Skip to main content
SpringerLink
Log in

Systems of Vector Quasi-equilibrium Problems and Their Applications

  • Chapter
  • First Online:
Variational Analysis and Generalized Differentiation in Optimization and Control

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 47))

Abstract

In this survey chapter, we present systems of various kinds of vector quasi-equilibrium problems and give existence theory for their solutions. Some applications to systems of vector quasi-optimization problems, quasi-saddle point problems for vector-valued functions and Debreu type equilibrium problems, also known as constrained Nash equilibrium problems, for vector-valued functions are presented. The investigations of this chapter are based on our papers: Ansari (J Math Anal Appl 341:1271–1283, 2008); Ansari et al. (J Global Optim 29:45–57, 2004); Ansari and Khan (Mathematical Analysis and Applications, edited by S. Nanda and G.P. Rajasekhar, Narosa, New Delhi, 2004, pp.1–13); and Ansari et al. (J Optim Theory Appl 127:27–44, 2005).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Al-Homidan, Q. H. Ansari and S. Schaible, Existence of solutions of systems of generalized implicit vector variational inequalities, J. Optim. Theory Appl., 134 (2007), 515–531.

    Article  MATH  MathSciNet  Google Scholar 

  2. Q. H. Ansari, A note on generalized vector variational-like inequalities, Optimization, 41 (1997), 197–205.

    Article  MATH  MathSciNet  Google Scholar 

  3. Q. H. Ansari, Vector equilibrium problems and vector variational inequalities, In [48], pp. 1–17, 2000.

    Google Scholar 

  4. Q. H. Ansari, Existence of solutions of systems of generalized implicit vector quasi-equilibrium problems, J. Math. Anal. Appl., 341 (2008), 1271–1283.

    Article  MATH  MathSciNet  Google Scholar 

  5. Q. H. Ansari, W. K. Chan and X. Q. Yang, The systems of vector quasi-equilibrium problems with applications, J. Global Optim., 29 (2004), 45–57.

    Article  MATH  MathSciNet  Google Scholar 

  6. Q. H. Ansari, W. K. Chan and X. Q. Yang, Weighted quasi-variational inequalities and constrained Nash equilibrium problems, Taiwanese J. Math., 10 (2006), 361–380.

    MATH  MathSciNet  Google Scholar 

  7. Q. H. Ansari and F. Flores-Bazán, Generalized vector quasi-equilibrium problems with applications, J. Math. Anal. Appl., 277 (2003), 246–256.

    Article  MATH  MathSciNet  Google Scholar 

  8. Q. H. Ansari and F. Flores-Bazan, Recession method for generalized vector equilibrium problems, J. Math. Anal. Appl., 321(1) (2006), 132–146.

    Article  MATH  MathSciNet  Google Scholar 

  9. Q. H. Ansari, A. Idzik and J. C. Yao, Coincidence and fixed point theorems with applications, Topolgical Meth. Nonlinear Anal., 15(1) (2000), 191–202.

    MATH  MathSciNet  Google Scholar 

  10. Q. H. Ansari and Z. Khan, System of generalized vector quasi-equilibrium problems with applications, in Mathematical Analysis and Applications, Edited by S. Nanda and G. P. Rajasekhar, Narosa, New Delhi, pp.1–13, 2004.

    Google Scholar 

  11. Q. H. Ansari, I. V. Konnov and J. C. Yao, On generalized vector equilibrium problems, Nonlinear Anal., 47 (2001), 543–554.

    Article  MATH  MathSciNet  Google Scholar 

  12. Q. H. Ansari, L. J. Lin and L. B. Su, Systems of simultaneous generalized vector quasiequilibrium problems and their applications, J. Optim. Theory Appl., 127(1) (2005), 27–44.

    Article  MATH  MathSciNet  Google Scholar 

  13. Q. H. Ansari, W. Oettli and D. Schläger, A generalization of vectorial equilibria, Math. Meth. Oper. Res., 46 (1997), 147–152.

