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

Yıl 2018, Cilt: 8 Sayı: 1, 144 - 154, 01.06.2018

- Z.yücedağ

Öz

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.

Anahtar Kelimeler

Kirchho type equation, discrete boundary value problem, Varitional methods, Ekeland variational principle, Mountain Pass theorem.

Kaynakça

• [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.