On the Solvability of Third-Order Three Point Systems of Differential Equations with Dependence on the First Derivative

Article

Abstract

This paper presents sufficient conditions for the solvability of the third order three point boundary value problem
\begin{aligned} \left\{ \begin{array}{c} -u^{\prime \prime \prime }(t)=f(t,\,v(t),\,v^{\prime }(t)) \\ -v^{\prime \prime \prime }(t)=h(t,\,u(t),\,u^{\prime }(t)) \\ u(0)=u^{\prime }(0)=0,u^{\prime }(1)=\alpha u^{\prime }(\eta ) \\ v(0)=v^{\prime }(0)=0,v^{\prime }(1)=\alpha v^{\prime }(\eta ). \end{array} \right. \end{aligned}
The arguments apply Green’s function associated to the linear problem and the Guo–Krasnosel’skiĭ theorem of compression-expansion cones. The dependence on the first derivatives is overcome by the construction of an adequate cone and suitable conditions of superlinearity/sublinearity near 0 and $$+\infty$$. Last section contains an example to illustrate the applicability of the theorem.

Keywords

Coupled systems Green functions Guo–Krasnosel’skiĭ fixed-point in cones Positive solution

Mathematics Subject Classification

34B15 34B18 34B27 34L30

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Authors and Affiliations

• Feliz Minhós
• 1
• Robert de Sousa
• 2
1. 1.Departamento de Matemática, Escola de Ciências e Tecnologia, Centro de Investigação em Matemática e Aplicações (CIMA), Instituto de Investigação e Formação AvançadaUniversidade de ÉvoraÉvoraPortugal
2. 2.Faculdade de Ciências e Tecnologia, Núcleo de Matemática e Aplicações (NUMAT)Universidade de Cabo Verde, Campus de PalmarejoPraiaCabo Verde

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