On the solvability of third-order three point systems of differential equations with dependence on the first derivative

TL;DR

This paper establishes sufficient conditions for the existence of solutions to a third-order three-point boundary value problem involving derivatives, using Green's functions and fixed point theorems, with an example demonstrating applicability.

Contribution

It introduces new solvability criteria for third-order systems with derivative dependence, employing cone construction and fixed point theorems, extending previous results to more complex boundary conditions.

Findings

Derived solvability conditions using Green's functions.

Applied Guo--Krasnosel'ski
2f theorem to establish existence.

Provided an example illustrating the theoretical results.

Abstract

This paper presents sufficient conditions for the solvability of the third order three point boundary value problem \begin{equation*} \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{equation*} The arguments apply Green's function associated to the linear problem and the Guo--Krasnosel'ski\u{\i} 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.