Existence of Solutions for Fourth-Order Three-Point Boundary Value Problems on the Half-Line via Upper and Lower Solutions
Abstract
In this paper, we study on a half-line and demonstrate the existence of unbounded or bounded solutions of the following three-point fourth-order boundary value problem: For all $\xi\in(0,+\infty)$, ${\Phi}''''(\xi)+p(\xi) g(\xi, {\Phi}(\xi), {\Phi}'(\xi), {\Phi}''(\xi),{\Phi}'''(\xi))=0$ with ${\Phi}''(0)= \Lambda$, ${\Phi}(\rho)=B_1$, ${\Phi}'(0)=B_2$, and ${\Phi}'''(+\infty)=\Omega$, where $\rho$ is fixed and $\rho\in(0,+\infty)$, and $g:[0,+\infty)\times \mathbb{R}^4\rightarrow\mathbb{R}$ provides the condition of Nagumo. In order to address this objective, we employ various mathematical techniques, including the upper and lower solution method, Schauder's fixed point theorem, and topological degree theory. By utilizing these methods, we establish sufficient conditions that guarantee the existence of at least one solution, as well as at least three solutions, for the aforesaid problem. To illustrate the significance of the obtained findings, we provide an example demonstrating the practical implications of the results herein.
Keywords
Three-point boundary value problem, lower and upper solution method, Schauder's fixed point theorem, topological degree theory, half-line
Supporting Institution
Office of Scientific Research Projects Coordination at Ege University
Thanks
This work was supported by the Office of Scientific Research Projects Coordination at Ege University, Grant number: 15-FEN-070.
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