The convergence relation between ordinary and delay-integro-differential equations (Q286373)

The authors consider an \(n\)-dimensional delay-integro-differential equation with multiple time-varying delays of the following form: NEWLINE\[NEWLINE \dot x(t)=f(t,x(t), x(t- \tau _1(t)),\dots,x(t- \tau _m(t)), \int(t,s,x(s))ds),~~t \geq t_0 \geq 0,\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINEx(t)= \Phi (t-t_0 ),~~t_0- \tau \leq t \leq t_0,~~ \tau_1 =\sup_{ t \geq t_0 } \tau _i (t),~~h =\sup_{ t \geq t_0 } h (t),~~ \tau =\max_{ i=1,2,\dots,m } \{\tau _1 ,~h \},NEWLINE\]NEWLINE satisfying some basic assumptions.NEWLINENEWLINEThey study the global exponential convergence properties of the solution of the above equation through the unique solution of the corresponding differential equation NEWLINE\[NEWLINEy'(t)= f(t,y(t), y(t),\dots, y(t),0),\,t \geq t_0,\,t(t_0)=\Phi (0).NEWLINE\]NEWLINENEWLINENEWLINEThey assume that the unique solution of the latter equation is globally uniformly exponentially convergent to the ball \(B_r\) and prove that the System (1) is globally uniformly exponentially convergent to the certain ball provided the delay is quite small. They illustrate their results with an example.