Uniform Decay Estimates For Solutions Of Retarded Integral Inequalities With Applications To Delay Differential Equations

This thesis is devoted to the studies of the uniform decay estimates of retarded integral inequalities as well as some applications.This paper contains two parts.In the first part,we first establish some uniform decay estimates for solutions of the following type of finite and infinite retarded integral inequalities:y(t)≤ E(t,τ)‖yτ‖+∫τtK1(t,s)‖ys‖ds+∫t∞ K2(t,s)‖ys‖ds+ρ,t≥τ≥ 0,We establish the uniform decay estimates for solutions of the above inequality under some reasonable assumptions of the factor E,the integral kernel K1 and K2.Furthermore,we get the the result of exponential decay.In the second part,we mainly use the theory of sectorial operators,the theory of pullback attractors and the inequality as above to study the dynamical behavior of some classical retarded differential equations.Firstly,the retarded scalar functional differential equation x=-a(t)x+B(t,xt)is considered,and the global asymptotic stability of 0 solution of the equation is proved under weaker conditions.Specially,we allow a(t)to be a function which may change sign.The results which we got contain many classic results as the special cases of them.Secondly,we consider the ODE system x=F0(t,x)+∑i=1m Fi(t,x(t-ri(t)))on Rn with superlinear nonlinearities Fi(0≤i≤m).The existence of a global pullback attractor of the system is established under appropriate dissipation conditions of F0.Thirdly,we concern the study of the dynamics of the functional cocycle system du/dt+Au=F(θtp,ut)in a Banach space with sublinear nonlinearity.In particular,the existence and uniqueness of a nonautonomous equilibrium solution Γ is obtained under the hyperbolicity assumption on operator A and some additional hypotheses,and the global asymptotic stability of Γ is also addressed.Furthermore,we discuss the long time behavior of the neural network system with multiple delays.Finally,in the case of infinite,under some appropriate conditions,we establish the existence and uniqueness of the periodic solution of the following equation:u’(t)+A(t)u(t)=f(t,ut),t>ι,uι=φ.In addition,we establish the dissipativity of the following functional differential equation under some appropriate state space du/dt-Δu=-u3+f(ut),t>0,u0=φ∈B.