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

Local completeness, Pareto efficiency and Mackey Bishop-Phelps cones

Elvio Accinelli and  Armando García ,

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.

Keywords: Banach disk, local completeness, generalized dynamical system, nuclear cone, Pareto efficiency.
Mathematics Subject Classification: Primary: 49J53; Secondary: 46N10.
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.  Google Scholar

[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.  Google Scholar

[3]

C. Bosch, A. 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.  Google Scholar

[4]

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

[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.  Google Scholar

[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. Google Scholar

[7]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

[19]

A. Petrusel, I. 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.  Google Scholar

[20]

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

[21]

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

[22]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

C. Bosch, A. 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.  Google Scholar

[4]

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

[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.  Google Scholar

[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. Google Scholar

[7]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

[19]

A. Petrusel, I. 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.  Google Scholar

[20]

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

[21]

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

[22]

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

[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.  Google Scholar

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