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Existence and asymptotic behaviour for solutions of dynamical equilibrium systems
1. | Laboratory LIBMA Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, 40000 Marrakech, Morocco, Morocco |
References:
[1] |
M. Ait Mansour, Z. Chbani and H. Riahi, Recession bifunction and solvability of noncoercive equilibrium problems, Comm. Appl. Anal., 7 (2003), 369-377. |
[2] |
H. Attouch and A. Damlamian, Strong solutions for parabolic variational inequalities, Nonlinear Anal., 2 (1978), 329-353.
doi: 10.1016/0362-546X(78)90021-4. |
[3] |
J.-B. Baillon, Un exemple concernant le comportement asymptotique de la solution du problème ${du}/{dt} +\partial\varphi(u) = 0$, J. Funct. Anal., 28 (1978), 369-376.
doi: 10.1016/0022-1236(78)90093-9. |
[4] |
J. B. Baillon and H. Brézis, Une remarque sur le comportement asymptotique des semi-groupes non linéaires, Houston J. Math., 2 (1976), 5-7. |
[5] |
V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, 2010.
doi: 10.1007/978-1-4419-5542-5. |
[6] |
E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145. |
[7] |
H. Brézis, Equations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier, 18 (1968), 115-175.
doi: 10.5802/aif.280. |
[8] |
H. Brézis, Inéquations variationnelles associées des opérateurs d'évolution, in Theory and applications of monotone operators, Proc. NATO Institute, Venice, (1968), 249-258. |
[9] |
H. Brézis, Opérateurs Maximaux Monotones Dans Les Espaces de Hilbert et Équations D'évolution, Lecture Notes, vol. 5, North-Holland, 1972. |
[10] |
H. Brézis, Analyse Fonctionnelle: Théorie et Applications, Masson, Paris, 1983. |
[11] |
H. Brézis, Asymptotic behavior of some evolution systems, Nonlinear Evolution Equations, Academic Press, New York, 40 (1978), 141-154. |
[12] |
F. Browder, Non-linear equations of evolution, Annals of Mathematics, Second Series, 80 (1964), 485-523.
doi: 10.2307/1970660. |
[13] |
F. Browder, Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, Nonlinear Functional Analysis, Symposia in Pure Math., 18, Part 2, F. Browder (Ed.), American Mathematical Society, Providence, RI, 1976. |
[14] |
R. E. Bruck, Asymptotic convergence of nonlinear contraction semigroups in Hilbert spaces, J. Funct. Anal., 18 (1975), 15-26.
doi: 10.1016/0022-1236(75)90027-0. |
[15] |
O. Chadli, Z. Chbani and H. Riahi, Recession methods for equilibrium problems and applications to variational and hemivariational inequalities, Discrete Contin. Dyn. Syst., 5 (1999), 185-196. |
[16] |
O. Chadli, Z. Chbani and H. Riahi, Equilibrium problems with generalized monotone bifunctions and Applications to Variational inequalities, J. Optim. Theory Appl., 105 (2000), 299-323.
doi: 10.1023/A:1004657817758. |
[17] |
Z. Chbani and H. Riahi, Variational principle for monotone and maximal bifunctions, Serdica Math. J., 29 (2003), 159-166. |
[18] |
P. L. Combettes and A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117-136. |
[19] |
S. Effati and M. Baymani, A new nonlinear neural network for solving convex nonlinear programming problems, Applied Mathematics and Computation, 168 (2005), 1370-1379.
