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SOLUTIONS FOR A DISCRETE BOUNDARY VALUE PROBLEM INVOLVING KIRCHHOFF TYPE EQUATION VIA VARIATIONAL METHODS

Year 2018, Volume: 8 Issue: 1, 144 - 154, 01.06.2018

Abstract

In this paper, Mountain Pass theorem is applied together with Ekeland variational principle, and we show the existence of nontrivial solutions for a discrete boundary value problem of p k -Kirchho -type in a nite dimensional Hilbert space.

References

  • [1] Agarwal, R. P., Perera, K. and O’Regan, D., (2005), Multiple positive solutions of singular discrete p−Laplacian problems via variational methods, Adv. Diff. Equ. 2, pp. 93–99.
  • [2] Moghadam, M.Khaleghi and Avci, M., (2017), Existence results to a nonlinear p(k)-Laplacian difference equation, Journal of Difference Equations and Appl.https://doi.org/10.1080/10236198.2017.1354991.
  • [3] Avci, M., (2017), On a nonlocal Neumann problem in Orlicz-Sobolev spaces, Journal of Nonlinear Functional Analysis, Vol. 2017 (2017), Article ID 42, 1-11.
  • [4] Avci, M., (2016), Existence results for anisotropic discrete boundary value problems, Electron. J. Differ. Equ. 148, pp. 1-11.
  • [5] Avci, M. and Pankov, A., (2015), Nontrivial solutions of discrete nonlinear equations with variable exponent, J.Math.Anal.Appl. 431, pp. 22-33.
  • [6] Cabada, A., Lannizzotto A. and Tersian, S., (2009), Multiple solutions for discrete boundary value problems, J. Math. Anal. Appl. 356, pp. 418-428.
  • [7] Cai, X. and Yu, J., (2006), Existence theorems for second-order discrete boundary value problems, J. Math. Anal. Appl. 320, pp. 649–661.
  • [8] Candito, P. and Giovannelli, N., (2008), Multiple solutions for a discrete boundary value problem involving the p−Laplacian, Comput. Math. Appl. 56, pp. 959-964.
  • [9] Chen, Y., Levine S. and Rao, M., (2006), Variable exponent linear growth functionals in image processing, SIAM J. Appl. Math. 66 (4), pp. 1383-1406.
  • [10] Ekeland, I., (1974), On the variational principle. J Math Anal Appl. 47, pp. 324-353. discrete boundary value problem for anisotropic equation, J. Math. Anal. Appl. 386, pp. 956-965.
  • [11] Galewski, M. and WieteskaI, R., (2013), Existence and multiplicity of positive solutions for discrete anisotropic equations, Turk. J. Math. doi:10.3906/mat-1303-6.
  • [12] Galewski, M., Heidarkhani, S. and Salari, A., Multiplicity results for discrete anisotropic equations, Discrete Contin. Dyn. Syst. Ser. B, to appear.
  • [13] Guiro, A., Nyanquini, I. and Ouaro, S., (2011), On the solvability of discrete nonlinear Neumann problems involving the p(x)−Laplacian, Adv. Diff. Equ. 32.
  • [14] Halsey, T. C., (1992), Electrorheological fluids, Science 258, pp. 761-766.
