## 2 Formulation in Banach Space
### Power Series Approximation  
Let us express \(K(x,t)\) as a power series in \(t\) at a point \(t_{0}\), namely  
\[K(x,t)=\sum_{k=0}^{\infty}a_{k}(x)(t-t_{0})^{k},\]  
where the coefficients \(a_{k}(x)\), \(k=0,1,\ldots,\) are continuous functions of \(x\) on \([a,b]\). We truncate this series and take the partial sum of the first \(n+1\) terms, viz.  
\[K_{n}(x,t)=\sum_{j=1}^{n+1}a_{j-1}(x)(t-t_{0})^{j-1}. \tag{15}\]  
We replace the kernel \(K(x,t)\) in (1) by (15) to obtain the degenerate integral equation  
\[\bar{u}_{n}(x)-\lambda\,\sum_{j=1}^{n+1}a_{j-1}(x)\int_{a}^{b}(t-t_{0})^{j-1} \bar{u}_{n}(t)dt=f(x),\quad x\in[a,b], \tag{16}\]  
where \(\bar{u}_{n}(x)\) indicates an approximate solution of (1). By defining the vectors  
\[G=\lambda\big{(}\begin{array}{ccccc}g_{1}&g_{2}&\ldots&g_{n+1}\end{array} \big{)}=\lambda\big{(}\begin{array}{ccccc}a_{0}(x)&a_{1}(x)&\ldots&a_{n}(x) \end{array}\big{)},\]  
and  
\[\Psi(\bar{u}_{n})=\left(\begin{array}{c}\Psi_{1}(\bar{u}_{n})\\ \Psi_{2}(\bar{u}_{n})\\ \vdots\\ \Psi_{n+1}(\bar{u}_{n})\end{array}\right),\quad\Psi_{j}(\bar{u}_{n})=\int_{a} ^{b}(t-t_{0})^{j-1}\bar{u}_{n}(t)dt,\quad j=1,2,\ldots,n+1,\]  
where \(\bar{u}_{n}=\bar{u}_{n}(x)\); Equation (16) may be written in the symbolic form  
\[T_{s}\bar{u}_{n}=\bar{u}_{n}-G\Psi(\bar{u}_{n})=f, \tag{17}\]  
wherein the operator \(T_{s}:X\to X\) and \(X=C[a,b]\).  
Furthermore, to facilitate the computation of \(\Psi(G)\) and \(\Psi(f)\), without resorting to numerical integration techniques, we can approximate the functions \(\{g_{j}(x)\}\) and \(f(x)\), provided they are analytic in \([a,b]\), by power series of the same type as above.  
Specifically, we can take  
\[f_{r}(x)=\sum_{l=1}^{r+1}\varphi_{l-1}(x-x_{0})^{l-1}, \tag{18}\]  
where \(\varphi_{l-1}\), \(l=1,2,\ldots,r+1\) are known constants, e.g., in a Taylor series expansion \(\varphi_{l-1}=\frac{1}{(l-1)!}f^{(l-1)}(x_{0})\). Substituting (18) into Equation (17), we obtain  
\[T_{s}\bar{u}_{n}=\bar{u}_{n}-G\Psi(\bar{u}_{n})=f_{r}, \tag{19}\]  
where \(f_{r}=f_{r}(x)\). For similar modifications, one can look at [9; 19].  
Likewise, we may also express the functions \(\{g_{j}(x)\}\) as partial sums of power series, namely  
\[g_{jr}(x)=\sum_{l=1}^{r+1}\gamma_{j(l-1)}(x-x_{0})^{l-1},\quad j=1,2,\ldots, n+1, \tag{20}\]  
with \(\gamma_{j(l-1)}\), \(l=1,2,\ldots,r+1\) being known constants. Using (20), we may write  
\[G_{r}=\lambda\big{(}\begin{array}{ccccc}g_{1r}&g_{2r}&\ldots&g_{(n+1)r} \end{array}\big{)}, \tag{21}\]and Equation (19) then becomes  
\[T_{ss}\bar{a}_{n}=\bar{a}_{n}-G_{r}\Psi(\bar{a}_{n})=f_{r}, \tag{22}\]  
where the operator \(T_{ss}:X\to X\) and \(X=C[a,b]\).  
Note that we will arrive at an equation similar to (22) if we approximate the kernel \(K(x,t)\) directly by a finite segment of a double power series, viz.  
\[K_{nn}(x,t)=\sum_{l=1}^{n+1}\sum_{k=1}^{n+1}a_{(l-1)(k-1)}(x-x_{0})^{l-1}(t-t_{ 0})^{k-1},\]  
where the coefficients \(a_{(l-1)(k-1)}\) are constants. This shows how involved computations can be handled efficiently by the present formulation.  
All three Equations (17), (19) and (22) are of the kind (3) and can be solved explicitly through Theorem 1.