January 2019, 6(1): 19-25. doi: 10.3934/jdg.2019002

Local completeness, Pareto efficiency and Mackey Bishop-Phelps cones

Facultad de Economía UASLP, Av. Pintores S7N, San Luis Potosí, CP 78280, México

* Corresponding author: Armando García

Received  August 2018 Revised  November 2018 Published  January 2019

Avoiding usual completeness hipothesis and working on the frame of locally complete spaces some Pareto optimization results are obtained. The Mackey Bishop-Phelps cones are defined and a characterization for the existence of Pareto efficiency respect to these cones is obtained.

Citation: Elvio Accinelli, Armando García. Local completeness, Pareto efficiency and Mackey Bishop-Phelps cones. Journal of Dynamics & Games, 2019, 6 (1) : 19-25. doi: 10.3934/jdg.2019002
References:
[1]

C. D. Aliprantis and R. Tourky, Cones and Duality, Graduate Studies in Mathematics AMS, 2007. doi: 10.1090/gsm/084.

[2]

J. P. Aubin and J. Siegel, Fixed points and satationary points of dissipative multivalued maps, Proceed. Amer. Math. Soc., 78 (1980), 391-398. doi: 10.1090/S0002-9939-1980-0553382-1.

[3]

C. BoschA. García and C. L. García, An extension of Ekeland's variational principle to locally complete spaces, J. Math. Anal. Appl., 328 (2007), 106-108. doi: 10.1016/j.jmaa.2006.05.012.

[4]

C. BoschA. GarcíaC. Gómez and S. Hernández, Equivalents to Ekeland's variational principle in locally complete spaces, Sci. Math. Jpn., 72 (2010), 283-287.

[5]

J. X. Fang, The variational principle and fixed point theorems in certain topological spaces, J. Math. Anal. Appl., 202 (1996), 398-412. doi: 10.1006/jmaa.1996.0323.

[6]

L. Hurwicz, Programing in linear spaces, in: K.J. Arrow, L. Hurwicz, H. Huzawa (Eds.), Studies in Linear and Nonlinear Programming, Stanford Univ. Press, Stanford, California, 1958, 38-102.

[7]

G. Isac, Sur l'existence de l'optimum de Pareto, Riv. Math. Univ. Parma, 9 (1983), 303-325.

[8]

G. Isac, Pareto optimization in infinite dimensional spaces. The importance of nuclear cones, J. Math. Anal. Appl., 182 (1994), 393-404. doi: 10.1006/jmaa.1994.1093.

[9]

G. Isac, The Ekeland's principle and the Pareto $ \epsilon $-efficiency, in: M. Tamiz (Ed.), In Multi-Objective Programming and Goal Programming, in: Lectures Notes in Econom. Math. Systems, Springer-Verlag, 432 (1996), 148-162.

[10]

G. Isac, Ekeland's principle and nuclear cones: A geometrical aspect, Math. Comput. Modelling, 26 (1997), 111-116. doi: 10.1016/S0895-7177(97)00223-9.

[11]

G. Isac, Nuclear cones in product spaces, Pareto efficiency and Ekeland-type variational principles in locally convex spaces, Optimization, 53 (2004), 253-268. doi: 10.1080/02331930410001720923.

[12]

G. Isac and A. O. Bahya, Full nuclear cones associated to normal cone. Application to Pareto efficiency, Appl. Math. Letters, 15 (2002), 633-639. doi: 10.1016/S0893-9659(02)80017-9.

[13]

H. Jarchow, Locally Convex Spaces, B.G. Teubner, Stuttgart, 1981.

[14]

H. W. Kuhn and A. W. Tucker, Nonlinear programming, in: J. Neymann (Ed), Proceeding of the 2nd Berkeley Simposium on Mathematical Statistics and Probability, University of California Press, Berkeley, CA, 1950,481-492.

[15]

A. Muntean, Some fixed point theorems for commuting multivalued operators, Seminar on fixed point theory Cluj-Napoca, 2 (2001), 71-79.

[16]

A. L. Peressini, Ordered Topological Vector Spaces, Harper and Raw, 1967.

[17]

P. Pérez-Carreras and J. Bonet, Barreled Locally Convex Spaces, North-Holland, Amsterdam, 1987.

[18]

A. Petrusel and I. A. Rus, An abstract point of view on iterative approximation schemes of fixed points for multivalued operators, J. Nonlinear Sci. Appl., 6 (2013), 97-107. doi: 10.22436/jnsa.006.02.05.

[19]

A. PetruselI. A. Rus and J. C. Yao, Well-posedness in the generalized sense of the fixed point problems for multivalued operators, Taiwanese J. Math., 11 (2007), 903-914. doi: 10.11650/twjm/1500404764.

