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The existence of bounded solutions for equations x′ = A(t)x + r(x, t) is proved, where the linear part is exponentially dichotomic and the nonlinear term r satisfies some weak conditions. Introduction. We shall deal with the nonlinear differential equations of the form (1) x′ = A(t)x+ r(x, t) where A : R → L(H) is a continuous function taking values in the space L(H) of bounded linear operators in a Hilbert space H and r : H × R→ H is a nonlinear continuous mapping. We shall assume that the linear equation x′ = A(t)x is exponentially dichotomic, hence there exists the Main Green Function G : R→ L(H) such that (2) ||G(t, s)|| ≤ Ne−α|t−s| for all t, s ∈ R, where N and α are some positive constants. The bounded solutions of (1) are the fixed points of the following operator S : BC(R, H)→ BC(R, H):

The existence of bounded solutions for differential equations in Hilbert spaces