In this paper we present results concerning the existence of solutions in certain abstract semilinear equations and the asymptomatic behavior of these solutions. We also study the sufficient conditions of the conditional stability for the zero solution. The basic techniques used are Lyapunov-like methods and fixed point theory for differential equations in Banach space. However, the improvement here is that it can be applied directly to non-linear equations and hence, we do not need to assume that the differential equation can be linearized.
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