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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 49J40 Variational methods including variational inequalities
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##### References:
 [1] Agarwal, R. P.: Difference equations and inequalities: theory, methods, and applications. (2000) · Zbl 0952.39001 [2] Ahlbrandt, C. D.; Peterson, A. C.: Discrete Hamiltonian systems: difference equations, continued fraction and Riccati equations. (1996) · 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] Atici, F. Merdivenci; 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) [10] Hale, J. K.: Ordinary differential equations. (1969) · 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) · Zbl 0556.34055 [13] Moser, J.: Stable and random motions in dynamical systems. (1973) · Zbl 0271.70009 [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) [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, No. 2, 146-160 (2005) · Zbl 1073.39009