Existence and uniqueness of positive solution for nonlinear difference equations involving p(k)-Laplacian operator

Moghadam, Mohsen Khaleghi; Wieteska, Renata

doi:10.2478/auom-2019-0008

Abstract

In this paper, we deal with the existence of at least one and of at least two positive solutions as well the uniqueness of a positive solution for an anisotropic discrete non-linear problem involving p(k)-Laplacian with Dirichlet boundary value conditions. The technical approach for the existence part is based on a local minimum theorem and on a two critical points theorem for differentiable functionals, and for uniqueness part is based on a Lipschitzian continuous condition on the nonlinearity term.

References

[1] G.A. Afrouzi, A. Hadjian and G.M. Bisci, Some remarks for one-dimensional mean curvature problems through a local minimization principle Adv. Nonlinear Anal. 2 (2013) 427–441.10.1515/anona-2013-0021
Search in Google Scholar Back to article
[2] G.A. Afrouzi, A. Hadjian and S. Heidarkhani, Non-trivial solutions for a two-point boundary value problem, Ann. Polo. Math. 108.1 (2013) 75–84.10.4064/ap108-1-6
Search in Google Scholar Back to article
[3] R.P. Agarwal, K. Perera, and D. O’Regan, Multiple positive solutions of singular discrete p-Laplacian problems via variational methods, Adv. Diff. Equ. 2 (2005) 93–99.10.1155/ADE.2005.93
Search in Google Scholar Back to article
[4] R. Avery and J. Henderson, Existence of three positive pseudo-symmetric solutions for a one dimensional discrete p-Laplacian, J. Differ. Equ. Appl. 10 (2004) 529–539.10.1080/10236190410001667959
Search in Google Scholar Back to article
[5] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381.10.1016/0022-1236(73)90051-7
Search in Google Scholar Back to article
[6] G. Bonanno, A Critical point theorem via the Ekeland variational principle, Nonl. Anal. TMA 75 (2012) 2992–3007.10.1016/j.na.2011.12.003
Search in Google Scholar Back to article
[7] G. Bonanno, P. Candito and G. D’Agu‘i, Variational methods on finite dimensional Banach spaces and discrete problems, Adv. Nonlinear Stud. 14 (2014) 915–939.10.1515/ans-2014-0406
Search in Google Scholar Back to article
[8] G. Bonanno and P. Candito, Infinitely many solutions for a class of discrete non-linear boundary value problems, Appl. Anal. 884 (2009) 605–616.10.1080/00036810902942242
Search in Google Scholar Back to article
[9] G. Bonanno, G. D’Agu‘i, Two non-zero solutions for elliptic Dirichlet problems, Z. Anal. Anwend. 35 (2016) 449–464.10.4171/ZAA/1573
Search in Google Scholar Back to article
[10] G. Bonanno, B. Di Bella and D. O’Regan, Non-trivial solutions for nonlinear fourth-order elastic beam equations, Compu. Math. Appl. 62 (2011) 1862–1869.10.1016/j.camwa.2011.06.029
Search in Google Scholar Back to article
[11] G. Bonanno, S. Heidarkhani and D. O’Regan, Nontrivial solutions for Sturm-Liouville systems via a local minimum theorem for functional, Bull. Aust. Math. Soc. to appear.
Search in Google Scholar Back to article
[12] G. Bonanno and G. Molica Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl. 2009 (2009) 670–675.10.1155/2009/670675
Search in Google Scholar Back to article
[13] C. Bereanu, P. Jebelean, C. Serban, Periodic and Neumann problems for discrete p(k)-Laplacian, J. Math. Anal. Appl. 399 (2013), 75–87.10.1016/j.jmaa.2012.09.047
Search in Google Scholar Back to article
[14] P. Candito and G. D’Aguì, Three solutions for a discrete nonlinear Neumann problem involving the p-Laplacian, Adv. Difference Equ. 2010, Art. ID 862016, 11 pp.10.1186/1687-1847-2010-862016
Search in Google Scholar Back to article
[15] P. Candito and N. Giovannelli, Multiple solutions for a discrete boundary value problem, Comput. Math. Appl. 56 (2008) 959–964.10.1016/j.camwa.2008.01.025
Search in Google Scholar Back to article
