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A NEW SYSTEM OF GENERALIZED NONLINEAR MIXED QUASIVARIATIONAL INEQUALITIES AND ITERATIVE ALGORITHMS IN HILBERT SPACES

Kim, Jong-Kyu   (DEPARTMENT OF MATHEMATICS, EDUCATION KYUNGNAM UNIVERSITYUU0000075  ); Kim, Kyung-Soo   (DEPARTMENT OF MATHEMATICS KYUNGNAM UNIVERSITYUU0000075  );
  • 초록

    We introduce a new system of generalized nonlinear mixed quasivariational inequalities and prove the existence and uniqueness of the solution for the system in Hilbert spaces. The main result of this paper is an extension and improvement of the well-known corresponding results in Kim-Kim [16], Noor [21]-[23] and Verma [24]-[26].


  • 주제어

    a system of generalized nonlinear mixed quasivariational inequalities .   iterative sequence with errors .   algorithm.  

  • 참고문헌 (26)

    1. C. Baiocchi and A. Caopelo, Variational and quasivariational inequalities, John Wiley & Sons, Inc., New York, 1984 
    2. A. Bensounssan and J. L. Lions, Impulse Control and Quasivariational Inequalities, Gauthiers-Villers, Bordas, Paris, 1984 
    3. H. Brezis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London, 1973 
    4. S. S. Chang, J. K. Kim, and K. H. Kim, On the existence and iterative approximation problems of solutions for set-valued variational inclusions in Banach spaces, J. Math. Anal. Appl. 268 (2002), no. 1, 89-108 
    5. S. S. Chang, J. K. Kim, and Y. M. Nam, Multivalued quasi-variational inclusions and multivalued accretive equations, Comput. Math. Appl. 48 (2004), no. 10-11, 1441-1452 
    6. Y. P. Fang, N. J. Huang, and J. K. Kim, A system of multi-valued generalized order complementarity problems in ordered metric spaces, Z. Anal. Anwendungen 22 (2003), no. 4, 779-788 
    7. Y. P. Fang, N. J. Huang, and J. K. Kim, Existence results for systems of vector equilibrium problems, J. Global Optim. 35 (2006), no. 1, 71-83 
    8. F. Giannessi and A. Maugeri, Variational Inequalities and Network Equilibrium Problems, Plenum Press, New York, 1995 
    9. N. J. Huang, Generalized nonlinear variational inclusions with noncompact valued mapping, Appl. Math. Lett. 9 (1996), no. 3, 25-29 
    10. N. J. Huang, On the generalized implicit quasivariational inequalities, J. Math. Anal. Appl. 216 (1997), no. 1, 197-210 
    11. N. J. Huang, Mann and Ishikawa type perturbed iterative algorithms for generalized nonlinear implicit quasivariational inclusions, Comput. Math. Appl. 35 (1998), no. 10, 1-7 
    12. N. J. Huang, A new completely general class of variational inclusions with noncompact valued mappings, Comput. Math. Appl. 35 (1998), no. 10, 9-14 
    13. N. J. Huang, Y. P. Liu, Y. Y. Tang, and M. R. Bai, The generalized set-valued strongly nonlinear implicit variational inequalities, Comput. Math. Appl. 37 (1999), no. 10, 29-36 
    14. N. J. Huang, M. R. Bai, Y. J. Cho, and S. M. Kang, Generalized nonlinear mixed quasi-variational inequalities, Comput. Math. Appl. 40 (2000), no. 2-3, 205-215 
    15. J. K. Kim and D. S. Kim, A new system of generalized nonlinear mixed variational inequalities in Hilbert spaces, J. Convex Anal. 11 (2004), no. 1, 235-243 
    16. J. K. Kim, Y. M. Nam, N. J. Huang, and H. Dong, On generalized vector variational inequalities with set-valued mappings, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 13 (2006), no. 2, 221-230 
    17. J. K. Kim, K. H. Kim, and K. S. Kim, Set-valued quasivariational inclusions and implicit resolvent equations in Banach spaces, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 11 (2004), no. 4, 491-502 
    18. J. Li, N. J. Huang, and J. K. Kim, On implicit vector equilibrium problems, J. Math. Anal. Appl. 283 (2003), no. 2, 501-512 
    19. M. A. Noor, Set-valued quasivariational inclusions, Korean J. Comput. Appl. Math. 7 (2000), no. 1, 101-113 
    20. M. A. Noor, K. I. Noor, and Th. M. Rassias, Set-valued resolvent equations and mixed variational inequalities, J. Math. Anal. Appl. 220 (1998), no. 2, 741-759 
    21. M. A. Noor, K. I. Noor, and Th. M. Rassias, Some aspects of variational inequalities, J. Comput. Appl. Math. 47 (1993), no. 3, 285-312 
    22. R. U. Verma, On a new system of nonlinear variational inequalities and associated iterative algorithms, Math. Sci. Res. Hot-Line 3 (1999), no. 8, 65-68 
    23. R. U. Verma, Iterative algorithms and a new system of nonlinear quasivariational inequalities, Adv. Nonlinear Var. Inequal. 4 (2001), no. 1, 117-124 
    24. R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM Publication Co., Philadelphia, 1989 
    25. H. Y. Lan, J. K. Kim, and N. J. Huang, On the generalized nonlinear quasi-variational inclusions involving non-monotone set-valued mappings, Nonlinear Funct. Anal. and Appl. 9 (2004), no. 3, 451-465 
    26. R. U. Verma, Projection methods, Algorithms and a new system of nonlinear variational inequalities, Compu. Math. Appl. 41 (2001), no. 7-8, 1025-1031 

