American Institute of Mathematical Sciences

Advanced
August  2012, 5(4): 729-739. doi: 10.3934/dcdss.2012.5.729

Fourth-order hemivariational inequalities

Gabriele Bonanno 1, and  Beatrice Di Bella 1,

1. 

Department of Science for Engineering and Architecture (Mathematics Section), Engineering Faculty, University of Messina, 98166 - Messina, Italy

Received  February 2011 Revised  May 2011 Published  November 2011

The aim of this paper is to investigate an ordinary fourth-order hemivariational inequality. By using non-smooth variational methods, infinitely many solutions satisfying this type of inequality, whenever the potential of the nonlinear term has a suitable growth condition or convenient oscillatory assumptions at zero or at infinity, are guaranteed. As a consequence, a multiplicity result for non-smooth fourth-order boundary value problems is pointed out.
Keywords: Fourth-order equations, variational-hemivariational inequalities., critical points.
Mathematics Subject Classification: Primary: 34B15; Secondary: 58E0.
Citation: Gabriele Bonanno, Beatrice Di Bella. Fourth-order hemivariational inequalities. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 729-739. doi: 10.3934/dcdss.2012.5.729
References:
[1]

Z. Bai and H. Wang, On positive solutions of some nonlinear fourth-order beam equations,, J. Math. Anal. Appl., 270 (2002), 357.  doi: 10.1016/S0022-247X(02)00071-9.  Google Scholar

[2]

G. Bonanno and B. Di Bella, A boundary value problem for fourth-order elastic beam equations,, J. Math. Anal. Appl., 343 (2008), 1166.  doi: 10.1016/j.jmaa.2008.01.049.  Google Scholar

[3]

G. Bonanno and B. Di Bella, A fourth-order boundary value problem for a Sturm-Liouville type equation,, Appl. Math. Comput., 217 (2010), 3635.  doi: 10.1016/j.amc.2010.10.019.  Google Scholar

[4]

G. Bonanno and B. Di Bella, Infinitely many solutions for fourth-order elastic beam equations,, NoDEA Nonlinear Differential Equations Appl., 18 (2011), 357.   Google Scholar

[5]

G. Bonanno and G. Molica Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities,, Bound. Value Probl., 2009 (6706).   Google Scholar

[6]

A. Cabada, J. Á. Cid and L. Sanchez, Positivity and lower and upper solutions for fourth order boundary value problems,, Nonlinear Anal., 67 (2007), 1599.  doi: 10.1016/j.na.2006.08.002.  Google Scholar

[7]

S. Carl, V. K. Le and D. Motreanu, "Nonsmoth Variational Problems and Their Inequalities. Comparison Principles and Applications,", Springer Monographs in Mathematics, (2007).   Google Scholar

[8]

K. C. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations,, J. Math. Anal. Appl., 80 (1981), 102.  doi: 10.1016/0022-247X(81)90095-0.  Google Scholar

[9]

F. H. Clarke, "Optimization and Nonsmooth Analysis," Second edition,, Classics Appl. Math., 5 (1990).   Google Scholar

[10]

E. De Giorgi, G. Buttazzo and G. Dal Maso, On the lower semicontinuity of certain integral functionals,, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 74 (1983), 274.   Google Scholar

[11]

M. R. Grossinho, L. Sanchez and S. A. Tersian, On the solvability of a boundary value problem for a fourth-order ordinary differential equation,, Appl. Math. Lett., 18 (2005), 439.   Google Scholar

[12]

T. Gyulov and G. Moroşanu, On a nonsmooth fourth order boundary value problem,, Nonlinear Anal., 67 (2007), 2800.  doi: 10.1016/j.na.2006.09.041.  Google Scholar

[13]

T. Gyulov and G. Moroşanu, On a class of boundary value problems involving the p-biharmonic operator,, J. Math. Anal. Appl., 367 (2010), 43.  doi: 10.1016/j.jmaa.2009.12.022.  Google Scholar

[14]

G. Han and Z. Xu, Multiple solutions of some nonlinear fourth-order beam equations,, Nonlinear Anal., 68 (2008), 3646.  doi: 10.1016/j.na.2007.04.007.  Google Scholar

[15]

