doi: 10.3934/naco.2021043
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Existence and well-posedness for excess demand equilibrium problems

1. 

Department of Mathematics, Teacher College, Cantho University, Cantho, Vietnam

2. 

University of Science, Ho Chi Minh City, Vietnam, Vietnam National University, Ho Chi Minh City, Vietnam

3. 

Department of Mathematics, Vo Truong Toan University, Hau Giang, Vietnam

4. 

Department of Mathematics, FPT University, Can Tho, Vietnam

* Corresponding author: Tran Quoc Duy

Received  February 2021 Revised  September 2021 Early access October 2021

In this paper, we study excess demand equilibrium problems in Euclidean spaces. Applying the Glicksberg's fixed point theorem, sufficient conditions for the existence of solutions for the reference problems are established. We introduce a concept of well-posedness, say Levitin–Polyak well-posedness in the sense of Painlevé–Kuratowski, and investigate sufficient conditions for such kind of well-posedness.

Citation: Lam Quoc Anh, Pham Thanh Duoc, Tran Quoc Duy. Existence and well-posedness for excess demand equilibrium problems. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021043
References:
[1]

L. Q. Anh and T. Q. Duy, Tykhonov well-posedness for lexicographic equilibrium problems, Optimization, 65 (2016), 1929-1948.  doi: 10.1080/02331934.2016.1209673.  Google Scholar

[2]

L. Q. AnhT. Q. Duy and D. V. Hien, Well-posedness for the optimistic counterpart of uncertain vector optimization problems, Ann. Oper. Res., 295 (2020), 517-533.  doi: 10.1007/s10479-020-03840-0.  Google Scholar

[3]

L. Q. AnhT. Q. Duy and P. Q. Khanh, Levitin–Polyak well-posedness for equilibrium problems with the lexicographic order, Positivity, 23 (2021), 1323-1349.  doi: 10.1007/s11117-021-00818-5.  Google Scholar

[4]

L. Q. AnhT. Q. DuyL. D. Muu and T. Van Tri, The Tikhonov regularization for vector equilibrium problems, Comput. Optim. Appl., 78 (2021), 769-792.  doi: 10.1007/s10589-020-00258-z.  Google Scholar

[5]

L. Q. Anh and N. V. Hung, Levitin–Polyak well-posedness for strong bilevel vector equilibrium problems and applications to traffic network problems with equilibrium constraints, Positivity, 23 (2018), 1223-1239.  doi: 10.1007/s11117-018-0569-2.  Google Scholar

[6]

K. J. Arrow, General economic equilibrium: purpose, analytic techniques, collective choice, Am. Econ. Rev., 64 (1974), 253-272.   Google Scholar

[7]

P. BoonmanL. Q. Anh and R. Wangkeeree, Levitin–Polyak well-posedness by perturbations of strong vector mixed quasivariational inequality problems, J. Nonlinear Convex Anal., 22 (2021), 1327-1352.   Google Scholar

[8]

J. M. Bottazzi and T. Hens, Excess demand functions and incomplete markets, J. Econ. Theory, 68 (1996), 49-63.  doi: 10.1007/s001990100200.  Google Scholar

[9]

D. L. Cai, M. Sofonea and Y. B. Xiao, Tykhonov well-posedness of a mixed variational problem, Optimization, 2020. doi: 10.1080/02331934.2020.1808646.  Google Scholar

[10]

J. ChenZ. Wan and L. Yuan, Existence of solutions and $\alpha $-well-posedness for a system of constrained set-valued variational inequalities, Numer. Alg., Control Optim., 3 (2013), 567-581.  doi: 10.3934/naco.2013.3.567.  Google Scholar

[11]

P. A. Chiappori and I. Ekeland, Disaggregation of excess demand functions in incomplete markets, J. Math. Econ., 31 (1999), 111-129.  doi: 10.1016/S0304-4068(98)00059-7.  Google Scholar

[12]

G. Debreu, Excess demand functions, J. Math. Econ., 1 (1974), 15-21.   Google Scholar

[13]

