Taiwanese Journal of Mathematics

An Existence Result for Discrete Anisotropic Equations

Shapour Heidarkhani, Ghasem A. Afrouzi, and Shahin Moradi

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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.

Article information

Taiwanese J. Math., Volume 22, Number 3 (2018), 725-739.

Received: 10 March 2017
Revised: 27 June 2017
Accepted: 1 August 2017
First available in Project Euclid: 8 September 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34B15: Nonlinear boundary value problems 39A10: Difference equations, additive 39A70: Difference operators [See also 47B39] 46E39: Sobolev (and similar kinds of) spaces of functions of discrete variables

discrete boundary value problem anisotropic problem one solution variational methods critical point theory


Heidarkhani, Shapour; Afrouzi, Ghasem A.; Moradi, Shahin. An Existence Result for Discrete Anisotropic Equations. Taiwanese J. Math. 22 (2018), no. 3, 725--739. doi:10.11650/tjm/170801. https://projecteuclid.org/euclid.twjm/1504836027

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  • 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.
  • G. Bonanno, Relations between the mountain pass theorem and local minima, Adv. Nonlinear Anal. 1 (2012), no. 3, 205–220.
  • G. Bonanno and P. Candito, Nonlinear difference equations investigated via critical point methods, Nonlinear Anal. 70 (2009), no. 9, 3180–3186.
  • 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.
  • 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.
  • A. Cabada, A. Iannizzotto and S. Tersian, Multiple solutions for discrete boundary value problems, J. Math. Anal. Appl. 356 (2009), no. 2, 418–428.
  • M. Galewski and S. Głąb, On the discrete boundary value problem for anisotropic equation, J. Math. Anal. Appl. 386 (2012), no. 2, 956–965.
  • M. Galewski, S. Heidarkhani and A. Salari, Multiplicity results for discrete anisotropic equations, Discrete Contin. Dyn. Syst. Ser. B, to appear.
  • M. Galewski and G. Molica Bisci, Existence results for one-dimensional fractional equations, Math. Methods Appl. Sci. 39 (2016), no. 6, 1480–1492.
  • M. Galewski and R. Wieteska, On the system of anisotropic discrete BVPs, J. Difference Equ. Appl. 19 (2013), no. 7, 1065–1081.
  • ––––, Existence and multiplicity of positive solutions for discrete anisotropic equations, Turkish J. Math. 38 (2014), no. 2, 297–310.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • G. Molica Bisci and D. Repov\us, Existence of solutions for $p$-Laplacian discrete equations, Appl. Math. Comput. 242 (2014), 454–461.
  • ––––, On sequences of solutions for discrete anisotropic equations, Expo. Math. 32 (2014), no. 3, 284–295.
  • B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), no. 1-2, 401–410.
  • R. Stegliński, On sequences of large solutions for discrete anisotropic equations, Electron. J. Qual. Theory Differ. Equ. 2015, no. 25, 1–10.
  • 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.
  • 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.