    Article  MATH  Google Scholar 

  14. Q. H. Ansari, S. Schaible and J. C. Yao, System of vector equlibrium problems and its applications, J. Optim. Theory Appl., 107 (2000), 547–557.

    Article  MATH  MathSciNet  Google Scholar 

  15. Q. H. Ansari, S. Schaible and J. C. Yao, The system of generalized vector equilibrium problems with applications, J. Global Optim., 22 (2002), 3–16.

    Article  MATH  MathSciNet  Google Scholar 

  16. Q. H. Ansari, S. Schaible and J. C. Yao, Generalized vector quasi-variational inequality problems over product sets, J. Global Optim., 32(4) (2005), 437–449.

    Article  MATH  MathSciNet  Google Scholar 

  17. Q. H. Ansari and A. H. Siddiqi, A generalized vector variational-like inequality and optimization over an efficient set, in Functional Analysis with Current Applications in Science, Technology and Industry, Edited by M. Brokate and A. H. Siddiqi, Pitman Research Notes in Mathematics, Vol.377, Addison Wesley Longman, Essex, UK, pp.177–191, 1998.

    MathSciNet  Google Scholar 

  18. Q. H. Ansari, A. H. Siddiqi and S. Y. Wu, Existence and duality of generalized vector equilibrium problems, J. Math. Anal. Appl., 259 (2001), 115–126.

    Article  MATH  MathSciNet  Google Scholar 

  19. Q. H. Ansari, X. Q. Yang and J. C. Yao, Existence and duality of implicit vector variational problems, Numer. Funct. Anal. Optim., 22(7–8) (2001), 815–829.

    Article  MATH  MathSciNet  Google Scholar 

  20. Q. H. Ansari and J. C. Yao, A fixed point theorem and its applications to the system of variational inequalities, Bull. Austral. Math. Soc., 59 (1999), 433–442.

    Article  MATH  MathSciNet  Google Scholar 

  21. Q. H. Ansari and J. C. Yao, An existence result for the generalized vector equilibrium problem, Appl. Math. Lett., 12(8) (1999), 53–56.

    Article  MATH  MathSciNet  Google Scholar 

  22. Q. H. Ansari and J. C. Yao, Systems of generalized variational inequalities and their applications, Appl. Anal., 76 (2000), 203–217.

    Article  MATH  MathSciNet  Google Scholar 

  23. Q. H. Ansari and J. C. Yao, On nondifferentiable and nonconvex vector optimization problems, J. Optim. Theory Appl., 106 (2000), 487–500.

    Article  MathSciNet  Google Scholar 

  24. Q. H. Ansari and J. C. Yao, On vector quasi-equilibrium problems, in Equilibrium Problems and Variational Models, Edited by P. Daniele, F. Giannessi and A. Maugeri, Kluwer, Dordrecht, 2002.

    Google Scholar 

  25. J.-P. Aubin, Mathematical Methods of Game and Economic Theory, North-Holland, Amsterdam, 1979.

    MATH  Google Scholar 

  26. J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.

    MATH  Google Scholar 

  27. C. Berge, Topological Space, Oliver&Boyd, Edinburgh, 1963.

    Google Scholar 

  28. M. Bianchi, N. Hadjisavvas and S. Schaible, Vector equilibrium problems with generalized monotone bifunctions, J. Optim. Theory Appl., 92 (1997), 527–542.

    Article  MATH  MathSciNet  Google Scholar 

  29. S. Chebbi and M. Florenzano, Maximal elements and equilibria for condensing correspondences, Nonlinear Anal., 38 (1999), 995–1002.

    Article  MathSciNet  Google Scholar 

  30. G. Y. Chen and B. D. Craven, A vector variational inequality and optimization over an efficient set, ZOR-Math. Meth. Oper. Res., 34 (1990), 1–12.

    Article  MATH  MathSciNet  Google Scholar 

  31. G. Y. Chen, C. J. Goh and X. Q. Yang, Existence of a solution for generalized vector variational inequalities, Optimization, 50(1) (2001), 1–15.

