Geodesic
Positive solutions of a fourth-order differential equation with integral boundary conditions
Mathematica Bohemica, Tome 148 (2023) no. 4, pp. 583-601
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We study the existence of positive solutions to the fourth-order two-point boundary value problem $$ \begin {cases} u^{\prime \prime \prime \prime }(t) + f(t,u(t))=0, 0 t 1,\\ u^{\prime }(0) = u^\prime (1) = u^{\prime \prime }(0) =0, u(0) = \alpha [u], \end {cases} $$ where $\alpha [u]=\int ^{1}_{0}u(t){\rm d}A(t)$ is a Riemann-Stieltjes integral with $A \geq 0$ being a nondecreasing function of bounded variation and $f \in \mathcal {C}([0,1] \times \mathbb {R}_{+}, \mathbb {R}_{+})$. The sufficient conditions obtained are new and easy to apply. Their approach is based on Krasnoselskii's fixed point theorem and the Avery-Peterson fixed point theorem.
We study the existence of positive solutions to the fourth-order two-point boundary value problem $$ \begin {cases} u^{\prime \prime \prime \prime }(t) + f(t,u(t))=0, 0 t 1,\\ u^{\prime }(0) = u^\prime (1) = u^{\prime \prime }(0) =0, u(0) = \alpha [u], \end {cases} $$ where $\alpha [u]=\int ^{1}_{0}u(t){\rm d}A(t)$ is a Riemann-Stieltjes integral with $A \geq 0$ being a nondecreasing function of bounded variation and $f \in \mathcal {C}([0,1] \times \mathbb {R}_{+}, \mathbb {R}_{+})$. The sufficient conditions obtained are new and easy to apply. Their approach is based on Krasnoselskii's fixed point theorem and the Avery-Peterson fixed point theorem.
DOI : 10.21136/MB.2022.0045-22
Classification : 34B10, 34B18
Keywords: boundary value problem; fixed point; positive solution; cone; existence \hbox {theorem} 
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Padhi, Seshadev; Graef, John R. Positive solutions of a fourth-order differential equation with integral boundary conditions. Mathematica Bohemica, Tome 148 (2023) no. 4, pp. 583-601. doi: 10.21136/MB.2022.0045-22

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