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Positive solutions for systems of generalized three-point nonlinear boundary value problems

Johnny Henderson, Sotiris K. Ntouyas, Ioannis K. Purnaras (2008)

Commentationes Mathematicae Universitatis Carolinae

Values of λ are determined for which there exist positive solutions of the system of three-point boundary value problems, u ' ' + λ a ( t ) f ( v ) = 0 , v ' ' + λ b ( t ) g ( u ) = 0 , for 0 < t < 1 , and satisfying, u ( 0 ) = β u ( η ) , u ( 1 ) = α u ( η ) , v ( 0 ) = β v ( η ) , v ( 1 ) = α v ( η ) . A Guo-Krasnosel’skii fixed point theorem is applied.

Positive solutions for third order multi-point singular boundary value problems

John R. Graef, Lingju Kong, Bo Yang (2010)

Czechoslovak Mathematical Journal

We study a third order singular boundary value problem with multi-point boundary conditions. Sufficient conditions are obtained for the existence of positive solutions of the problem. Recent results in the literature are significantly extended and improved. Our analysis is mainly based on a nonlinear alternative of Leray-Schauder.

Positive solutions of a fourth-order differential equation with integral boundary conditions

Seshadev Padhi, John R. Graef (2023)

Mathematica Bohemica

We study the existence of positive solutions to the fourth-order two-point boundary value problem u ' ' ' ' ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ' ( 0 ) = u ' ( 1 ) = u ' ' ( 0 ) = 0 , u ( 0 ) = α [ u ] , where α [ u ] = ∫ 0 1 u ( t ) d A ( t ) is a Riemann-Stieltjes integral with A ≥ 0 being a nondecreasing function of bounded variation and f ∈ 𝒞 ( [ 0 , 1 ] × ℝ + , 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.

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