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In this paper we prove that the exponential dichotomy for evolution equations in Banach spaces is not destroyed, if we perturb the equation by``small'' unbounded linear operator. This is done by employing a skew-product semiflow technique and a perturbation principle from linear operator theory. Finally, we apply these results to partial parabolic equations and functional differential equations. AcademicPress, Inc.

This paper is primarily concerned with the Dynamical Spectrum for time dependent linear differential equations in Banach spaces. We give a characterization of the Dynamical Spectrum which is an extension of the Sacker-Sell Theorem. Also we define the Lyapunov exponents, who measure the decay rate of the solutions of a linear differential equations" we investigate the relation between the Dynamical Spectrum, the Spectral Subbundles associated with the corresponding spectral intervals and the Lyapunov exponents. These problems are treated in the unified setting of a Linear Skew-Product Semiflow. Finally we present some examples of Linear Skew-Prodnct Semiflow arising from time dependent functional differential equations and parabolic partial differential equations.

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In this paper we apply Rothe's type fixed-point theorem to prove the controllability of the following semilinear impulsive nonautonomous systems of differential equationsare continuous matrices of dimension n × n and n × m, respectively, the control function u belongs to C(0, τ ; R m ) andUnder additional conditions we prove the following statement: if the linearź(t) = A(t)z(t) + B(t)u(t) is controllable on [0, τ ], then the semilinear impulsive system is also controllable on [0, τ ]. Moreover, we could exhibit a control steering the nonlinear system from an initial state z 0 to a final state z 1 at time τ > 0.