Uniform decay estimates for solutions of a class of retarded integral inequalities

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:271

Article

Li, Desheng¹  Liu, Qiang¹  Ju, Xuewei² 

[1] Tianjin Univ, Sch Math, Tianjin 300072, Peoples R China

[2] Civil Aviat Univ China, Dept Math, Tianjin 300300, Peoples R China

关键词: Retarded integral inequality;    Delay differential equation;    Asymptotic stability;    Exponential asymptotic stability;    Pullback attractor;    Nonautonomous equilibrium solution;   

DOI  :  10.1016/j.jde.2020.08.017

来源: Elsevier

【 摘 要 】

Some uniform decay estimates are established for solutions of the following type of retarded integral inequalities: y(t) <= E(t, tau)parallel to y(tau)parallel to + integral(t)(tau) K-1(t, s)parallel to y(s)parallel to ds + integral(infinity)(t) K-2(t, s)parallel to y(s)parallel to ds + rho, t >= tau >= 0. As a simple example of application, the retarded scalar functional differential equation (x)over dot = -a(t)x + B(t, x(t)) is considered, and the global asymptotic stability of the equation is proved under weaker conditions. Another example is the ODE system (x)over dot = F-0(t, x) + Sigma(m)(i=1) F-i(t, x(t - r(i)(t))) on R-n with superlinear nonlinearities F-i (0 <= i <= m). The existence of a global pullback attractor of the system is established under appropriate dissipation conditions. The third example for application concerns the study of the dynamics of the functional cocycle system du/dt + Au = F(theta(t) p, u(t)) in a Banach space X with sublinear nonlinearity. In particular, the existence and uniqueness of a nonautonomous equilibrium solution Gamma is obtained under the hyperbolicity assumption on operator A and some additional hypotheses, and the global asymptotic stability of Gamma is also addressed. (C) 2020 Elsevier Inc. All rights reserved.

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