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The value of a minimax problem involving impulse control
Local completeness, Pareto efficiency and Mackey Bishop-Phelps cones
Facultad de Economía UASLP, Av. Pintores S7N, San Luis Potosí, CP 78280, México |
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.
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. Google Scholar |
[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. 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. |
[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. Google Scholar |
[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. 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. |
[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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