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Homoclinic orbits and subharmonics for nonlinear second order difference equations. (English) Zbl 1120.39007

The authors derive the existence of a nontrivial homoclinic orbit as limit of subharmonic solutions for scalar nonlinear second order self-adjoint difference equations of the form \[ \Delta[p(t)\Delta u(t-1)]+q(t)u(t)=f(t,u(t)), \] where \(p,q\) and \(f\) are \(T\)-periodic functions (in time). Moreover, the precise assumptions read as follows: (1) \(p(t)>0\), (2) \(q(t)<0\), (3) \(\lim_{x\to 0}\frac{f(t,x)}{x}=0\), and (4) \(xf(t,x)\leq\beta\int_0^xf(t,s)\,ds<0\) for some constant \(\beta>2\).
The basic proof technique is to embed the above problem into a variational framework based on Ekeland’s principle and the application of an appropriate Mountain Pass lemma.

MSC:

39A14 Partial difference equations
37C29 Homoclinic and heteroclinic orbits for dynamical systems
49J40 Variational inequalities
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[1] Agarwal, R. P., Difference Equations and Inequalities: Theory, Methods, and Applications (2000), Marcel Dekker, Inc. · Zbl 0952.39001
[2] Ahlbrandt, C. D.; Peterson, A. C., Discrete Hamiltonian Systems: Difference Equations, Continued Fraction and Riccati Equations (1996), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0860.39001
[3] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381 (1973) · Zbl 0273.49063
[4] Merdivenci Atici, F.; Guseinov, G. Sh., Positive periodic solutions for nonlinear difference equations with periodic coefficients, J. Math. Anal., 232, 166-182 (1999) · Zbl 0923.39010
[5] Benci, V.; Giannoni, F., Homoclinic orbits on compact manifolds, J. Math. Anal. Appl., 157, 568-576 (1991) · Zbl 0737.58052
[6] Butler, G. J., Integral average and the oscillation of second order ordinary differential equations, SIAM J. Math. Anal., 11, 190-200 (1980) · Zbl 0424.34033
[7] Coti-Zalati, V.; Ekeland, I.; Séré, E., A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 288, 133-160 (1990) · Zbl 0731.34050
[8] Ding, Y.; Girardi, M., Infinitely many homoclinic orbits of a hamiltonian system with symmetry, Nonlinear Anal., 38, 391-415 (1999) · Zbl 0938.37034
[9] Gao, Z. M.; Yu, J. S., The existence of periodic and subharmonic solutions for second order superlinear difference equations, Sci. China Ser. A, 33, 226-235 (2003), (in Chinese)
[10] Hale, J. K., Ordinary Differential Equations (1969), Wiley Interscience: Wiley Interscience New York · Zbl 0186.40901
[11] Kwong, M. K., On certain comparison theorems for second order linear oscillation, Proc. Amer. Math. Soc., 84, 539-542 (1982) · Zbl 0494.34022
[12] Nehari, Z., Asymptotic behavior of second order differential equations with integrable coefficients, Trans. Amer. Math. Soc., 282, 577-588 (1984)
[13] Moser, J., Stable and random Motions in Dynamical Systems (1973), Princeton University Press: Princeton University Press Princeton
[14] M. Ma, Z. Guo, Homoclinic orbits for second order self-adjoint difference equations, J. Math. Anal. Appl. 323 (1) 513-521; M. Ma, Z. Guo, Homoclinic orbits for second order self-adjoint difference equations, J. Math. Anal. Appl. 323 (1) 513-521 · Zbl 1107.39022
[15] Poincaré, H., Les éthodes Nouvelles de la Mécanique Céleste (1899), Gauthier-Viua: Gauthier-Viua Paris · JFM 30.0834.08
[16] Rabinowitz, P. H., Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh, 114A, 33-38 (1990) · Zbl 0705.34054
[17] Rabinowitz, P. H.; Tanaka, K., Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206, 473-479 (1991) · Zbl 0707.58022
[18] Yu, J. S.; Guo, Z. M.; Zou, X. F., Positive periodic solutions of second order self-adjoint difference equations, J. London Math. Soc., 71, 2, 146-160 (2005) · Zbl 1073.39009
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