doi: 10.1016/j.amc.2004.10.028. |
[20] |
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Dunod, 1974. |
[21] |
N. Hadjisavvas and H. Khatibzadeh, Maximal monotonicity of bifunctions, Optimization, 59 (2010), 147-160.
doi: 10.1080/02331930801951116. |
[22] |
A. Haraux, Nonlinear Evolution Equations-Global Behavior of Solutions, Lecture note in Mathematics, 841, Springer-Verlag, 1981. |
[23] |
J. J. Hopfield and D. W. Tank, Neural computation of decisions in optimization problems, Biol. Cybern., 52 (1985), 141-152. |
[24] |
F. Li, Delayed Lagrangian neural networks for solving convex programming problems, Neurocomputing, 73 (2010), 2266-2273.
doi: 10.1016/j.neucom.2010.01.009. |
[25] |
U. Mosco, Implicit variational problems and quasivariational inequalities, Lecture Notes in Mathematics, 543 Springer, Berlin, (1976), 83-156. |
[26] |
A. Moudafi, A recession notion for a class of monotone bivariate functions, Serdica Math. J., 26 (2000), 207-220. |
[27] |
A. Moudafi, Proximal point algorithm extended to equilibrium problems, J. Nat. Geom., 15 (1999), 91-100. |
[28] |
Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591-597.
doi: 10.1090/S0002-9904-1967-11761-0. |
[29] |
R. T. Rockafellar, Saddle-points and convex analysis, In Differential Games and Related Topics, (Kuhn, H.W., Szegö, G.P. eds.), North-Holland, (1971), 109-127. |
[30] |
R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equation, Math. Surveys Monogr. 49, Amer. Math. Soc., 1997. |
[31] |
Y. Xia and J. Wang, A recurrent neural network for solving nonlinear convex programs subject to linear constraints, IEEE Transactions on Neural Networks, 16 (2005), 379-386.
doi: 10.1109/TNN.2004.841779. |
[32] |
G. X. Z. Yuan, KKM Theory and Applications in Nonlinear Analysis, Marcel Dekker Inc, New-York, 1999. |
[33] |
E. Zeidler, Nonlinear Functional Analysis and Its Applications, II B: Nonlinear Monotone Operators, Springer-Verlag, 1990.
doi: 10.1007/978-1-4612-0985-0. |
[34] |
E. Zeidler, Nonlinear Functional Analysis and Its Applications, III: Variational Methods and Optimization, Springer-Verlag, 1985. |
show all references
References:
[1] |
M. Ait Mansour, Z. Chbani and H. Riahi, Recession bifunction and solvability of noncoercive equilibrium problems, Comm. Appl. Anal., 7 (2003), 369-377. |
[2] |
H. Attouch and A. Damlamian, Strong solutions for parabolic variational inequalities, Nonlinear Anal., 2 (1978), 329-353.
doi: 10.1016/0362-546X(78)90021-4. |
[3] |
J.-B. Baillon, Un exemple concernant le comportement asymptotique de la solution du problème ${du}/{dt} +\partial\varphi(u) = 0$, J. Funct. Anal., 28 (1978), 369-376.
doi: 10.1016/0022-1236(78)90093-9. |
[4] |
J. B. Baillon and H. Brézis, Une remarque sur le comportement asymptotique des semi-groupes non linéaires, Houston J. Math., 2 (1976), 5-7. |
[5] |
V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, 2010.
doi: 10.1007/978-1-4419-5542-5. |
[6] |
E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145. |
[7] |
H. Brézis, Equations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier, 18 (1968), 115-175.
doi: 10.5802/aif.280. |
[8] |
H. Brézis, Inéquations variationnelles associées des opérateurs d'évolution, in Theory and applications of monotone operators, Proc. NATO Institute, Venice, (1968), 249-258. |
[9] |
H. Brézis, Opérateurs Maximaux Monotones Dans Les Espaces de Hilbert et Équations D'évolution, Lecture Notes, vol. 5, North-Holland, 1972. |
[10] |
H. Brézis, Analyse Fonctionnelle: Théorie et Applications, Masson, Paris, 1983. |
[11] |
H. Brézis, Asymptotic behavior of some evolution systems, Nonlinear Evolution Equations, Academic Press, New York, 40 (1978), 141-154. |
[12] |
F. Browder, Non-linear equations of evolution, Annals of Mathematics, Second Series, 80 (1964), 485-523.
doi: 10.2307/1970660. |
[13] |
F. Browder, Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, Nonlinear Functional Analysis, Symposia in Pure Math., 18, Part 2, F. Browder (Ed.), American Mathematical Society, Providence, RI, 1976. |
[14] |
R. E. Bruck, Asymptotic convergence of nonlinear contraction semigroups in Hilbert spaces, J. Funct. Anal., 18 (1975), 15-26.
