【作者】 刘强；

【导师】 李德生；

【作者基本信息】 天津大学 ， 数学， 2021， 博士

【摘要】 本文主要研究了两类新的时滞积分不等式解的一致衰减估计及其应用方面的一些问题,全文共分两大部分.第一部分主要研究具有有限或无穷时滞的下述积分不等式:y(t)≤ E(t,τ)‖yτ‖+∫τt K1(t,s)‖ys‖ds+∫t∞ K2(t,s)‖ys‖ds+ρ,?t≥ τ≥ 0.我们在对衰减因子E和积分核K1,K2的一些合理的假设条件下建立了上述不等式解的一致衰减估计,并进一步获得了指数衰减的结果.第二部分主要应用上述积分不等式和扇形算子理论、拉回吸引子理论等工具研究了几类典型的时滞微分方程的动力学行为.首先,我们讨论了下述纯量泛函微分方程:x=-a(t)x+B(t,xt),在更弱的条件下讨论了 0解的全局渐近稳定性;特别地,a(t)甚至可以允许变号,所得结果包含了很多与此相关的经典结果作为特殊情形.其次,我们考虑了 Rn上的超线性常微分系统x=F0(t,x)+(?)Fi(t,x(t-ri(t)))的全局动力学行为,在F0满足适当的耗散条件下证明了系统全局拉回吸引子的存在性.这一结果将文献中关于该方程次线性情形的相关工作推广到了超线性的情形.再次,我们研究了 Banach空间上具有次线性项的泛函cocycle系统du/dt+Au=F(θtp,ut),在主算子A是双曲的假设下证明了非自治稳态解的存在唯一性和全局渐近稳定性.在此基础上,讨论了神经网络系统的长时间行为.最后,我们讨论了具有无穷时滞的非自治泛函微分方程u’(t)+A(t)u(t)=f(t,ut),t>ι,uι=φ,在适当条件下证明了周期解的存在唯一性.此外,作为积分不等式应用的另一个例子,研究了具有无穷时滞的方程du/dt-Δu=-u3+f(ut),t>0,u0=φ耗散性问题.更多还原

【Abstract】 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.更多还原