Study of Third-Order Three-Point Boundary Value Problem with Dependence on the First-Order Derivative

Abstract

Under certain conditions on the nonlinearity f and by using Leray–Schauder nonlinear alternative and the Banach contraction theorem, we prove the existence and uniqueness of nontrivial solution of the following third-order three-point boundary value problem (BVP1):

$$\displaystyle\begin{array}{rcl} & & \left \{\begin{array}{c} {u}^{{\prime\prime\prime}} + f\left (t,u\left (t\right ),{u}^{{\prime}}\left (t\right )\right ) = 0,\ \ \ t \in \left (0,1\right ) \\ \alpha {u}^{{\prime}}\left (1\right ) =\beta u\left (\eta \right ), u\left (0\right ) = {u}^{{\prime}}\left (0\right ) = 0 \end{array} \right. \\ & & \begin{array}{c} \text{where} \beta, \text{ }\alpha \in \mathbb{R}_{+}^{{\ast}},\text{ }0 <\eta < 1; \end{array} \\ \end{array}$$

then we study the positivity by applying the well-known Guo–Krasnosel’skii fixed-point theorem. The interesting point lies in the fact that the nonlinear term is allowed to depend on the first-order derivative u ^′.