T. Q. Duy, Levitin–Polyak well-posedness in set optimization concerning Pareto efficiency, Positivity, 2021. doi: 10.1007/s11117-021-00851-4.  Google Scholar

[14]

I. L. Glicksberg, A further generalization of the kakutani fixed point theorem, with application to nash equilibrium points, Proc. Am. Math. Soc., 3 (1952), 170-174.  doi: 10.2307/2032478.  Google Scholar

[15]

M. Gupta and M. Srivastava, Well-posedness and scalarization in set optimization involving ordering cones with possibly empty interior, J. Glob. Optim., 73 (2019), 447-463.  doi: 10.1007/s10898-018-0695-1.  Google Scholar

[16]

Y. Han and X.-H. Gong, Levitin–Polyak well-posedness of symmetric vector quasi-equilibrium problems, Optimization, 64 (2015), 1537-1545.  doi: 10.1080/02331934.2014.886037.  Google Scholar

[17]

R. HuM. Sofonea and Y. B. Xiao, A Tykhonov-type well-posedness concept for elliptic hemivariational inequalities, Z. Angew. Math. Phys., 71 (2020), 1-17.  doi: 10.1007/s00033-020-01337-1.  Google Scholar

[18]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Volume I: Theory, Kluwer, Dordrecht, 1997. doi: 10.1007/978-1-4615-6359-4.  Google Scholar

[19]

X. X. Huang and X. Q. Yang, Generalized Levitin–Polyak well-posedness in constrained optimization, SIAM J. Optim., 17 (2006), 243-258.  doi: 10.1137/040614943.  Google Scholar

[20] G. Kassay and V. D. Rădulescu, Equilibrium Problems and Applications, Academic Press, London, 2018.   Google Scholar
[21]

P. Q. KhanhL. M. Luu and T. T. M. Son, On the stability and Levitin–Polyak well-posedness of parametric multiobjective generalized games, Vietnam J. Math., 44 (2016), 857-871.  doi: 10.1007/s10013-016-0189-8.  Google Scholar

[22]

S. Khoshkhabar-amiranloo, Characterizations of generalized levitin–polyak well-posed set optimization problems, Optim. Lett., 13 (2019), 147-161.  doi: 10.1007/s11590-018-1258-6.  Google Scholar

[23] K. Kuratowski, Topology, Academic Press, London, 1968.   Google Scholar
[24]

C. S. Lalitha and P. Chatterjee, Levitin–Polyak well-posedness for constrained quasiconvex vector optimization problems, J. Glob. Optim., 59 (2014), 191-205.  doi: 10.1007/s10898-013-0103-9.  Google Scholar

[25]

E. S. Levitin and B. T. Polyak, On the convergence of minimizing sequences in conditional extremum problems, Dokl. Akad. Nauk SSSR, 168 (1966), 997-1000.   Google Scholar

[26]

R. R. Mantel, On the characterization of aggregate excess demand, J. Econ. Theory, 7 (1974), 348-353.   Google Scholar

[27]

T. Momi, Excess demand function around critical prices in incomplete markets, J. Math. Econ., 46 (2010), 293-302.  doi: 10.1016/j.jmateco.2009.11.013.  Google Scholar

[28]

J. K.H. Quah, The existence of equilibrium when excess demand obeys the weak axiom, J. Math. Econ., 44 (2008), 337-343.  doi: 10.1016/j.jmateco.2007.05.010.  Google Scholar

[29]

H. Sonnenschein, Do Walras' identity and continuity characterize the class of community excess demand functions?, J. Econ. Theory, 6 (1973), 345-354.  doi: 10.1016/0022-0531(73)90066-5.  Google Scholar

[30]

G. Tian, On the existence of price equilibrium in economies with excess demand functions, Econ. Theory Bull., 4 (2016), 5-16.  doi: 10.1007/s40505-015-0091-7.  Google Scholar

[31]

A. N. Tykhonov, On the stability of the functional optimization problem, USSR Comput. Math. Math. Phys., 6 (1966), 28-33.   Google Scholar

[32]

G. Virmani and M. Srivastava, Levitin-Polyak well-posedness of constrained inverse quasivariational inequality, Numer. Func. Anal. Optim., 38 (2017), 91-109.  doi: 10.1080/01630563.2016.1232728.  Google Scholar

