differential equation; Banach space; existence; uniqueness; boundedness; bounded solution; derivative of the norm of a linear mapping; fixed point
The properties of solutions of the nonlinear differential equation $x'=A(s)x+f(s,x)$ in a Banach space and of the special case of the homogeneous linear differential equation $x'=A(s)x$ are studied. Theorems and conditions guaranteeing boundedness of the solution of the nonlinear equation are given on the assumption that the solutions of the linear homogeneous equation have certain properties.
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