Nonself KKM Maps and Corresponding Theorems in Hadamard Manifolds

Parin Chaipunya, Poom Kumam

Abstract

In this paper, we consider the KKM maps defined for a nonself map and the correlated intersection theorems in Hadamard manifolds. We also study some applications of the intersection results. Our outputs improved the results of Raj and Somasundaram [17, V. Sankar Raj and S. Somasundaram, KKM-type theorems for best proximity points, Appl. Math. Lett., 25(3): 496–499, 2012.].


Keywords

KKM Maps; Hadamard manifolds; generalized equilibrium problems; best proximity points

Subject classification

53C22; 53C25; 47H04

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References

M. Bianchi and S. Schaible, Generalized monotone bifunctions and equilibrium problems, J. Optimization Theory Appl. 90, no. 1 (1996), 31-43. https://doi.org/10.1007/BF02192244

G. Bigi, A. Capata and G. Kassay, Existence results for strong vector equilibrium problems and their applications, Optimization 61, no. 4-6 (2012), 567-583. https://doi.org/10.1080/02331934.2010.528761

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, The Mathematics Student 63, no. 1-4 (1994), 123-145.

F. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Mathematische Annalen 177, no. 4 (1968), 283-301. https://doi.org/10.1007/BF01350721

M. Castellani and M. Giuli, On equivalent equilibrium problems, J. Optim. Theory Appl. 147, no. 1 (2010), 157-168. https://doi.org/10.1007/s10957-010-9703-4

V. Colao, G. López, G. Marino and V. Martín-Márquez, Equilibrium problems in Hadamard manifolds, J. Math. Anal. Appl. 388, no. 1 (2012), 61-77. https://doi.org/10.1016/j.jmaa.2011.11.001

K. Fan, A generalization of Tychonoff's fixed point theorem, Math. Ann. 142 (1961), 305-310. https://doi.org/10.1007/BF01353421

K. Fan, A minimax inequality and applications, Inequalities, III (Proceedings of the Third Symposium, University of California, Los Angeles, CA, 1969; dedicated to the memory of Theodore S. Motzkin), pages 103-113, 1972.

A. N. Iusem and W. Sosa, New existence results for equilibrium problems, Nonlinear Anal. Theory Methods Appl. 52, no. 2 (2003), 621-635. https://doi.org/10.1016/S0362-546X(02)00154-2

L.-J. Lin and H. I. Chen, Coincidence theorems for families of multimaps and their applications to equilibrium problems, Abstr. Appl. Anal. 2003, no. 5 (2003), 295-309. https://doi.org/10.1155/S1085337503210034

S. Németh, Variational inequalities on Hadamard manifolds, Nonlinear Anal. Theory Methods Appl. 52, no. 5 (2003), 1491-1498. https://doi.org/10.1016/S0362-546X(02)00266-3

C. P. Niculescu and I. Roventa, Fan's inequality in geodesic spaces, Applied Mathematics Letters 22, no. 10 (2009), 1529-1533. https://doi.org/10.1016/j.aml.2009.03.020

W. Oettli, A remark on vector-valued equilibria and generalized monotonicity, Acta Math. Vietnam. 22, no. 1 (1997), 213-221.

A. Papadopoulos, Metric Spaces, Convexity and Nonpositive Curvature, IRMA lectures in mathematics and theoretical physics, European Mathematical Society, 2005. https://doi.org/10.4171/010

S. Park, Some coincidence theorems on acyclic multifunctions and applications to kkm theory, Fixed Point Theory and Applications (1992), pp. 248-277.

S. Park, Ninety years of the brouwer fixed point theorem, Vietnam J. Math. 27 (1997), 187-222.

V. Sankar Raj and S. Somasundaram, KKM-type theorems for best proximity points, Appl. Math. Lett. 25, no. 3 (2012), 496-499. https://doi.org/10.1016/j.aml.2011.09.044

K.-T. Sturm, Probability measures on metric spaces of nonpositive curvature, In: Heat kernels and analysis on manifolds, graphs, and metric spaces. Lecture notes from a quarter program on heat kernels, random walks, and analysis on manifolds and graphs, April 16-July 13, 2002, Paris, France, pp. 357-390. Providence, RI: American Mathematical Society (AMS), 2003. https://doi.org/10.1090/conm/338/06080

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