Bishop–Phelps cones given by an equation in Banach spaces

References

• Bishop E, Phelps RR. A proof that every Banach space is subreflexive. Bull Am Math Soc. 1961;67:97–98.
   Web of Science ®Google Scholar
• Phelps RR. Support cones in Banach spaces and their applications. Adv Math. 1974; 13:1–19.
   Web of Science ®Google Scholar
• Bednarczuk EM. Bishop–Phelps cones and convexity: applications to stability of vector optimization problems. INRIA Rapport de recherche 2806; 1996.
   Google Scholar
• Hyers DH, Isac G, Rassias T. Topics in nonlinear analysis and applications. River Edge (NJ): World Scientific Publishing Co., Inc.; 1997.
   Google Scholar
• Jahn J. A generalization of a theorem of Arrow-Barankin-Blackwell. SIAM J Control Optim. 1988;26:999–1005.
   Web of Science ®Google Scholar
• Jahn J. Bishop–Phelps cones in optimization. Int J Optim: Theory Methods Appl. 2009; 1:123–139.
   Google Scholar
• Kasimbeyli R. A nonlinear cone separation theorem and scalarization in nonconvex vector optimization. SIAM J Optim. 2010;20:1591–1619.
   Web of Science ®Google Scholar
• Kasimbeyli N, Kasimbeyli R. A representation theorem for Bishop–Phelps cones. Pac J Optim. 2017;13:55–74.
   Web of Science ®Google Scholar
• Penot J-P. Variational analysis for the consumer theory. J Optim Theory Appl. 2013; 159:769–794.
   Web of Science ®Google Scholar
• Ng KF, Zheng XY. Existence of efficient points in vector optimization and generalized Bishop–Phelps theorem. J Optim Theory Appl. 2002;115:29–47.
   Web of Science ®Google Scholar
• Ng KF, Zheng XY. On the density of positive proper efficient points in a normed space. Optim Theory Appl. 2003;119:105–122.
   Web of Science ®Google Scholar
• Petschke M. On a theorem of arrow, Barankin and Blackwell. SIAM J Control Optim. 1990;28:395–401.
   Web of Science ®Google Scholar
• Song W. Characterizations of some remarkable classes of cones. J Math Anal Appl. 2003;279:308–316.
   Web of Science ®Google Scholar
• Alizadeh F, Goldfarb D. Second-order cone programming. Math Program Ser B. 2003; 95:3–51.
   Web of Science ®Google Scholar
• Gajardo P, Seeger A. Equilibrium problems involving the Lorentz cone. J Global Optim. 2013;58:321–340.
   Web of Science ®Google Scholar
• Ha TXD, Jahn J. Properties of Bishop–Phelps cones. J Nonlinear Convex Anal. 2017; 18:415–429.
   Web of Science ®Google Scholar
• James RC. Characterizations of reflexivity. Studia Math. 1963/64;23:205–216.
   Google Scholar
• Krasnosel'skii MA. Positive solutions of operator equations. Groningen (The Netherlands): P. Noordhoff Ltd.; 1964. Translated from Russian.
   Google Scholar
• Bakhtin IA. Positive cones in Banach spaces. Voronezh (Russia): Voronezh State Pedagogical Institute; 1975. Russian.
   Google Scholar
• Clarkson JA. Uniformly convex spaces. Trans Am Math Soc. 1936;40:396–414.
   Google Scholar
• Day MM. Strict convexity and smoothness of normed spaces. Trans Am Math Soc. 1955;78:516–528.
   Google Scholar
• Day MM. Reflexive Banach spaces not isomorphic to uniformly convex spaces. Bull Am Math Soc. 1941;47:313–317.
   Google Scholar
• James RC. Orthogonality and linear functionals in normed linear spaces. Trans Am Math Soc. 1947;61:265–292.
   Web of Science ®Google Scholar
• Petryshyn WV. A characterization of strict convexity of Banach spaces and other uses of duality mappings. J Funct Anal. 1970;6:282–291.
   Google Scholar
• Raj VS, Eldred AA. A characterization of strictly convex spaces and applications. J Optim Theory Appl. 2014;160:703–710.
   Web of Science ®Google Scholar
• Smul'jan V. On some geometrical properties of the unit sphere in spaces of the type (B). Mat Sb. 1939;48:90–94.
   Google Scholar
• Taylor AE. The extension of linear functionals. Duke Math J. 1939;5:538–547.
   Google Scholar
• Dunford N, Schwartz JT. Linear operators. 1: general theory. New York (NY): Interscience; 1958. (Pure and Applied Mathematics; vol. 7).
   Google Scholar
• Rudin W. Functional analysis. New York (NY): McGraw-Hill; 1991.
   Google Scholar
• Phelps RR. Uniqueness of Hahn–Banach extensions and unique best approximation. Trans Am Math Soc. 1960;95:238–255.
   Google Scholar
• Chiang Y, Pan S, Chen J-S. A merit function method for infinite-dimensional SOCCPs. J Math Anal Appl. 2011;383:159–178.
   Web of Science ®Google Scholar
• Yang C-Y, Chang Y-L, Chen J-S. Analysis of nonsmooth vector-valued functions associated with infinite-dimensional second-order cones. Nonlinear Anal. 2011;74:5766–5783.
   Web of Science ®Google Scholar
• LaSalle JP. The ‘bang-bang’ principle. IFAC Proc Vol. 1960;1(1):503–507.
   Google Scholar
• Schmidt EJP. The ‘bang-bang’ principle for the time-optimal problem in boundary control of the heat equation. SIAM J Control Optim. 1980;18:101–107.
   Web of Science ®Google Scholar
• Chui CK, Deutsch F, Ward JD. Constrained best approximation in Hilbert space. Constr Approx. 1990;6:35–64.
   Web of Science ®Google Scholar