[20]

M. Petschke, On a theorem of Arrow, Barankin and Blackwell, SIAM J. Control Opt., 28 (1990), 395-401. doi: 10.1137/0328021.

[21]

P. R. Phelps, Convex Functions, Monotone Operators and Differentiability, Second edition. Lecture Notes in Mathematics, 1364. Springer-Verlag, Berlin, 1993.

[22]

J. H. Qiu, Ekeland's variational principle in locally complete spaces, Math. Nach., 257 (2003), 55-58. doi: 10.1002/mana.200310077.

[23]

J. H. Qiu, Local completeness, drop theorem and Ekeland's variational principle, J. Math. Anal. Appl., 311 (2005), 23-39. doi: 10.1016/j.jmaa.2004.12.045.

show all references

References:
[1]

C. D. Aliprantis and R. Tourky, Cones and Duality, Graduate Studies in Mathematics AMS, 2007. doi: 10.1090/gsm/084.

[2]

J. P. Aubin and J. Siegel, Fixed points and satationary points of dissipative multivalued maps, Proceed. Amer. Math. Soc., 78 (1980), 391-398. doi: 10.1090/S0002-9939-1980-0553382-1.

[3]

C. BoschA. García and C. L. García, An extension of Ekeland's variational principle to locally complete spaces, J. Math. Anal. Appl., 328 (2007), 106-108. doi: 10.1016/j.jmaa.2006.05.012.

[4]

C. BoschA. GarcíaC. Gómez and S. Hernández, Equivalents to Ekeland's variational principle in locally complete spaces, Sci. Math. Jpn., 72 (2010), 283-287.

[5]

J. X. Fang, The variational principle and fixed point theorems in certain topological spaces, J. Math. Anal. Appl., 202 (1996), 398-412. doi: 10.1006/jmaa.1996.0323.

[6]

L. Hurwicz, Programing in linear spaces, in: K.J. Arrow, L. Hurwicz, H. Huzawa (Eds.), Studies in Linear and Nonlinear Programming, Stanford Univ. Press, Stanford, California, 1958, 38-102.

[7]

G. Isac, Sur l'existence de l'optimum de Pareto, Riv. Math. Univ. Parma, 9 (1983), 303-325.

[8]

G. Isac, Pareto optimization in infinite dimensional spaces. The importance of nuclear cones, J. Math. Anal. Appl., 182 (1994), 393-404. doi: 10.1006/jmaa.1994.1093.

[9]

G. Isac, The Ekeland's principle and the Pareto $ \epsilon $-efficiency, in: M. Tamiz (Ed.), In Multi-Objective Programming and Goal Programming, in: Lectures Notes in Econom. Math. Systems, Springer-Verlag, 432 (1996), 148-162.

[10]

G. Isac, Ekeland's principle and nuclear cones: A geometrical aspect, Math. Comput. Modelling, 26 (1997), 111-116. doi: 10.1016/S0895-7177(97)00223-9.

[11]

G. Isac, Nuclear cones in product spaces, Pareto efficiency and Ekeland-type variational principles in locally convex spaces, Optimization, 53 (2004), 253-268. doi: 10.1080/02331930410001720923.

[12]

G. Isac and A. O. Bahya, Full nuclear cones associated to normal cone. Application to Pareto efficiency, Appl. Math. Letters, 15 (2002), 633-639. doi: 10.1016/S0893-9659(02)80017-9.

[13]

H. Jarchow, Locally Convex Spaces, B.G. Teubner, Stuttgart, 1981.

[14]

H. W. Kuhn and A. W. Tucker, Nonlinear programming, in: J. Neymann (Ed), Proceeding of the 2nd Berkeley Simposium on Mathematical Statistics and Probability, University of California Press, Berkeley, CA, 1950,481-492.

[15]

A. Muntean, Some fixed point theorems for commuting multivalued operators, Seminar on fixed point theory Cluj-Napoca, 2 (2001), 71-79.

[16]

A. L. Peressini, Ordered Topological Vector Spaces, Harper and Raw, 1967.

[17]

P. Pérez-Carreras and J. Bonet, Barreled Locally Convex Spaces, North-Holland, Amsterdam, 1987.

[18]

A. Petrusel and I. A. Rus, An abstract point of view on iterative approximation schemes of fixed points for multivalued operators, J. Nonlinear Sci. Appl., 6 (2013), 97-107. doi: 10.22436/jnsa.006.02.05.

[19]

A. PetruselI. A. Rus and J. C. Yao, Well-posedness in the generalized sense of the fixed point problems for multivalued operators, Taiwanese J. Math., 11 (2007), 903-914. doi: 10.11650/twjm/1500404764.

[20]

M. Petschke, On a theorem of Arrow, Barankin and Blackwell, SIAM J. Control Opt., 28 (1990), 395-401. doi: 10.1137/0328021.

