Region of existence of multiple solutions for a class of 4-point BVPs

TL;DR

This paper establishes the existence of multiple solutions for a class of 4-point boundary value problems using monotone iterative techniques, maximum principles, and Newton's quasilinearization, with a focus on computational simplicity.

Contribution

It introduces a novel approach allowing the use of simple iterative methods for 4-point BVPs with a Lipschitz source term depending on x, u, and u', and provides conditions for convergence.

Findings

Proved existence of solutions under new conditions.

Developed monotone iterative method for well-ordered solutions.

Verified results with two example problems.

Abstract

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.