Existence results for nonlinear periodic boundary-value problems. (English) Zbl 1178.34024

Graef, John R.; Kong, Lingju

Proc. Edinb. Math. Soc., II. Ser. 52, No. 1, 79-95 (2009).

Consider a second order scalar equation of the form
\[ -(p(t)u')'+q(t)u-w(t)f(t,u) =0 \]
subject to the boundary conditions
\[ u(0)=u(T), \, p(0)u'(0)=p(1)u'(1), \]
where \(1/p, q, w \in L^1(0,T), \, p,w >0, \, q \geq 0, \, q \not\equiv 0\) a.e. in \((0,T)\), and \(f\) is a continuous function defined in \([0,T]\times {\mathbb R}\). The authors provide sufficient conditions for the existence of non-trivial, positive and negative solutions when \(f\) may change sign and is not necessarily bounded from below. More precisely, restrictions are imposed on the behaviour of the quotient \(f(t,x)/ x\) near zero and \(\pm \infty\) and on the smallest eigenvalue of the associated linear problem. Applications are also given to some related eigenvalue problem. The proofs are performed in the framework of fixed point theory in cones.

Reviewer: Anna Capietto (Torino)

MSC:

34B15	Nonlinear boundary value problems for ordinary differential equations
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34C25	Periodic solutions to ordinary differential equations
47H10	Fixed-point theorems
47N20	Applications of operator theory to differential and integral equations

Keywords:

periodic boundary value problem; cones; degree theory