp(x_{2})+\rho\exp(x_{4}-x_{3})-\eta,\] \[0 = \gamma\exp(x_{2}+x_{3})-\eta.\]  
has a unique solution \(\mathbf{x}^{\star}\in\partial\Omega\cap\ker(L)\) and by definition 2.3 of [36], we conclude that  
\[\deg \Big{(}JQN(\mathbf{x}^{T}),\Omega\cap\ker(L),\mathbf{0}^{T}\Big{)}= \deg\Big{(}\Phi(\mathbf{x}^{T}),\Omega\cap\ker(L),\mathbf{0}^{T}\Big{)}\] \[=\deg\Big{(}(\overline{\varphi}-\overline{\beta}\exp(x_{4})- \overline{h},\overline{\psi}+\overline{\beta}\exp(x_{1}+x_{4}-x_{2}),\overline {\rho}-\overline{\gamma}\exp(x_{2}),\overline{\gamma}\exp(x_{2}+x_{3}-x_{4})- \overline{\eta}),\Omega\cap\ker(L),\mathbf{0}^{T}\Big{)}\] \[=\text{sgn}\Bigg{|}