[33]

P. T. VuiL. Q. Anh and R. Wangkeeree, Levitin–Polyak well-posedness for set optimization problems involving set order relations, Positivity, 23 (2019), 599-616.  doi: 10.1007/s11117-018-0627-9.  Google Scholar

[34]

K. C. Wong, Excess demand functions, equilibrium prices, and existence of equilibrium, Econ. Theory, 10 (1997), 39-54.  doi: 10.1007/s001990050145.  Google Scholar

show all references

References:
[1]

L. Q. Anh and T. Q. Duy, Tykhonov well-posedness for lexicographic equilibrium problems, Optimization, 65 (2016), 1929-1948.  doi: 10.1080/02331934.2016.1209673.  Google Scholar

[2]

L. Q. AnhT. Q. Duy and D. V. Hien, Well-posedness for the optimistic counterpart of uncertain vector optimization problems, Ann. Oper. Res., 295 (2020), 517-533.  doi: 10.1007/s10479-020-03840-0.  Google Scholar

[3]

L. Q. AnhT. Q. Duy and P. Q. Khanh, Levitin–Polyak well-posedness for equilibrium problems with the lexicographic order, Positivity, 23 (2021), 1323-1349.  doi: 10.1007/s11117-021-00818-5.  Google Scholar

[4]

L. Q. AnhT. Q. DuyL. D. Muu and T. Van Tri, The Tikhonov regularization for vector equilibrium problems, Comput. Optim. Appl., 78 (2021), 769-792.  doi: 10.1007/s10589-020-00258-z.  Google Scholar

[5]

L. Q. Anh and N. V. Hung, Levitin–Polyak well-posedness for strong bilevel vector equilibrium problems and applications to traffic network problems with equilibrium constraints, Positivity, 23 (2018), 1223-1239.  doi: 10.1007/s11117-018-0569-2.  Google Scholar

[6]

K. J. Arrow, General economic equilibrium: purpose, analytic techniques, collective choice, Am. Econ. Rev., 64 (1974), 253-272.   Google Scholar

[7]

P. BoonmanL. Q. Anh and R. Wangkeeree, Levitin–Polyak well-posedness by perturbations of strong vector mixed quasivariational inequality problems, J. Nonlinear Convex Anal., 22 (2021), 1327-1352.   Google Scholar

[8]

J. M. Bottazzi and T. Hens, Excess demand functions and incomplete markets, J. Econ. Theory, 68 (1996), 49-63.  doi: 10.1007/s001990100200.  Google Scholar

[9]

D. L. Cai, M. Sofonea and Y. B. Xiao, Tykhonov well-posedness of a mixed variational problem, Optimization, 2020. doi: 10.1080/02331934.2020.1808646.  Google Scholar

[10]

J. ChenZ. Wan and L. Yuan, Existence of solutions and $\alpha $-well-posedness for a system of constrained set-valued variational inequalities, Numer. Alg., Control Optim., 3 (2013), 567-581.  doi: 10.3934/naco.2013.3.567.  Google Scholar

[11]

P. A. Chiappori and I. Ekeland, Disaggregation of excess demand functions in incomplete markets, J. Math. Econ., 31 (1999), 111-129.  doi: 10.1016/S0304-4068(98)00059-7.  Google Scholar

[12]

G. Debreu, Excess demand functions, J. Math. Econ., 1 (1974), 15-21.   Google Scholar

[13]

T. Q. Duy, Levitin–Polyak well-posedness in set optimization concerning Pareto efficiency, Positivity, 2021. doi: 10.1007/s11117-021-00851-4.  Google Scholar

[14]

I. L. Glicksberg, A further generalization of the kakutani fixed point theorem, with application to nash equilibrium points, Proc. Am. Math. Soc., 3 (1952), 170-174.  doi: 10.2307/2032478.  Google Scholar

[15]

M. Gupta and M. Srivastava, Well-posedness and scalarization in set optimization involving ordering cones with possibly empty interior, J. Glob. Optim., 73 (2019), 447-463.  doi: 10.1007/s10898-018-0695-1.  Google Scholar

