×

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.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
47H11 Degree theory for nonlinear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Cabada, A.; Grossinho, M. R.; Minhós, F., On the solvability of some discontinuous third order nonlinear differential equations with two point boundary conditions, J. Math. Anal. Appl., 285, 174-190 (2003) · Zbl 1048.34033
[2] Cabada, A.; Pouso, R. L., Extremal solutions of strongly nonlinear discontinuous second-order equations with nonlinear functional boundary conditions, Nonlinear Anal., 42, 1377-1396 (2000) · Zbl 0964.34016
[3] Cabada, A.; Pouso, R. L., Existence results for the problem \((\varphi(u^\prime))^\prime = f(t, u, u^\prime)\) with nonlinear boundary conditions, Nonlinear Anal., 35, 221-231 (1999) · Zbl 0920.34029
[6] Du, Z.; Ge, W.; Lin, X., Existence of solutions for a class of third-order nonlinear boundary value problems, J. Math. Anal. Appl., 294, 104-112 (2004) · Zbl 1053.34017
[7] Grossinho, M.; Minhós, F., Existence result for some third order separated boundary value problems, Nonlinear Anal., 47, 2407-2418 (2001) · Zbl 1042.34519
[12] Habets, P.; Pouso, R. L., Examples of the nonexistence of a solution in the presence of upper and lower solutions, ANZIAM J., 44, 591-594 (2003) · Zbl 1048.34036
[13] Mawhin, J., Topological Degree Methods in Nonlinear Boundary Value Problems, Regional Conf. Ser. in Math., vol. 40 (1979), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0414.34025
[16] Nagumo, M., Über die differentialgleichung \(y'' = f(t, y, y^\prime)\), Proc. Phys.-Math. Soc. Japan, 19, 861-866 (1937)
[17] Nagumo, M., On principally linear elliptic differential equations of the second order, Osaka Math. J., 6, 207-229 (1954) · Zbl 0057.08201
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.