Instability of nonautonomous state-dependent delay integro-differential systems
Consider the system with state-dependent time delay \[ \begin{multlined} u^\prime(t)=A(t)\left( u(t)+\int_{t_0}^t G(t,s)u(s)\,ds\right) \\+B(t)\left( u(t-r(t,u(t)))+\int_{t_0}^t G(t,s)u(s-r(s,u(s)))\,ds \right), \end{multlined} \] \[ u_{t_0}=\phi, \] where \(G(t,s)\) is continuous on \([t_0,\infty)\times [t_0,\infty)\), \(r(t,u(t))\) is a nonnegative bounded delay function, \(A,\,B\in C([t_0-\tau,\infty],\mathbb{R}^{n\times n})\) for \(\tau>0\), \(\phi\in C([-\tau,\,0],\mathbb{R}^n)\). The authors derive sufficient conditions for the instability of the zero solution based for the concept of \((h,k)\) dichotomies.
scientific article; zbMATH DE number 5116302