Lurie-type abstract functional differential equations (Q2785866)

The paper deals with stability of solutions of abstract functional differential equations NEWLINE\[NEWLINE{du\over dt}= Au+ L(t,u_t)+ f(t,u_t),\quad t\geq t_0\geq 0,\quad u=\varphi(t),\quad t\in[t_0-r,t_0],NEWLINE\]NEWLINE where \(u= (u_1,\dots,u_n)^T\), \(u_t= u(t+\theta)\), \(-r\leq\theta\leq 0\), \(A\) is a linear operator on \(X^n\), \(X\) is a real Banach space. The authors use Lyapunov functionals and comparison theorems to obtain stability criteria on partial variables for the above problem. Further, these results are used to study the stability of the Lurie problem. Finally, the existence of periodic and almost periodic solutions of the Lurie problem with parameter is discussed.