Minimax theorems for functions involving a real variable and applications

Fixed Point Theory

Volumn 9, Issue 1, 2008, Pages 275-291

  Ricceri, Biagio ^a  

^a UNIVERSITY OF CATANIA   (Italy)

Author keywords

Bifurcation; Boundary value problems; Connectedness; Critical points; Integral functionals on lp and sobolev spaces; Local and global minima; Minimax theorems; Multiplicity; Nonlinear equations; Potential operators; Singular set

Indexed keywords

EID: 70350627517     PISSN: 15835022     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Conference Paper

References (75)

1. E. Asplund and V. Pták, A minimax inequality for operators and a related numerical range, Acta Math., 126 (1971), 53-62.
2. Z. B. Bai and W. G. Ge, Existence of three positive solutions for some second-order boundary value problems, Comput. Math. Appl., 48 (2004), 699-707.
3. G. Barletta and R. Livrea, Existence of three periodic solutions for a non autonomous system, Matematiche, 57 (2002), 205-215.
4. G. Bonanno, Existence of three solutions for a two point boundary value problem, Appl. Math. Lett., 13 (2000), 53-57.
5. G. Bonanno, A minimax inequality and its applications to ordinary differential equations, J. Math. Anal. Appl., 270 (2002), 210-229.
6. G. Bonanno, Multiple solutions for a Neumann boundary value problem, J. Nonlinear Convex Anal., 4 (2003), 287-290.
7. G. Bonanno, Some remarks on a three critical points theorem, Nonlinear Anal., 54 (2003), 651-665.
8. G. Bonanno and P. Candito, Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian, Arch. Math. (Basel), 80 (2003), 424-429.
9. G. Bonanno and R. Livrea, Multiplicity theorems for the Dirichlet problem involving the p-Laplacian, Nonlinear Anal., 54 (2003), 1-7.
10. G. Bonanno and P. Candito, On a class of nonlinear variational- hemivariational inequalities, Appl. Anal., 83 (2004), 1229-1244.
11. O. Yu. Borenshtein and V. S. Shul'man, A minimax theorem, Math. Notes, 50 (1991), 752-754.
12. B. E. Breckner and Cs. Varga, A multiplicity result for gradient-type systems with nondifferentiable term, Acta Math. Hungar., 118 (2008), 85-104.
13. B. E. Breckner, A. Horváth and Cs. Varga, A multiplicity result for a special class of gradient-type systems with non-differentiable term, Nonlinear Anal., to appear.
14. F. Cammaroto, A. Chinnì and B. Di Bella, Multiple solutions for a two point boundary value problem, J. Math. Anal. Appl, 323 (2006), 530-534.
15. F. Cammaroto, A. Chinnì and B. Di Bella, Multiple solutions for a quasilinear elliptic variational system on strip-like domains, Proc. Edinb. Math. Soc, 50 (2007), 597-603.
16. F. Cammaroto, A. Chinnì and B. Di Bella, Multiplicity results for nonlinear Schrödinger equation, Glasg. Math. J., 49 (2007), 423-429.
17. F. Cammaroto, A. Chinnì and B. Di Bella, Multiple solutions for a Dirichlet problem involving the p-Laplacian, Dynam. Systems Appl., 16 (2007), 673-679.
18. G. Cordaro, On a minimax problem of Ricceri, J. Inequal. Appl., 6 (2001), 261-285.
19. G. Cordaro, Three periodic solutions to an eigenvalue problem for a class of second order Hamiltonian systems, Abstr. Appl. Anal., 18 (2003), 1037-1045.
20. G. Cordaro, Further results related to a minimax problem of Ricceri, J. Inequal. Appl., (2005), 523-533.
21. F. Faraci and A. Iannizzotto, An extension of a multiplicity theorem by Ricceri with an application to a class of quasilinear equations, Studia Math., 172 (2006), 275-287.
22. F. Faraci, A. Iannizzotto, H. Lisei and Cs. Varga, A multiplicty result for hemivariational inequalities, J. Math. Anal. Appl., 330 (2007), 683-698.
