October  2014, 19(8): 2535-2547. doi: 10.3934/dcdsb.2014.19.2535

Multiple periodic solutions to a discrete $p^{(k)}$ - Laplacian problem

1. 

Institute of Mathematics, Technical University of Lodz, Wolczanska 215, 90-924 Lodz, Poland

Received  November 2013 Revised  November 2013 Published  August 2014

We investigate the existence of multiple periodic solutions to the anisotropic discrete system. We apply the linking method and a new three critical point theorem which we provide.
Citation: Marek Galewski, Renata Wieteska. Multiple periodic solutions to a discrete $p^{(k)}$ - Laplacian problem. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2535-2547. doi: 10.3934/dcdsb.2014.19.2535
References:
[1]

R. P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, New York, 1992.

[2]

C. Bereanu, P. Jebelean and C. Şerban, Periodic and Neumann problems for discrete $p(\cdot )-$Laplacian, J. Math. Anal. Appl., 399 (2013), 75-87. doi: 10.1016/j.jmaa.2012.09.047.

[3]

C. Bereanu, P. Jebelean and C. Şerban, Ground state and mountain pass solutions for discrete $p(\cdot )-$Laplacian, Bound. Value Probl., 104 (2012), 1-13. doi: 10.1186/1687-2770-2012-104.

[4]

G. Bonanno and P. Candito, Nonlinear difference equations investigated via critical point methods, Nonlinear Anal., Theory Methods Appl. A, 70 (2009), 3180-3186. doi: 10.1016/j.na.2008.04.021.

[5]

G. Bonanno and P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities, J. Differ. Eqs, 244 (2008), 3031-3059. doi: 10.1016/j.jde.2008.02.025.

[6]

G. Bonanno and S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. Anal., 89 (2010), 1-10. doi: 10.1080/00036810903397438.

[7]

A. Cabada and S. Tersian, Multiplicity of solutions of a two point boundary value problem for a fourth-order equation, Appl. Math. Comput., 219 (2013), 5261-5267. doi: 10.1016/j.amc.2012.11.066.

[8]

Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math., 66 (2006), 1383-1406. doi: 10.1137/050624522.

[9]

S. N. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-3110-1.

[10]

X. L. Fan and H. Zhang, Existence of solutions for $p(x)-$Lapacian dirichlet problem, Nonlinear Anal., Theory Methods Appl., 52 (2003), 1843-1852. doi: 10.1016/S0362-546X(02)00150-5.

[11]

M. Galewski and R. Wieteska, A note on the multiplicity of solutions to anisotropic discrete BVP's, Appl. Math. Lett., 26 (2013), 524-529. doi: 10.1016/j.aml.2012.11.002.

[12]

A. Guiro, I. Nyanquini and S. Ouaro, On the solvability of discrete nonlinear Neumann problems involving the $p(x)-$Laplacian, Adv. Difference Equ., 32 (2011), 14 pp.

[13]

P. Harjulehto, P. Hästö, U. V. Le and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal., 72 (2010), 4551-4574. doi: 10.1016/j.na.2010.02.033.

[14]

B. Kone and S. Ouaro, Weak solutions for anisotropic discrete boundary value problems, J. Difference Equ. Appl., 17 (2011), 1537-1547. doi: 10.1080/10236191003657246.

[15]

V. Lakshmikantham and D. Trigiante, Theory of Difference Equations: Numerical Methods and Applications, Academic Press, New York, 1988.

[16]

S. Liu, Multiple periodic solutions for nonlinear difference systems involving the p-Laplacian, J. Difference Equ. Appl., 17 (2011), 1591-1598. doi: 10.1080/10236191003730480.

[17]

N. Marcu and G. Molica Bisci, Existence and multiplicity results for nonlinear discrete inclusions, Electron. J. Differential Equations, (2012), 1-13.

[18]

M. Mihăilescu, V. Rădulescu and S. Tersian, Eigenvalue problems for anisotropic discrete boundary value problems. J. Difference Equ. Appl., 15 (2009), 557-567. doi: 10.1080/10236190802214977.

[19]

G. Molica Bisci and D. Repovs, On some variational algebraic problems, Adv. Nonlinear Analysis, 2 (2013), 127-146.

[20]

G. Molica Bisci and D. Repovs, Nonlinear Algebraic Systems with discontinuous terms, J. Math. Anal. Appl., 398 (2013), 846-856. doi: 10.1016/j.jmaa.2012.09.046.

[21]

M. Růžička, Electrorheological Fluids: Modelling and Mathematical Theory, Lecture Notes in Mathematics, Vol. 1748, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.

[22]

P. Stehlík, On variational methods for periodic discrete problems, J. Difference Equ. Appl., 14 (2008), 259-273. doi: 10.1080/10236190701483160.

