Abstract
To prove the existence and uniqueness (or only the existence) of a bounded solution of a weakly nonlinear differential equation of the nth order, we use both the contraction mapping principle and the Tikhonov fixed-point theorem. We obtain an important qualitative estimate for the smallness of a nonlinear perturbation preserving the basic characteristics of the behavior of a linear equation (absolute stability or exponential dichotomy) under the passage to the nonlinear equation.
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