## 2 Generalized Banach Spaces  
We present some standard concepts that are needed in what follows to make the paper as self-contained as possible. More details on generalized Banach spaces can be found in [5; 6; 7; 14], and the references therein. **Definition 2.1**. : _A generalized Banach space is a triplet \((x,E,/\cdot/)\) such that_  
_(i) \(X\) is a linear space over \(\mathbb{R}\left(\mathbb{C}\right)\)._  
_(ii) \(E=(E,K,\left\|\cdot\right\|)\) is a partially ordered Banach space, i.e.,_  
_(ii\({}_{1}\)) \((E,\left\|\cdot\right\|)\) is a real Banach space,_  
_(ii\({}_{2}\)) \(E\) is partially ordered by a closed convex cone \(K\),_  
_(iii\({}_{3}\)) The norm \(\left\|\cdot\right\|\) is monotone on \(K\)._  
_(iii) The operator \(/\cdot:X\to K\) satisfies_  
\(/x/=0\)__\(\Leftrightarrow\)__\(x=0\)_, \(/\theta x/=\left|\theta\right|/x/\)_,_  
\(/x+y/\leq/x/+/y/\) _for each \(x,y\in X\), \(\theta\in\mathbb{R}(\mathbb{C})\)._  
_(iv) \(X\) is a Banach space with respect to the induced norm \(\left\|\cdot\right\|_{i}:=\left\|\cdot\right\|\cdot/\cdot/\)._  
**Remark 2.2**. : _The operator \(/\cdot/\) is called a generalized norm. In view of (iii) and (ii\({}_{3}\)), \(\left\|\cdot\right\|_{i}\) is a real norm. In the rest of this paper all topological concepts will be understood with respect to this norm._Let \(L\left(X^{j},Y\right)\) stand for the space of \(j\)-linear symmetric and bounded operators from \(X^{j}\) to \(Y\), where \(X\) and \(Y\) are Banach spaces. For \(X,Y\) partially ordered \(L_{+}\left(X^{j},Y\right)\) stands for the subset of monotone operators \(P\) such that  
\[0\leq a_{i}\leq b_{i}\Rightarrow P\left(a_{1},...,a_{j}\right)\leq P\left(b_{1 },...,b_{j}\right). \tag{2.1}\]  
**Definition 2.3**. : _The set of bounds for an operator \(Q\in L\left(X,X\right)\) on a generalized Banach space \(\left(X,E,/\cdot\right)\) is defined to be:_  
\[B\left(Q\right):=\left\{P\in L_{+}\left(E,E\right)\text{, }/Qx/\leq P\left/x \right/\text{ for each }x\in X\right\}. \tag{2.2}\]  
_Let \(D\subset X\) and \(T:D\to D\) be an operator. If \(x_{0}\in D\) the sequence \(\left\{x_{n}\right\}\) given by_  
\[x_{n+1}:=T\left(x_{n}\right)=T^{n+1}\left(x_{0}\right) \tag{2.3}\]  
_is well-defined. We write in case of convergence_  
\[T^{\infty}\left(x_{0}\right):=\lim\left(T^{n}\left(x_{0}\right)\right)=\lim_{ n\rightarrow\infty}x_{n}. \tag{2.4}\]  
We need some auxiliary results on inequations. **Lemma 2.4**. : _Let \(\left(E,K,\left\|\cdot\right\|\right)\) be a partially ordered Banach space, \(\xi\in K\) and \(M,N\in L_{+}\left(E,E\right)\)._  
_(i) Suppose there exists \(r\in K\) such that_  
\[R\left(r\right):=\left(M+N\right)r+\xi\leq r \tag{2.5}\]  
_and_  
\[\left(M+N\right)^{k}r\to 0\text{ \ as \ }k\rightarrow\infty. \tag{2.6}\]  
_Then, \(b:=R^{\infty}\left(0\right)\) is well-defined, satisfies the equation \(t=R\left(t\right)\) and is the smaller than any solution of the inequality \(R\left(s\right)\leq s\)._  
_(ii) Suppose there exists \(q\in K\) and \(\theta\in\left(0,1\right)\) such that \(R\left(q\right)\leq\theta q\), then there exists \(r\leq q\) satisfying (i)._  
**Proof. ** (i) Define sequence \(\left\{b_{n}\right\}\) by \(b_{n}=R^{n}\left(0\right)\). Then, we have by Equantion (2.5) that \(b_{1}=R\left(0\right)=\xi\leq r\Rightarrow b_{1}\leq r\). Suppose that \(b_{k}\leq r\) for each \(k=1,2,...,n\). Then, we have by Equantion (2.5) and the inductive hypothesis that \(b_{n+1}=R^{n+1}\left(0\right)=R\left(R^{n}\left(0\right)\right)=R\left(b_{n} \right)=\left(M+N\right)b_{n}+\xi\leq\left(M+N\right)r+\xi\leq r\Rightarrow b _{n+1}\leq r\). Hence, sequence \(\left\{b_{n}\right\}\) is bounded above by \(r\). Set \(P_{n}=b_{n+1}-b_{n}\).