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.

[2]

H. Attouch and A. Damlamian, Strong solutions for parabolic variational inequalities,, Nonlinear Anal., 2 (1978), 329. 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. 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.

[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.

[7]

H. Brézis, Equations et inéquations non linéaires dans les espaces vectoriels en dualité,, Ann. Inst. Fourier, 18 (1968), 115. doi: 10.5802/aif.280.

[8]

H. Brézis, Inéquations variationnelles associées des opérateurs d'évolution,, in Theory and applications of monotone operators, (1968), 249.

[9]

H. Brézis, Opérateurs Maximaux Monotones Dans Les Espaces de Hilbert et Équations D'évolution,, Lecture Notes, (1972).

[10]

H. Brézis, Analyse Fonctionnelle: Théorie et Applications,, Masson, (1983).

[11]

H. Brézis, Asymptotic behavior of some evolution systems,, Nonlinear Evolution Equations, 40 (1978), 141.

[12]

F. Browder, Non-linear equations of evolution,, Annals of Mathematics, 80 (1964), 485. doi: 10.2307/1970660.

[13]

F. Browder, Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces,, Nonlinear Functional Analysis, 18 (1976).

[14]

R. E. Bruck, Asymptotic convergence of nonlinear contraction semigroups in Hilbert spaces,, J. Funct. Anal., 18 (1975), 15. 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.

[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. doi: 10.1023/A:1004657817758.

[17]

Z. Chbani and H. Riahi, Variational principle for monotone and maximal bifunctions,, Serdica Math. J., 29 (2003), 159.

[18]

P. L. Combettes and A. Hirstoaga, Equilibrium programming in Hilbert spaces,, J. Nonlinear Convex Anal., 6 (2005), 117.

[19]

S. Effati and M. Baymani, A new nonlinear neural network for solving convex nonlinear programming problems,, Applied Mathematics and Computation, 168 (2005), 1370. 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. doi: 10.1080/02331930801951116.

[22]

A. Haraux, Nonlinear Evolution Equations-Global Behavior of Solutions,, Lecture note in Mathematics, 841 (1981).

[23]

J. J. Hopfield and D. W. Tank, Neural computation of decisions in optimization problems,, Biol. Cybern., 52 (1985), 141.

[24]

F. Li, Delayed Lagrangian neural networks for solving convex programming problems,, Neurocomputing, 73 (2010), 2266. doi: 10.1016/j.neucom.2010.01.009.

[25]

U. Mosco, Implicit variational problems and quasivariational inequalities,, Lecture Notes in Mathematics, 543 (1976), 83.

[26]

A. Moudafi, A recession notion for a class of monotone bivariate functions,, Serdica Math. J., 26 (2000), 207.

[27]

A. Moudafi, Proximal point algorithm extended to equilibrium problems,, J. Nat. Geom., 15 (1999), 91.

[28]

Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings,, Bull. Amer. Math. Soc., 73 (1967), 591. doi: 10.1090/S0002-9904-1967-11761-0.

[29]

R. T. Rockafellar, Saddle-points and convex analysis,, In Differential Games and Related Topics, (1971), 109.

[30]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equation,, Math. Surveys Monogr. 49, 49 (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. doi: 10.1109/TNN.2004.841779.

[32]

G. X. Z. Yuan, KKM Theory and Applications in Nonlinear Analysis,, Marcel Dekker Inc, (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.

[2]

H. Attouch and A. Damlamian, Strong solutions for parabolic variational inequalities,, Nonlinear Anal., 2 (1978), 329. 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. 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.

[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.

[7]

H. Brézis, Equations et inéquations non linéaires dans les espaces vectoriels en dualité,, Ann. Inst. Fourier, 18 (1968), 115. doi: 10.5802/aif.280.

[8]

H. Brézis, Inéquations variationnelles associées des opérateurs d'évolution,, in Theory and applications of monotone operators, (1968), 249.

[9]

H. Brézis, Opérateurs Maximaux Monotones Dans Les Espaces de Hilbert et Équations D'évolution,, Lecture Notes, (1972).

[10]

H. Brézis, Analyse Fonctionnelle: Théorie et Applications,, Masson, (1983).

[11]

H. Brézis, Asymptotic behavior of some evolution systems,, Nonlinear Evolution Equations, 40 (1978), 141.

[12]

F. Browder, Non-linear equations of evolution,, Annals of Mathematics, 80 (1964), 485. doi: 10.2307/1970660.

[13]

F. Browder, Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces,, Nonlinear Functional Analysis, 18 (1976).

[14]

R. E. Bruck, Asymptotic convergence of nonlinear contraction semigroups in Hilbert spaces,, J. Funct. Anal., 18 (1975), 15. 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.

[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. doi: 10.1023/A:1004657817758.

[17]

Z. Chbani and H. Riahi, Variational principle for monotone and maximal bifunctions,, Serdica Math. J., 29 (2003), 159.

[18]

P. L. Combettes and A. Hirstoaga, Equilibrium programming in Hilbert spaces,, J. Nonlinear Convex Anal., 6 (2005), 117.

[19]

S. Effati and M. Baymani, A new nonlinear neural network for solving convex nonlinear programming problems,, Applied Mathematics and Computation, 168 (2005), 1370. 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. doi: 10.1080/02331930801951116.

[22]

A. Haraux, Nonlinear Evolution Equations-Global Behavior of Solutions,, Lecture note in Mathematics, 841 (1981).

[23]

J. J. Hopfield and D. W. Tank, Neural computation of decisions in optimization problems,, Biol. Cybern., 52 (1985), 141.

[24]

F. Li, Delayed Lagrangian neural networks for solving convex programming problems,, Neurocomputing, 73 (2010), 2266. doi: 10.1016/j.neucom.2010.01.009.

[25]

U. Mosco, Implicit variational problems and quasivariational inequalities,, Lecture Notes in Mathematics, 543 (1976), 83.

[26]

A. Moudafi, A recession notion for a class of monotone bivariate functions,, Serdica Math. J., 26 (2000), 207.

[27]

A. Moudafi, Proximal point algorithm extended to equilibrium problems,, J. Nat. Geom., 15 (1999), 91.

[28]

Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings,, Bull. Amer. Math. Soc., 73 (1967), 591. doi: 10.1090/S0002-9904-1967-11761-0.

[29]

R. T. Rockafellar, Saddle-points and convex analysis,, In Differential Games and Related Topics, (1971), 109.

[30]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equation,, Math. Surveys Monogr. 49, 49 (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. doi: 10.1109/TNN.2004.841779.

[32]

G. X. Z. Yuan, KKM Theory and Applications in Nonlinear Analysis,, Marcel Dekker Inc, (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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