CIASSIFICATIONS OF FONC TIONAL DIFFERENTIAL EQUATIONS BY PROCESSES

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Let B = B((-*, 0] ; X), where X is a complete metric space. Consider functional differential equations with infinite delaydu/dt = AIOTA(t) + L(t, IOTA_t),where A is an infinitesimal generator of a compact semigroup of bounded lineat operators and IOTA_t is an element of B defined by IOTA_t (s) =IOTA_t (t + s), s epsilon (-*, 0).This phase space has two types, that is, one is a uniform fading memory space and the other is a fading memory space. Furthermore, if we consider an evolution equations which is a genaralization of partial differential equations, above equation must be considered a functional partial differential equation with infinite delay. In this report, we have the followings :(i) Stability theoris are characterized by processes. We have a new concept of process which is a generalization of processes and we called it a quasi-process.(ii) Nonlinear oscillations are characterized by processes. In particular, the existence of an almost periodic integral of almost periodic processes is given.Inaba has given an example of a flow on a non-compact surface without minimal set. This result has given important informations to solve a problem (ii).