[23]

Y. Tian, Z. Du and W. Ge, Existence results for discrete Sturm-Liouville problem via variational methods, J. Difference Equ. Appl., 13 (2007), 467-478. doi: 10.1080/10236190601086451.

[24]

M. Willem, Minimax Theorem, Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.

[25]

V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710.

show all references

References:
[1]

R. P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, New York, 1992.

[2]

C. Bereanu, P. Jebelean and C. Şerban, Periodic and Neumann problems for discrete $p(\cdot )-$Laplacian, J. Math. Anal. Appl., 399 (2013), 75-87. doi: 10.1016/j.jmaa.2012.09.047.

[3]

C. Bereanu, P. Jebelean and C. Şerban, Ground state and mountain pass solutions for discrete $p(\cdot )-$Laplacian, Bound. Value Probl., 104 (2012), 1-13. doi: 10.1186/1687-2770-2012-104.

[4]

G. Bonanno and P. Candito, Nonlinear difference equations investigated via critical point methods, Nonlinear Anal., Theory Methods Appl. A, 70 (2009), 3180-3186. doi: 10.1016/j.na.2008.04.021.

[5]

G. Bonanno and P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities, J. Differ. Eqs, 244 (2008), 3031-3059. doi: 10.1016/j.jde.2008.02.025.

[6]

G. Bonanno and S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. Anal., 89 (2010), 1-10. doi: 10.1080/00036810903397438.

[7]

A. Cabada and S. Tersian, Multiplicity of solutions of a two point boundary value problem for a fourth-order equation, Appl. Math. Comput., 219 (2013), 5261-5267. doi: 10.1016/j.amc.2012.11.066.

[8]

Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math., 66 (2006), 1383-1406. doi: 10.1137/050624522.

[9]

S. N. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-3110-1.

[10]

X. L. Fan and H. Zhang, Existence of solutions for $p(x)-$Lapacian dirichlet problem, Nonlinear Anal., Theory Methods Appl., 52 (2003), 1843-1852. doi: 10.1016/S0362-546X(02)00150-5.

[11]

M. Galewski and R. Wieteska, A note on the multiplicity of solutions to anisotropic discrete BVP's, Appl. Math. Lett., 26 (2013), 524-529. doi: 10.1016/j.aml.2012.11.002.

[12]

A. Guiro, I. Nyanquini and S. Ouaro, On the solvability of discrete nonlinear Neumann problems involving the $p(x)-$Laplacian, Adv. Difference Equ., 32 (2011), 14 pp.

[13]

P. Harjulehto, P. Hästö, U. V. Le and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal., 72 (2010), 4551-4574. doi: 10.1016/j.na.2010.02.033.

[14]

B. Kone and S. Ouaro, Weak solutions for anisotropic discrete boundary value problems, J. Difference Equ. Appl., 17 (2011), 1537-1547. doi: 10.1080/10236191003657246.

[15]

V. Lakshmikantham and D. Trigiante, Theory of Difference Equations: Numerical Methods and Applications, Academic Press, New York, 1988.

[16]

S. Liu, Multiple periodic solutions for nonlinear difference systems involving the p-Laplacian, J. Difference Equ. Appl., 17 (2011), 1591-1598. doi: 10.1080/10236191003730480.

[17]

N. Marcu and G. Molica Bisci, Existence and multiplicity results for nonlinear discrete inclusions, Electron. J. Differential Equations, (2012), 1-13.

[18]

M. Mihăilescu, V. Rădulescu and S. Tersian, Eigenvalue problems for anisotropic discrete boundary value problems. J. Difference Equ. Appl., 15 (2009), 557-567. doi: 10.1080/10236190802214977.

[19]

G. Molica Bisci and D. Repovs, On some variational algebraic problems, Adv. Nonlinear Analysis, 2 (2013), 127-146.

[20]

G. Molica Bisci and D. Repovs, Nonlinear Algebraic Systems with discontinuous terms, J. Math. Anal. Appl., 398 (2013), 846-856. doi: 10.1016/j.jmaa.2012.09.046.

[21]

M. Růžička, Electrorheological Fluids: Modelling and Mathematical Theory, Lecture Notes in Mathematics, Vol. 1748, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.

[22]

P. Stehlík, On variational methods for periodic discrete problems, J. Difference Equ. Appl., 14 (2008), 259-273. doi: 10.1080/10236190701483160.

[23]

Y. Tian, Z. Du and W. Ge, Existence results for discrete Sturm-Liouville problem via variational methods, J. Difference Equ. Appl., 13 (2007), 467-478. doi: 10.1080/10236190601086451.

[24]

M. Willem, Minimax Theorem, Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.

[25]

V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 675-710.

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