Existence result for a third-order ODE with nonlinear boundary conditions in presence of a sign-type Nagumo control. (English) Zbl 1089.34017

Summary: The authors provide an existence and location result for the third-order nonlinear differential equation \[ u^{\prime \prime \prime }\left( t\right) =f\left( t,u\left( t\right) ,u^{\prime }\left( t\right) ,u^{\prime \prime }\left( t\right) \right) , \] where \(f:\left[ a,b\right] \times \mathbb{R}^{3}\rightarrow \mathbb{R}\) is a continuous function, and two types of boundary conditions: \[ u\left( a\right) = A,\text{ }\phi \left( u^{\prime }\left( b\right) ,u^{\prime \prime }\left( b\right) \right) =0,\text{ }u^{\prime \prime }\left( a\right) =B, \] or \[ u\left( a\right) = A,\text{ }\psi \left( u^{\prime }\left( a\right) ,u^{\prime \prime }\left( a\right) \right) =0,\text{ }u^{\prime \prime }\left( b\right) =C, \] with \(\phi \), \(\psi :\mathbb{R}^{2}\rightarrow \mathbb{R}\) continuous functions, monotonous in the second variable and \(A,\) \(B,\) \(C\in \mathbb{R}\). They assume that \(f\) satisfies a sign-type Nagumo condition which allows an asymmetric unbounded behaviour on the nonlinearity and they use Leray-Schauder degree theory and lower and upper solutions technique.


34B15 Nonlinear boundary value problems for ordinary differential equations
47H11 Degree theory for nonlinear operators
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