Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 51, pp. 1-17.

Title: Existence and uniqueness results for fourth-order four-point BVP arising
in bridge design in the presence of reverse ordered upper and lower solutions
       
Authors: Nazia Urus (IIT Patna, India)
         Amit K. Verma (IIT Patna, India)

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,
$$\displaylines{
- y^{(4)}( s)-\lambda y''( s)=\mathcal{F}( s, y( s)), \quad   s\in(0,1),\cr
 y(0)=0,\quad  y(1)= \delta_1  y(\eta_1)+\delta_2  y(\eta_2),\\
 y''(0)=0,\quad  y''(1)= \delta_1  y''(\eta_1)+\delta_2  y''(\eta_2),
}$$
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-order
four-point non-linear BVPs.

Submitted March 17, 2023. Published August 04, 2023.
Math Subject Classifications: 34B10, 34B15, 34B16, 34B27, 34B60.
Key Words: Monotone iterative technique; upper solutions; lower solutions;
    fourth-order; non-linear; four-point; Green's function.