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\(F\)-implicit complementarity problems in Banach spaces. (English) Zbl 1135.90412

Summary: The \(F\)-implicit complementarity problem (F-ICP) and \(F\)-implicit variational inequality problem (F-IVIP) are introduced and studied. The equivalence between (F-ICP) and (F-IVIP) is presented under certain assumptions. Furthermore, we derive some new existence theorems of solutions for (F-ICP) and (F-IVIP) by using the Fan-Knaster-Kuratowski-Mazurkiewicz theorem [K. Fan, Math. Ann. 142, 305–310 (1961; Zbl 0093.36701)] and T. C. Lin’s theorem [Bull. Aust. Math. Soc. 34, 107–117 (1986; Zbl 0597.47038)] under some suitable assumptions without the monotonicity.

MSC:

90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
49J40 Variational inequalities
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
90C48 Programming in abstract spaces
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References:

[1] Ahmad, K., Kazmi K. R. and N. Rehman: Fixed-point technique for implicit complementarity problem in Hilbert lattice. J. Optim. Theory Appl. 93 (1997), 67 - 72. · Zbl 0899.90156
[2] Chang, S. S. and N. J. Huang: Generalized strongly nonlinear quasi- complementarity problems in Hilbert spaces. J. Math. Anal. Appl. 158 (1991), 194 - 202. · Zbl 0739.90067
[3] Cottle, R. W.: Nonlinear programs with positively bounded Jacobians. SIAM J. Appl. Math. 14 (1966), 147 - 158. · Zbl 0158.18903
[4] Cottle, R. W., Pang, J. S. and R. E. Stone, The Linear Complementarity Problems. New York: Acad. Press 1992. · Zbl 0757.90078
[5] Fan, K.: A generalization of Tychonoff ’s fixed point theorem. Math. Ann. 142 (1961), 305 - 310. · Zbl 0093.36701
[6] Fang, Y. P. and N. J. Huang: Variational-like inequalities with generalized monotone mappings in Banach spaces. J. Optim. Theory Appl. 118 (2003), 327 - 338. · Zbl 1041.49006
[7] Fang, Y. P., Huang, N. J. and J. K. Kim: A system of multi-valued generalized order complementarity problems in ordered Banach spaces. Z. Anal. Anw. 22 (2003)(4), 779 - 788. N. J. Huang and J. Li · Zbl 1140.90498
[8] Giannessi, F. and A. Maugeri: Variational Inequalities and Network Equilib- rium Problems. New York: Plenum 1995.
[9] Harker, P. T. and J. S. Pang: Finite-dimensional variational inequalities and nonlinear complementarity problems: A survey of theory, algorithms and ap- plications. Math. Program. 48B (1990), 161 - 220. · Zbl 0734.90098
[10] Huang, N. J.: Generalized nonlinear variational inclusions with noncompact valued mapping. Appl. Math. Lett. 9 (1996)(3), 25 - 29. · Zbl 0851.49009
[11] Huang, N. J.: A new completely general class of variational inclusions with noncompact valued mappings. Comput. Math. Appl. 35 (1998)(6), 9 - 14. · Zbl 0999.47056
[12] Huang, N. J. and C. X. Deng: Auxiliary principle and iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities. J. Math. Anal. Appl. 256 (2001), 345 - 359. · Zbl 0972.49008
[13] Huang, N. J., Gao, C. J. and X. P. Huang: Exceptional family of elements and feasibility for nonlinear complementarity problems. J. Global Optim. 25 (2003), 337 - 344.
[14] Huang, N. J., Li, J. and H. B. Thompson: Implicit vector equilibrium problems with applications. Math. Comput. Modelling 37 (2003), 1343 - 1356. · Zbl 1080.90086
[15] Isac, G.: A special variational inequality and the implicit complementarity problem. J. Fac. Sci. Univ. Tokyo 37 (1990), 109 - 127. · Zbl 0702.49008
[16] Isac, G.: Complementarity Problems. Lect. Notes Math. 1528. Berlin: Springer- Verlag 1992. · Zbl 0795.90072
[17] Isac, G.: Topological Methods in Complementarity Theory. Dordrecht: Kluwer Acad. Publ. 2000. · Zbl 0954.90056
[18] Isac, G. and D. Goeleven: The implicit general order complementarity problem, models and iterative methods. Ann. Oper. Res. 44 (1993), 63 - 92. · Zbl 0816.46013
[19] Karamardian, S.: General complementarity problem. J. Optim. Theory Appl. 8 (1971), 161 - 168. · Zbl 0218.90052
[20] Karamardian, S.: Complementarity over cones with monotone and pseu- domonotone maps. J. Optim. Theory Appl. 18 (1976), 445 - 454. · Zbl 0304.49026
[21] Li, J., Huang, N. J. and J. K. Kim: On implicit vector equilibrium problems. J. Math. Anal. Appl. 283 (2003), 501 - 512. · Zbl 1137.90715
[22] Lin, T. C.: Convex sets, fixed points, variational and minimax inequalities. Bull. Austral. Math. Soc. 34 (1986), 107 - 117. · Zbl 0597.47038
[23] Yao, J. C.: Existence of generalized variational inequalities. Oper. Res. Lett. 15 (1994), 35 - 40. · Zbl 0874.49012
[24] Yin, H. Y., Xu, C. X. and Z. X. Zhang: The F -complementarity problem and its equivalence with the least element problem (in Chinese). Acta Math. Scinica 44 (2001), 679 - 686. · Zbl 1022.90034
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