Dissipativity Analysis Of General Linear Methods For Coupled System Of Nonlinear Functional Differential And Functional Equations

Let X be a real(or complex)Hilbert space,<·,·>and ‖·‖ are the inner product in X and the corresponding inner product norm respectively,Consider the initial value problem of a class of nonlinear functional differential and functional equations in X as follows where τ＞ 0 are given constant delay,φ,ψ are given continuous function,satisfying the consistency condition:ψ(0)=g(0,φ(0),φ(-τ),ψ(-τ)).Mapping f:[0,+∞)×X × X × X→X and g:[0,+∞)×X × X × X→X continuous function,and for all t≥0,y,u,v,w∈X,f and g satisfy:Re(f(t,u,v,w),u>≤γ1+α‖u‖2+,β1‖v‖2+β2‖w‖2,‖g(t,u,v,w)‖2≤γ2+u ‖u‖2+Lv‖v‖2+Lw‖w‖2,here coefficient are β1,β2,γ1,γ2,Lu,Lv,Lw are non-negative real constands,α≤ 0.The main results in this paper are as follows:First,when[α+β1+β2(Lv+Lu)/1-Lw]h≤p/2,(k,p,0)-algebraically stable general linear meth-ods can inherit the dissipativity of the system.Secend,numerical experiment with 2 stages 2 steps multistep Runge-Kutta meth-ods have been carried out and the results also prove the correctness of the theoretical analysis.