Multiple periodic solutions for two classes of nonlinear difference systems involving classical \((\phi_1,\phi_2)\)-Laplacian
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Authors
Xingyong Zhang
- Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan 650500, P. R. China.
Liben Wang
- Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan 650500, P. R. China.
Abstract
In this paper, we investigate the
existence of multiple periodic solutions for two classes of nonlinear difference systems involving
\((\phi_1,\phi_2)\)-Laplacian. First, by using an important critical point theorem due to B. Ricceri, we establish an existence theorem of three periodic solutions for
the first nonlinear difference system with \((\phi_1,\phi_2)\)-Laplacian and two parameters. Moreover, for the second nonlinear difference
system with \((\phi_1,\phi_2)\)-Laplacian, by using the Clark's Theorem, we obtain a
multiplicity result of periodic solutions under a symmetric
condition. Finally, two examples are given to verify
our theorems.
Share and Cite
ISRP Style
Xingyong Zhang, Liben Wang, Multiple periodic solutions for two classes of nonlinear difference systems involving classical \((\phi_1,\phi_2)\)-Laplacian, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 8, 4381--4397
AMA Style
Zhang Xingyong, Wang Liben, Multiple periodic solutions for two classes of nonlinear difference systems involving classical \((\phi_1,\phi_2)\)-Laplacian. J. Nonlinear Sci. Appl. (2017); 10(8):4381--4397
Chicago/Turabian Style
Zhang, Xingyong, Wang, Liben. "Multiple periodic solutions for two classes of nonlinear difference systems involving classical \((\phi_1,\phi_2)\)-Laplacian." Journal of Nonlinear Sciences and Applications, 10, no. 8 (2017): 4381--4397
Keywords
- Difference systems
- periodic solutions
- multiplicity
- variational approach.
MSC
References
-
[1]
G. Bonanno, P. Candito, Nonlinear difference equations investigated via critical point methods, Nonlinear Anal., 70 (2009), 3180–3186.
-
[2]
P. Candito, N. Giovannelli, Multiple solutions for a discrete boundary value problem involving the p-Laplacian, Comput. Math. Appl., 56 (2008), 959–964.
-
[3]
H.-Y. Deng, X.-Y. Zhang, H. Fang, Existence of periodic solutions for a class of discrete systems with classical or bounded \((\phi_1,\phi_2)\)-Laplacian, J. Nonlinear Sci. Appl., 10 (2017), 535–559.
-
[4]
Z.-M. Guo, J.-S. Yu, Existence of periodic and subharmonic solutions for second-order superlinear difference equations, Sci. China Ser. A, 46 (2003), 506–515.
-
[5]
Z.-M. Guo, J.-S. Yu, The existence of periodic and subharmonic solutions of subquadratic second order difference equations, J. London Math. Soc., 68 (2003), 419–430.
-
[6]
T.-S. He, W.-G. Chen, Periodic solutions of second order discrete convex systems involving the p-Laplacian, Appl. Math. Comput., 206 (2008), 124–132.
-
[7]
X.-F. He, P. Chen, Homoclinic solutions for second order discrete p-Laplacian systems, Adv. Difference Equ., 2011 (2011), 16 pages.
-
[8]
Y.-K. Li, T.-W. Zhang, Infinitely many periodic solutions for second-order (p, q)-Laplacian differential systems, Nonlinear Anal., 74 (2011), 5215–5221.
-
[9]
X.-Y. Lin, X.-H. Tang, Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems, J. Math. Anal. Appl., 373 (2011), 59–72.
-
[10]
J. Q. Liu, A generalized saddle point theorem, J. Differential Equations, 82 (1989), 372–385.
-
[11]
J. Mawhin, Periodic solutions of second order nonlinear difference systems with \(\phi\)-Laplacian: a variational approach, Nonlinear Anal., 75 (2012), 4672–4687.
-
[12]
J. Mawhin , Periodic solutions of second order Lagrangian difference systems with bounded or singular \(\phi\)-Laplacian and periodic potential, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1065–1076.
-
[13]
J. Mawhin, M. Willem, Critical point theory and Hamiltonian systems, Applied Mathematical Sciences, Springer- Verlag, New York (1989)
-
[14]
D. Paşca, Periodic solutions of a class of nonautonomous second order differential systems with (q, p)-Laplacian, Bull. Belg. Math. Soc. Simon Stevin, 17 (2010), 841–850.
-
[15]
D. Paşca, C.-L. Tang, Some existence results on periodic solutions of nonautonomous second-order differential systems with (q, p)-Laplacian, Appl. Math. Lett., 23 (2010), 246–251.
-
[16]
P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (1986)
-
[17]
B. Ricceri, A further refinement of a three critical points theorem, Nonlinear Anal., 74 (2011), 7446–7454.
-
[18]
M. Schechter , Minimax systems and critical point theory, Birkhäuser Boston, Inc., Boston, MA (2009)
-
[19]
X. H. Tang, X.-Y. Lin, Existence and multiplicity of homoclinic solutions for second-order discrete Hamiltonian systems with subquadratic potential, J. Difference Equ. Appl., 17 (2011), 1617–1634.
-
[20]
X. H. Tang, X.-Y. Zhang, Periodic solutions for second-order discrete Hamiltonian systems, J. Difference Equ. Appl., 17 (2011), 1413–1430.
-
[21]
Y. Wang, X.-Y. Zhang, Multiple periodic solutions for a class of nonlinear difference systems with classical or bounded \((\phi_1,\phi_2)\)-Laplacian, Adv. Difference Equ., 2014 (2014), 33 pages.
-
[22]
Y.-F. Xue, C.-L. Tang, Existence of a periodic solution for subquadratic second-order discrete Hamiltonian system, Nonlinear Anal., 67 (2007), 2072–2080.
-
[23]
X.-X. Yang, H.-B. Chen, Periodic solutions for autonomous (q, p)-Laplacian system with impulsive effects, J. Appl. Math., 2011 (2011), 19 pages.
-
[24]
X.-X. Yang, H.-B. Chen, Periodic solutions for a nonlinear (q, p)-Laplacian dynamical system with impulsive effects, J. Appl. Math. Comput., 40 (2012), 607–625.
-
[25]
E. Zeidler, Nonlinear functional analysis and its applications, II/B, Nonlinear monotone operators, Translated from the German by the author and Leo F. Boron, Springer-Verlag, New York (1990)
-
[26]
X.-Y. Zhang, Notes on periodic solutions for a nonlinear discrete system involving the p-Laplacian, Bull. Malays. Math. Sci. Soc., 37 (2014), 499–509.
-
[27]
X.-Y. Zhang, X.-H. Tang, Existence of solutions for a nonlinear discrete system involving the p-Laplacian, Appl. Math., 57 (2012), 11–30.
-
[28]
Q.-F. Zhang, X. H. Tang, Q.-M. Zhang, Existence of periodic solutions for a class of discrete Hamiltonian systems, Discrete Dyn. Nat. Soc., 2011 (2011), 14 pages.
-
[29]
X.-Y. Zhang, Y. Wang, Homoclinic solutions for a class of nonlinear difference systems with classical \((\varphi_1,\varphi_2)\)-Laplacian, Adv. Difference Equ., 2015 (2015), 24 pages.
-
[30]
Z. Zhou, J.-S. Yu, Z.-M. Guo, Periodic solutions of higher-dimensional discrete systems, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 1013–1022.