A Variational Inequality Approach to a Class of Environmental Equilibrium Problems

Affiliation(s)

School of Mathematical Sciences, Rochester Institute of Technology, Rochester, USA.

Department of Mathematics and Computer Science, University of Catania, Catania, Italy.

School of Mathematical Sciences, Rochester Institute of Technology, Rochester, USA.

Department of Mathematics and Computer Science, University of Catania, Catania, Italy.

ABSTRACT

In this note we consider a class of environmental games recently proposed in the literature and investigate them by using the powerful tools of variational inequalities. We also consider the case where some data of the problem can depend on a parameter and analyze the regularity of the solution with respect to the parameter. In view of applications to time-dependent or random models, we also introduce the variational inequality formulation in Lebesgue spaces.

In this note we consider a class of environmental games recently proposed in the literature and investigate them by using the powerful tools of variational inequalities. We also consider the case where some data of the problem can depend on a parameter and analyze the regularity of the solution with respect to the parameter. In view of applications to time-dependent or random models, we also introduce the variational inequality formulation in Lebesgue spaces.

Cite this paper

B. Jadamba and F. Raciti, "A Variational Inequality Approach to a Class of Environmental Equilibrium Problems,"*Applied Mathematics*, Vol. 3 No. 11, 2012, pp. 1723-1728. doi: 10.4236/am.2012.311238.

B. Jadamba and F. Raciti, "A Variational Inequality Approach to a Class of Environmental Equilibrium Problems,"

References

[1] M. Breton, G. Zaccour and M. Zahaf, “A Game-Theoretic Formulation of Joint Implementation of Environmental Projects,” European Journal of Operational Research, Vol. 168, No. 1, 2005, pp. 221-239. doi:10.1016/j.ejor.2004.04.026

[2] M. Tidball and G. Zaccour, “An Environmental Game with Coupling Con-straints,” Environmental Modeling and Assessment, Vol. 10, No. 2, 2005, pp. 153-158. doi:10.1007/s10666-005-5254-8

[3] F. Giannessi and A. Maugeri, Eds., “Variational Inequalities and Network Equilibrium Problems,” Plenum Press, New York, 1995.

[4] F. Raciti, “Equilibrium Conditions and Vector Variational Inequalities: A Complex Relation,” Journal of Global Optimization, Vol. 40, No. 1-3, 2008, pp. 353-360. doi:10.1007/s10898-007-9202-9

[5] J. Gwinner and F. Raciti, “Random Equilibrium Problems on Networks,” Mathematical Computer Modelling, Vol. 43, No. 7-8, 2006, pp. 880-891. doi:10.1016/j.mcm.2005.12.007

[6] A. O. Caruso, A. A. Khan and F. Raciti, “Continuity Results for Some Classes of Variational Inequalities and Applications to Time-Dependent Equilibrium Problems,” Numerical Functional Analysis and Optimization, Vol. 30 No. 11-12, 2009, pp. 1272-1288. doi:10.1080/01630560903381696

[7] A. Causa and F. Raciti, “Lipschitz Continuity Results for a Class of Varia-tional Inequalities: A Geometric Approach,” Journal of Optimization Theory and Applications, Vol. 145, No. 2, 2010, pp. 235-248. doi:10.1007/s10957-009-9622-4

[8] A. Causa and F. Raciti, “Some Remarks on the Walras Equilibrium Problem in Lebesgue Spaces,” Optimization Letters, Vol. 5, No. 1, 2011, pp. 99-112. doi:10.1007/s11590-010-0193-y

[9] J. Gwinner and F. Raciti, “On a Class of Random Variational Inequalities on Random Sets,” Numerical Functional Analysis and Opti-mization, Vol. 27, No. 5-6, 2006, pp. 619-636.

[10] J. Gwinner and F. Raciti, “On Monotone Variational In-equalities with Random Data,” Journal of Mathematical Inequalities, Vol. 3, No. 3, 2009, pp. 443-453.

[11] J. Gwinner and F. Raciti, “Some Equilibrium Problems under Uncertainty and Random Variational Inequalities, Annals of Operations Research, Vol. 200, No. 1, 2012, pp. 299-319. doi:10.1007/s10479-012-1109-2

[12] P. T. Harker, “Generalized Nash Games and Quasi-Variational Inequalities,” European Journal of Operational Research, Vol. 54, No. 1, 1991, pp. 81-94. doi:10.1016/0377-2217(91)90325-P

[13] A. Maugeri and F. Raciti, “On Existence Theorems for Monotone and Nonmonotone Variational Inequalities,” Journal of Convex Analysis, Vol. 16, No. 3-4, 2009, pp. 899-911.

