Abstract
In this paper, we first prove a new result on quasi-variational inequality problems; then, we use it to show the existence of solutions to quasi-intersection problems. In particular, we establish several results on the existence of solutions to generalized quasi-equilibrium problems and fixed point problems concerning separately scalar weakly lower and upper semi-continuous mappings. The results in this paper generalize some well-known existence theorems from the literature.
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References
Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis, 3rd edn. Springer, Berlin (2006)
Browder, F.E.: The fixed point theory of multi-valued mappings in topological vector spaces. Math. Ann. 177, 283–301 (1968)
Duong, T.T.T.: Mixed generalized quai-equilibrium problems. J. Glob. Optim. 56(2), 647–667 (2013)
Duong, T.T.T., Tan, N.X.: On the existence of solutions to generalized quasi-equilibrium problems of type I and related problemms. Ad. Non. Var. Inequalities 13(1), 29–47 (2010)
Duong, T.T.T., Tan, N.X.: On the existence of solutions to generalized quasi-equilibrium. J. Glob. Optim. 52(4), 711–728 (2012)
Fan, K.: Fixed-point and minimax theorems in locally convex topological linear spaces. Proc. Nat. Acad. Sci. U.S.A. 38, 121–126 (1952)
Heyde, F., Schrage, C.: Continuity concepts for set-valued functions and a fundamental duality formula for set-valued optimization. J. Math. Anal. Appl. 397(2), 772–784 (2013)
Lin, L.J., Tu, C.I.: The studies of systems of variational inclusions problems and variational disclusions problems with applications. Nonlinear Anal. 69, 1981–1998 (2008)
Li, S.J., Chen, C.R.: Stability of weak vector variational inequality. Non. Anal. 70, 1528–1535 (2009)
Sion, M.: On general minimax theorems. Pac. J. Math. 8(1), 171–176 (1958)
Tan, N.X.: On the existence of solutions of quasi-variational inclusion problems. J. Optim. Theory Appl. 123, 619–638 (2004)
Tan, N.X., Hoa, N.Q.: Quasi-equilibrium problems and fixed point theorems of l.s.c mappings. Adv. Non. Var. Inequalities 19(2), 52–63 (2016)
Tan, N.X.: Quasi-equilibrium problems and fixed point theorems of separately l.s.c and u.s.c mappings. Numer. Funct. Anal. Optim. 39, 233–255 (2018)
Acknowledgments
The second author of the paper is a member of the project “Joint study in Analysis and Geometry” conducted at the International Center for Research and Training in Mathematics, Institute of Mathematics, VAST, ICRTM01-2019.05; he would like to thank other members of the project for useful discussions. This work was supported by the Vietnam Academy of Science and Technologies under the code NVCC1.14/19-19.
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Dedicated to Hoang Tuy on the occasion of his 90th birthday
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Duong, T.T.T., Tan, N.X. Quasi-Intersection Problems and Fixed Point Theorems Concerning Separately Scalar Weakly l.s.c and u.s.c Mappings. Acta Math Vietnam 45, 311–328 (2020). https://doi.org/10.1007/s40306-019-00344-3
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DOI: https://doi.org/10.1007/s40306-019-00344-3
Keywords
- Quasi-intersection problems
- Separately scalar weakly upper and separately scalar weakly lower semi-continuous mappings
- Fixed point theorems
- Quasi-equilibrium problems