Local completeness, Pareto efficiency and Mackey Bishop-Phelps cones

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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 and 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. 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.
[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.
[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. 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.
[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. 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.
[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.
[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. 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.
[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.

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