Geodesic
An existence result for a $(p(x), q(x))$-Kirchhoff type system with Dirichlet boundary conditions via topological degree method
Eurasian mathematical journal, Tome 15 (2024) no. 2, pp. 75-91.

Voir la notice de l'article provenant de la source Math-Net.Ru

This paper focuses on the existence of at least one weak solution for a nonlocal elliptic system of $(p(x), q(x))$-Kirchhoff type with Dirichlet boundary conditions. The results are obtained by applying the topological degree method of Berkovits applied to an abstract Hammerstein equation associated to our system and also by the theory of the generalized Sobolev spaces. 
@article{EMJ_2024_15_2_a5,
     author = {S. Yacini and C. Allalou and K. Hilal},
     title = {An existence result for a $(p(x), q(x))${-Kirchhoff} type system with {Dirichlet} boundary conditions via topological degree method},
     journal = {Eurasian mathematical journal},
     pages = {75--91},
     publisher = {mathdoc},
     volume = {15},
     number = {2},
     year = {2024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EMJ_2024_15_2_a5/}
}
TY  - JOUR
AU  - S. Yacini
AU  - C. Allalou
AU  - K. Hilal
TI  - An existence result for a $(p(x), q(x))$-Kirchhoff type system with Dirichlet boundary conditions via topological degree method
JO  - Eurasian mathematical journal
PY  - 2024
SP  - 75
EP  - 91
VL  - 15
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/EMJ_2024_15_2_a5/
LA  - en
ID  - EMJ_2024_15_2_a5
ER  - 
%0 Journal Article
%A S. Yacini
%A C. Allalou
%A K. Hilal
%T An existence result for a $(p(x), q(x))$-Kirchhoff type system with Dirichlet boundary conditions via topological degree method
%J Eurasian mathematical journal
%D 2024
%P 75-91
%V 15
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/EMJ_2024_15_2_a5/
%G en
%F EMJ_2024_15_2_a5
S. Yacini; C. Allalou; K. Hilal. An existence result for a $(p(x), q(x))$-Kirchhoff type system with Dirichlet boundary conditions via topological degree method. Eurasian mathematical journal, Tome 15 (2024) no. 2, pp. 75-91. http://geodesic.mathdoc.fr/item/EMJ_2024_15_2_a5/

[1] A. Abbassi, C. Allalou, A. Kassidi, “Existence results for some nonlinear elliptic equations via topological degree methods”, Journal of Elliptic and Parabolic Equations, 7 (2021), 121–136 | DOI | MR | Zbl

[2] Y. Akdim, E. Azroul, A. Benkirane, “Existence of solutions for quasilinear degenerate elliptic equations”, Electronic Journal of Differential Equations (EJDE), 2001, 71, 19 pp. | MR | Zbl

[3] C. Allalou, K. Hilal, S. Yacini, “Existence of weak solution for $p$-Kirchhoff type problem via topological degree”, Journal of Elliptic and Parabolic Equations, 9:2 (2023), 673–686 | DOI | MR | Zbl

[4] H. Benchira, A. Matallah, M. E.O. El Mokhtar, K. Sabri, “The existence result for a$p$-Kirchhoff-type problem involving critical Sobolev exponent”, Journal of Function Spaces, 2023, no. 1, 3247421 | MR

[5] J. Berkovits, “Extension of the Leray-Schauder degree for abstract Hammerstein type mappings”, Journal of differential equations, 234:1 (2007), 289–310 | DOI | MR | Zbl

[6] J. Berkovits, O. Lehto, “On the degree theory for nonlinear mappings of monotone type”, Thesis, Annales Academiae Scientiarum Fennicae, Series A: Chemica, 1986 | MR

[7] Y. Chen, S. Levine, M. Rao, “Variable exponent, linear growth functionals in image restoration”, SIAM journal on Applied Mathematics, 66:4 (2006), 1383–1406 | DOI | MR | Zbl

[8] Y. J. Cho, Y. Q. Chen, Topological degree theory and applications, Chapman and Hall/CRC, 2006 | MR | Zbl

[9] E. Cabanillas, A. G. Aliaga, W. Barahona, G. Rodriguez, “Existence of solutions for a class of $p(x)$-Kirchhoff type equation via topological methods”, J. Adv. Appl. Math. and Mech., 2:4 (2015), 64–72 | MR | Zbl

[10] N. T. Chung, “Multiple solutions for a class of $p(x)$-Kirchhoff type problems with Neumann boundary conditions”, Advances in Pure and Applied Mathematics, 4:2 (2013), 165–177 | DOI | MR | Zbl

[11] G. Dai, R. Hao, “Existence of solutions for a $p(x)$-Kirchhoff-type equation”, Journal of Mathematical Analysis and Applications, 359:1 (2009), 275–284 | DOI | MR | Zbl