  • [15] Heidarkhani, S., Afrouzi, G.A., Moradi, S. and Caristi, G., Existence of multiple solutions for a perturbed discrete anisotropic equation, J. Differ. Equ. Appl., doi: 10.1080/10236198.2017.1337108.
  • [16] Heidarkhani, S., Afrouzi, G.A., Henderson, J., Moradi,S. and G. Caristi, Variational approaches to p−Laplacian discrete problems of Kirchhoff-type, J. Differ. Equ. Appl., https://doi.org/10.1080/10236198.2017.1306061.
  • [17] Heidarkhani, S., Caristi, G. and Salari, A., Perturbed Kirchhoff-type p-Laplacian discrete problems, Collect. Math., doi: 10.1007/s13348-016-0180-4.
  • [18] Heidarkhani, S., Afrouzi, G.A., Caristi, G., Henderson, J. and Moradi, S., (2016)A variational approach to difference equations, J. Differ. Equ. Appl. 22, pp. 1761-1776.
  • [19] Jebelean, P. and S¸erban, C., (2011), Ground state periodic solutions for difference equations with discrete p−Laplacian, Appl. Math. Comput. 217, pp. 9820-9827.
  • [20] Kirchhoff, G., (1883), Mechanik, Teubner, Leipzig.
  • [21] Kon´e, B. and Ouaro, S., (2010), Weak solutions for anisotropic discrete boundary value problems, J. Diff. Equ. Appl. 17 (10), pp. 1-11.
  • [22] Mashiyev, R. A., Yucedag, Z. and Ogras, S., (2011), Existence and multiplicity of solutions for a Dirichlet problem involving the discrete p(x)-Laplacian operator, E. J. Qualitative Theory of Diff. Equ. 67, pp. 1-10.
  • [23] Mashiyev, R. A., Cekic, B., Avci, M. and Yucedag, Z., (2012), Existence and multiplicity of weak solutions for nonuniformly elliptic equations with nonstandard growth condition, Complex Variables and Elliptic Equations, 57, No. 5, pp. 579–595.
  • [24] Mih˘ailescu, M., R˘adulescu, V. and Tersian, S., (2009), Eigenvalue problems for anisotropic discrete boundary value problems, J. Differ. Equ. Appl. 15 (2009), pp. 557–56.
  • [25] Moghadam, M. K., Heidarkhaniand, S., Henderson, J., (2014), Infinitely many solutions for perturbed difference equations, J. Differ. Equ. Appl., vol. 20, no. 7, pp. 1055-1068.
  • [26] R˚uˇziˇcka, M., (2000), Electrorheological fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, 1748, Springer-Verlag, Berlin.
  • [27] Willem, M., (1996), Minimax Theorems, Birkhauser, Basel.
  • [28] Y¨ucedag, Z., (2014), Existence of solutions for anisotropic discrete boundary value problem of Kirchhoff type, Int. J. Differ. Equ. Appl. 13(1), pp. 1-15.
  • [29] Zhang, X. and Tang, X., (2012), Existence of solutions for a nonlinear discrete system involving the p−Laplacian. Appl. Math. Praha 57, No. 1, pp. 11-30.
  • [30] Zhikov, V. V., (1987), Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv. 9, pp. 33–66.
Year 2018, Volume: 8 Issue: 1, 144 - 154, 01.06.2018

