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Double resonance with Landesman-Lazer conditions for planar systems of ordinary differential equations. (English) Zbl 1227.34037

This paper deals with the periodic boundary value problem associated to a planar first order system of the form \(u'=F(t,u)\), \(u(0)=u(t)\), with
\[ F(t,u)=-(1-\gamma(t,u)) J \nabla H_1 (u) - J \nabla H_2 (u) + r(t,u), \]
where \(J\) is the standard simplex matrix, \(H_1\) and \(H_2\) are two positively homogeneous functions of order two, \(\gamma\) and \(r\) are \(L^2\)-Carathéodory functions satisfying \(0 \leq \gamma(t,u) \leq 1\), \(|r(t,u)| \leq \eta(t)\) for a suitable \(\eta \in L^2\), for a.e. \(t\) and for every \(u\). In order to state the main result in this paper, let \(\phi,\psi\) be such that
\[ J \phi' = \nabla H_1(\phi), \, J \psi' = \nabla H_2(\psi), \]
respectively. Denote by \(\tau_{\phi},\tau_{\psi}\) the minimal periods of \(\phi,\psi\), respectively.
The main result (Theorem 2.1) reads as follows: “Assume the existence of \(N \in\mathbb N\) such that \(T/(N+1) \leq \tau_\psi \leq \tau_\phi < T/N.\) Suppose that, for every \(\theta\), the following relations are satisfied
\[ \liminf_{\lambda \to +\infty,\;\omega \to \theta}[\langle J F(t,\lambda \phi(t+\omega) ) \mid \phi(t+\omega) \rangle - 2 \lambda H_1(\phi(t)) ] \, dt > 0, \]
\[ \liminf_{\lambda \to +\infty,\;\omega \to \theta}[2 \lambda H_2(\psi(t)) - \langle J F(t,\lambda \psi(t+\omega) )\mid \psi(t+\omega) \rangle]\,dt > 0. \]
Then the given periodic BVP has at least one solution.”
The situation considered above is known as “double resonance”, and the integral conditions are known as “Landesman-Lazer”-type conditions. Section 2 contains also some interesting remarks which compare the above theorem with the result due to H. Brezis and L. Nirenberg [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 5, 225–326 (1978; Zbl 0386.47035)] and with the notion of \(\Gamma\)-convergence (cf. the book by G. Dal Maso [An introduction to \(\Gamma\)-convergence. Basel: Birkhäuser (1993; Zbl 0816.49001)]). In Section 3, the authors show that their main theorem generalizes to planar systems the result by C. Fabry [J. Differ. Equations 116, No. 2, 405–418 (1995; Zbl 0816.34014)] on the existence of periodic solutions to a second order scalar equation of the form \(x''+g(t,x)=0\); the notion of “resonance” is stated in this case in relation with the Dancer-Fučik spectrum.
In Section 4, another application is obtained for planar systems which are, roughly speaking, asymptotically controlled by piecewise linear functions. In Section 5, the authors consider the particular case when the nonlinearity interacts only with one resonant Hamiltonian, while, in Section 6, results are given for the case when double resonance occurs with two multiples of the same Hamiltonian function. Second order equations with damping are treated in Section 7. The proofs are performed in the framework of degree theory, together with some phase-plane analysis arguments based on the notion of rotation number.

MSC:

34C25 Periodic solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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