23. F. Faraci, A. Iannizzotto, P. Kupán and Cs. Varga, Existence and multiplicity results for hemivariational inequalities two parameters, Nonlinear Anal., 67 (2007), 2654-2669.
24. F. Faraci and A. Iannizzotto, Multiplicity theorems for discrete boundary value problems, Aequationes Math., 74 (2007), 111-118.
25. F. Faraci and A. Iannizzotto, A conjecture for finding non-convex Chebyshev sets in Hilbert spaces, SIAM J. Optim., 19 (2008), 211-216.
26. F. Faraci and A. Iannizzotto, Bifurcation for second order Hamiltonian systems with periodic boundary conditions, Abstr. Appl. Anal., 2008, Art. ID 756934, 13 pp.
27. E. Giner, Lower bounds for integral junctionals: a variational property, preprint.
28. A. Kristaly, Multiplicity results for an eigenvalue problem for hemivariational inequalities in strip-like domains, Set-Valued Anal., 13 (2005), 85-103. (Pubitemid 40669843)
29. A. Kristaly, Existence of two nontrivial solutions for a class of quasilinear elliptic variational systems on strip-like domains, Proc. Edinb. Math. Soc, 48 (2005), 465-477.
30. N, Math. Nachr., 278 (2005), 1756-1765.
31. A. Kristály, A double eigenvalue problem for Schrödinger equations involving sublinear nonlinearities at infinity, Electron. J. Differential Equations 2007, No 42, 11 pp. (electronic).
32. A. Kristály and Cs. Varga, Multiple solutions for elliptic problems with singular and sublinear potentials, Proc. Amer. Math. Soc, 135 (2007), 2121-2126.
33. A. Kristaly, Multiple solutions for a sublinear Schrödinger equation, NoDEA Nonlinear Differential Equations Appl, 14 (2007), 291-301.
34. A. Kristály, H. Lisei and Cs. Varga, Multiple solutions for p-Laplacian type equations, Nonlinear Anal, 68 (2008), 1375-1381.
35. A. Kristály and V. Rǎdulescu, Sublinear eigenvalue problems on compact Riemannian manifolds with applications in Emden-Fowler equations, preprint.
36. A. Kristály, M. Mihǎilescu and V. Rǎdulescu, Two nontrivial solutions for a nonhomogeneous Neumann problem: an Orlicz-Sobolev setting, Proc. Roy. Soc. Edinburgh Sect. A, to appear.
37. C. Li and C.-L. Tang, Three solutions for a class of quasilinear elliptic systems involving the (p, q)-Laplacian, Nonlinear Anal., to appear.
38. H. Lisei, Cs. Varga and A. Horváth, Multiplicity results for a class of quasilinear problems on unbounded domains, Arch. Math. (Basel), 90 (2008), 256-266.
39. H. Lisei, G. Moroşanu and Cs. Varga, Multiplicity results for double eigenvalue problems involving the p-Laplacian, Taiwanese J. Math., to appear.
40. X.-L. Liu and W.-T. Li, Existence and multiplicity of solutions for fourth-order boundary value problems with parameters, J. Math. Anal. Appl., 327 (2007), 362-375.
41. Q. Liu, Existence of three solutions for p (x)-Laplacian equations, Nonlinear Anal., 68 (2008), 2119-2127.
42. R. Livrea, Existence of three solutions for a quasilinear two point boundary value problem, Arch. Math. (Basel), 79 (2002), 288-298.
43. S. A. Marano and D. Motreanu, On a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems, Nonlinear Anal., 48 (2002), 37-52.
44. M. Mihǎilescu, Existence and multiplicity of solutions for a Neumann problem involving the p (x)-Laplace operator, Nonlinear Anal., 67 (2007), 1419-1425.
45. O. Naselli, On a class of functions with equal infima over a domain and its boundary, J. Optim. Theory Appl, 91 (1996), 81-90.
46. O. Naselli, On the solution set of an equation of the type f (t, Φ (u) (t)) = 0, Set-Valued Anal., 4 (1996), 1399-405.
47. B. Ricceri, Some topological mini-max theorems via an alternative principle for multifunctions, Arch. Math. (Basel), 60 (1993), 367-377.