[16] J. Chu and D. Jiang, Eigenvalues and discrete boundary value problems for the one-dimensional p-Laplacian, J. Math. Anal. Appl. 305 (2005) 452–465.10.1016/j.jmaa.2004.10.055
Search in Google Scholar Back to article
[17] G. D’Aguì, J. Mawhin and A. Sciammetta, Positive solutions for a discrete two point nonlinear boundary value problem with p-Laplacian, Math. Anal. Appl. 447 (2017) 383–397.10.1016/j.jmaa.2016.10.023
Search in Google Scholar Back to article
[18] F. Faraci and A. Iannizzotto, Multiplicity theorems for discrete boundary problems, Aequationes Math. 74 (2007) 111–118.10.1007/s00010-006-2855-5
Search in Google Scholar Back to article
[19] M. Galewski and G.M. Bisci, Existence results for one-dimensional fractional equations, Math. Meth. Appl. Sci. 39 (2016) 1480–149210.1002/mma.3582
Search in Google Scholar Back to article
[20] M. Galewski, G. Molica Bisci and R. Wieteska, Existence and multiplicity of solutions to discrete inclusions with the p(k) -Laplacian problem, J. Difference Equ. Appl. 21(10) (2015) 887–903.10.1080/10236198.2015.1056177
Search in Google Scholar Back to article
[21] M. Galewski and R. Wieteska, Existence and multiplicity of positive solutions for discrete anisotropic equations, Turk. J. Math. 38 (2014) 297–31010.3906/mat-1303-6
Search in Google Scholar Back to article
[22] M. Galewski, R. Wieteska, Existence and multiplicity results for boundary value problems connected with the discrete p(·) -Laplacian on weighted finite graphs, Appl. Math. Comput. 290 (2016) 376–391.10.1016/j.amc.2016.06.016
Search in Google Scholar Back to article
[23] S. Heidarkhani and M. Khaleghi Moghadam, Existence of Three solutions for Perturbed nonlinear difference equations, Opuscula Math. 344 (2014), 747–761.10.7494/OpMath.2014.34.4.747
Search in Google Scholar Back to article
[24] J. Henderson and H.B. Thompson, Existence of multiple solutions for second order discrete boundary value problems, Comput. Math. Appl. 43 (2002), 1239–1248.10.1016/S0898-1221(02)00095-0
Search in Google Scholar Back to article
[25] M. Khaleghi Moghadam, M. Avci, Existence results to a nonlinear p(k)-Laplacian difference equation, J. Difference Equ. Appl. Vol. 23, No. 10 (2017), 1652–166910.1080/10236198.2017.1354991
Search in Google Scholar Back to article
[26] M. Khaleghi Moghadam and J. Henderson, Triple solutions for a dirichlet boundary value problem involving a perturbed discrete p(k) -laplacian operator, Open Math., 15(2017) 1075–1089.10.1515/math-2017-0090
Search in Google Scholar Back to article
[27] M. Khaleghi Moghadam, S. Heidarkhani and J. Henderson, Infinitely many solutions for perturbed difference equations, J. Difference Equ. Appl.207 (2014), 1055–1068.10.1080/10236198.2014.884219
Search in Google Scholar Back to article
[28] M. Khaleghi Moghadam, L. Li and S. Tersian, Existence of three solutions for a discrete anisotropic boundary value problem, Bull. Iranian Math. Sco. (to appear).
Search in Google Scholar Back to article
[29] P. Lindqvist, On the equation div(|u|p−2u) + |u|p−2u = 0, Proc. Amer. Math. Soc. 109 (1990), 157–164.10.1090/S0002-9939-1990-1007505-7
Search in Google Scholar Back to article
[30] Y. Li and L. Lu, Existence of positive solutions of p-Laplacian difference equations, Appl. Math. Lett. 19 (2006) 1019–1023.10.1016/j.aml.2005.10.020
Search in Google Scholar Back to article
[31] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000) 401–410.10.1016/S0377-0427(99)00269-1
Search in Google Scholar Back to article
[32] J. Smejda and R. Wieteska, On the dependence on paramereters for second order discrete boundary value problems with the p(k) -Laplacian, Opuscula Math. 344 (2014) 851–870.10.7494/OpMath.2014.34.4.851
Search in Google Scholar Back to article

Existence and uniqueness of positive solution for nonlinear difference equations involving p(k)-Laplacian operator

Abstract

Paradigm

My account