# American Institute of Mathematical Sciences

March  2014, 3(1): 1-14. doi: 10.3934/eect.2014.3.1

## 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

Received  February 2013 Revised  January 2014 Published  February 2014

In this paper, we give an existence result for the following dynamical equilibrium problem: $\langle \frac{du}{dt},v-u(t)\rangle+F(u(t),v)\geq 0 \;\; \forall v\in K$ and for $a.e. \;t \geq 0$, where $K$ is a closed convex set in a Hilbert space and $F:K \times K \rightarrow \mathbb{R}$ is a monotone bifunction. We introduce a class of demipositive bifunctions and use it to study the asymptotic behaviour of the solution $u(t)$ when $t\rightarrow\infty$. We obtain weak convergence of $u(t)$ to some solution $x\in K$ of the equilibrium problem $F(x,y)\geq 0$ for every $y\in K$. Our applications deal with the asymptotic behaviour of the dynamical convex minimization and dynamical system associated to saddle convex-concave bifunctions. We then present a new neural model for solving a convex programming problem.
Citation: Zaki Chbani, Hassan Riahi. Existence and asymptotic behaviour for solutions of dynamical equilibrium systems. Evolution Equations & Control Theory, 2014, 3 (1) : 1-14. doi: 10.3934/eect.2014.3.1
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar [6] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145.  Google Scholar [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.  Google Scholar [8] H. Brézis, Inéquations variationnelles associées des opérateurs d'évolution, in Theory and applications of monotone operators, Proc. NATO Institute, Venice, (1968), 249-258. Google Scholar [9] H. Brézis, Opérateurs Maximaux Monotones Dans Les Espaces de Hilbert et Équations D'évolution, Lecture Notes, vol. 5, North-Holland, 1972. Google Scholar [10] H. Brézis, Analyse Fonctionnelle: Théorie et Applications, Masson, Paris, 1983.  Google Scholar [11] H. Brézis, Asymptotic behavior of some evolution systems, Nonlinear Evolution Equations, Academic Press, New York, 40 (1978), 141-154.  Google Scholar [12] F. Browder, Non-linear equations of evolution, Annals of Mathematics, Second Series, 80 (1964), 485-523. doi: 10.2307/1970660.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] Z. Chbani and H. Riahi, Variational principle for monotone and maximal bifunctions, Serdica Math. J., 29 (2003), 159-166.  Google Scholar [18] P. L. Combettes and A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117-136.  Google Scholar [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.  Google Scholar [20] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Dunod, 1974. Google Scholar [21] N. Hadjisavvas and H. Khatibzadeh, Maximal monotonicity of bifunctions, Optimization, 59 (2010), 147-160. doi: 10.1080/02331930801951116.  Google Scholar [22] A. Haraux, Nonlinear Evolution Equations-Global Behavior of Solutions, Lecture note in Mathematics, 841, Springer-Verlag, 1981.  Google Scholar [23] J. J. Hopfield and D. W. Tank, Neural computation of decisions in optimization problems, Biol. Cybern., 52 (1985), 141-152.  Google Scholar [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.  Google Scholar [25] U. Mosco, Implicit variational problems and quasivariational inequalities, Lecture Notes in Mathematics, 543 Springer, Berlin, (1976), 83-156.  Google Scholar [26] A. Moudafi, A recession notion for a class of monotone bivariate functions, Serdica Math. J., 26 (2000), 207-220.  Google Scholar [27] A. Moudafi, Proximal point algorithm extended to equilibrium problems, J. Nat. Geom., 15 (1999), 91-100.  Google Scholar [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.  Google Scholar [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.  Google Scholar [30] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equation, Math. Surveys Monogr. 49, Amer. Math. Soc., 1997.  Google Scholar [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.  Google Scholar [32] G. X. Z. Yuan, KKM Theory and Applications in Nonlinear Analysis, Marcel Dekker Inc, New-York, 1999.  Google Scholar [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.  Google Scholar [34] E. Zeidler, Nonlinear Functional Analysis and Its Applications, III: Variational Methods and Optimization, Springer-Verlag, 1985.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar [6] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145.  Google Scholar [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.  Google Scholar [8] H. Brézis, Inéquations variationnelles associées des opérateurs d'évolution, in Theory and applications of monotone operators, Proc. NATO Institute, Venice, (1968), 249-258. Google Scholar [9] H. Brézis, Opérateurs Maximaux Monotones Dans Les Espaces de Hilbert et Équations D'évolution, Lecture Notes, vol. 5, North-Holland, 1972. Google Scholar [10] H. Brézis, Analyse Fonctionnelle: Théorie et Applications, Masson, Paris, 1983.  Google Scholar [11] H. Brézis, Asymptotic behavior of some evolution systems, Nonlinear Evolution Equations, Academic Press, New York, 40 (1978), 141-154.  Google Scholar [12] F. Browder, Non-linear equations of evolution, Annals of Mathematics, Second Series, 80 (1964), 485-523. doi: 10.2307/1970660.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] Z. Chbani and H. Riahi, Variational principle for monotone and maximal bifunctions, Serdica Math. J., 29 (2003), 159-166.  Google Scholar [18] P. L. Combettes and A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117-136.  Google Scholar [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.  Google Scholar [20] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Dunod, 1974. Google Scholar [21] N. Hadjisavvas and H. Khatibzadeh, Maximal monotonicity of bifunctions, Optimization, 59 (2010), 147-160. doi: 10.1080/02331930801951116.  Google Scholar [22] A. Haraux, Nonlinear Evolution Equations-Global Behavior of Solutions, Lecture note in Mathematics, 841, Springer-Verlag, 1981.  Google Scholar [23] J. J. Hopfield and D. W. Tank, Neural computation of decisions in optimization problems, Biol. Cybern., 52 (1985), 141-152.  Google Scholar [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.  Google Scholar [25] U. Mosco, Implicit variational problems and quasivariational inequalities, Lecture Notes in Mathematics, 543 Springer, Berlin, (1976), 83-156.  Google Scholar [26] A. Moudafi, A recession notion for a class of monotone bivariate functions, Serdica Math. J., 26 (2000), 207-220.  Google Scholar [27] A. Moudafi, Proximal point algorithm extended to equilibrium problems, J. Nat. Geom., 15 (1999), 91-100.  Google Scholar [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.  Google Scholar [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.  Google Scholar [30] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equation, Math. Surveys Monogr. 49, Amer. Math. Soc., 1997.  Google Scholar [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.  Google Scholar [32] G. X. Z. Yuan, KKM Theory and Applications in Nonlinear Analysis, Marcel Dekker Inc, New-York, 1999.  Google Scholar [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.  Google Scholar [34] E. Zeidler, Nonlinear Functional Analysis and Its Applications, III: Variational Methods and Optimization, Springer-Verlag, 1985.  Google Scholar
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