## 3 Main Theoretical Results
### Monotone-Iterative Technique
_The function \(\Delta\in C(\mathbb{R}\times\mathbb{R},\mathbb{R})\), and_ \[\Delta(x_{2},y_{2})-\Delta(x_{1},y_{1})\leq\mu(x_{2}-x_{1})+ \delta(y_{1}-y_{2}),\] \[\qquad\text{for}\ \ x_{1},x_{2},y_{1},y_{2}\in\mathbb{R}:\ \varepsilon(0) \leq x_{1}\leq x_{2}\leq w(0),\ \varepsilon(a)\leq y_{1}\leq y_{2}\leq\varpi(a)\] _holds, where \(\mu,\delta>0\)._ _Then there exist sequences \(\{\varepsilon^{(n)}(\cdot)\}_{n=0}^{\infty}\) and \(\{\varpi^{(n)}(\cdot)\}_{n=0}^{\infty}\), such that:_ _The sequence \(\{\varepsilon^{(k)}(\sigma)\}_{k=0}^{\infty}:\ \varepsilon^{(n+1)}( \sigma)=\Omega(\varepsilon^{(n)})(\sigma)\) for \(\sigma\in[0,a],\ \ n=0,1,2,\ldots\), consists of mild lower solutions of the GPDE (4) on \([0,a]\), it is increasing and it approaches uniformly on \([0,a]\) to \(\chi\in C([0,a])\) and the limit function \(\chi\) is a mild solution of the GPDE (4) on \([0,a]\), where the operator \(\Omega\) is defined by (7)._ _The sequence \(\{\varpi^{(k)}(\sigma)\}_{k=0}^{\infty}:\ \varpi^{(n+1)}(\sigma)=\Omega( \varpi^{(n)})(\sigma)\) for \(\sigma\in[0,a],\ \ n=0,1,2,\ldots\), consists of mild upper solutions of the GPDE (4) on \([0,a]\), it is decreasing and it approaches uniformly on \([0,a]\) to \(\mathcal{W}\in C([0,a])\) and the limit function \(\mathcal{W}\) is a mild solution of the GPDE (4) on \([0,a]\), where the operator \(\Omega\) is defined by (7)._ _The inequalities \(\varepsilon^{(k)}(\sigma)\leq\varpi^{(k)}(\sigma)\) and \(\chi(\sigma)\leq\mathcal{W}(\sigma)\) are satisfied for \(t\in[0,a]\), \(k=1,2,\ldots\), i.e., the inequalities_ \[\varepsilon^{(0)}(\sigma)\leq\varepsilon^{(1)}(\sigma)\leq\varepsilon^{(2)}( \sigma)\leq\cdots\leq\chi(\sigma)\leq\mathcal{W}(\sigma)\leq\ldots\varpi^{(2 )}(\sigma)\leq\varpi^{(1)}(\sigma)\leq\varpi^{(0)}(\sigma)\] _hold._  
**Proof. ** We will use the operator \(\Omega\) given by Equation (7) with constants \(\lambda,\mu\) defined in conditions 2 and 3 of Theorem 2. By induction we will prove the monotonicity of the sequence \(\{\varepsilon^{(k)}(\sigma)\}_{k=0}^{\infty}\). Let \(n=1\). The function \(\varsigma^{(0)}(\cdot)\) is a mild lower solution of GPDE (4). Thus, we have  
\[ \begin{split}\varsigma^{(0)}(\sigma)&\leq\Big{(} \varepsilon^{(0)}(0)-\frac{1}{\mu}\Delta(\varsigma^{(0)}(0),\varsigma^{(0)}(T ))\Big{)}e^{(\xi-1)\frac{\sigma}{\xi}}E_{\gamma}(-\lambda\big{(}\frac{\sigma} {\xi}\big{)}^{\gamma})\\ &\quad+\frac{1}{\xi^{\gamma}\Gamma(\gamma)}\int_{0}^{\sigma}( \sigma-\tau)^{\gamma-1}e^{(\xi-1)(\frac{\sigma-\tau}{\xi})}E_{\gamma,\gamma} \big{(}-\lambda\big{(}\frac{\sigma-\tau}{\xi}\big{)}^{\gamma}\big{)}\Big{(} \vartheta(\tau,\varsigma^{(0)}(\tau))+\lambda\varsigma^{(0)}(\tau)\Big{)}\ d\tau\\ &=\varsigma^{(1)}(\sigma),\qquad\sigma\in[0,a].\end{split} \]  
Then, from condition 2, we obtain  
\[ \begin{split}&\int_{0}^{\sigma}(\sigma-\tau)^{\gamma-1}e^{(\xi-1) \big{(}\frac{\sigma-\tau}{\xi}\big{)}}E_{\gamma,\gamma}\Big{(}-\lambda\Big{(} \frac{\sigma-\tau}{\xi}\Big{)}^{\gamma}\Big{)}\Big{(}\vartheta\Big{(}\tau, \varsigma^{(0)}(\tau)\Big{)}+\mu\varsigma^{(0)}(\tau)\Big{)}\ d\tau\\ &=\int_{0}^{\sigma}(\sigma-\tau)^{\gamma-1}e^{(\xi-1)\big{(} \frac{\sigma-\tau}{\xi}\big{)}}E_{\gamma,\gamma}\Big{(}-\lambda\Big{(}\frac{ \sigma-\tau}{\xi}\Big{)}^{\gamma}\Big{)}\Big{(}\vartheta\Big{(}\tau, \varsigma^{(1)}(\tau)\Big{)}+\lambda\varsigma^{(1)}(\tau)\Big{)}d\tau\\ &+\int_{0}^{\sigma}(\sigma-\tau)^{\gamma-1}e^{(\xi-1)\big{(} \frac{\sigma-\tau}{\xi}\big{)}}E_{\gamma,\gamma}\Big{(}-\lambda\Big{(}\frac{ \sigma-\tau}{\xi}\Big{)}^{\gamma}\Big{)}\\ &\quad\times\Big{(}\vartheta(\tau,\varsigma^{(0)}(\tau))-\vartheta (\tau,\varsigma^{(1)}(\tau))+\lambda(\varsigma^{(0)}(\tau)-\varsigma^{(1)}( \tau))\Big{)}d\tau\\ &\leq\int_{0}^{\sigma}(\sigma-\tau)^{\gamma-1}e^{(\xi-1)\big{(} \frac{\sigma-\tau}{\xi}\big{)}}E_{\gamma,\gamma}\Big{(}-\lambda\Big{(}\frac{ \sigma-\tau}{\xi}\Big{)}^{\gamma}\Big{)}\Big{(}\vartheta(\tau,\varsigma^{(1)}( \tau))+\lambda\varsigma^{(1)}(\tau)\Big{)}d\tau.\end{split}  \tag{9}\]From condition 3 we obtain the inequality  
\[