[12] N. C. Eddine, A. Ouannasser, “Multiple solutions for nonlinear generalized-Kirchhoff type potential in unbounded domains”, Filomat, 37:13 (2023), 4317–4334 | DOI | MR

[13] N. C. Eddine, M. A. Ragusa, “Generalized critical Kirchhoff-type potential systems with Neumann Boundary conditions”, Applicable Analysis, 101:11 (2022), 3958–3988 | DOI | MR | Zbl

[14] X. L. Fan, D. Zhao, “On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$”, J. Math. Anal. Appl., 263:2 (2001), 424–446 | DOI | MR | Zbl

[15] P. Harjulehto, P. Hästö, M. Koskenoja, S. Varonen, “The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values”, Potential Analysis, 25 (2006), 205–222 | DOI | MR | Zbl

[16] G. Kirchhoff, Vorlesungen uber Mechanik, Teubner, Leipzig, 1883

[17] I. S. Kim, S. J. Hong, “A topological degree for operators of generalized $(S_+)$ type”, Fixed Point Theory and Applications, 2015, 1–16 | MR

[18] O. Kováčik, J. Rákosníik, “On spaces $L^{p(x)}$ and $W^{1,p(x)}$”, Czechoslovak Math. J., 41:4 (1991), 592–618 | DOI | MR | Zbl

[19] M. Massar, M. Talbi, N. Tsouli, “Multiple solutions for nonlocal system of $(p(x),q(x))$-Kirchhoff type”, Applied Mathematics and Computation, 242 (2014), 216–226 | DOI | MR | Zbl

[20] J. Simon, “Régularité de la solution d'uneequation non linaire dans $\mathbb{R}^N$” (Besančon, France, June 1977), Journées d'Analyse Non Linéaire, Springer, Berlin–Heidelberg, 2006, 205–227 | MR

[21] M. Ružička, Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Mathematics, 2000 | MR

[22] S. G. Samko, “Density of $C_0^\infty(\mathbb{R}^N)$ in the generalized Sobolev spaces $W^{m,p(x)}(\mathbb{R}^N)$”, Doklady Mathematics, 60:3 (1999), 382–385 | MR | Zbl

[23] A. Taghavi, H. Ghorbani, “Existence of a solution for a nonlocal elliptic system of $(p(x),q(x))$-Kirchhoff type”, Advances in Pure and Applied Mathematics, 9:3 (2018), 221–233 | DOI | MR | Zbl

[24] S. A. Temghart, A. Kassidi, C. Allalou, A. Abbassi, “On an elliptic equation of Kirchhoff type problem via topological degree”, Nonlinear Studies, 28:4 (2021), 1179–1193 | MR

[25] S. Yacini, C. Allalou, K. Hilal, A. Kassidi, “Weak solutions to Kirchhoff type problems via topological degree”, Adv. Math. Mod. Appl., 6 (2021), 309–321

[26] S. Yacini, A. Abbassi, C. Allalou, A. Kassidi, “On a nonlinear equation p(x)-elliptic problem of Neumann type by topological degree method”, International Conference on Partial Differential Equations and Applications, Modeling and Simulation, Springer International Publishing, Cham, 2021, 404–417 | MR

[27] S. Yacini, C. Allalou, K. Hilal, “Weak solution to $p(x)$-Kirchhoff type problems under no-flux boundary condition by topological degree”, Boletim da Sociedade Paranaense de Matemática, 41 (2023), 1–12 | MR

[28] S. Yacini, M. El. Ouaarabi, C. Allalou, K. Hilal, “$p(x)$-Kirchhoff-type problem with no-flux boundary conditions and convection”, Nonautonomous Dynamical Systems, 10:1 (2023), 2023–0105 | DOI | MR

[29] Z. Yucedag, “Existence of weak solutions for $p(x)$-Kirchhoff-type equation”, Int. J. Pure Appl. Math., 92 (2014), 61–71 | DOI | Zbl

[30] E. Zeidler, Nonlinear functional analysis and its applications, Springer Science and Business Media, v. II/B, Nonlinear monotone operators, 2013 | MR

[31] D. Zhao, W. J. Qiang, X. L. Fan, “On generalized Orlicz spaces $L^{p(x)}(\Omega)$”, J. Gansu Sci., 9:2 (1997), 1–7

[32] V. V. Zhikov, “Averaging of functionals of the calculus of variations and elasticity theory”, Mathematics of the USSR-Izvestiya, 29:1 (1987), 33 | DOI | MR | Zbl

[33] V. V. Zhikov, “Averaging of functionals of the calculus of variations and elasticity theory”, Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 50:4 (1986), 675–710 | MR