## 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)

### Citations:

Zbl 0386.47035; Zbl 0816.49001; Zbl 0816.34014
Full Text:

### References:

 [1] Brezis, H.; Nirenberg, L., Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Scuola Norm. Sup. Pisa, 5, 225-326 (1978) · Zbl 0386.47035 [2] Cañada, A.; Drábek, P., On semilinear problems with nonlinearities depending only on derivatives, SIAM J. Math. Anal., 27, 543-557 (1996) · Zbl 0852.34018 [3] Capietto, A.; Mawhin, J.; Zanolin, F., Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc., 329, 41-72 (1992) · Zbl 0748.34025 [4] Capietto, A.; Wang, Z., Periodic solutions of Liénard equations at resonance, Differential Integral Equations, 16, 605-624 (2003) · Zbl 1039.34034 [5] Capietto, A.; Wang, Z., Periodic solutions of Liénard equations with asymmetric nonlinearities at resonance, J. Lond. Math. Soc., 68, 119-132 (2003) · Zbl 1044.34002 [6] Dancer, E. N., Boundary-value problems for weakly nonlinear ordinary differential equations, Bull. Austral. Math. Soc., 15, 321-328 (1976) · Zbl 0342.34007 [7] de Figueiredo, D., The Dirichlet Problem for Nonlinear Elliptic Equations: a Hilbert Space Approach, Lecture Notes in Math., vol. 446 (1975), Springer: Springer Berlin · Zbl 0312.35032 [8] Drábek, P., Landesman-Lazer type condition and nonlinearities with linear growth, Czechoslovak Math. J., 40, 70-86 (1990) · Zbl 0705.34009 [9] Drábek, P., Landesman-Lazer condition for nonlinear problems with jumping nonlinearities, J. Differential Equations, 85, 186-199 (1990) · Zbl 0699.34019 [10] Fabry, C., Landesman-Lazer conditions for periodic boundary value problems with asymmetric nonlinearities, J. Differential Equations, 116, 405-418 (1995) · Zbl 0816.34014 [11] Fabry, C.; Fonda, A., Periodic solutions of nonlinear differential equations with double resonance, Ann. Mat. Pura Appl., 157, 99-116 (1990) · Zbl 0729.34025 [12] Fabry, C.; Fonda, A., Nonlinear equations at resonance and generalized eigenvalue problems, Nonlinear Anal., 18, 427-444 (1992) · Zbl 0756.34079 [13] Fabry, C.; Fonda, A., Nonlinear resonance in asymmetric oscillators, J. Differential Equations, 147, 58-78 (1998) · Zbl 0915.34033 [14] Fabry, C.; Fonda, A., Periodic solutions of perturbed isochronous Hamiltonian systems at resonance, J. Differential Equations, 214, 299-325 (2005) · Zbl 1080.34020 [15] Fonda, A., Positively homogeneous Hamiltonian systems in the plane, J. Differential Equations, 200, 162-184 (2004) · Zbl 1068.34035 [16] Fonda, A., Topological degree and generalized asymmetric oscillators, Topol. Methods Nonlinear Anal., 28, 171-188 (2006) · Zbl 1105.37036 [17] Fonda, A.; Mawhin, J., Planar differential systems at resonance, Adv. Differential Equations, 11, 1111-1133 (2006) · Zbl 1155.34020 [18] Frederickson, P. O.; Lazer, A. C., Necessary and sufficient damping in a second order oscillator, J. Differential Equations, 5, 262-270 (1969) · Zbl 0167.07902 [19] Fučik, S., Solvability of Nonlinear Equations and Boundary Value Problems (1980), Reidel: Reidel Boston · Zbl 0453.47035 [20] Habets, P.; Sanchez, L., A two-point problem with nonlinearity depending only on the derivative, SIAM J. Math. Anal., 28, 1205-1211 (1997) · Zbl 0886.34015 [21] Iannacci, R.; Nkashama, M. N., Nonlinear boundary value problems at resonance, Nonlinear Anal., 11, 455-473 (1987) · Zbl 0676.35023 [22] Kannan, R.; Nagle, R. K.; Pothoven, K. L., Remarks on the existence of solutions of $$x'' + x + \arctan x^\prime = p(t), x(0) = x(\pi) = 0$$, Nonlinear Anal., 22, 793-796 (1994) · Zbl 0802.34021 [23] Krasnosel’skiĭ, A. M., On bifurcation points of equations with Landesman-Lazer type nonlinearities, Nonlinear Anal., 18, 1187-1199 (1992) · Zbl 0783.47070 [24] Krasnosel’skiĭ, M. A., Translations Along Trajectories of Differential Equations (1968), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1398.34003 [25] Landesman, E.; Lazer, A. C., Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech., 19, 609-623 (1970) · Zbl 0193.39203 [26] Lazer, A. C., A second look at the first result of Landesman-Lazer type, Electron. J. Differ. Equ. Conf., 5, 113-119 (2000) · Zbl 0970.34033 [27] Lazer, A. C.; Leach, D. E., Bounded perturbations of forced harmonic oscillators at resonance, Ann. Mat. Pura Appl., 82, 49-68 (1969) · Zbl 0194.12003 [28] Mawhin, J., Landesman-Lazer’s type problems for nonlinear equations, Conf. Semin. Mat. Univ. Bari, 147, 1-22 (1977) · Zbl 0436.47050 [29] Nečas, J., On the range of nonlinear operators with linear asymptotes which are not invertible, Comment. Math. Univ. Carolin., 14, 63-72 (1973) · Zbl 0257.47032 [30] Omari, P.; Zanolin, F., A note on nonlinear oscillations at resonance, Acta Math. Sinica (N.S.), 3, 351-361 (1987) · Zbl 0648.34040 [31] Rebelo, C., A note on uniqueness of Cauchy problems associated to planar Hamiltonian systems, Port. Math., 57, 415-419 (2000) · Zbl 0979.37032 [32] Ruiz, D., Resonant semilinear problems with nonlinear term depending on the derivative, J. Math. Anal. Appl., 295, 163-173 (2004) · Zbl 1058.34017
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