X.-L. Liu and W.-T. Li, Existence and multiplicity of solutions for fourth-order boundary values problems with three parameters,, Math. Comput. Modelling, 46 (2007), 525.  doi: 10.1016/j.mcm.2006.11.018.  Google Scholar

[16]

X.-L. Liu and W.-T. Li, Existence and multiplicity of solutions for fourth-order boundary values problems with parameters,, J. Math. Anal. Appl., 327 (2007), 362.  doi: 10.1016/j.jmaa.2006.04.021.  Google Scholar

[17]

S. A. Marano and D. Motreanu, Infinitely many critical points of non-differentiable functions and applications to a Neumann type problem involving the p-Laplacian,, J. Differential Equations, 182 (2002), 108.  doi: 10.1006/jdeq.2001.4092.  Google Scholar

[18]

D. Motreanu and P. D. Panagiotopoulos, "Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities,", Nonconvex Optim. Appl., 29 (1999).   Google Scholar

[19]

D. Motreanu and V. Rădulescu, "Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems,", Nonconvex Optimization and Applications, 67 (2003).   Google Scholar

show all references

References:
[1]

Z. Bai and H. Wang, On positive solutions of some nonlinear fourth-order beam equations,, J. Math. Anal. Appl., 270 (2002), 357.  doi: 10.1016/S0022-247X(02)00071-9.  Google Scholar

[2]

G. Bonanno and B. Di Bella, A boundary value problem for fourth-order elastic beam equations,, J. Math. Anal. Appl., 343 (2008), 1166.  doi: 10.1016/j.jmaa.2008.01.049.  Google Scholar

[3]

G. Bonanno and B. Di Bella, A fourth-order boundary value problem for a Sturm-Liouville type equation,, Appl. Math. Comput., 217 (2010), 3635.  doi: 10.1016/j.amc.2010.10.019.  Google Scholar

[4]

G. Bonanno and B. Di Bella, Infinitely many solutions for fourth-order elastic beam equations,, NoDEA Nonlinear Differential Equations Appl., 18 (2011), 357.   Google Scholar

[5]

G. Bonanno and G. Molica Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities,, Bound. Value Probl., 2009 (6706).   Google Scholar

[6]

A. Cabada, J. Á. Cid and L. Sanchez, Positivity and lower and upper solutions for fourth order boundary value problems,, Nonlinear Anal., 67 (2007), 1599.  doi: 10.1016/j.na.2006.08.002.  Google Scholar

[7]

S. Carl, V. K. Le and D. Motreanu, "Nonsmoth Variational Problems and Their Inequalities. Comparison Principles and Applications,", Springer Monographs in Mathematics, (2007).   Google Scholar

[8]

K. C. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations,, J. Math. Anal. Appl., 80 (1981), 102.  doi: 10.1016/0022-247X(81)90095-0.  Google Scholar

[9]

F. H. Clarke, "Optimization and Nonsmooth Analysis," Second edition,, Classics Appl. Math., 5 (1990).   Google Scholar

[10]

E. De Giorgi, G. Buttazzo and G. Dal Maso, On the lower semicontinuity of certain integral functionals,, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 74 (1983), 274.   Google Scholar

[11]

M. R. Grossinho, L. Sanchez and S. A. Tersian, On the solvability of a boundary value problem for a fourth-order ordinary differential equation,, Appl. Math. Lett., 18 (2005), 439.   Google Scholar

[12]

T. Gyulov and G. Moroşanu, On a nonsmooth fourth order boundary value problem,, Nonlinear Anal., 67 (2007), 2800.  doi: 10.1016/j.na.2006.09.041.  Google Scholar

[13]

T. Gyulov and G. Moroşanu, On a class of boundary value problems involving the p-biharmonic operator,, J. Math. Anal. Appl., 367 (2010), 43.  doi: 10.1016/j.jmaa.2009.12.022.  Google Scholar

[14]

G. Han and Z. Xu, Multiple solutions of some nonlinear fourth-order beam equations,, Nonlinear Anal., 68 (2008), 3646.  doi: 10.1016/j.na.2007.04.007.  Google Scholar

[15]

X.-L. Liu and W.-T. Li, Existence and multiplicity of solutions for fourth-order boundary values problems with three parameters,, Math. Comput. Modelling, 46 (2007), 525.  doi: 10.1016/j.mcm.2006.11.018.  Google Scholar

[16]

