# On the Existence of Solutions of Nonlinear Fredholm Integral Equations from Kantorovich's Technique
## 4 Applications
### Application 1
Then,  
\[[\mathfrak{F}^{\prime}(x)]^{-1}z(s)=y(s)=z(s)+\cos(\pi s)\left(\frac{3}{5}\int _{0}^{1}x(t)^{2}\sin(\pi t)y(t)dt\right).\]0  
and  
\[[\mathfrak{F}^{\prime}(x)]^{-1}z(s)=y(s)=z(s)+\cos(\pi s)\left(\frac{3}{5}\int _{0}^{1}x(t)^{2}\sin(\pi t)y(t)dt\right).\]1  
From \(\frac{\alpha_{n+1}}{\gamma_{n+1}}=\frac{\alpha_{n}^{2}(\gamma_{n}\ell^{\prime }(t_{n})-\ell(t_{n}))}{\gamma_{n}^{2}(\alpha_{n}\ell^{\prime}(t_{n})-\ell(t_{ n}))}\), it follows  
\[[\mathfrak{F}^{\prime}(x)]^{-1}z(s)=y(s)=z(s)+\cos(\pi s)\left(\frac{3}{5}\int _{0}^{1}x(t)^{2}\sin(\pi t)y(t)dt\right).\]2  
\[[\mathfrak{F}^{\prime}(x)]^{-1}z(s)=y(s)=z(s)+\cos(\pi s)\left(\frac{3}{5}\int _{0}^{1}x(t)^{2}\sin(\pi t)y(t)dt\right).\]3\[[\mathfrak{F}^{\prime}(x)]^{-1}z(s)=y(s)=z(s)+\cos(\pi s)\left(\frac{3}{5}\int _{0}^{1}x(t)^{2}\sin(\pi t)y(t)dt\right).\]4  
where \(P=0.7718\), \(Q=0.7858\), \(U=0.0985\) and \(V=0.0967\), and then taking into account that \(\gamma_{n+1}=(R-r)+\alpha_{n+1}\), we obtain  
\[[\mathfrak{F}^{\prime}(x)]^{-1}z(s)=y(s)=z(s)+\cos(\pi s)\left(\frac{3}{5}\int _{0}^{1}x(t)^{2}\sin(\pi t)y(t)dt\right).\]5  
where \(\delta_{n}=\left\{\frac{(R-r)V^{2^{n}}}{P-V^{2^{n}}}\right\}\) and \(\varepsilon_{n}=\left\{\frac{(R-r)U^{2^{n}}}{Q-U^{2^{n}}}\right\}\) for all \(n\geq 0\). In Table 1, we can see the a priori error estimates that lead to the well-known quadratic convergence of [PERSON]'s method. Now, taking into account the exact solution  
\[[\mathfrak{F}^{\prime}(x)]^{-1}z(s)=y(s)=z(s)+\cos(\pi s)\left(\frac{3}{5}\int _{0}^{1}x(t)^{2}\sin(\pi t)y(t)dt\right).\]6  
of equation (10), we compare the obtained results with those given by other authors when different numerical methods are applied to solve (10). In Table 2, we show the real errors for \(n=10\) and \(n=20\) when the adapted [PERSON]'s method is used in [13] to solve (10) and some points of the interval involved are chosen. In Table 3, we show the real errors when a combination of [PERSON]'s method and quadrature methods [14] and an iterative scheme based on the homotopy analysis method [12] are applied. Notice that \(\|x^{*}-x_{n}\|\leq\varepsilon_{n}\), since \(\|x^{*}-x_{n}\|\leq r-t_{n}\), so we already improve the results obtained in Tables 2 and 3 by other authors with four iterations of [PERSON]'s method,  
\[[\mathfrak{F}^{\prime}(x)]^{-1}z(s)=y(s)=z(s)+\cos(\pi s)\left(\frac{3}{5}\int _{0}^{1}x(t)^{2}\sin(\pi t)y(t)dt\right).\]7  
and the stopping criterion \(\|x_{n}-x_{n-1}\|<10^{-32}\). Finally, although we have already guaranteed that the numerical approximation given by \(x_{4}(s)\) to the solution \(\psi(s)\) of equation (10) is of at least order \(10^{-17}\), we see in Table 4 that this approximation is, in fact, of at least order \(10^{-30}\). \begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{n}\) & \(\delta_{\mathbf{n}}\) & \(\varepsilon_{\mathbf{n}}\) \\ \hline
0 & \(1.3272\times 10^{-1}\) & \(1.3272\times 10^{-1}\) \\
1 & \(1.1368\times 10^{-2}\) & \(1.1577\times 10^{-2}\) \\
2 & \(1.0513\times 10^{-4}\) & \(1.1095\times 10^{-4}\) \\
3 & \(9.2090\times 10^{-9}\) & \(1.0444\times 10^{-8}\) \\
4 & \(7.0677\times 10^{-17}\) & \(9.2563\times 10^{-17}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: A priori error estimates. \begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{t}\) & \(\mathbf{n=10}\) & \(\mathbf{n=20}\) \\ \hline
0.0 & \(5.44\times 10^{-8}\) & \(3.19\times 10^{-16}\) \\
0.2 & \(4.40\times 10^{-8}\) & \(2.22\times 10^{-16}\) \\
0.4 & \(1.68\times 10^{-8}\) & \(1.11\times 10^{-16}\) \\
0.6 & \(1.68\times 10^{-8}\) & \(1.11\times 10^{-16}\) \\
0.8 & \(4.40\times 10^{-8}\) & \(2.22\times 10^{-16}\) \\
1.0 & \(5.44\times 10^{-8}\) & \(3.19\times 10^{-16}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Real errors for \(n=10\) and \(n=20\) when the adapted [PERSON]’s method given in [13] is applied.