Electron. J. Differential Equations, Vol. 2023 (2023), No. 51, pp. 1-17.

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

Nazia Urus, Amit K. Verma

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

Submitted March 17, 2023. Published August 4, 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.
DOI: https://doi.org/10.58997/ejde.2023.51

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Nazia Urus
Department of Mathematics
IIT Patna, India
email: nazia.pma17@iitp.ac.in
Amit K. Verma
Department of Mathematics
IIT Patna, India
email: akverma@iitp.ac.in

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