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.
Monotone Iterative and Upper–Lower Solution Techniques for Solving the Nonlinear ψ-Caputo Fractional Boundary Value Problem
