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Existence of solutions to a generalized system. (English) Zbl 1178.49009
Summary: In this paper, we consider a generalized system in real Banach spaces. Using Brouwer’s fixed-point theorem, we establish some existence theorems for generalized system without monotonicity. Further, we extend the concept of $C$-strong pseudomonotonicity for a bifunction and extend Minty’s lemma for a generalized system. Furthermore, using Minty’s lemma and KKM-Fan lemma, we establish an existence theorem for a generalized system with monotonicity in real reflexive Banach spaces.

49J40Variational methods including variational inequalities
47N10Applications of operator theory in optimization, convex analysis, programming, economics
Full Text: DOI
[1] Chen, G.-Y., Hou, S.H.: Existence of solutions for vector variational inequalities. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria, Mathematical Theories. Nonconvex Optimization and Applications, vol. 38, pp. 73--86. Kluwer Academic, Dordrecht (2000) · Zbl 1012.49007
[2] Fang, Y.-P., Huang, N.-J.: Strong vector variational inequalities in Banach spaces. Appl. Math. Lett. 19, 362--368 (2006) · Zbl 1138.49300 · doi:10.1016/j.aml.2005.06.008
[3] Giannessi, F.: Theorem of alternative, quadratic programs and complementarity problems. In: Cottle, R.W., Giannessi, F., Lions, J.L. (eds.) Variational Inequalities and Complementarity Problems, pp. 151--186. Wiley, New York (1980) · Zbl 0484.90081
[4] Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123--145 (1994) · Zbl 0888.49007
[5] Rouhani, B.D., Tarafdar, E., Watson, P.J.: Existence of solutions to some equilibrium problems. J. Optim. Theory Appl. 126, 97--107 (2005) · Zbl 1093.90083 · doi:10.1007/s10957-005-2660-7
[6] Brouwer, L.E.J.: Zur invarianz des n-dimensional gebietes. Math. Ann. 72, 55--56 (1912) · Zbl 43.0479.03 · doi:10.1007/BF01456889
[7] Fan, K.: A generalization of Tychonoff’s fixed-point theorem. Math. Ann. 142, 305--310 (1961) · Zbl 0093.36701 · doi:10.1007/BF01353421