## Nontrivial periodic solutions for asymptotically linear resonant difference problem.(English)Zbl 1103.39005

By combining minimax methods and Morse theory, the authors obtain results for the existence of nontrivial periodic solutions of second order difference equations of the form $\Delta^2 x_{n-1}+F^\prime(n,x_n)=0,\quad n\in \mathbb{Z}.$ where $$\Delta$$ is the forward difference operator, $$F\in C^2\left(\mathbb{Z}\times R^N, \mathbb{R}\right)$$ such that $$F(n,z)$$ is $$T$$-periodic with respect to the first variable, and $$F^\prime(n,z)$$ denotes the gradient of $$F$$ with respect to the second variable. Furthermore, they assume that $$F^\prime$$ satisfies the following asymptotically linear growth conditions. \begin{aligned} &F^\prime(n,z)=A_\infty(n)z+o(| z| )\quad \text{as} \quad | z| \rightarrow \infty, \tag{H1}\\ &F^\prime(n,z)=A_0(n)z+o(| z| )\quad \text{as} \quad | z| \rightarrow 0,\tag{H2}\end{aligned} where $$A_\infty(n),~A_0(n)$$ are $$N\times N$$ symmetric and $$T$$-periodic matrices.

### MSC:

 39A11 Stability of difference equations (MSC2000) 39A10 Additive difference equations
Full Text:

### References:

 [1] Agarwal, R.P., Difference equations and inequalities: theory, methods, and applications, (2000), Marcel Dekker New York · Zbl 0952.39001 [2] Agarwal, R.P.; Perera, K.; Regan, D.O., Multiple positive solutions of singular and nonsingular discrete problems via variational methods, J. nonlinear anal., 58, 69-73, (2004) · Zbl 1070.39005 [3] Atici, F.M.; Cabada, A., Existence and uniqueness results for discrete second order periodic boundary value problems, Comput. math. appl., 45, 1417-1427, (2003) · Zbl 1057.39008 [4] Atici, F.M.; Guseinov, G.Sh., Positive periodic solutions for nonlinear difference equations with periodic coefficients, J. math. anal. appl., 232, 166-182, (1999) · Zbl 0923.39010 [5] Ahlbrandt, C.D.; Peterson, A.C., Discrete Hamiltonian system: difference equations, continued fractions, and Riccati equations, (1996), Kluwer Acad. Publ. London · Zbl 0860.39001 [6] Amann, H.; Zehnder, E., Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. sc. norm. sup. Pisa, 7, 539-603, (1980) · Zbl 0452.47077 [7] Bartsch, T.; Li, S.J., Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear anal., 28, 419-441, (1997) · Zbl 0872.58018 [8] Chang, K.C., Critical point theory and its applications, (1980), Science and Technical Press Shanghai, China [9] Chang, K.C., Infinite dimensional Morse theory and multiple solution problems, (1993), BirkhĂ¤user Boston [10] Fei, G.H.; Kim, S.K.; Wang, T.X., Periodic solutions of classical Hamiltonian systems without palais – smale condition, J. math. anal. appl., 267, 665-678, (2002) · Zbl 0994.37032 [11] Guo, Z.M.; Yu, J.S., The existence of periodic and subharmonic solutions to subquadratic second order difference equations, J. London math. soc., 68, 419-430, (2003) · Zbl 1046.39005 [12] Guo, Z.M.; Yu, J.S., Existence of periodic and subharmonic solutions for second-order superlinear difference equation, Sci. China ser. A, 46, 506-515, (2003) · Zbl 1215.39001 [13] Kelley, W.G.; Peterson, A.C., Difference equations: an introduction with application, (1991), Academic Press Boston [14] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer New York · Zbl 0676.58017 [15] Rabinowitz, P.H., Minimax methods in critical point theory with applications to differential equations, CBMS reg. conf. ser. marh., vol. 65, (1986), Amer. Math. Soc. Providence, RI · Zbl 0609.58002 [16] Su, J.B., Existence and multiplicity results for classes of elliptic resonant problems, J. math. anal. appl., 273, 565-579, (2002) · Zbl 1121.35059 [17] Su, J.B., Multiplicity results for asymptotically linear elliptic problems at resonance, J. math. anal. appl., 278, 397-408, (2003) · Zbl 1290.35109 [18] Wu, X.P.; Tang, C.L., Periodic solutions of nonautonomous second order Hamiltonian systems with even-type potentials, Nonlinear anal., 55, 759-769, (2003) · Zbl 1030.37043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.