## 2 Preliminaries
### Topological Degree Theory Notation an Results  
**Definition 1**.: _Let \(L:\operatorname{Dom}\,L\subset X\to Y\) be a linear operator defined on the normed vector spaces \(X\) and \(Y\). If \(\operatorname{Im}\,L\) and \(\operatorname{Ker}\,L\) are such that_  
\[\dim(\operatorname{Ker}\,L)=\operatorname{codim}(\operatorname{Im}\,L)< \infty,\qquad\operatorname{Im}\,L\text{ is closed in }Y, \tag{18}\]  
_are satisfied, \(L\) is a Fredholm operator of index zero._  
**Proposition 1**.: _Let \(L\) be a Fredholm mapping of index zero from \(X\) to \(Y\), then_  
1. _There exist_ \(P:X\to X\) _and_ \(Q:Y\to Y\) _such that_ \(P\) _and_ \(Q\) _are continuous projectors satisfying_ \(\operatorname{Im}\,P=\operatorname{Ker}\,L\) _and_ \(\operatorname{Im}\,L=\operatorname{Ker}\,Q=\operatorname{Im}\,(I-Q)\)_._
2. _The restriction of_ \(L\) _to operator_ \(\operatorname{Dom}\,L\cap\operatorname{Ker}\,P\) _defines an invertible operator_ \(L_{P}\)_, i.e.,_ \(L_{P}:=L|_{\operatorname{Dom}\,L\cap\operatorname{Ker}\,P}:(I-P)X\to \operatorname{Im}\,L\) _and_ \(K_{P}=L_{P}^{-1}\) _is well defined._
3. _The sets_ \(\operatorname{Im}\,Q\) _and_ \(\operatorname{Ker}\,L\) _are isomorphic, there exists_ \(J:\operatorname{Im}\,Q\to\operatorname{Ker}\,L\) _such that_ \(J\) _is an isomorphism._  
**Definition 2**.: _Let \(X\) and \(Y\) be two normed vector spaces and \(\Omega\subset X\) be an open bounded set; and \(L\) a Fredholm mapping of index zero from \(X\) to \(Y\) with \(P,Q\), and \(K_{P}\) the operators satisfying the assertions (i)-(iii) of Proposition 1. The operator \(N:X\to Y\) is called \(L\)-compact on \(\overline{\Omega}\) if the following assertions are satisfied: (a) \(N\) is continuous, (b) \(QN(\overline{\Omega})\) is bounded, and (c) the operator \(K_{P}(I-Q)N\) is compact on the set \(\overline{\Omega}\)._  
**Definition 3**.: _Let us consider \(\Omega\subset\mathbb{R}^{n},\ f:\Omega\to\mathbb{R}^{n},\ y\in\mathbb{R}^{n}\) and \(N_{f}=\{x\in\Omega:I_{f}(x)=0\}\), with \(I_{f}(x)\) the Jacobian of \(f\) at \(x\), such that \(\Omega\) is open and bounded, \(f\in C^{1}(\Omega,\mathbb{R}^{n})\cap C(\overline{\Omega},\mathbb{R}^{n})\), and \(y\in\mathbb{R}^{n}\backslash f(\partial\Omega\cup N_{f})\). Then, \(\deg\{f,\Omega,y\}=\sum_{x\in f^{-1}(y)}\operatorname{sgn}_{f}(x)\), defines the degree \(\deg\{f,\Omega,y\}\), with the agreement \(\sum\limits_{\varnothing}=0\)._  
**Theorem 2**.: _Assume that \(X\) and \(Y\) are Banach spaces, \(\Omega\subset X\) is open bounded, \(L\) is a Fredholm mapping of index zero from \(X\) to \(Y\), and \(N\) is an \(L\)-compact operator on \(\overline{\Omega}\). If the following hypotheses are met:_  
Figure 1: Schematic dependencies of model.  
_For each_ \((\lambda,x)\in(0,1)\times(\partial\Omega\cap\operatorname{Dom}\ L)\) _the relation_ \(Lx\
eq\lambda Nx\) _is satisfied; for each_ \((\lambda,x)\in(0,1)\times(\partial\Omega\cap\operatorname{Dom}\ L)\)_._ _For each_ \(x\in\partial\Omega\cap\operatorname{Ker}\ L\) _the relation_ \(QNx\
eq 0\) _is satisfied;_ _\((\text{C}_{3})\ \deg(IQN,\Omega\cap\operatorname{Ker}\ L,0)\
eq 0\)_._  
_Then, there exists at least one \(x\in\operatorname{Dom}\ L\cap\overline{\Omega}\) satisfying the operator equation \(Lx=Nx\)._