# On the Existence of Solutions of Nonlinear Fredholm Integral Equations from Kantorovich's Technique
## 4 Applications
### Application 2  
Secondly, we consider the following nonlinear integral Fredholm equation,  
\[x(s)=s+\frac{1}{2}\int_{-\frac{1}{2}}^{\frac{1}{2}}\mathrm{e}^{st}x(t)^{\frac{10 }{3}}\,dt, \tag{11}\]  
with \(s\in\left[-\frac{1}{2},\frac{1}{2}\right]\). Observe that, in this case, kernel \(\mathfrak{K}(s,t)=\mathrm{e}^{st}\) is not separable. In addition, the corresponding function \(\mathfrak{K}(x)\) defined in (2) and associated with (11) is defined in \(\Omega=\mathcal{C}\left[-\frac{1}{2},\frac{1}{2}\right]\). From Equation (11), we see that \(x_{0}(s)=s\) is a reasonable choice of starting point for [PERSON]'s method. In addition, condition (6) of Theorem 4 is satisfied, since \(|\alpha|pS\|x_{0}\|^{p-1}=0.1670<1\), and the auxiliary scalar function \(\phi(t)\) involved in our study is  
\[\phi(t)=\frac{1}{8}\left(-8t+2^{\frac{2}{3}}\sinh\frac{1}{4}(1+2t)^{\frac{10}{ 3}}\right),\]  
that has two positive real zeros \(r=0.0842\) and \(R=0.4921\) As a consequence of Theorem 4, Equation (11) then has a solution \(x^{*}(s)\) in \(\overline{B(x_{0},0.0842)}\) and it is unique in \(B(x_{0},0.4921)\). As kernel \(\mathfrak{K}(s,t)=\mathrm{e}^{st}\) is not separable, the application of [PERSON]'s method for solving (11) is not easy. Taking into account this fact, we first use [PERSON]'s series to approximate \(\mathfrak{K}(s,t)=\mathrm{e}^{st}\). So,  
\[\mathfrak{K}(s,t)=\mathrm{e}^{st}=\widetilde{\mathfrak{K}}(s,t)+\mathfrak{R}( \epsilon,s,t);\quad\widetilde{\mathfrak{K}}(s,t)=\sum_{i=0}^{j-1}\frac{s^{i} \,t^{i}}{i!},\quad\mathfrak{R}(\epsilon,s,t)=\frac{\mathrm{e}^{s\epsilon}}{j! }s^{j}\,t^{i},\]  
where \(\epsilon\in(\min\{0,t\},\max\{0,t\})\), and consider the integral equation  
\[x(s)=s+\frac{1}{2}\int_{-\frac{1}{2}}^{\frac{1}{2}}\widetilde{\mathfrak{K}}(s,t)x(t)^{\frac{10}{3}}\,dt,\quad s\in\left[-\frac{1}{2},\frac{1}{2}\right]. \tag{12}\]  
Next, we take into account the following relation that is satisfied by \(x^{*}(s)\) and \(\widetilde{x}(s)\), that are respectively solutions of (11) and (12): \begin{table}
\begin{tabular}{c c c c} \hline \hline \(t\) & \(\psi(t)\) & \(x_{4}(t)\) & \(\|\psi(t)-x_{4}(t)\|\) \\ \hline
0.0 & \(7.54\times 10^{-2}\) & \(7.54\times 10^{-2}\) & \(2.87\times 10^{-30}\) \\
0.2 & \(6.48\times 10^{-1}\) & \(6.48\times 10^{-1}\) & \(2.32\times 10^{-30}\) \\
0.4 & \(9.74\times 10^{-1}\) & \(9.74\times 10^{-1}\) & \(8.87\times 10^{-31}\) \\
0.6 & \(9.27\times 10^{-1}\) & \(9.27\times 10^{-1}\) & \(8.87\times 10^{-31}\) \\
0.8 & \(5.26\times 10^{-1}\) & \(5.26\times 10^{-1}\) & \(2.32\times 10^{-30}\) \\
1.0 & \(-7.54\times 10^{-2}\) & \(-7.54\times 10^{-2}\) & \(2.87\times 10^{-30}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Real errors. \begin{table}
\begin{tabular}{c c c} \hline \hline \(t\) & [14]-Errors & [12]-Errors \\ \hline
0.0 & \(4.98\times 10^{-2}\) & \(5.53\times 10^{-15}\) \\
0.2 & \(4.03\times 10^{-2}\) & \(4.55\times 10^{-15}\) \\
0.4 & \(1.53\times 10^{-2}\) & \(1.77\times 10^{-15}\) \\
0.6 & \(1.53\times 10^{-2}\) & \(1.77\times 10^{-15}\) \\
0.8 & \(4.03\times 10^{-2}\) & \(4.55\times 10^{-15}\) \\
1.0 & \(1.53\times 10^{-2}\) & \(5.53\times 10^{-15}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: 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. \[\|x^{*}(s)-\widetilde{x}(s)\|\leq\frac{|\sigma|T(\rho^{*})^{\frac{10}{3}}}{1- \frac{10}{3}|\sigma|\widetilde{S}(|\rho^{*}-\bar{\rho}|+\bar{\rho})^{\frac{2}{3} }},\]  
where \(\widetilde{S}=\max_{s\in\left[-\frac{1}{2},\frac{1}{2}\right]}\int_{-\frac{1}{2 }}^{\frac{1}{2}}\left|\widetilde{\mathfrak{R}}(s,t)\right|dt\), \(T=\max_{\epsilon\in\left[-\frac{1}{2},\frac{1}{2}\right]}\left(\max_{s\in \left[-\frac{1}{2},\frac{1}{2}\right]}\int_{-\frac{1}{2}}^{\frac{1}{2}}| \mathfrak{R}(\epsilon,s,t)|\,dt\right)\), \(\rho^{*}\geq\|x^{*}(s)\|\) and \(\bar{\rho}\geq\|\widetilde{x}(s)\|\) after taking norms in (11) and (12).