Second order boundary value problems with sign-changing nonlinearities and nonhomogeneous boundary conditions. (English) Zbl 1249.34055

Graef, J. R.; Kong, L.; Kong Q.; Yang, Bo

Math. Bohem. 136, No. 4, 337-356 (2011).

The paper studies the differential equation \[ u''+g(t)f(t,u)=0, \quad t\in (0,1), \] subject to the multi-point nonhomogeneous conditions \[ u(0)=\alpha u(\xi )+\lambda, \quad u(1)=\beta u(\eta )+\mu, \] where \(f\:[0,1]\times \mathbb {R}\to \mathbb {R}\) and \(g\:[0,1]\to [0,\infty )\) are continuous with \(g(t)\not \equiv 0\) on \([0,1]\), \(\xi , \eta \in [0,1]\), \(\alpha \), \(\beta \), \(\lambda \), \(\mu \in [0,\infty ),\) \(\alpha (1-\xi )<1\), \(\beta \eta <1\) and \((1-\alpha )(1-\beta \eta )+(1-\beta )\alpha \xi >0\). The authors apply the topological degree theory and derive several new criteria for the existence of nontrivial solutions of the above boundary value problem provided the nonlinear term \(f\) is a sign-changing function and not necessarily bounded from below. Some of the existence conditions are determined by the relationship between the behavior of the quotient \(f(t,x)/x\) for \(x\) near 0 and \(\pm \infty \) and the smallest positive characteristic value of a related linear operator. Illustrative examples are given here, as well.

Reviewer: Irena Rachůnková (Olomouc)

MSC:

34B10	Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
34B08	Parameter dependent boundary value problems for ordinary differential equations
47N20	Applications of operator theory to differential and integral equations

Keywords:

nontrivial solution; multi-point nonhomogeneous conditions; existence; Leray-Schauder degree; cone