Fourth-order hemivariational inequalities

Previous Article
Multiplicity results to elliptic problems in $\mathbb{R}^N$
DCDS-S Home
This Issue
Next Article
Multiple solutions for a Navier boundary value problem involving the $p$--biharmonic operator

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

Abstract
Related Papers

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]	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
[3]	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
[4]	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
[5]	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
[6]	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
[7]	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
[8]	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
[9]	Haitao Che, Haibin Chen, Yiju Wang. On the M-eigenvalue estimation of fourth-order partially symmetric tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2018153
[10]	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
[11]	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
[12]	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
[13]	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
[14]	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
[15]	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
[16]	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
[17]	Bertram Düring, Daniel Matthes, Josipa Pina Milišić. A gradient flow scheme for nonlinear fourth order equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 935-959. doi: 10.3934/dcdsb.2010.14.935
[18]	Leszek Gasiński, Nikolaos S. Papageorgiou. Nonlinear hemivariational inequalities with eigenvalues near zero. Conference Publications, 2005, 2005 (Special) : 317-326. doi: 10.3934/proc.2005.2005.317
[19]	Leszek Gasiński. Existence results for quasilinear hemivariational inequalities at resonance. Conference Publications, 2007, 2007 (Special) : 409-418. doi: 10.3934/proc.2007.2007.409
[20]	Siegfried Carl. Comparison results for a class of quasilinear evolutionary hemivariational inequalities. Conference Publications, 2007, 2007 (Special) : 221-229. doi: 10.3934/proc.2007.2007.221

2018 Impact Factor: 0.545

American Institute of Mathematical Sciences