Journal of Applied Nonlinear Dynamics
Nonlinear Integral Inequalities WITH Parameter and Applications
Journal of Environmental Accounting and Management 7(4) (2018) 425436  DOI:10.5890/JAND.2018.12.009
Taoufik Ghrissi, M. A. Hammami
Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax Route Soukra, BP 1171, 3000 Sfax, Tunisia
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Abstract
We discuss the problem of establishing dissipative estimates for certain differential equations for which the usual methods do not work. The aim of this paper are some new nonlinear integral inequalities leading to suitable uniform (with respect to time and the arameter ε ) bounds on the solutions to problems x(t) = f(t,x(t))+gε(t,x(t)), t ≥ 0 where f, gε: R+ × Rn → Rn are supposed to be piecewise continuous in time, locally Lipschitz in x, for any fixed ε ≥ 0. These problems are seen as perturbation to x(t) = f(t,x(t)), t ≥ 0. Furthermore, some examples are given to illustrate the applicability of the obtained results.
References

[1]  Zelik, S. (2007), Spatially nondecayingsolutions of the 2D NavierStokes equation in a strip, Glasg. Math. J., 525588. 

[2]  Zelik, S. (2008), Weak Spatially nondecaying solutions of the 3D NavierStokes equations in cylindrical domains, in: C. Bardos, A. Fursikov(Eds.), instability in models connected with floods flows, in: International Math. Series, vol. 56, Springer, New York. 

[3]  Ayels, O. and Penteman, P. (1998), A new asymptotic stability criterion for non linear time varying differential equations, IEEE Trans. Aut. Contr, 43, 968971. 

[4]  Bay, N.S. and Phat, V.N. (1999), Stability of nonlinear difference time varying systems with delays, Vietnam J. of Math, 4, 129136. 

[5]  BenAbdallah, A., Ellouze, I., and Hammami, M.A. (2009), Practical stability of nonlinear timevarying cascade systems, J. Dyn. Control Sys., 15, 4562. 

[6]  Corless, M. (1990), Guaranteed Rates of Exponential Convergence for Uncertain Systems, Journal of Opti mization Theory and Applications, 64, 481494. 

[7]  Corless, M. and Leitmann, G. (1988), Controller Design for Uncertain Systems via Lyapunov Functions, Proceedings of the 1988 American Control Conference, Atlanta, Georgia. 

[8]  Garofalo, F. and Leitmann, G. (1989), Guaranteeing Ultimate Boundedness and Exponential Rate of Con vergence for a Class of Nominally Linear Uncertain Systems, Journal of Dynamic Systems, Measurement, and Control, 111, 584588. 

[9]  Hahn, W. (1967), Stability of Motion, Springer, New York. 

[10]  Khalil, H. (2002), Nonlinear Systems, Prentice Hall. 

[11]  Gronwall, T.H. (1919), Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. Math., 20(2), 293296. 

[12]  Bainov, D. and Simenov, P. (1992), Integral inequalities and applications, Kluwer Academic Publishers, Dordrecht. 

[13]  Bellman, R. (1943), The stability of solutions of linear differential equations, Duke Math. J., 10, 643647. 

[14]  Pata, V. (2011), Uniform estimates of Gronwall types, Journal of Mathematical Analysis and Applications, 264270. 

[15]  Gatti, S., Pata, V., and Zelik, S. (2009), A Gronwalltype lemma with parameter and dissipative estimates for PDEs, Nonlinear Analysis, 23372343. 

[16]  Pata, V. (2011), Uniform estimates of Gronwall types, Journal of Mathematical Analysis and Applications, 264270. 

[17]  Chepyzhov, V.V., Pata, V., and Vishik, M.I. (2009), Averaging of 2D NavierStokes equations with singularly oscilating forces, Nonlinearity, 22, 351370. 

[18]  Temam, R. (1988), InfiniteDimensional Systems in Mechanics and Physics, Springer, New York. 