    Article  MATH  MathSciNet  Google Scholar 

  32. G. Y. Chen and X. Q. Yang, Vector complementarity problem and its equivalence with weak minimal element in ordered spaces, J. Math. Anal. Appl., 153 (1990), 136–158.

    Article  MATH  MathSciNet  Google Scholar 

  33. G. Y. Chen, X. Q. Yang and H. Yu, A nonlinear scalarization function and generalized vector quasi-equilibrium problems, J. Global Optim., 32 (2005), 451–466.

    Article  MATH  MathSciNet  Google Scholar 

  34. F. H. Clarke, Optimization and Nonsmooth Analysis, SIAM, Philadelphia, PA, 1990.

    MATH  Google Scholar 

  35. G. Cohen and F. Chaplais, Nested monotony for variational inequalities over a product of spaces and convergence of iterative algorithms, J. Optim. Theory Appl., 59 (1988), 360–390.

    Article  MathSciNet  Google Scholar 

  36. B. D. Craven, Nondifferentiable optimization by smooth approximation, Optimization, 17 (1986), 3–17.

    Article  MATH  MathSciNet  Google Scholar 

  37. A. Daniilidis and N. Hadjisavvas, Existence theorems for vector variational inequalities, Bull. Austral. Math. Soc., 54 (1996), 473–481.

    Article  MATH  MathSciNet  Google Scholar 

  38. G. Debreu, A social equilibrium existence theorem, Proc. Natl. Acad. Sci. U.S.A., 38 (1952), 886–893.

    Article  MATH  MathSciNet  Google Scholar 

  39. P. Deguire, K. K. Tan and G. X.-Z. Yuan, The study of maximal element, fixed point for L S -majorized mappings and their applications to minimax and variational inequalities in the product topological spaces, Nonlinear Anal., 37 (1999), 933–951.

    Article  MathSciNet  Google Scholar 

  40. N. H. Dien, Some remarks on variational-like and quasivariational-like inequalities, Bull. Austral. Math. Soc., 46 (1992), 335–342.

    Article  MATH  MathSciNet  Google Scholar 

  41. X. P. Ding, Existence of solutions for quasi-equilibrium problems in noncompact topological spaces, Comput. Math. Appl., 39 (2000), 13–21.

    Article  MATH  Google Scholar 

  42. X. P. Ding and E. Tarafdar, Generalized vector variational-like inequalities without monotonicity, in [48], pp.113–124, 2000.

    Google Scholar 

  43. F. Ferro, A minimax theorem for vector-valued functions, J. Optim. Theory Appl., 60(1) (1989), 19–31.

    Article  MATH  MathSciNet  Google Scholar 

  44. F. Flores-Bazán and F. Flores-Bazán, Vector equilibrium problems under asymptotic analysis, J. Global Optim., 26 (2003), 141–166.

    Article  MATH  MathSciNet  Google Scholar 

  45. J. Fu, Simultaneous vector variational inequalities and vector implicit complementarity problem, J. Optim. Theory Appl., 93 (1997), 141–151.

    Article  MATH  MathSciNet  Google Scholar 

  46. J. Y. Fu, Generalized vector quasi-equilibrium problems, Math. Meth. Oper. Res., 52 (2000), 57–64.

    Article  MATH  Google Scholar 

  47. F. Giannessi, Theorems of the alternative, quadratic programs and complementarity problems, in Variational Inequalities and Complementarity Problems, Edited by R. W. Cottle, F. Giannessi and J. L. Lions, Wiley, New York, pp.151–186, 1980.

    Google Scholar 

  48. F. Giannessi (Ed.), Vector Variational Inequalities and Vector Equilibria. Mathematical Theories, Kluwer, Dordrecht, 2000.

    Google Scholar 

  49. X. H. Gong, Efficiency and Henig efficiency for vector equilibrium problems, J. Optim. Theory Appl., 108 (2001), 139–154.

