On a class of nonlinear discrete problems of Kirchhoff type

Mohammed Barghouthe; Abdesslem Ayoujil; Mohammed Berrajaa

doi:10.24193/subbmath.2025.4.10

Authors

Mohammed Barghouthe Department of Mathematics, Faculty of Sciences, Mohammed I University, Oujda, Morocco https://orcid.org/0000-0002-5714-0157
Abdesslem Ayoujil Deptartment of Mathematics, Regional Centre of Trades Education and Training, Oujda, Morocco https://orcid.org/0000-0002-0559-3242
Mohammed Berrajaa Department of Mathematics, Faculty of Sciences, Mohammed I University, Oujda, Morocco

DOI:

https://doi.org/10.24193/subbmath.2025.4.10

Keywords:

Anisotropic problem, discrete boundary value problem, variational methods, Kirchhoff-type problems

Abstract

In view of variational methods and critical points theory, we study the existence of solutions for a discrete boundary value problem, which is
a discrete variant of a continuous \((p_1(x), p_2(x))\)-Kirchhoff-type problem with a real parameter \(\lambda>0\).

References

Agarwal, R.P., Perera K., D. O'Regan D., Multiple positive solutions of singular and nonsingular discrete problems via variational methods. Nonlinear Anal. 58 (2004), no 1-2, 69-73.

Arosio, A., Panizzi, S., On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 (1996), no. 1, 305-330.

Avci, M., Existence and multiplicity of solutions for Dirichlet problems involving the p(x)-Laplace operator, Electron. J. Differential Equations, 2013 (2013), no. 14 1-9.

Avci, M., Ayazoglu, R., Solutions of Nonlocal (p₁(x), p₂(x))-Laplacian Equations, J. Partial Differ. Equ. 14 (2013), Article ID 364251.

Avci, M., On a nonlocal Neumann problem in Orlicz-Sobolev spaces, J. Nonlinear Funct. Anal. 2017 (2017), no. 42, 1-11.

Cai, X., Yu, J., Existence theorems for second-order discrete boundary value problems, J. Math. Anal. Appl. 320 (2006), no. 2 649-661.

Cavalcanti, M.M., Domingos Cavalcanti, V.N., Soriano, J.A., Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differential Equations, 6 (2001), no. 6, 701-730.

Chen, Y., Levine, S., Rao, M., Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), no. 4, 1383-1406.

Chung, N., Multiplicity results for a class of p(x)-Kirchhoff type equations with combined nonlinearities, Electron. J. Qual. Theory Differ. Equ. 2012 (2012), no. 42, 1-13.

D'Ancona, P., Spagnolo, S., Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math. 108 (1992), no. 1, 247-262.

Dreher, M., The Kirchhoff equation for the p-Laplacian, Rend. Semin. Mat. Univ. Politec. Torino. 64 (2006), no. 2, 217-238.

Ekeland, I., On the variational principle, J. Math. Anal. Appl. 47 (1974), no. 2, 324-353.

Galewski, M., Wieteska, R., On the system of anisotropic discrete BVPs, J. Difference Equ. Appl. 19 (2013), no. 7, 1065-1081.

Kirchhoff, G., Vorlesungen über mathematische Physik, BG Teubner, 1891.

Mihăilescu, M., Rădulescu, V., Tersian, S., Eigenvalue problems for anisotropic discrete boundary value problems, J. Difference Equ. Appl. 15 (2009), no. 6, 557-56.

Ruzicka, M., Flow of shear dependent electrorheological fluids, Comptes Rendus de l'Académie des Sciences-Series I-Mathematics. 329 (1999), no. 5, 393-398.

Ruzicka, M., Electrorheological fluids: modeling and mathematical theory, Springer, 2007.

Willem, M., Linking theorem, Minimax Theorems, Birkhäuser, Boston, 1996.

Yücedağ, Z., Existence of solutions for anisotropic discrete boundary value problems of Kirchhoff type, J. Difference Equ. Appl. 13 (2014), no. 1, 1-15.

Yücedağ, Z., Solutions for a discrete boundary value problem involving kirchhoff type equation via variational methods, TWMS J. Appl. Eng. Math. 8 (2018), no. 1, 144-154.

Zhikov, V.V.E., Averaging of functionals of the calculus of variations and elasticity theory, Izv. Math. 29 (1987), no. 1, 33-66.