Individual stability and instability for evolutionary processes (Q682113)

The authors consider an evolutionary process defined by a linear bounded operator \(\Phi(t,t_0)\), \(t\geq t_0\), on a Banach space \(X\). They are interested to study the convergence to zero or the divergence of a single trajectory \(\Phi(t,t_0)x\), when \(t\to +\infty\). Under some technical assumptions, the main result states that if there exists \(k(t_0)\) such that \[ \left( \int_t^\infty | | \Phi(\tau,t_0)x| | ^p\, d\tau \right)^{1/p}\leq k(t_0) | | \Phi(t,t_0)x| | \] for each \(t\geq t_0\), then the trajectory converges to zero with exponential decay. A similar result is given for the divergence of the trajectory \(\Phi(t,t_0)x\).