    Article  MATH  MathSciNet  Google Scholar 

  50. N. Hadjisavvas and S. Schaible, From scalar to vector equilibrium problems in the quasimonotone case, J. Optim. Theory Appl., 96 (1998), 297–309.

    Article  MATH  MathSciNet  Google Scholar 

  51. M. A. Hanson, On sufficiency of the Kuhn–Tucker conditions, J. Math. Anal. Appl., 80 (1982), 545–550.

    Article  MathSciNet  Google Scholar 

  52. T. Husain and E. Tarafdar, Simultaneous variational inequalities, minimization problems and related results, Math. Japonica, 39 (1994), 221–231.

    MATH  MathSciNet  Google Scholar 

  53. V. Jeykumar, W. Oettli and M. Natividad, A solvability theorem for a class of quasiconvex mappings with applications to optimization, J. Math. Anal. Appl., 179 (1993), 537–546.

    Article  MathSciNet  Google Scholar 

  54. S. Kakutani, A generalization of Brouwer’s fixed point theorem, Duke Math. J., 8 (1941), 457–459.

    Article  MATH  MathSciNet  Google Scholar 

  55. W. K. Kim and K. K. Tan, On generalized vector quasi-variational inequalities, Optimization, 46 (1999), 185–198.

    Article  MATH  MathSciNet  Google Scholar 

  56. I. V. Konnov and J.-C. Yao, On the generalized vector variational inequality problem, J. Math. Anal. Appl., 206 (1997), 42–58.

    Article  MATH  MathSciNet  Google Scholar 

  57. I. V. Konnov and J.-C. Yao, Existence of solutions for generalized vector equilibrium problems, J. Math. Anal. Appl., 233 (1999), 328–335.

    Article  MATH  MathSciNet  Google Scholar 

  58. S. Kum and G. M. Lee, Remarks on implicit vector variational inequalities, Taiwanese J. Math., 6(3) (2002), 369–382.

    MATH  MathSciNet  Google Scholar 

  59. G. M. Lee, On relations between vector variational inequality and vector optimization problem, in Progress in Optimization, II: Contributions from Australasia, Edited by X. Q. Yang, A. I. Mees, M. E. Fisher and L. S. Jennings, Kluwer, Dordrecht, pp.167–179, 2000.

    Google Scholar 

  60. G. M. Lee and D. S. Kim, Existence of solutions for vector optimization problems, J. Math. Anal. Appl., 220 (1998), 90–98.

    Article  MATH  MathSciNet  Google Scholar 

  61. G. M. Lee, D. S. Kim and B. S. Lee, On noncooperative vector equilibrium, Indian J. Pure Appl. Math., 27 (1996), 735–739.

    MATH  MathSciNet  Google Scholar 

  62. G. M. Lee, D. S. Kim and B. S. Lee, Generalized vector variational inequality, Appl. Math. Lett., 9(1) (1996), 39–42.

    Article  MATH  MathSciNet  Google Scholar 

  63. G. M. Lee, D. S. Kim and B. S. Lee, On vector quasivariational-like inequality, Bull. Korean Math. Soc., 33 (1996), 45–55.

    MATH  MathSciNet  Google Scholar 

  64. G. M. Lee and S. Kum, On implicit vector variational inequalities, J. Optim. Theory Appl., 104(2) (2000), 409–425.

    Article  MATH  MathSciNet  Google Scholar 

  65. G. M. Lee, B. S. Lee and S.-S. Change, On vector quasivariational inequalities, J. Math. Anal. Appl., 203 (1996), 626–638.

    Article  MATH  MathSciNet  Google Scholar 

  66. Z. F. Li and S. Y. Wang, A type of minimax inequality for vector-valued mappings, J. Math. Anal. Appl., 227 (1998), 68–80.

    Article  MATH  MathSciNet  Google Scholar 

  67. L. J. Lin, Existence theorems of simultaneous equilibrium problems and generalized vector quasi-saddle points, J. Global Optim., 32 (2005), 613–632.

