Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems pp 245-272 | Cite as

# Non-Symmetric Perturbation of Symmetric Eigenvalue Problems

## Abstract

In this Chapter we establish the influence of an arbitrary small perturbation for several classes of symmetric hemivariational eigenvalue inequalities with constraints. If the symmetric problem has infinitely many solutions we show that the number of solutions of the perturbed problem tends to infinity if the perturbation approaches zero with respect to an appropriate topology. This is a very natural phenomenon that occurs often in concrete situations. We illustrate it with the following elementary example: consider on the real axis the equation sin *x* = 1/2. This is a “symmetric” problem (due to the periodicity) with infinitely many solutions. Let us now consider an arbitrary non-symmetric “small” perturbation of the above equation. For instance, the equation sin x = 1/2 + *εx* ^{2} has finitely many solutions, for any *ε* ≠ 0. However, the number of solutions of the perturbed equation becomes greater and greater if the perturbation (that is, |ε|) is smaller and smaller. In contrast with this elementary example, our proofs rely on powerful tools such as topological methods in nonsmooth critical point theory. For different perturbation results and their applications we refer to [1], [15], [20] (see also [9] for a nonsmooth setting) in the case of elliptic equations, [8] for variational inequalities and [3], [5], [6], [14], [16], [17], [18] for various perturbations of hemivariational inequalities. This abstract developments are motivated by important appications in Mechanics (see [12], [13]).

## Keywords

Variational Inequality Eigenvalue Problem Variational Method Generalize Gradient Real Hilbert Space## Preview

Unable to display preview. Download preview PDF.

## References

- [1]A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications,
*Trans. Amer. Math. Soc.*267 (1981), 1–32.MathSciNetzbMATHCrossRefGoogle Scholar - [2]M. Bocea, D. Motreanu and P. D. Panagiotopoulos, Multiple solutions for a double eigenvalue hemivariational inequality on a sphere-like type manifold,
*Nonlinear Analysis*,*T.M.A.*42A (2000), 737–749.MathSciNetzbMATHGoogle Scholar - [3]M. Bocea, P. D. Panagiotopoulos and V. Râdulescu, A perturbation result for a double eigenvalue hemivariational inequality with constraints and applications,
*J. Global Optimiz.*14 (1999), 137–156.zbMATHCrossRefGoogle Scholar - [4]K. C. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations,
*J. Math. Anal. Appl.*80 (1981), 102–129.MathSciNetzbMATHCrossRefGoogle Scholar - [5]C. Ciulcu, D. Motreanu and V. Râdulescu, Multiplicity of solutions for a class of non-symmetric eigenvalue hemivariational inequalities,
*Math. Methods Appl. Sciences*,in press.Google Scholar - [6]F. Cirstea and V. Râdulescu, Multiplicity of solutions for a class of non-symmetric eigenvalue hemivariational inequalities,
*J. Global Optimiz.*17 (1/4) (2000), 43–54.zbMATHCrossRefGoogle Scholar - [7]F. H. Clarke,
*Optimization and Nonsmooth Analysis*, Willey, New York, 1983.zbMATHGoogle Scholar - [8]M. Degiovanni and S. Lancelotti, Perturbations of even non-smooth functionals,
*Differential Integral Equations*8 (1995), 981–992.MathSciNetzbMATHGoogle Scholar - [9]M. Degiovanni and V. Râdulescu, Perturbations of non-smooth symmetric nonlinear eigenvalue problems,
*C.R. Acad. Sci. Paris*329 (1999), 281–286.zbMATHCrossRefGoogle Scholar - [10]D. Goeleven, D. Motreanu and P. D. Panagiotopoulos, Multiple solutions for a class of hemivariational inequalities involving periodic energy functionals,
*Math. Methods Appl. Sciences*, 20 (1997), 548–568.MathSciNetGoogle Scholar - [11]D. Motreanu and P. D. Panagiotopoulos, Double eigenvalue problems for hemivariational inequalities,
*Arch. Rat. Mech. Anal.*140 (1997), 225–251.MathSciNetzbMATHCrossRefGoogle Scholar - [12]P. D. Panagiotopoulos,
*Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functionals*, Birkhäuser-Verlag, Boston, Basel, 1985.Google Scholar - [13]P. D. Panagiotopoulos,
*Ilemivariational Inequalities: Applications to Mechanics and Engineering*, Springer-Verlag, New York, Boston, Berlin, 1993.Google Scholar - [14]P. D. Panagiotopoulos and V. Râdulescu, Perturbations of hemivariational inequalities with constraints and applications, J.
*Global Optinniz*, 12, 285–297 (1998).zbMATHCrossRefGoogle Scholar - [15]P. Rabinowitz,
*Minimax Methods in Critical Point Theory with Applications to Differential Equations*, CBMS Reg. Conf. Ser. Math. 65, Amer. Math. Soc., Providence, R.I., 1986.Google Scholar - [16]V. Râdulescu, Perturbations of hemivariational inequalities with constraints,
*Revue Roumaine Math. Pures Appl.**44*(1999), 455–461.zbMATHGoogle Scholar - [17]V. Râdulescu, Perturbations of eigenvalue problems with constraints for hemivariational inequalities,
*From Convexity to Nonconvexity*,*volume dedicated to the memory of Prof. G. Fichera*, Nonconvex Optim. Appl., 55, Kluwer Acad. Publ., Dordrecht, 2001 (Gilbert, Pardalos, Eds. ), 243–253.Google Scholar - [18]V. Râdulescu, Perturbations of symmetric hemivariational inequalities, in
*Nonsmooth/Nonconvex Mechanics with Applications in Engineering*, Editions Ziti, Thessaloniki, 2002 (C. Baniotopoulos, Ed. ), 61–72.Google Scholar - [19]E. H. Spanier,
*Algebraic Topology*, McGraw-Hill, New York, 1966.zbMATHGoogle Scholar - [20]M. Struwe,
*Variational Methods*, Springer-Verlag, Berlin, Heidelberg, 1990.Google Scholar