 저자의 다른 논문

  • Kim, Jong-Kyu (31)

    1. 1985 "ERGODIC THEOREMS FOR AN ASYMPTOTICALLY NONEXPANSIVE SEMI-GROUP IN HILBERT SPACES" The Pusan Kyongnam mathematical journal 1 (): 47~54    
    2. 1996 "NONLINEAR ERGODIC THEOREMS FOR ALMOST-ORBITS OF ASYMPTOTICALLY NONEXPANSIVE SEMIGROUPS IN BANACH SPACES" The Pusan Kyongnam mathematical journal 12 (2): 191~202    
    3. 1997 "ASYMPTOTIC BEHAVIOR OF AN ASYMPTOTICALLY NONEXPANSIVE SEMIGROUP IN BANACH SPACES WITH OPIAL'S CONDITION" Communications of the Korean Mathematical Society = 대한수학회논문집 12 (2): 237~250    
    4. 1998 "DIFFERENTIAL EQUATIONS ON CLOSED SUBSETS OF A PROBABILISTIC NORMED SPACE" Korean journal of computational & applied mathematics 5 (1): 223~233    
    5. 1999 "EXISTENCE OF ERGODIC RETRACTIONS FOR ASYMPTOTICALLY NONEXPANSIVE SEMIGROUPS WITH NONCONVEX AND NONCLOSED DOMAINS" East Asian mathematical journal 15 (2): 301~312    
    6. 2001 "NONLINEAR ERGODIC THEOREMS FOR ALMOST-ORBITS OF ASYMPTOTICALLY NONEXPANSIVE TYPE MAPPINGS IN BANACH SPACES" Bulletin of the Korean Mathematical Society = 대한수학회보 38 (3): 587~603    
    7. 2001 "NECESSARY AND SUFFICIENT CONDITIONS FOR THE CONVERGENCE OF MANN ITERATIVE SEQUENCES WITH ERRORS IN BANACH SPACES" The Korean journal of computational & applied mathematics, 한국전산응용수학술지 Series A 8 (3): 1027~1033    
    8. 2002 "CONVERGENCE THEOREMS AND STABILITY PROBLEMS OF THE MODIFIED ISHIKAWA ITERATIVE SEQUENCES FOR STRICTLY SUCCESSIVELY HEMICONTRACTIVE MAPPINGS" Bulletin of the Korean Mathematical Society = 대한수학회보 39 (3): 455~469    
    9. 2003 "NECESSARY AND SUFFICIENT CONDITIONS FOR CONVERGENCE OF ISHIKAWA ITERATIVE SCHEMES WITH ERRORS TO φ-HEMICONTRACTIVE MAPPINGS" Communications of the Korean Mathematical Society = 대한수학회논문집 18 (2): 251~261    
    10. 2004 "ALMOST STABILITY OF ISHIKAWA ITERATIVE SCHEMES WITH ERRORS FOR φ-STRONGLY QUASI-ACCRETIVE AND φ-HEMICONTRACTIVE OPERATORS" Communications of the Korean Mathematical Society = 대한수학회논문집 19 (2): 267~281    
  • Kim, Kyung-Soo (3)

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