WELL ORDERED MONOTONE ITERATIVE TECHNIQUE FOR NONLINEAR SECOND ORDER FOUR POINT DIRICHLET BVPS

Amit Verma,Nazia Urus

doi:10.3846/mma.2022.14198

Abstract

In this article, we develop a monotone iterative technique (MI-technique) with lower and upper (L-U) solutions for a class of four-point Dirichlet nonlinear boundary value problems (NLBVPs), defined as, ... where ..., ... the non linear term ...is continuous function in x, one sided Lipschitz in ψ and Lipschitz in . To show the existence result, we construct Green’s function and iterative sequences for the corresponding linear problem. We use quasilinearization to construct these iterative schemes. We prove maximum principle and establish monotonicity of sequences of lower solution and upper solution such that... Then under certain sufficient conditions we prove that these sequences converge uniformly to the solution ψ(x) in a specific region where

Highlights

In real-life, there are many applications of multi-point (m-point) boundary value problems (BVPs), e.g., suspension bridge
We prove maximum principle and establish monotonicity of sequences of lower solution (lm(x))m and upper solution (um(x))m such that lm(x) ≤ um(x)
We have focused on the existence of solution of second-order four-point Dirichlet nonlinear boundary value problems (NLBVPs)

Summary

Introduction

In real-life, there are many applications of multi-point (m-point) boundary value problems (BVPs), e.g., suspension bridge. By using FP theorem Liu et al [22], studied the existence of positive solutions for second order problem ψ (x) + a(x)f (ψ(x)) = 0 with BCs (1.2). In article [4], authors concentrated on the existence, nonexistence, and multiplicity of positive solutions for the non resonance problem (1.1)-(1.2), where. To study the existence of a solution we have explored an iterative process for a class of four-point Dirichlet NLBVPs with nonlinear. L-U solutions are defined, some assumptions are considered on the source term F , and we establish the main result on the existence of a solution for λ = 0.

Preliminaries

Green’s function and maximum principle

The nonlinear problem

The existence theorem

Numerical verifications

Conclusions