[21]

P. R. Phelps, Convex Functions, Monotone Operators and Differentiability, Second edition. Lecture Notes in Mathematics, 1364. Springer-Verlag, Berlin, 1993.

[22]

J. H. Qiu, Ekeland's variational principle in locally complete spaces, Math. Nach., 257 (2003), 55-58. doi: 10.1002/mana.200310077.

[23]

J. H. Qiu, Local completeness, drop theorem and Ekeland's variational principle, J. Math. Anal. Appl., 311 (2005), 23-39. doi: 10.1016/j.jmaa.2004.12.045.

[1]

Viviane Baladi, Sébastien Gouëzel. Banach spaces for piecewise cone-hyperbolic maps. Journal of Modern Dynamics, 2010, 4 (1) : 91-137. doi: 10.3934/jmd.2010.4.91

[2]

Florian Caro, Bilal Saad, Mazen Saad. Study of degenerate parabolic system modeling the hydrogen displacement in a nuclear waste repository. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 191-205. doi: 10.3934/dcdss.2014.7.191

[3]

Guolin Yu. Global proper efficiency and vector optimization with cone-arcwise connected set-valued maps. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 35-44. doi: 10.3934/naco.2016.6.35

[4]

Ji Gao. On the generalized pythagorean parameters and the applications in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 557-567. doi: 10.3934/dcdsb.2007.8.557

[5]

J. F. Padial. Existence and estimate of the location of the free-boundary for a non local inverse elliptic-parabolic problem arising in nuclear fusion. Conference Publications, 2011, 2011 (Special) : 1176-1185. doi: 10.3934/proc.2011.2011.1176

[6]

Zhaoquan Xu, Jiying Ma. Monotonicity, asymptotics and uniqueness of travelling wave solution of a non-local delayed lattice dynamical system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5107-5131. doi: 10.3934/dcds.2015.35.5107

[7]

Chui-Jie Wu, Hongliang Zhao. Generalized HWD-POD method and coupling low-dimensional dynamical system of turbulence. Conference Publications, 2001, 2001 (Special) : 371-379. doi: 10.3934/proc.2001.2001.371

[8]

Fengming Ma, Yiju Wang, Hongge Zhao. A potential reduction method for the generalized linear complementarity problem over a polyhedral cone. Journal of Industrial & Management Optimization, 2010, 6 (1) : 259-267. doi: 10.3934/jimo.2010.6.259

[9]

Ren-You Zhong, Nan-Jing Huang. Strict feasibility for generalized mixed variational inequality in reflexive Banach spaces. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 261-274. doi: 10.3934/naco.2011.1.261

[10]

Giselle A. Monteiro, Milan Tvrdý. Generalized linear differential equations in a Banach space: Continuous dependence on a parameter. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 283-303. doi: 10.3934/dcds.2013.33.283

[11]

Tomás Caraballo, Francisco Morillas, José Valero. On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 51-77. doi: 10.3934/dcds.2014.34.51

[12]

Fuchen Zhang, Xiaofeng Liao, Guangyun Zhang, Chunlai Mu, Min Xiao, Ping Zhou. Dynamical behaviors of a generalized Lorenz family. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3707-3720. doi: 10.3934/dcdsb.2017184

[13]

P.K. Newton. The dipole dynamical system. Conference Publications, 2005, 2005 (Special) : 692-699. doi: 10.3934/proc.2005.2005.692

[14]

Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A smoothing Newton method for generalized Nash equilibrium problems with second-order cone constraints. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 1-18. doi: 10.3934/naco.2012.2.1

[15]

Stefano Galatolo. Global and local complexity in weakly chaotic dynamical systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1607-1624. doi: 10.3934/dcds.2003.9.1607

[16]

Farida Benmakrouha, Christiane Hespel, Edouard Monnier. Drawing the output of dynamical systems by juxtaposing local outputs. Conference Publications, 2011, 2011 (Special) : 145-154. doi: 10.3934/proc.2011.2011.145

[17]

Jonathan P. Desi, Evelyn Sander, Thomas Wanner. Complex transient patterns on the disk. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1049-1078. doi: 10.3934/dcds.2006.15.1049

[18]

Tomás Caraballo, Stefanie Sonner. Random pullback exponential attractors: General existence results for random dynamical systems in Banach spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6383-6403. doi: 10.3934/dcds.2017277

[19]

Piotr Gwiazda, Agnieszka Świerczewska-Gwiazda, Aneta Wróblewska. Generalized Stokes system in Orlicz spaces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2125-2146. doi: 10.3934/dcds.2012.32.2125

[20]

Dorota Bors, Robert Stańczy. Dynamical system modeling fermionic limit. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 45-55. doi: 10.3934/dcdsb.2018004

 Impact Factor: 

Article outline

[Back to Top]