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Alhussein Ma Ahmed, Mutasim Abdalmonim Alsiddig, Tarteel Abdalgader, Khalid Ahmed Abbakar, Badradeen A. A. Adam, and Haroun M. M. Suliman, Multiple positive solutions of discrete third-order three-point BVP with sign-changing Green’s function, Eur. J. Math. Appl. 5 (2025), Article ID 10.

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Volume 5 (2025), Article ID 10

https://doi.org/10.28919/ejma.2025.5.10

Published: 08/05/2025

Abstract:

In this article,by using the Leggett-Williams fixed point theorem we research the multiple Positive Solutions for the following third-order three-point boundary value problem (BVP):
\begin{equation*}
\left\{\begin{array}{ll}\Delta^{3}u(t-1) = a(t)f (t,u(t)) ,\quad t \in [1,T-2]_\mathbb{Z},\\ u(T)=\Delta^{2}u(0)=\Delta u(T-1)-\Delta^{2}u(\eta)=0 \end{array}\right.
\end{equation*}
where $T>6$ is an integer and $f:[1,T-2]_\mathbb{Z}\times[0,+\infty)\rightarrow[0,+\infty)$ is continuous. $a: [0,T-2]_\mathbb{Z}\rightarrow(0,\infty) $, and $\eta$ satisfies the condition:
$F_{0}$ $\eta\in[\frac{T-1}{2},T-2]$. if $T$ is an odd number or $\eta\in[\frac{T-2}{2},T-2]$. if $T$ is an even number\\ The emphasis is mainly that although the corresponding Green’s function is sign-changing, we still obtain the existence of at least $2n-1$ positive solutions for arbitrary positive integer $m$ under suitable conditions on $f$.

How to Cite:

Alhussein Ma Ahmed, Mutasim Abdalmonim Alsiddig, Tarteel Abdalgader, Khalid Ahmed Abbakar, Badradeen A. A. Adam, and Haroun M. M. Suliman, Multiple positive solutions of discrete third-order three-point BVP with sign-changing green’s function, Eur. J. Math. Appl. 5 (2025), Article ID 10. https://doi.org/10.28919/ejma.2025.5.10