Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ON SOLVABILITY OF GENERALIZED NONLINEAR VARIATIONAL-LIKE INEQUALITIES

  • Zhang, Lili (Department of Applied Mathematics Dalian University of Technology) ;
  • Liu, Zeqing (Department of Mathematics Liaoning Normal University) ;
  • Kang, Shin-Min (Department of Mathematics and Research Institute of Natural Science Gyeongsang National University)
  • Published : 2008.01.31
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Abstract

In this paper, we introduce and study a new class of generalized nonlinear variational-like inequalities. By employing the auxiliary principle technique we suggest an iterative algorithm to compute approximate solutions of the generalized nonlinear variational-like inequalities. We discuss the convergence of the iterative sequences generated by the algorithm in Banach spaces and prove the existence of solutions and convergence of the algorithm for the generalized nonlinear variational-like inequalities in Hilbert spaces, respectively. Our results extend, improve and unify several known results due to Ding, Liu et al, and Zeng, and others.

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References

  1. Q. H. Ansari and J. C. Yao, Iterative schemes for solving mixed variational-like inequalities, J. Optim. Theory Appl. 108 (2001), no. 3, 527-541 https://doi.org/10.1023/A:1017531323904
  2. M. S. R. Chowdhury and K. K. Tan, Generalized variational inequalities for quasimonotone operators and applications, Bull. Polish Acad. Sci. Math. 45 (1997), no. 1, 25-54
  3. X. P. Ding, General algorithm of solutions for nonlinear variational inequalities in Banach space, Comput. Math. Appl. 34 (1997), no. 9, 131-137 https://doi.org/10.1016/S0898-1221(97)00194-6
  4. X. P. Ding, Generalized quasi-variational-like inclusions with nonconvex functionals, Appl. Math. Comput. 122 (2001), no. 3, 267-282 https://doi.org/10.1016/S0096-3003(00)00027-8
  5. X. P. Ding, Existence and algorithm of solutions for nonlinear mixed variational-like inequalities in Banach spaces, J. Comput. Appl. Math. 157 (2003), no. 2, 419-434 https://doi.org/10.1016/S0377-0427(03)00421-7
  6. X. P. Ding and E. Tarafdar, Existence and uniqueness of solutions for a general nonlinear variational inequality, Appl. Math. Lett. 8 (1995), no. 1, 31-36 https://doi.org/10.1016/0893-9659(94)00106-M
  7. X. P. Ding and J. C. Yao, Existence and algorithm of solutions for mixed quasivariational-like inclusions in Banach spaces, Comput. Math. Appl. 49 (2005), no. 5-6, 857-869 https://doi.org/10.1016/j.camwa.2004.05.013
  8. R. Glowinski, J. Lions, and R. Tremolieres, Numerical Analysis of variational inequalities, North-Holland, Amsterdam, 1987
  9. Z. Liu, Z. An, S. M. Kang, and J. S. Ume, Convergence and stability of the three-step iterative schemes for a class of general quasivariational-like inequalities, Int. J. Math. Math. Sci. 70 (2004), no. 69-72, 3849-3857
  10. Z. Liu, J. H. Sun, S. H. Shim, and S. M. Kang, On solvability of general nonlinear variational-like inequalities in reflexive Banach spaces, Int. J. Math. Math. Sci. (2005), no. 9, 1415-1420 https://doi.org/10.1155/IJMMS.2005.1415
  11. Z. Liu, J. S. Ume, and S. M. Kang, Generalized nonlinear variational-like inequalities in reflexive Banach spaces, J. Optim. Theory Appl. 126 (2005), no. 1, 157-174 https://doi.org/10.1007/s10957-005-2666-1
  12. A. H. Siddiqi and Q. H. Ansani, An algorithm for a class of quasivariational inequalities, J. Math. Anal. Appl. 145 (1990), no. 2, 413-418 https://doi.org/10.1016/0022-247X(90)90409-9
  13. Q. H. Wu, Y. Hao, Z. Liu, and S. M. Kang, Sensitivity analysis of solutions for parametric general quasivariational-like inequalities, Int. J. Pure Appl. Math. 21 (2005), no. 1, 121-130
  14. L. C. Zeng, Iterative algorithm for finding approximate solutions of a class of mixed variational-like inequalities, Acta Math. Appl. Sin. Engl. Ser. 20 (2004), no. 3, 477-486 https://doi.org/10.1007/s10255-004-0185-8