On the Frictional Contact Problem of p(x)-Kirchhoff Type

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-20-2022-52
Published: Oct 7, 2022
Cite as:
Eugenio Cabanillas Lapa, Willy Barahona Martinez, On the Frictional Contact Problem of p(x)-Kirchhoff Type, Int. J. Anal. Appl., 20 (2022), 52.

Main Article Content

Eugenio Cabanillas Lapa, Willy Barahona Martinez

Abstract

In this article we consider a class of frictional contact problem of p(x)-Kirchhoff type, on a bounded domain Ω⊆R2. Using an abstract Lagrange multiplier technique and the Schauder’s fixed point theorem we establish the existence of weak solutions. Furthermore, we also obtain the uniqueness of the solution assuming that the datum f1 satisfies a suitable monotonicity condition.

Article Details

References

  1. G.A. Afrouzi, M. Mirzapour, Eigenvalue Problems for p(x)-Kirchhoff Type Equations, Electron. J. Differ. Equ. 2013 (2013), 253. https://www.emis.de/journals/EJDE/2013/253/abstr.html.
  2. G.A. Afrouzi, N.T. Chung, Z. Naghizadeh, Multiple Solutions for P(x)-Kirchhoff Type Problems With Robin Boundary Conditions, Electron. J. Differ. Equ. 2022 (2022), 24. https://ejde.math.txstate.edu/Volumes/2022/24/abstr.html.
  3. M.M. Boureanu, A. Matei, M. Sofonea, Nonlinear Problems with p(·)-Growth Conditions and Applications to Antiplane Contact Models, Adv. Nonlinear Stud. 14 (2014), 295–313. https://doi.org/10.1515/ans-2014-0203.
  4. L.E. Cabanillas, L. Huaringa, A Nonlocal p(x)&q(x) Elliptic Transmission Problem With Dependence on the Gradient, Int. J. Appl. Math, 34 (2021), 93-108. https://doi.org/10.12732/ijam.v34i1.4.
  5. M. Chivu Cojocaru, A. Matei, Well-Posedness for a Class of Frictional Contact Models via Mixed Variational Formulations, Nonlinear Anal.: Real World Appl. 47 (2019), 127–141. https://doi.org/10.1016/j.nonrwa.2018.10.009.
  6. N.T. Chung, Multiple Solutions for a Class of p(x)-Kirchhoff Type Problems With Neumann Boundary Conditions, Adv. Pure Appl. Math. 4 (2013), 165-177. https://doi.org/10.1515/apam-2012-0034.
  7. G. Dai, R. Ma, Solutions for a p(x)-Kirchhoff Type Equation With Neumann Boundary Data, Nonlinear Anal.: Real World Appl. 12 (2011), 2666–2680. https://doi.org/10.1016/j.nonrwa.2011.03.013.
  8. L. Diening, P. Harjulehto, P. Hästö, et al. Lebesgue and Sobolev Spaces with Variable Exponents, Springer Berlin Heidelberg, Berlin, Heidelberg, 2011. https://doi.org/10.1007/978-3-642-18363-8.
  9. X.L. Fan, J.S. Shen, D. Zhao, Sobolev Embedding Theorems for Spaces Wk ,p(x) (Ω), J. Math. Anal. Appl. 262 (2001), 749–760. https://doi.org/10.1006/jmaa.2001.7618.
  10. X.L. Fan, Q.H. Zhang, Existence of Solutions for p(x)-Laplacian Dirichlet Problem, Nonlinear Anal.: Theory Methods Appl. 52 (2003), 1843–1852. https://doi.org/10.1016/s0362-546x(02)00150-5.
  11. X. Fan, D. Zhao, On the Spaces L p(x) and Wm,p(x) , J. Math. Anal. Appl. 263 (2001), 424–446. https://doi.org/10.1006/jmaa.2000.7617.
  12. W. Han, M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, American Mathematical Society, Providence, R.I, 2002.
  13. M.K. Hamdani, N.T. Chung, D.D. Repovš, New Class of Sixth-Order Nonhomogeneous p(x)-Kirchhoff Problems With Sign-Changing Weight Functions, Adv. Nonlinear Anal. 10 (2021), 1117–1131. https://doi.org/10.1515/anona-2020-0172.
  14. S. Hüeber, A. Matei, B.I. Wohlmuth, Efficient Algorithms for Problems with Friction, SIAM J. Sci. Comput. 29 (2007), 70–92. https://doi.org/10.1137/050634141.
  15. E.J. Hurtado, O.H. Miyagaki, R.S. Rodrigues, Multiplicity of Solutions to Class of Nonlocal Elliptic Problems With Critical Exponents, Math. Methods Appl. Sci. 45 (2021), 3949–3973. https://doi.org/10.1002/mma.8025.
  16. C. Vetro, Variable Exponent p(x)-Kirchhoff Type Problem With Convection, J. Math. Anal. Appl. 506 (2022), 125721. https://doi.org/10.1016/j.jmaa.2021.125721.
  17. M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2002.
  18. A. Matei, M. Sofonea, Variational Inequalities With Applications: A Study of Antiplane Frictional Contact Problems, Springer, New York, 2009.
  19. N. Tsouli, M. Haddaoui, E.M. Hssini, Multiple Solutions for a Critical p(x)-Kirchhoff Type Equations, Bol. Soc. Paran. Mat. 38 (2019), 197–211. https://doi.org/10.5269/bspm.v38i4.37697.