Stability and instability of a semilinear functional differential equation with finite delay in Banach spaces (Q629750)

The paper is concerned with the functional differential equation \[ u'(t)=Lu_t + F(t,u_t), \quad t\geq 0, \] where \(L:C([-r,0];E)\rightarrow E\) is a linear continuous operator, \(E\) is a Banach space, \(r>0\) and \(F:[0,\infty)\times C([-r,0];E) \rightarrow E\) is a completely continuous operator, with \(F(t,0)=0\). Stability and instability of the null solution of the above equation is studied by using the variation of constants formula involving the semigroup associated with the linear problem \[ u'(t)=Lu(t), \quad t\geq 0; \;u_0=\phi \in C([-r,0);E). \]