    Article  MATH  MathSciNet  Google Scholar 

  68. L.-J. Lin, System of generalized vector quasi-equilibrium problems with applications to fixed point theorems for a family of nonexpansive multivalued Maps, J. Global Optim., 34 (2006), 15–32.

    Article  MathSciNet  Google Scholar 

  69. L.-J. Lin and Q. H. Ansari, Collective fixed points and maximal elements with applications to abstract economies, J. Math. Anal. Appl., 296 (2004), 455–472.

    Article  MATH  MathSciNet  Google Scholar 

  70. L.-J. Lin and H.-W. Hsu, Existence theorems for systems of generalized vector quasiequilibrium problems and optimization problems, J. Optim. Theory Appl., 130 (2006), 461–475.

    Article  MATH  MathSciNet  Google Scholar 

  71. L.-J. Lin and Y. H. Liu, Existence theorems for systems of generalized vector quasi-equilibrium problems and mathematical programs with equilibrium constraint, J. Optim. Theory Appl., 130 (2006), 461–475.

    Article  MATH  MathSciNet  Google Scholar 

  72. L.-J. Lin and S. Park, On some generalized quasi-equilibrium problems, J. Math. Anal. Appl., 224 (1998), 167–181.

    Article  MATH  MathSciNet  Google Scholar 

  73. L. J. Lin and Y. L. Tasi, On vector quasi-saddle points of set-valued maps, in Generalized Convexity/Monotonicity, Edited by D. T. Luc and N. Hadjisavvs, Kluwer, Dordrecht, pp.311–319, 2004.

    Google Scholar 

  74. L. J. Lin and Z. T. Yu, On some equilibrium problems for multimaps, J. Comput. Appl. Math., 129 (2001), 171–183.

    Article  MATH  MathSciNet  Google Scholar 

  75. Z. Lin and J. Yu, The existence of solutions for the systems of generalized vector quasi-equilibrium problems, Appl. Math. Lett., 18 (2005), 415–422.

    Article  MATH  MathSciNet  Google Scholar 

  76. D. T. Luc, Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, Vol.319, Springer, Berlin, 1989.

    Google Scholar 

  77. D. T. Luc and C. A. Vargas, A saddle point theorem for set-valued maps, Nonlinear Anal. Theory Meth. Appl., 18 (1992), 1–7.

    Article  MATH  Google Scholar 

  78. S. R. Mohan and S. K. Neogy, On invex sets and preinvex functions, J. Math. Anal. Appl., 189 (1995), 901–908.

    Article  MATH  MathSciNet  Google Scholar 

  79. S. B. Nadler, Jr., Multi-valued contraction mappings, Pacific J. Math., 30 (1969), 475–488.

    MATH  MathSciNet  Google Scholar 

  80. J. F. Nash, Equilibrium point in n-person games, Proc. Natl. Acad. Sci. U.S.A., 36 (1950), 48–49.

    Article  MATH  MathSciNet  Google Scholar 

  81. J. F. Nash, Non-cooperative games, Ann. Math., 54 (1951), 286–295.

    Article  MathSciNet  Google Scholar 

  82. J. F. Nash, Two-persons cooperative games, Econometrica, 21 (1953), 128–140.

    Article  MATH  MathSciNet  Google Scholar 

  83. W. Oettli, A remark on vector-valued equilibria and generalized monotonicity, Acta Math. Vietnam., 22(1) (1997), pp.213–221.

    MATH  MathSciNet  Google Scholar 

  84. W. Oettli and D. Schläger, Generalized vectorial equilibria and generalized monotonicity, in Functional Analysis with current Applications in Science, Technology and Industry, Edited by M. Brokate and A. H. Siddiqi, Pitman Research Notes in Mathematics Series, Vol.377, Longman, Essex, pp.145–154, 1998.

    Google Scholar 

  85. W. Oettli and D. Schläger, Existence of equilibria for monotone multivalued mappings, Math. Meth. Oper. Res., 48 (1998), 219–228.