[16]

Y. Han and X.-H. Gong, Levitin–Polyak well-posedness of symmetric vector quasi-equilibrium problems, Optimization, 64 (2015), 1537-1545.  doi: 10.1080/02331934.2014.886037.  Google Scholar

[17]

R. HuM. Sofonea and Y. B. Xiao, A Tykhonov-type well-posedness concept for elliptic hemivariational inequalities, Z. Angew. Math. Phys., 71 (2020), 1-17.  doi: 10.1007/s00033-020-01337-1.  Google Scholar

[18]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Volume I: Theory, Kluwer, Dordrecht, 1997. doi: 10.1007/978-1-4615-6359-4.  Google Scholar

[19]

X. X. Huang and X. Q. Yang, Generalized Levitin–Polyak well-posedness in constrained optimization, SIAM J. Optim., 17 (2006), 243-258.  doi: 10.1137/040614943.  Google Scholar

[20] G. Kassay and V. D. Rădulescu, Equilibrium Problems and Applications, Academic Press, London, 2018.   Google Scholar
[21]

P. Q. KhanhL. M. Luu and T. T. M. Son, On the stability and Levitin–Polyak well-posedness of parametric multiobjective generalized games, Vietnam J. Math., 44 (2016), 857-871.  doi: 10.1007/s10013-016-0189-8.  Google Scholar

[22]

S. Khoshkhabar-amiranloo, Characterizations of generalized levitin–polyak well-posed set optimization problems, Optim. Lett., 13 (2019), 147-161.  doi: 10.1007/s11590-018-1258-6.  Google Scholar

[23] K. Kuratowski, Topology, Academic Press, London, 1968.   Google Scholar
[24]

C. S. Lalitha and P. Chatterjee, Levitin–Polyak well-posedness for constrained quasiconvex vector optimization problems, J. Glob. Optim., 59 (2014), 191-205.  doi: 10.1007/s10898-013-0103-9.  Google Scholar

[25]

E. S. Levitin and B. T. Polyak, On the convergence of minimizing sequences in conditional extremum problems, Dokl. Akad. Nauk SSSR, 168 (1966), 997-1000.   Google Scholar

[26]

R. R. Mantel, On the characterization of aggregate excess demand, J. Econ. Theory, 7 (1974), 348-353.   Google Scholar

[27]

T. Momi, Excess demand function around critical prices in incomplete markets, J. Math. Econ., 46 (2010), 293-302.  doi: 10.1016/j.jmateco.2009.11.013.  Google Scholar

[28]

J. K.H. Quah, The existence of equilibrium when excess demand obeys the weak axiom, J. Math. Econ., 44 (2008), 337-343.  doi: 10.1016/j.jmateco.2007.05.010.  Google Scholar

[29]

H. Sonnenschein, Do Walras' identity and continuity characterize the class of community excess demand functions?, J. Econ. Theory, 6 (1973), 345-354.  doi: 10.1016/0022-0531(73)90066-5.  Google Scholar

[30]

G. Tian, On the existence of price equilibrium in economies with excess demand functions, Econ. Theory Bull., 4 (2016), 5-16.  doi: 10.1007/s40505-015-0091-7.  Google Scholar

[31]

A. N. Tykhonov, On the stability of the functional optimization problem, USSR Comput. Math. Math. Phys., 6 (1966), 28-33.   Google Scholar

[32]

G. Virmani and M. Srivastava, Levitin-Polyak well-posedness of constrained inverse quasivariational inequality, Numer. Func. Anal. Optim., 38 (2017), 91-109.  doi: 10.1080/01630563.2016.1232728.  Google Scholar

[33]

P. T. VuiL. Q. Anh and R. Wangkeeree, Levitin–Polyak well-posedness for set optimization problems involving set order relations, Positivity, 23 (2019), 599-616.  doi: 10.1007/s11117-018-0627-9.  Google Scholar

[34]

K. C. Wong, Excess demand functions, equilibrium prices, and existence of equilibrium, Econ. Theory, 10 (1997), 39-54.  doi: 10.1007/s001990050145.  Google Scholar

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