This first part in a series of articles deals with the behavior of trajectories of a semiflow \(F_t= F(\cdot,t)\): \(U\times \mathbb{R}\to E\) \((E\) is a Banach space, \(U\) a suitable neighborhood of 0 in \(E\), \(F(\cdot, 0)=\text{Id}\), and \(F_t\circ F_s= F_{t+s}\) whenever \(F_t\), \(F_s\) and \(F_{t+s}\) are all defined) in a neighborhood of a fixed point \(u_0\) of \(F(\cdot,t)\) \((F(u_0,t) =u_0)\). A new abstract version of the linearization principle is formulated and proved: if some modest continuity conditions for \(F(\cdot,t)\) are satisfied and if the linearized systems of the semiflow have eigenvalues all with negative real parts, then the fixed point \(u_0\) is locally asymptotically stable, and, for some neighborhood of the fixed point, the global existence of solutions holds.
This new version is a generalization of the earlier result by M. Potier-Ferry [Arch. Ration. Mech. Anal. 77, 301-320 (1981; Zbl 0497.35006)] and more convenient in some applications (which will be considered in Part II). This general principle is applied to the equation \[ {du\over dt}=A(u)u+g(u); \] the checking of the assumption is based on Sobolevskij’s classical results on abstract parabolic equations in Banach spaces.