QUALITATIVE THEOR OF SOLUTION OF DIFFERENTIAL EQUATIONS WITH DELAYS

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Let B = B((-∞, 0] ; X), where X is a complete metric space. Consider functional differential equations with infinite delay.du/dt= Au(t) + L(t, u_t),where A is an infinitesimal generator of a compact semigroup of bounded lineat operators and u_t is an element of B defined by u_t(s) = u(t + s), s ∈ (-∞, 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 generalization of partial differential equations, above equation must be considered a functional partial differential equation with infinite delay. In this report, we have the following for the above equations.(i) Equivalent relationship between ρ-stabilities and BC-stabilities are shown.(ii) BC-total stability for a limiting equation implies that of the given euqation. BC-uniform asymptotic stabilituy also has the same property, under the uniqueness of the solution for the initial value problem.(iii) BC-total stability and BC-uniform asymtotic stability implies the existence of almost periodic solution of almost periodic euquations under the assumption of the existence of a bounded solution.