Abstract

References

  • [1] Agarwal, R. P., Perera, K. and O’Regan, D., (2005), Multiple positive solutions of singular discrete p−Laplacian problems via variational methods, Adv. Diff. Equ. 2, pp. 93–99.
  • [2] Moghadam, M.Khaleghi and Avci, M., (2017), Existence results to a nonlinear p(k)-Laplacian difference equation, Journal of Difference Equations and Appl.https://doi.org/10.1080/10236198.2017.1354991.
  • [3] Avci, M., (2017), On a nonlocal Neumann problem in Orlicz-Sobolev spaces, Journal of Nonlinear Functional Analysis, Vol. 2017 (2017), Article ID 42, 1-11.
  • [4] Avci, M., (2016), Existence results for anisotropic discrete boundary value problems, Electron. J. Differ. Equ. 148, pp. 1-11.
  • [5] Avci, M. and Pankov, A., (2015), Nontrivial solutions of discrete nonlinear equations with variable exponent, J.Math.Anal.Appl. 431, pp. 22-33.
  • [6] Cabada, A., Lannizzotto A. and Tersian, S., (2009), Multiple solutions for discrete boundary value problems, J. Math. Anal. Appl. 356, pp. 418-428.
  • [7] Cai, X. and Yu, J., (2006), Existence theorems for second-order discrete boundary value problems, J. Math. Anal. Appl. 320, pp. 649–661.
  • [8] Candito, P. and Giovannelli, N., (2008), Multiple solutions for a discrete boundary value problem involving the p−Laplacian, Comput. Math. Appl. 56, pp. 959-964.
  • [9] Chen, Y., Levine S. and Rao, M., (2006), Variable exponent linear growth functionals in image processing, SIAM J. Appl. Math. 66 (4), pp. 1383-1406.
  • [10] Ekeland, I., (1974), On the variational principle. J Math Anal Appl. 47, pp. 324-353. discrete boundary value problem for anisotropic equation, J. Math. Anal. Appl. 386, pp. 956-965.
  • [11] Galewski, M. and WieteskaI, R., (2013), Existence and multiplicity of positive solutions for discrete anisotropic equations, Turk. J. Math. doi:10.3906/mat-1303-6.
  • [12] Galewski, M., Heidarkhani, S. and Salari, A., Multiplicity results for discrete anisotropic equations, Discrete Contin. Dyn. Syst. Ser. B, to appear.
  • [13] Guiro, A., Nyanquini, I. and Ouaro, S., (2011), On the solvability of discrete nonlinear Neumann problems involving the p(x)−Laplacian, Adv. Diff. Equ. 32.
  • [14] Halsey, T. C., (1992), Electrorheological fluids, Science 258, pp. 761-766.
  • [15] Heidarkhani, S., Afrouzi, G.A., Moradi, S. and Caristi, G., Existence of multiple solutions for a perturbed discrete anisotropic equation, J. Differ. Equ. Appl., doi: 10.1080/10236198.2017.1337108.
  • [16] Heidarkhani, S., Afrouzi, G.A., Henderson, J., Moradi,S. and G. Caristi, Variational approaches to p−Laplacian discrete problems of Kirchhoff-type, J. Differ. Equ. Appl., https://doi.org/10.1080/10236198.2017.1306061.
  • [17] Heidarkhani, S., Caristi, G. and Salari, A., Perturbed Kirchhoff-type p-Laplacian discrete problems, Collect. Math., doi: 10.1007/s13348-016-0180-4.
  • [18] Heidarkhani, S., Afrouzi, G.A., Caristi, G., Henderson, J. and Moradi, S., (2016)A variational approach to difference equations, J. Differ. Equ. Appl. 22, pp. 1761-1776.
  • [19] Jebelean, P. and S¸erban, C., (2011), Ground state periodic solutions for difference equations with discrete p−Laplacian, Appl. Math. Comput. 217, pp. 9820-9827.
  • [20] Kirchhoff, G., (1883), Mechanik, Teubner, Leipzig.
  • [21] Kon´e, B. and Ouaro, S., (2010), Weak solutions for anisotropic discrete boundary value problems, J. Diff. Equ. Appl. 17 (10), pp. 1-11.
  • [22] Mashiyev, R. A., Yucedag, Z. and Ogras, S., (2011), Existence and multiplicity of solutions for a Dirichlet problem involving the discrete p(x)-Laplacian operator, E. J. Qualitative Theory of Diff. Equ. 67, pp. 1-10.
  • [23] Mashiyev, R. A., Cekic, B., Avci, M. and Yucedag, Z., (2012), Existence and multiplicity of weak solutions for nonuniformly elliptic equations with nonstandard growth condition, Complex Variables and Elliptic Equations, 57, No. 5, pp. 579–595.
  • [24] Mih˘ailescu, M., R˘adulescu, V. and Tersian, S., (2009), Eigenvalue problems for anisotropic discrete boundary value problems, J. Differ. Equ. Appl. 15 (2009), pp. 557–56.
  • [25] Moghadam, M. K., Heidarkhaniand, S., Henderson, J., (2014), Infinitely many solutions for perturbed difference equations, J. Differ. Equ. Appl., vol. 20, no. 7, pp. 1055-1068.
  • [26] R˚uˇziˇcka, M., (2000), Electrorheological fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, 1748, Springer-Verlag, Berlin.
  • [27] Willem, M., (1996), Minimax Theorems, Birkhauser, Basel.
  • [28] Y¨ucedag, Z., (2014), Existence of solutions for anisotropic discrete boundary value problem of Kirchhoff type, Int. J. Differ. Equ. Appl. 13(1), pp. 1-15.
  • [29] Zhang, X. and Tang, X., (2012), Existence of solutions for a nonlinear discrete system involving the p−Laplacian. Appl. Math. Praha 57, No. 1, pp. 11-30.
  • [30] Zhikov, V. V., (1987), Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv. 9, pp. 33–66.
There are 30 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

- Z.yücedağ This is me

Publication Date June 1, 2018
Published in Issue Year 2018 Volume: 8 Issue: 1

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