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

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

Marek Galewski 1, and  Renata Wieteska 1,

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.
Keywords: non-smooth problems, Discrete boundary value problem, local linking., three critical point theorem.
Mathematics Subject Classification: Primary: 39A10; Secondary: 34B15, 65Q2.
Citation: Marek Galewski, Renata Wieteska. Multiple periodic solutions to a discrete $p^{(k)}$ - Laplacian problem. Discrete & 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

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