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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 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 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
[14] 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 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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