HYBRID-TYPE SET-VALUED VARIATIONAL-LIKE INEQUALITIES IN REFLEXIVE BANACH SPACES

  • Published : 2009.09.30

Abstract

In this paper, we introduce a relaxed hybrid-type$\eta$-f-${\alpha}$-pseudomo-notonicity. By using the KKM-technique, we establish some existence results for set-valued variational-like inequalities with $\eta-f-\alpha$-pseudomonotone, relaxed $\eta-f-\alpha$-pseudomonotone, Fan-KKM Theorem.

Keywords

References

  1. M.R. Bai, S.Z. Zhou and G.Y. Ni, Variational like inequalities with relaxed ${\eta}-{\alpha}$ - pseudomonotone mappings in Banach spaces, Appl. Math. Lett. 19 (2006), 547-554. https://doi.org/10.1016/j.aml.2005.07.010
  2. Y.Q. Chen, On the semimonotone operator theory and applications, J. Math. Anal. Appl. 231 (1999), 177-192. https://doi.org/10.1006/jmaa.1998.6245
  3. R.W. Cottle and J.C. Yao, Pseudomonotone complementarity problems in Hilbert spaces, J. Optim. Theo. Appl. 78 (1992), 281-295.
  4. K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), 519-537. https://doi.org/10.1007/BF01458545
  5. Y.P. Fang and N.J. Huang, Variational like inequalities with generalized monotone mappings in Banach spaces, J. Optim. Theo. & Appl. 118 (2003), 327-338. https://doi.org/10.1023/A:1025499305742
  6. N.El. Farouq, Pseudomonotone variational inequalities: converges of proximal method, J. Optim. Theo. & Appl. 109 (2001), 311-326. https://doi.org/10.1023/A:1017562305308
  7. D. Goeleven and D. Motreanu, Eigenvalue and dynamic problems for variational and hemivariational inequalities, Comm. Appl. Nonlinear Anal. 3 (1996), 1-21.
  8. P. Hartman and G. Stampacchia, On some nonlinear elleptic differential function equations, Acta Math. 115 (1966), 271-310. https://doi.org/10.1007/BF02392210
  9. N.J. Huang, M.R. Bai, Y.J. Cho and S.M. Kang, Generalized nonlinear mixed quasivariational inequalities, Comput. Math. Appl. 40 (2000), 205-215. https://doi.org/10.1016/S0898-1221(00)00154-1
  10. M.K. Kang, N.J. Huang and B.S. Lee, Generalized pseudomonotone set-valued variationallike inequalities, Indian J. Math. 45(3) (2003), 251-264.
  11. S. Karamardian and S. Schaible, Seven kinds of monotone maps, J. Optim. Theo. Appl. 66 (1990), 37-46. https://doi.org/10.1007/BF00940531
  12. S. Karamardian, S. Schaible and J.P. Cronzeix, Characterization of general monotone map, J. Optim. Theo. Appl. 76 (1993), 399-413. https://doi.org/10.1007/BF00939374
  13. D.T. Luc, Existence results for densely pseudomonotone variational inequalities, J. Math. Anal. Appl. 254 (2001), 309-320. https://doi.org/10.1006/jmaa.2000.7279
  14. R.U. Verma, On generalized variational inequalities involving relaxed Lipschitz and relaxed monotone operators, J. Math. Anal. Appl. 118 (2003), 327-338.