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An existence result for discrete anisotropic equations. (English) Zbl 1401.39002

Summary: A critical point result is exploited in order to prove that a class of discrete anisotropic boundary value problems possesses at least one solution under an asymptotical behaviour of the potential of the nonlinear term at zero. Some recent results are extended and improved. Some examples are presented to demonstrate the applications of our main results.

MSC:

39A10 Additive difference equations
34B15 Nonlinear boundary value problems for ordinary differential equations
39A70 Difference operators
46E39 Sobolev (and similar kinds of) spaces of functions of discrete variables
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[1] L.-H. Bian, H.-R. Sun and Q.-G. Zhang, Solutions for discrete \(p\)-Laplacian periodic boundary value problems via critical point theory, J. Difference Equ. Appl. 18 (2012), no. 3, 345–355. · Zbl 1247.39004
[2] G. Bonanno, Relations between the mountain pass theorem and local minima, Adv. Nonlinear Anal. 1 (2012), no. 3, 205–220. · Zbl 1277.35170
[3] G. Bonanno and P. Candito, Nonlinear difference equations investigated via critical point methods, Nonlinear Anal. 70 (2009), no. 9, 3180–3186. · Zbl 1166.39006
[4] G. Bonanno, P. Candito and G. D’Agu\`\i, Variational methods on finite dimensional Banach spaces and discrete problems, Adv. Nonlinear Stud. 14 (2014), no. 4, 915–939. · Zbl 1312.39002
[5] G. Bonanno, P. Jebelean and C. Şerban, Three solutions for discrete anisotropic periodic and Neumann problems, Dynam. Systems Appl. 22 (2013), no. 2-3, 183–196. · Zbl 1303.39003
[6] A. Cabada, A. Iannizzotto and S. Tersian, Multiple solutions for discrete boundary value problems, J. Math. Anal. Appl. 356 (2009), no. 2, 418–428. · Zbl 1169.39008
[7] M. Galewski and S. Głąb, On the discrete boundary value problem for anisotropic equation, J. Math. Anal. Appl. 386 (2012), no. 2, 956–965. · Zbl 1233.39004
[8] M. Galewski, S. Heidarkhani and A. Salari, Multiplicity results for discrete anisotropic equations, Discrete Contin. Dyn. Syst. Ser. B, to appear. · Zbl 1374.34061
[9] M. Galewski and G. Molica Bisci, Existence results for one-dimensional fractional equations, Math. Methods Appl. Sci. 39 (2016), no. 6, 1480–1492. · Zbl 1381.34012
[10] M. Galewski and R. Wieteska, On the system of anisotropic discrete BVPs, J. Difference Equ. Appl. 19 (2013), no. 7, 1065–1081. · Zbl 1274.39008
[11] ——–, Existence and multiplicity of positive solutions for discrete anisotropic equations, Turkish J. Math. 38 (2014), no. 2, 297–310. · Zbl 1297.39014
[12] S. Heidarkhani, G. A. Afrouzi, J. Henderson, S. Moradi and G. Caristi, Variational approaches to \(p\)-Laplacian discrete problems of Kirchhoff-type, J. Difference Equ. Appl. 23 (2017), no. 5, 917–938. · Zbl 1371.39002
[13] S. Heidarkhani, G. A. Afrouzi, S. Moradi and G. Caristi, Existence of multiple solutions for a perturbed discrete anisotropic equation, J. Difference Equ. Appl., to appear. · Zbl 1353.35153
[14] J. Henderson and H. B. Thompson, Existence of multiple solutions for second-order discrete boundary value problems, Comput. Math. Appl. 43 (2002), no. 10-11, 1239–1248. · Zbl 1005.39014
[15] E. M. Hssini, Multiple solutions for a discrete anisotropic \((p_{1}(k),p_{2}(k))\)-Laplacian equations, Electron. J. Differential Equations 2015 (2015), no. 195, 10 pp. · Zbl 1321.39008
[16] L. Jiang and Z. Zhou, Three solutions to Dirichlet boundary value problems for \(p\)-Laplacian difference equations, Adv. Difference Equ. 2008, Art. ID 345916, 10 pp.
[17] A. Kristály, M. Mihăilescu and V. Rădulescu, Discrete boundary value problems involving oscillatory nonlinearities: Small and large solutions, J. Difference Equ. Appl. 17 (2011), no. 10, 1431–1440.
[18] H. Liang and P. Weng, Existence and multiple solutions for a second-order difference boundary value problem via critical point theory, J. Math. Anal. Appl. 326 (2007), no. 1, 511–520. · Zbl 1112.39008
[19] M. Mihăilescu, V. Rădulescu and S. Tersian, Eigenvalue problems for anisotropic discrete boundary value problems, J. Difference Equ. Appl. 15 (2009), no. 6, 557–567. · Zbl 1181.47016
[20] M. K. Moghadam, S. Heidarkhani and J. Henderson, Infinitely many solutions for perturbed difference equations, J. Difference Equ. Appl. 20 (2014), no. 7, 1055–1068. · Zbl 1291.39002
[21] G. Molica Bisci and V. D. Rădulescu, Bifurcation analysis of a singular elliptic problem modelling the equilibrium of anisotropic continuous media, Topol. Methods Nonlinear Anal. 45 (2015), no. 2, 493–508. · Zbl 1367.35029
[22] G. Molica Bisci and D. Repov\us, Existence of solutions for \(p\)-Laplacian discrete equations, Appl. Math. Comput. 242 (2014), 454–461. · Zbl 1334.39019
[23] ——–, On sequences of solutions for discrete anisotropic equations, Expo. Math. 32 (2014), no. 3, 284–295. · Zbl 06339867
[24] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), no. 1-2, 401–410. · Zbl 0946.49001
[25] R. Stegliński, On sequences of large solutions for discrete anisotropic equations, Electron. J. Qual. Theory Differ. Equ. 2015, no. 25, 1–10.
[26] Y. Tian, Z. Du and W. Ge, Existence results for discrete Sturm-Liouville problem via variational methods, J. Difference Equ. Appl. 13 (2007), no. 6, 467–478. · Zbl 1129.39007
[27] D.-B. Wang and W. Guan, Three positive solutions of boundary value problems for \(p\)-Laplacian difference equations, Comput. Math. Appl. 55 (2008), no. 9, 1943–1949. · Zbl 1147.39008
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