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

• Feliz Minhós
• Robert de Sousa
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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