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Multiple periodic solutions to a discrete $p^{(k)}$ - Laplacian problem
1. | Institute of Mathematics, Technical University of Lodz, Wolczanska 215, 90-924 Lodz, Poland |
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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