X.-L. Liu and W.-T. Li, Existence and multiplicity of solutions for fourth-order boundary values problems with parameters,, J. Math. Anal. Appl., 327 (2007), 362.  doi: 10.1016/j.jmaa.2006.04.021.  Google Scholar

[17]

S. A. Marano and D. Motreanu, Infinitely many critical points of non-differentiable functions and applications to a Neumann type problem involving the p-Laplacian,, J. Differential Equations, 182 (2002), 108.  doi: 10.1006/jdeq.2001.4092.  Google Scholar

[18]

D. Motreanu and P. D. Panagiotopoulos, "Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities,", Nonconvex Optim. Appl., 29 (1999).   Google Scholar

[19]

D. Motreanu and V. Rădulescu, "Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems,", Nonconvex Optimization and Applications, 67 (2003).   Google Scholar

[1]

Stanisław Migórski, Biao Zeng. Convergence of solutions to inverse problems for a class of variational-hemivariational inequalities. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4477-4498. doi: 10.3934/dcdsb.2018172
[2]

Zhenhai Liu, Van Thien Nguyen, Jen-Chih Yao, Shengda Zeng. History-dependent differential variational-hemivariational inequalities with applications to contact mechanics. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020044
[3]

Zijia Peng, Cuiming Ma, Zhonghui Liu. Existence for a quasistatic variational-hemivariational inequality. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020058
[4]

Jaime Angulo Pava, Carlos Banquet, Márcia Scialom. Stability for the modified and fourth-order Benjamin-Bona-Mahony equations. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 851-871. doi: 10.3934/dcds.2011.30.851
[5]

Feliz Minhós, João Fialho. On the solvability of some fourth-order equations with functional boundary conditions. Conference Publications, 2009, 2009 (Special) : 564-573. doi: 10.3934/proc.2009.2009.564
[6]

Benoît Pausader. The focusing energy-critical fourth-order Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1275-1292. doi: 10.3934/dcds.2009.24.1275
[7]

Lili Ju, Xinfeng Liu, Wei Leng. Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1667-1687. doi: 10.3934/dcdsb.2014.19.1667
[8]

O. Chadli, Z. Chbani, H. Riahi. Recession methods for equilibrium problems and applications to variational and hemivariational inequalities. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 185-196. doi: 10.3934/dcds.1999.5.185
[9]

José A. Carrillo, Ansgar Jüngel, Shaoqiang Tang. Positive entropic schemes for a nonlinear fourth-order parabolic equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 1-20. doi: 10.3934/dcdsb.2003.3.1
[10]

Baishun Lai, Qing Luo. Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 227-241. doi: 10.3934/dcds.2011.30.227
[11]

Chunhua Jin, Jingxue Yin, Zejia Wang. Positive periodic solutions to a nonlinear fourth-order differential equation. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1225-1235. doi: 10.3934/cpaa.2008.7.1225
[12]

Haitao Che, Haibin Chen, Yiju Wang. On the M-eigenvalue estimation of fourth-order partially symmetric tensors. Journal of Industrial & Management Optimization, 2020, 16 (1) : 309-324. doi: 10.3934/jimo.2018153
[13]

Carlos Banquet, Élder J. Villamizar-Roa. On the management fourth-order Schrödinger-Hartree equation. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020037
[14]

Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831
[15]

A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044
[16]

Pablo Álvarez-Caudevilla, Jonathan D. Evans, Victor A. Galaktionov. Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3913-3938. doi: 10.3934/dcds.2018170
[17]

Zhilin Yang, Jingxian Sun. Positive solutions of a fourth-order boundary value problem involving derivatives of all orders. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1615-1628. doi: 10.3934/cpaa.2012.11.1615
[18]

Maria-Magdalena Boureanu. Fourth-order problems with Leray-Lions type operators in variable exponent spaces. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 231-243. doi: 10.3934/dcdss.2019016
[19]

Luca Calatroni, Bertram Düring, Carola-Bibiane Schönlieb. ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 931-957. doi: 10.3934/dcds.2014.34.931
[20]

Amine Laghrib, Abdelkrim Chakib, Aissam Hadri, Abdelilah Hakim. A nonlinear fourth-order PDE for multi-frame image super-resolution enhancement. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 415-442. doi: 10.3934/dcdsb.2019188

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences