Abstract
In this paper we study the connection between the metric projection operator P K : B → K, where B is a reflexive Banach space with dual space B* and K is a non–empty closed convex subset of B, and the generalized projection operators ∏ K : B → K and π K : B* → K. We also present some results in non–reflexive Banach spaces.
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Goebel, K., Reich, S.: Uniformly Convex Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekker, Inc., 1984
Johnson, G. A.: Nonconvex set which has the unique nearest point property. J. Approx. Theory, 51, 289–332 (1987)
Li, J. L.: Characteristics of the metric projection operator in Banach spaces and its applications. to appear
Li, J. L., Rhoades, B.: An approximation of solutions of variational inequalities in Banach spaces. to appear
Vainberg, M. M.: Variational Methods and Method of Monotone Operators, Wiley, New York, 1973
Xu, Z. B., Roach, G. F.: Charcteristic inequalities of uniformly convex and uniformly smooth Banach spaces. J. Math. Anal. Appl., 157, 189–210 (1991)
Xu, Z. B., Roach, G. F.: On the uniform continuity of metric projections in Banach spaces. Approx. Theory and Its Appl., 8(3), 11–20 (1992)
Isac, G.: Complementarity problem, Lecture Notes in Math., 1528, Spring–Verlag, Berlin, Heidelberg, New York, 1992
Isac, G., Sehgal, V. M., Singh, S. P.: An alternate version of a variational inequality. Indian J. of Math., 41(1), 25–31 (1999)
Li, J. L.: On the existence of solutions of variational inequalities in Banach spaces. J. Math. Anal. Appl., 295, 115–126 (2004)
Singh, S. P.: Ky Fan’s best approximation theorems, Proceeding of the National Academy of Sciences, India, LXVII, Part I, 1997
Tan, K. K., Wu, Z., Yuan, X. Z.: Equilibrium existence theorems with closed preferences. To appear
Wen, S., Cao, Z. J.: The generalized decomposition theorem in Banach spaces and its applications. Journal of Approximation Theory, 129(2), 167–181 (2004)
Takahashi, W.: Nonlinear Functional Analysis, Yokohama Publishers, 2000
Alber, Ya.: Generalized projection operators in Banach spaces: properties and applications, Functional Differential Equations, Proceedings of the Israel Seminar in Ariel, 1, 1–21, 1994
Alber, Ya.: Metric and generalized projection operators in Banach spaces: properties and applications, In: “Theory and Applications of Nonlinear Operators of Accretive and Monotone Type” (A. Kartsatos, Ed.), 15–50, Marcel Dekker, Inc., 1996
Alber, Ya., Iusem, A., Solodov, M.: Minimization of nonsmooth convex functionals in Banach spaces. Journal of Convex Analysis, 4(2), 235–254 (1997)
Alber, Ya., Burachik, R., Iusem, A.: A proximal point method for nonsmooth convex optimization problems in Banach spaces. Abstract and Applied Analysis, 2(1–2), 97–120 (1997)
Alber, Ya.: Decomposition theorem in Banach spaces. Fields Institute Communications, 25, 77–93 (2000)
Alber, Ya., Guerre–Delabriere, S.: On the projection methods for fixed point problems. Analysis, 21, 17–39 (2001)
Alber, Ya., Iusem, A.: Extension of subgradient techniques for nonsmooth optimization in Banach spaces. Set–Valued Analysis, 9(4), 315–335 (2001)
Alber, Ya.: Proximal projection method for variational inequalities and Cesro averaged approximations. Computers and Mathematics with Applications, 43, 1107–1124 (2002)
Alber, Ya.: On average convergence of the iterative projection methods. Taiwanese Journal of Mathematics, 6(3), 323–341 (2002)
Alber, Ya., Nashed, M.: Iterative–projection regularization of unstable variational inequalities. Analysis, 24, 19–39 (2004)
Fan, K.: A generalization of Tychonoff’s fixed point theorem. Math. Ann., 142, 305–310 (1961)
Li, J. L.: The generalized projection operator on reflexive Banach spaces and its applications. J. Math. Anal. Appl., 306, 55–71 (2005)
Kazmi, K. R.: Mann and Ishikawa type perturbed iterative algorithms for generalized quasivariational inclusions. J. Math. Anal. Appl., 209, 572–584 (1997)
Lions, J. L.: Quelques methodes de resolution des problemes aux limites non lineaires, Dunod Gauthier– Villars, Paris, 1969
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Alber, Y., Li, J.L. The Connection Between the Metric and Generalized Projection Operators in Banach Spaces. Acta Math Sinica 23, 1109–1120 (2007). https://doi.org/10.1007/s10114-005-0718-y
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DOI: https://doi.org/10.1007/s10114-005-0718-y
Keywords
- Banach spaces
- normalized duality mappings
- metric and generalized projection operators
- variational inequalities
- minimization problems
- closed and convex subsets and cones