Existence and uniqueness results for fourth-order four-point BVP arising in bridge design in the presence of reverse ordered upper and lower solutions

Abstract

In this article, we establish the existence of solutions for a fourth-order four-point non-linear boundary value problem (BVP) which arises in bridge design, $$\displaylines{ - y^{(4)}( s)-\lambda y''( s)=\mathcal{F}( s, y( s)), \quad s\in(0,1),\cry(0)=0,\quad y(1)= \delta_1 y(\eta_1)+\delta_2 y(\eta_2),\cr y''(0)=0,\quad y''(1)= \delta_1 y''(\eta_1)+\delta_2 y''(\eta_2), }$$ where \(\mathcal{F} \in C([0,1]\times \mathbb{R},\mathbb{R})\), \(\delta_1, \delta_2>0\), \(0<\eta_1\le \eta_2 <1\), \(\lambda=\zeta_1+\zeta_2 \), where \(\zeta_1\) and \(\zeta_2\) are the real constants. We have explored all gathered \(0<\zeta_1<\zeta_2\), \(\zeta_1<0<\zeta_2\), and \( \zeta_1<\zeta_2<0 \). We extend the monotone iterative technique and establish the existence results with reverse ordered upper and lower solutions to fourth-orderfour-point non-linear BVPs. For more information see https://ejde.math.txstate.edu/Volumes/2023/51/abstr.html