American Institute of Mathematical Sciences

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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 and 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), 357-368. doi: 10.1016/S0022-247X(02)00071-9.

[2]

G. Bonanno and B. Di Bella, A boundary value problem for fourth-order elastic beam equations, J. Math. Anal. Appl., 343 (2008), 1166-1176. doi: 10.1016/j.jmaa.2008.01.049.

[3]

G. Bonanno and B. Di Bella, A fourth-order boundary value problem for a Sturm-Liouville type equation, Appl. Math. Comput., 217 (2010), 3635-3640. doi: 10.1016/j.amc.2010.10.019.

[4]

G. Bonanno and B. Di Bella, Infinitely many solutions for fourth-order elastic beam equations, NoDEA Nonlinear Differential Equations Appl., 18 (2011), 357-368.

[5]

G. Bonanno and G. Molica Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl., 2009, Art. ID 670675, 20 pp.

[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), 1599-1612. doi: 10.1016/j.na.2006.08.002.

[7]

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

[8]

K. C. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129. doi: 10.1016/0022-247X(81)90095-0.

[9]

F. H. Clarke, "Optimization and Nonsmooth Analysis," Second edition, Classics Appl. Math., 5, SIAM, Philadelphia, 1990.

[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), 274-282.

[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), 439-444.

[12]

T. Gyulov and G. Moroşanu, On a nonsmooth fourth order boundary value problem, Nonlinear Anal., 67 (2007), 2800-2814. doi: 10.1016/j.na.2006.09.041.

[13]

T. Gyulov and G. Moroşanu, On a class of boundary value problems involving the p-biharmonic operator, J. Math. Anal. Appl., 367 (2010), 43-57. doi: 10.1016/j.jmaa.2009.12.022.

[14]

G. Han and Z. Xu, Multiple solutions of some nonlinear fourth-order beam equations, Nonlinear Anal., 68 (2008), 3646-3656. doi: 10.1016/j.na.2007.04.007.

[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), 525-534. doi: 10.1016/j.mcm.2006.11.018.

[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), 362-375. doi: 10.1016/j.jmaa.2006.04.021.

[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), 108-120. doi: 10.1006/jdeq.2001.4092.

[18]

D. Motreanu and P. D. Panagiotopoulos, "Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities," Nonconvex Optim. Appl., 29, Kluwer Academic Publishers, Dordrecht, 1999.

[19]

D. Motreanu and V. Rădulescu, "Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems," Nonconvex Optimization and Applications, 67, Kluwer Academic Publishers, Dordrecht, 2003.

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), 357-368. doi: 10.1016/S0022-247X(02)00071-9.

[2]

G. Bonanno and B. Di Bella, A boundary value problem for fourth-order elastic beam equations, J. Math. Anal. Appl., 343 (2008), 1166-1176. doi: 10.1016/j.jmaa.2008.01.049.

[3]

G. Bonanno and B. Di Bella, A fourth-order boundary value problem for a Sturm-Liouville type equation, Appl. Math. Comput., 217 (2010), 3635-3640. doi: 10.1016/j.amc.2010.10.019.

[4]

G. Bonanno and B. Di Bella, Infinitely many solutions for fourth-order elastic beam equations, NoDEA Nonlinear Differential Equations Appl., 18 (2011), 357-368.

[5]

G. Bonanno and G. Molica Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl., 2009, Art. ID 670675, 20 pp.

[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), 1599-1612. doi: 10.1016/j.na.2006.08.002.

[7]

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

[8]

K. C. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129. doi: 10.1016/0022-247X(81)90095-0.

[9]

F. H. Clarke, "Optimization and Nonsmooth Analysis," Second edition, Classics Appl. Math., 5, SIAM, Philadelphia, 1990.

[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), 274-282.

[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), 439-444.

[12]

T. Gyulov and G. Moroşanu, On a nonsmooth fourth order boundary value problem, Nonlinear Anal., 67 (2007), 2800-2814. doi: 10.1016/j.na.2006.09.041.

[13]

T. Gyulov and G. Moroşanu, On a class of boundary value problems involving the p-biharmonic operator, J. Math. Anal. Appl., 367 (2010), 43-57. doi: 10.1016/j.jmaa.2009.12.022.

[14]

G. Han and Z. Xu, Multiple solutions of some nonlinear fourth-order beam equations, Nonlinear Anal., 68 (2008), 3646-3656. doi: 10.1016/j.na.2007.04.007.

[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), 525-534. doi: 10.1016/j.mcm.2006.11.018.

[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), 362-375. doi: 10.1016/j.jmaa.2006.04.021.

[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), 108-120. doi: 10.1006/jdeq.2001.4092.

[18]

D. Motreanu and P. D. Panagiotopoulos, "Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities," Nonconvex Optim. Appl., 29, Kluwer Academic Publishers, Dordrecht, 1999.

[19]

D. Motreanu and V. Rădulescu, "Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems," Nonconvex Optimization and Applications, 67, Kluwer Academic Publishers, Dordrecht, 2003.

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