doi: 10.1016/0022-1236(75)90027-0. |
[15] |
O. Chadli, Z. Chbani and H. Riahi, Recession methods for equilibrium problems and applications to variational and hemivariational inequalities, Discrete Contin. Dyn. Syst., 5 (1999), 185-196. |
[16] |
O. Chadli, Z. Chbani and H. Riahi, Equilibrium problems with generalized monotone bifunctions and Applications to Variational inequalities, J. Optim. Theory Appl., 105 (2000), 299-323.
doi: 10.1023/A:1004657817758. |
[17] |
Z. Chbani and H. Riahi, Variational principle for monotone and maximal bifunctions, Serdica Math. J., 29 (2003), 159-166. |
[18] |
P. L. Combettes and A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117-136. |
[19] |
S. Effati and M. Baymani, A new nonlinear neural network for solving convex nonlinear programming problems, Applied Mathematics and Computation, 168 (2005), 1370-1379.
doi: 10.1016/j.amc.2004.10.028. |
[20] |
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Dunod, 1974. |
[21] |
N. Hadjisavvas and H. Khatibzadeh, Maximal monotonicity of bifunctions, Optimization, 59 (2010), 147-160.
doi: 10.1080/02331930801951116. |
[22] |
A. Haraux, Nonlinear Evolution Equations-Global Behavior of Solutions, Lecture note in Mathematics, 841, Springer-Verlag, 1981. |
[23] |
J. J. Hopfield and D. W. Tank, Neural computation of decisions in optimization problems, Biol. Cybern., 52 (1985), 141-152. |
[24] |
F. Li, Delayed Lagrangian neural networks for solving convex programming problems, Neurocomputing, 73 (2010), 2266-2273.
doi: 10.1016/j.neucom.2010.01.009. |
[25] |
U. Mosco, Implicit variational problems and quasivariational inequalities, Lecture Notes in Mathematics, 543 Springer, Berlin, (1976), 83-156. |
[26] |
A. Moudafi, A recession notion for a class of monotone bivariate functions, Serdica Math. J., 26 (2000), 207-220. |
[27] |
A. Moudafi, Proximal point algorithm extended to equilibrium problems, J. Nat. Geom., 15 (1999), 91-100. |
[28] |
Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591-597.
doi: 10.1090/S0002-9904-1967-11761-0. |
[29] |
R. T. Rockafellar, Saddle-points and convex analysis, In Differential Games and Related Topics, (Kuhn, H.W., Szegö, G.P. eds.), North-Holland, (1971), 109-127. |
[30] |
R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equation, Math. Surveys Monogr. 49, Amer. Math. Soc., 1997. |
[31] |
Y. Xia and J. Wang, A recurrent neural network for solving nonlinear convex programs subject to linear constraints, IEEE Transactions on Neural Networks, 16 (2005), 379-386.
doi: 10.1109/TNN.2004.841779. |
[32] |
G. X. Z. Yuan, KKM Theory and Applications in Nonlinear Analysis, Marcel Dekker Inc, New-York, 1999. |
[33] |
E. Zeidler, Nonlinear Functional Analysis and Its Applications, II B: Nonlinear Monotone Operators, Springer-Verlag, 1990.
doi: 10.1007/978-1-4612-0985-0. |
[34] |
E. Zeidler, Nonlinear Functional Analysis and Its Applications, III: Variational Methods and Optimization, Springer-Verlag, 1985. |
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