Existence of Solutions for a Class of Nonlinear Neumann BVPs in the Presence of Upper and Lower Solutions

In this article, we explore the monotone iterative technique (MI-technique) to study the existence of solutions for a class of nonlinear Neumann 4-point, boundary value problems (BVPs) defined as, \begin{eqnarray*} \begin{split} -\z^{(2)}(\y)=\x(\y,\z,\z^{(1)}),\quad 0<\y<1,\\ \z^{(1)}(0)=\lambda \z^{(1)}(\beta_1 ),\quad \z^{(1)}(1)=\delta \z^{(1)}(\beta_2), \end{split} \end{eqnarray*} where $ 0<\beta_1 \leq \beta_2 <1$ and $\lambda$, $\delta\in (0,1)$. The nonlinear term $ \x(\y,\z,\z^{(1)}): \Omega\rightarrow \mathbb{R} $, where $\Omega =[0,1]\times \mathbb{R}^2 $, is Lipschitz in $ \z^{(1)}(\y)$ and one sided Lipschitz in $ \z(\y)$. Using lower solution $l(\y)$ and upper solutions $u(\y)$, we develop MI-technique, which is based on quasilinearization. To construct the sequences of upper and lower solutions which are monotone, we prove maximum principle as well as anti maximum principle. Then under certain assumptions, we prove that these sequence converges uniformly to the solution $ \z(\y)$ in the specific region, where $ \frac{\partial\x}{\partial\z}<0 $ or $ \frac{\partial\x}{\partial\z}>0 $. To demonstrate that the proposed technique is effective, we compute the solution of the nonlinear multi-point BVPs. We don’t require sign restriction which is very common and very strict condition.