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.  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. 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.  Google Scholar

[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.   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. 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.  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. 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.  Google Scholar

[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.   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. 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.  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

[1]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[2]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[3]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002

[4]

Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021003

[5]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[6]

Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020377

[7]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[8]

Shin-Ichiro Ei, Masayasu Mimura, Tomoyuki Miyaji. Reflection of a self-propelling rigid disk from a boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 803-817. doi: 10.3934/dcdss.2020229

[9]

Mauricio Achigar. Extensions of expansive dynamical systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020399

[10]

Dmitry Dolgopyat. The work of Sébastien Gouëzel on limit theorems and on weighted Banach spaces. Journal of Modern Dynamics, 2020, 16: 351-371. doi: 10.3934/jmd.2020014

[11]

Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269

[12]

Gui-Qiang Chen, Beixiang Fang. Stability of transonic shock-fronts in three-dimensional conical steady potential flow past a perturbed cone. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 85-114. doi: 10.3934/dcds.2009.23.85

[13]

The Editors. The 2019 Michael Brin Prize in Dynamical Systems. Journal of Modern Dynamics, 2020, 16: 349-350. doi: 10.3934/jmd.2020013

[14]

Nitha Niralda P C, Sunil Mathew. On properties of similarity boundary of attractors in product dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021004

[15]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

[16]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[17]

Roland Schnaubelt, Martin Spitz. Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 155-198. doi: 10.3934/eect.2020061

[18]

Liam Burrows, Weihong Guo, Ke Chen, Francesco Torella. Reproducible kernel Hilbert space based global and local image segmentation. Inverse Problems & Imaging, 2021, 15 (1) : 1-25. doi: 10.3934/ipi.2020048

[19]

Claudio Bonanno, Marco Lenci. Pomeau-Manneville maps are global-local mixing. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1051-1069. doi: 10.3934/dcds.2020309

[20]

Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345

 Impact Factor: 

Metrics

  • PDF downloads (113)
  • HTML views (299)
  • Cited by (0)

Other articles
by authors

[Back to Top]