    Article  MATH  Google Scholar 

  86. J. S. Pang, Asymmetric variational inequality problems over product sets: Applications and iterative methods, Math. Prog., 31 (1985), 206–219.

    Article  MATH  Google Scholar 

  87. M. Patriksson, Nonlinear Programming and Variational Inequality Problems, Kluwer, Dordrecht, 1999.

    MATH  Google Scholar 

  88. W. V. Petryshyn and P. M. Fitzpatrick, Fixed point theorems of multivalued noncompact inward maps, J. Math. Anal. Appl., 46 (1974), 756–767.

    Article  MATH  MathSciNet  Google Scholar 

  89. W. V. Petryshyn and P. M. Fitzpatrick, Fixed point theorems of multivalued noncompact acyclic mappings, Pacific J. Math., 54 (1974), 17–23.

    MATH  MathSciNet  Google Scholar 

  90. R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970.

    MATH  Google Scholar 

  91. H. H. Schaefer, Topological Vector Spaces, Springer, Berlin, 1971.

    Google Scholar 

  92. M. Sion, On general minimax theorem, Pacific J. Math., 8 (1958), 171–176.

    MATH  MathSciNet  Google Scholar 

  93. W. Song, Vector equilibrium problems with set-valued mappings, in [48], pp.403–422, 2000.

    Google Scholar 

  94. N. X. Tan and P. N. Tinh, On the existence of equilibrium points of vector functions, Numer. Funct. Anal. Optim., 19 (1998), 141–156.

    Article  MATH  MathSciNet  Google Scholar 

  95. T. Tanaka, Generalized semicontinuity and existence theorems for cone saddle points, Appl. Math. Optim., 36 (1997), 313–322.

    Article  MATH  MathSciNet  Google Scholar 

  96. T. Weir and V. Jeyakumar, A class of nonconvex functions and mathematical programming, Bull. Austral. Math. Soc., 38 (1988), 177–189.

    Article  MATH  MathSciNet  Google Scholar 

  97. X. Q. Yang, Vector variational inequality and its duality, Nonlinear Anal. Theory Meth. Appl., 21 (1993), 869–877.

    Article  MATH  Google Scholar 

  98. G. X.-Z. Yuan, KKM Theory and Applications in Nonlinear Analysis, Marcel Dekker, New York, 1999.

    MATH  Google Scholar 

  99. G. X.-Z. Yuan, G. Isac, K.-K. Tan and J. Yu, The study of minimax inequalities, abstract economies and applications to variational inequalities and Nash equilibra, Acta Appl. Math., 54 (1998), 135–166.

    Article  MATH  MathSciNet  Google Scholar 

  100. E. Zeidler, Nonlinear Functional Analysis and Its Applications, I: Fixed Point Theorems, Springer, New York, 1986.

    MATH  Google Scholar 

  101. L. C. Zeng and J. C. Yao, An existence result for generalized vector equilibrium problems without pseudomonotonicity, Appl. Math. Lett., 19 (2006) 1320–1326.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

Authors are grateful to the referees for their valuable comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Aligarh Muslim University, Aligarh, 202 002, India

    Qamrul Hasan Ansari

  2. Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, 80424, Taiwan

    Jen-Chih Yao

Authors
  1. Qamrul Hasan Ansari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jen-Chih Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qamrul Hasan Ansari .

Editor information

Editors and Affiliations

  1. School of Mathematics & Statistics, Division of Information Technology, University of South Australia, Mawson Lakes Blvd., Mawson Lakes, 5095, South Australia, Australia

    Regina S. Burachik

  2. , Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, 80424, Taiwan R.O.C.

    Jen-Chih Yao

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Ansari, Q.H., Yao, JC. (2010). Systems of Vector Quasi-equilibrium Problems and Their Applications. In: Burachik, R., Yao, JC. (eds) Variational Analysis and Generalized Differentiation in Optimization and Control. Springer Optimization and Its Applications, vol 47. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-0437-9_1

Download citation

Publish with us

Policies and ethics

Search

Navigation