  The aim of this article is to prove the existence of solution and compute the
region of existence for a class of 4-point BVPs defined as, \begin{eqnarray*}
&&-u''(x)=\psi(x,u,u'), \quad 0<x<1, &&u'(0)=\lambda_{1}u(\xi), \quad
u'(1)=\lambda_{2} u(\eta), \end{eqnarray*} where $I=[0,1]$, $0<\xi\leq\eta<1$
and $\lambda_1,\lambda_2> 0$. The non linear source term $\psi\in
C(I\times\mathbb{R}\times\mathbb{R},\mathbb{R})$ is one sided Lipschitz in $u$
with Lipschitz constant $L_1$ and Lipschitz in $u'$ with Lipschitz function
$L_2(x)$, where $L_2(x):I\rightarrow \mathbb{R}^+ $ such that $ L_2(0)=0$ and
$L_2'(x)\geq 0$. The novelty in this paper allows us to use simplest form of
computational iteration and existence is achieved with a restriction that $L_2$
depends on $x$. We develop monotone iterative technique in well ordered and
reverse ordered cases. We prove maximum anti-maximum principle under certain
assumptions and use it to show the monotonic behavior of sequences of upper and
lower solutions. The conditions derived in this paper are sufficient and have
been verified for two examples. The method involves Newton's quasilinearization
which involves a parameter $k$ which is equivalent to $\dfrac{\partial
\psi}{\partial u}$. Our aim is to find a range of $k$ so that the iterative
technique is convergent.
