Existence and asymptotic behaviour for solutions of dynamical equilibrium systems

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

Existence and asymptotic behaviour for solutions of dynamical equilibrium systems

Zaki Chbani ^1, and Hassan Riahi ^1,

1.	Laboratory LIBMA Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, 40000 Marrakech, Morocco, Morocco

Received February 2013 Revised January 2014 Published February 2014

Abstract
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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.

Keywords: Cauchy problem, saddle point problem., convex minimisation, demipositive bifunction, asymptotic behaviour, Monotone bifunction.

Mathematics Subject Classification: Primary: 37N40, 46N10, 49J40, 90C3.

Citation: Zaki Chbani, Hassan Riahi. Existence and asymptotic behaviour for solutions of dynamical equilibrium systems. Evolution Equations and 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.
[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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