48. p-spaces of vector-valued functions, C. R. Acad. Sci. Paris, Série I, 318 (1994), 337-342.
49. B. Ricceri, On the integrable selections of certain multifunctions, Set-Valued Anal., 4 (1996), 91-99.
50. B. Ricceri, More on a variational property of integral functionals, J. Optim. Theory Appl, 94 (1997), 757-763.
51. B. Ricceri, On a topological minimax theorem and its applications, in "Minimax theory and applications", B. Ricceri and S. Simons eds., 191-216, Kluwer Academic Publishers, 1998.
52. B. Ricceri, On a three critical points theorem, Arch. Math. (Basel), 75 (2000), 220-226.
53. B. Ricceri, Further considerations on a variational property of integral functionals, J. Optim. Theory Appl., 106 (2000), 677-681.
54. B. Ricceri, A further improvement of a minimax theorem of Borenshtein and Shul'man, J. Nonlinear Convex Anal., 2 (2001), 279-283.
55. B. Ricceri, Sublevel sets and global minima of coercive functionals and local minima of their perturbations, J. Nonlinear Convex Anal., 5 (2004), 157-168.
56. B. Ricceri, A general multiplicity theorem for certain nonlinear equations in Hilbert spaces, Proc. Amer. Math. Soc, 133 (2005), 3255-3261.
57. B. Ricceri, A multiplicity theorem for the Neumann problem, Proc. Amer. Math. Soc., 134 (2006), 1117-1124.
58. B. Ricceri, Minimax theorems for limits of parametrized functions having at most one local minimum lying in a certain set, Topology Appl., 153 (2006), 3308-3312.
59. B. Ricceri, Uniqueness properties of functionals with Lipschitzian derivative, Port. Math. (N. S.), 63 1 (2006), 393-400.
60. B. Ricceri, On the existence and uniqueness of minima and maxima on spheres of the integral functional of the calculus of variations, J. Math. Anal. Appl., 324 (2006), 1282-1287.
61. 0 (0, l), J. Nonlinear Convex Anal., 7 (2006), 525-528.
62. 2 functional around non-critical points is well-posed, Proc. Amer. Math. Soc, 135 (2007), 2187-2191.
63. B. Ricceri, Recent advances in minimax theory and applications, in Pareto Optimality, Game Theory and Equilibria, Springer, to appear.
64. B. Ricceri, On the singular set of certain potential operators in Hilbert spaces, in Progr. Nonlinear Differential Equations Appl., 75, 377-391, Birkhäuser, 2007.
65. B. Ricceri, Well-posedness of constrained minimization problems via saddle-points, J. Global Optim., 40 (2008), 389-397.
66. R. S. Sadyrkhanov, On infinite dimensional features of proper and closed mappings, Proc. Amer. Math. Soc, 98 (1986), 643-648.
67. J. Saint Raymond, Connexité des sous-niveaux des fonctionnelles intégrates, Rend. Circ. Mat. Palermo, 44 (1995), 162-168.
68. J. Saint Raymond, On a minimax theorem, Arch. Math. (Basel), 74 (2000), 432-437.
69. R. Salvati, Multiple solutions for a mixed boundary value problem, Math. Sci. Res. J., 7 (2003), 275-283.
70. L. K. Shilgba, Multiplicity of periodic solutions for a boundary eigenvalue problem, Dyn. Syst., 20 (2005), 223-232.
71. Y. Tian and W. Ge, Multiple solutions for certain nonlinear second-order systems, J. Appl. Math. Comput., 25 (2007), 353-361. (Pubitemid 47393351)
72. I. G. Tsar'kov, Nonunique solvability of certain differential equations and their connection with geometric approximation theory, Math. Notes, 75 (2004), 259-271.
73. X. Wu, Saddle-point characterization and multiplicity of periodic solutions of nonautonomous second-order systems, Nonlinear Anal., 58 (2004), 899-907.
74. G. Zhang, W. Zhang and S. Liu, Multiplicity result for a discrete eigenvalue problem with discontinuous nonlinearities, J. Math. Anal. Appl., 328 (2007), 362-375.
75. N, Nonlinear Anal., 67 (2007), 2232-2239.