## 5 Example  
We will illustrate the application of the suggested above algorithm for successive approximations on a particular example. Consider the GPDE  
\[ \begin{split}&({}_{0}^{C}\mathcal{D}^{0.3,0.5}u)(t)=\frac{\sin(u(t))} {t+1}\text{ for }t\in(0,4],\\ & u(0)-u(4)=0,\end{split}  \tag{16}\]  
with \(\theta(t,x)=\frac{\sin(x)}{t+1}\), \(\Delta(x,y)=x-y\), \(\gamma=0.3,\xi=0.5\). Note that GPDE (16) is a periodic boundary value problem. The function \(\theta(\sigma,x)\) satisfies condition 2 of Theorem 2 with \(\lambda=1\). The function \(\Delta(x,y)\) satisfies condition 3 with \(\mu=\delta=1\). The fractional operator \(\Omega\), defined by (7), in this case is given by the following integral equation  
\[(\Omega u)(\sigma)=u(4)e^{-t}E_{0,3}(-\big{(}2\sigma\big{)}^{0.3})\\ +\frac{1}{0.5^{0.3}\Gamma(0.3)}\int_{0}^{\sigma}(\sigma-\tau)^{-0.7 }\varepsilon^{\tau-\sigma}E_{0.3,0,3}(-\big{(}2(\sigma-\tau)\big{)}^{0.3}) \Big{(}\frac{\sin(u(\tau))}{\tau+1}+u(\tau)\Big{)}d\tau,\\ \sigma\in[0,4]. \tag{17}\]  
The function \(u(t)\equiv 0\) is a mild solution of GPDE (16). According to Theorem 1 zero is a solution of GPDE (16). Consider the function \(\varpi^{(0)}(\sigma)=e^{-\sigma}\). Applying the inequality \(\frac{1}{0.5^{0.3}\Gamma(0.3)}(\sigma-\tau)^{-0.7}\varepsilon^{\tau}\):  
\[E_{0,3,0,3}(-\big{(}2(\sigma-\tau)\big{)}^{0.3})\Big{(}\frac{\sin(e^{-\tau})}{ \tau+1}+e^{-\tau}\Big{)}\leq 0.012\text{ for }\tau\in[0,4]\text{ (see Figure \ref{fig:1}) we obtain}\\ e^{-\sigma}\geq e^{-\sigma-4}E_{0,3}(-\big{(}2\sigma\big{)}^{0.3})\\ +\frac{1}{0.5^{0.3}\Gamma(0.3)}\int_{0}^{\sigma}(\sigma-\tau)^{-0.7 }e^{\tau-\sigma}E_{0,3,0,3}(-\big{(}2(\sigma-\tau)\big{)}^{0.3})\Big{(}\frac{ \sin(e^{-\tau})}{\tau+1}+e^{-\tau}\Big{)}d\tau,\\ \sigma\in[0,4]. \tag{18}\]  
or (see Figure 2)  
\[1 \geq e^{-4}E_{0,3}(-\big{(}2\sigma\big{)}^{0.3})+0.012 \tag{19}\] \[\geq e^{-4}E_{0,3}(-\big{(}2\sigma\big{)}^{0.3})\] \[\quad+\frac{1}{0.5^{0.3}\Gamma(0.3)}\int_{0}^{\sigma}(\sigma-\tau )^{-0.7}\varepsilon^{\tau}E_{0,3,0,3}(-\big{(}2(\sigma-\tau)\big{)}^{0.3}) \Big{(}\frac{\sin(e^{-\tau})}{\tau+1}+e^{-\tau}\Big{)}d\tau,\\ \sigma\in[0,4].\]  
Furthermore, \(\varpi^{(0)}(0)-\varpi^{(0)}(4)=1-e^{-4}>0\) and therefore, the function \(\varpi^{(0)}(\sigma)\) is a mild upper solution of GPDE (16). For \(n\geq 0\) the approximation \(\omega^{(n+1)}(t)\) is obtained on \([0,4]\) by  
\[ \begin{split}&\omega^{(n+1)}(\sigma)=\omega^{(n)}(4)e^{-\sigma}E_{ 0,3}(-\big{(}2\sigma\big{)}^{0,3})\\ &+\frac{1}{0.5^{0.3}\Gamma(0.3)}\int_{0}^{\sigma}(\sigma-\tau)^{ -0.7}e^{(\tau-\sigma)}E_{0,3,0,3}(-\big{(}2(\sigma-\tau)\big{)}^{0,3})\Big{(} \frac{\sin(\omega^{(n)}(\tau))}{\tau+1}+\omega^{(n)}(\tau)\Big{)}\ d\tau.\end{split}  \tag{20}\]  
Then the inequalities \(\omega^{(0)}(\sigma)=e^{-\sigma}\geq\omega^{(1)}(\sigma)\geq\omega^{(2)}( \sigma)\geq\omega^{(3)}(\sigma)\geq\omega^{(4)}(\sigma)\geq\ldots\geq 0\) hold (see Table 1 and Figure 3). The sequence of successive approximations \(\{w^{(n)}(\sigma)\}\) is a decreasing sequence and, according to Theorem 2 and Corollary 2, it is convergent and its limit is 0, i.e., the sequence is approaching the solution of GPDE (16). Consider the function \(\varsigma^{(0)}(\sigma)=-e^{-\sigma},\;\sigma\in[0,4]\). From inequality (19) it follows  
\[-e^{-\sigma}\leq-e^{-\sigma-4}E_{0,3}(-\big{(}2\sigma\big{)}^{0, 3})\\ +\frac{1}{0.5^{0.3}\Gamma(0.3)}\int_{0}^{\sigma}(\sigma-\tau)^{-0. 7}e^{\tau-\sigma}E_{0,3,0,3}(-\big{(}2(\sigma-\tau)\big{)}^{0,3})\Big{(}\frac {\sin(-e^{-\tau})}{\tau+1}-e^{-\tau}\Big{)}d\tau, \tag{21}\]  
\[\sigma\in[0,4],\]  
and \(\varsigma^{(0)}(0)-\varsigma^{(0)}(4)=-1+e^{-4}<0\). Therefore the function \(\varsigma^{(0)}(\sigma)=-e^{-\sigma}\) is a mild lower solution of (16). We construct an increasing sequence \(\{\varsigma^{(n)}(t)\}\) which, according to Theorem 2 and Corollary 2, is approaching 0, i.e., the sequence is approaching the solution of GPDE (16).