[14] J. B. Rosen, “Existence and Uniqueness of Equilibrium Points for Concave N-Person Games,” Econometrica, Vol. 33, No. 3, 1965, pp. 520-534. doi:10.2307/1911749

[15] F. Facchinei, A. Fischer and V. Piccialli, “On Generalized Nash Games and Variational Inequalities,” Operations Research Letters, Vol. 35, No. 2, 2007, pp. 159-164. doi:10.1016/j.orl.2006.03.004

[16] P. Falsaperla, F. Raciti and L. Scrimali, “A Variational Inequality Model of the Spatial Price Network Problem with Uncertain Data,” Optimization and Engineering, Vol. 13, No. 3, 2012, pp. 417-434. doi:10.1007/s11081-011-9158-y

[1] M. Breton, G. Zaccour and M. Zahaf, “A Game-Theoretic Formulation of Joint Implementation of Environmental Projects,” European Journal of Operational Research, Vol. 168, No. 1, 2005, pp. 221-239. doi:10.1016/j.ejor.2004.04.026

[2] M. Tidball and G. Zaccour, “An Environmental Game with Coupling Con-straints,” Environmental Modeling and Assessment, Vol. 10, No. 2, 2005, pp. 153-158. doi:10.1007/s10666-005-5254-8

[3] F. Giannessi and A. Maugeri, Eds., “Variational Inequalities and Network Equilibrium Problems,” Plenum Press, New York, 1995.

[4] F. Raciti, “Equilibrium Conditions and Vector Variational Inequalities: A Complex Relation,” Journal of Global Optimization, Vol. 40, No. 1-3, 2008, pp. 353-360. doi:10.1007/s10898-007-9202-9

[5] J. Gwinner and F. Raciti, “Random Equilibrium Problems on Networks,” Mathematical Computer Modelling, Vol. 43, No. 7-8, 2006, pp. 880-891. doi:10.1016/j.mcm.2005.12.007

[6] A. O. Caruso, A. A. Khan and F. Raciti, “Continuity Results for Some Classes of Variational Inequalities and Applications to Time-Dependent Equilibrium Problems,” Numerical Functional Analysis and Optimization, Vol. 30 No. 11-12, 2009, pp. 1272-1288. doi:10.1080/01630560903381696

[7] A. Causa and F. Raciti, “Lipschitz Continuity Results for a Class of Varia-tional Inequalities: A Geometric Approach,” Journal of Optimization Theory and Applications, Vol. 145, No. 2, 2010, pp. 235-248. doi:10.1007/s10957-009-9622-4

[8] A. Causa and F. Raciti, “Some Remarks on the Walras Equilibrium Problem in Lebesgue Spaces,” Optimization Letters, Vol. 5, No. 1, 2011, pp. 99-112. doi:10.1007/s11590-010-0193-y

[9] J. Gwinner and F. Raciti, “On a Class of Random Variational Inequalities on Random Sets,” Numerical Functional Analysis and Opti-mization, Vol. 27, No. 5-6, 2006, pp. 619-636.

[10] J. Gwinner and F. Raciti, “On Monotone Variational In-equalities with Random Data,” Journal of Mathematical Inequalities, Vol. 3, No. 3, 2009, pp. 443-453.

[11] J. Gwinner and F. Raciti, “Some Equilibrium Problems under Uncertainty and Random Variational Inequalities, Annals of Operations Research, Vol. 200, No. 1, 2012, pp. 299-319. doi:10.1007/s10479-012-1109-2

[12] P. T. Harker, “Generalized Nash Games and Quasi-Variational Inequalities,” European Journal of Operational Research, Vol. 54, No. 1, 1991, pp. 81-94. doi:10.1016/0377-2217(91)90325-P

[13] A. Maugeri and F. Raciti, “On Existence Theorems for Monotone and Nonmonotone Variational Inequalities,” Journal of Convex Analysis, Vol. 16, No. 3-4, 2009, pp. 899-911.

[14] J. B. Rosen, “Existence and Uniqueness of Equilibrium Points for Concave N-Person Games,” Econometrica, Vol. 33, No. 3, 1965, pp. 520-534. doi:10.2307/1911749

[15] F. Facchinei, A. Fischer and V. Piccialli, “On Generalized Nash Games and Variational Inequalities,” Operations Research Letters, Vol. 35, No. 2, 2007, pp. 159-164. doi:10.1016/j.orl.2006.03.004

[16] P. Falsaperla, F. Raciti and L. Scrimali, “A Variational Inequality Model of the Spatial Price Network Problem with Uncertain Data,” Optimization and Engineering, Vol. 13, No. 3, 2012, pp. 417-434. doi:10.1007/s11081-011-9158-y