We establish an averaging principle on the real semi-axis for semilinear equation x′ = ε(Ax + f(t) + F(t, x)) (1) with unbounded closed linear operator A and asymptotically Poisson stable (in particular, asymptotically stationary, asymptotically periodic, asymptotically quasi-periodic, asymptotically almost periodic, asymptotically almost automorphic, asymptotically recurrent) coefficients (see [1] for details). Under some conditions we prove that there exists at least one solution, which possesses the same asymptotically recurrence property as the coefficients, in a small neighborhood of the stationary solution to the averaged equation, and this solution converges to the stationary solution of averaged equation uniformly on the real semi-axis when the small parameter approaches to zero. Consider the following differential equation x′(t) = Ax(t) + f(t) + F(t, x(t)), (2) where f ∈ C(R+,B), F ∈ C(R+ × B,B) and A : D(A) → B is a linear operator acting from D(A) ⊆ B to B.We will consider a differential equation (2) when the linear operator A is an infinitesimal operator which generates a C0-semigroup {U(t)}t≥0. Definition. A semigroup of operators {U(t)}t≥0 is said to be hyperbolic if there is a projection P and constants N, ν > 0 such that each U(t) commutes with P, U(t) : ImQ → ImQ is invertible and for every x ∈ E |U(t)Px| ≤ Ne−νt|x|, for t ≥ 0; |UQ(t)x| ≤ Neνt|x|, for t < 0; where Q := I −P and, for t < 0, UQ(t) := [U(−t)Q]−1. Denote by Ψ the family of all decreasing, positive bounded functions ψ : R+ → R+ with lim t→+∞ ψ(t) = 0. Below we will use the following conditions: (G1): F(t, 0) = 0 for any t ≥ 0; (G2): there exists a positive constant L such that |F(t, x1) − F(t, x2)| ≤ L|x1 − x2| for any x1,x2 ∈ B and t ∈ R+; (G3): there exists functions ¯ F ∈ C(B,B) and ω : R+ × R+ → R+ (respectively, an element ¯ f ∈ B and function ω ∈ Ψ) such that 1 T  t+T t [F, x) − ¯ F(x)]dt ≤ ω(T, r) (respectively, 1 T t+T t [f(s) − ¯ f]dt  ≤ ω(T) )(s and ω(·, r) ∈ Ψ for any t ∈ R+, T > 0, r > 0 and x ∈ B[0, r].The standard form of (2) is x′(t) = ε(Ax(t) + f(t) + F(t, x(t))). (3) We will consider also the following equations x′(t) = Ax(t) + f( t ε ) + F( t ε ,x(t))), (4) where fε(t) := f( t ε ) (respectively, Fε(t, x) := F( t ε ,x)) for any t ∈ R+ (respectively, for any (t, x) ∈ R+×B), ε ∈ (0, ε0] and ε0 is some fixed small positive number. Along with equations (3)-(4) we will consider the following averaged differential equation x′(t) = Ax(t) + ¯ f + ¯ F(x(t)). (5) Theorem. Suppose that the following conditions hold: 1. −A is a sectorial hyperbolic operator; 2. the function F satisfies conditions (G1)-(G3); 3. the functions f and F are Lagrange stable; 4. L < ν 2N , where N and ν there are the numbers figuring in the Definition. Then there exists a positive number ε0 such that for any 0 < ε ≤ ε0 1. equation (4) has a unique solution ψε ∈ Cb(R+,B) with Pψε(0) = 0;2. if the function f ∈ Cb(R+, B) is asymptotically stationary (respectively, τ -periodic, quasi-periodic, Bohr almost periodic, Bohr almost automorphic, Birkhoff recurrent, positively Lagrange stable), then equation (1) has a unique solution φε ∈ Cb(R+,B) with Pφε(0) = 0 which is asymptotically stationary (respectively, τ -periodic, quasi-periodic, Bohr almost periodic, Bohr almost automorphic, Birkhoff recurrent, positively Lagrange stable); 3. lim ε→0 sup t∈R+ |ψε(t) − ¯ ψ| = 0, where ¯ ψ is a unique stationary solution of equation (5). 

We establish an averaging principle on the real semi-axis for semilinear equation x′ = ε(Ax + f(t) + F(t, x)) (1) with unbounded closed linear operator A and asymptotically Poisson stable (in particular, asymptotically stationary, asymptotically periodic, asymptotically quasi-periodic, asymptotically almost periodic, asymptotically almost automorphic, asymptotically recurrent) coefficients (see [1] for details). Under some conditions we prove that there exists at least one solution, which possesses the same asymptotically recurrence property as the coefficients, in a small neighborhood of the stationary solution to the averaged equation, and this solution converges to the stationary solution of averaged equation uniformly on the real semi-axis when the small parameter approaches to zero. Consider the following differential equation x′(t) = Ax(t) + f(t) + F(t, x(t)), (2) where f ∈ C(R+,B), F ∈ C(R+ × B,B) and A : D(A) → B is a linear operator acting from D(A) ⊆ B to B.We will consider a differential equation (2) when the linear operator A is an infinitesimal operator which generates a C0-semigroup {U(t)}t≥0. Definition. A semigroup of operators {U(t)}t≥0 is said to be hyperbolic if there is a projection P and constants N, ν > 0 such that each U(t) commutes with P, U(t) : ImQ → ImQ is invertible and for every x ∈ E |U(t)Px| ≤ Ne−νt|x|, for t ≥ 0; |UQ(t)x| ≤ Neνt|x|, for t < 0; where Q := I −P and, for t < 0, UQ(t) := [U(−t)Q]−1. Denote by Ψ the family of all decreasing, positive bounded functions ψ : R+ → R+ with lim t→+∞ ψ(t) = 0. Below we will use the following conditions: (G1): F(t, 0) = 0 for any t ≥ 0; (G2): there exists a positive constant L such that |F(t, x1) − F(t, x2)| ≤ L|x1 − x2| for any x1,x2 ∈ B and t ∈ R+; (G3): there exists functions ¯ F ∈ C(B,B) and ω : R+ × R+ → R+ (respectively, an element ¯ f ∈ B and function ω ∈ Ψ) such that 1 T  t+T t [F, x) − ¯ F(x)]dt ≤ ω(T, r) (respectively, 1 T t+T t [f(s) − ¯ f]dt  ≤ ω(T) )(s and ω(·, r) ∈ Ψ for any t ∈ R+, T > 0, r > 0 and x ∈ B[0, r].The standard form of (2) is x′(t) = ε(Ax(t) + f(t) + F(t, x(t))). (3) We will consider also the following equations x′(t) = Ax(t) + f( t ε ) + F( t ε ,x(t))), (4) where fε(t) := f( t ε ) (respectively, Fε(t, x) := F( t ε ,x)) for any t ∈ R+ (respectively, for any (t, x) ∈ R+×B), ε ∈ (0, ε0] and ε0 is some fixed small positive number. Along with equations (3)-(4) we will consider the following averaged differential equation x′(t) = Ax(t) + ¯ f + ¯ F(x(t)). (5) Theorem. Suppose that the following conditions hold: 1. −A is a sectorial hyperbolic operator; 2. the function F satisfies conditions (G1)-(G3); 3. the functions f and F are Lagrange stable; 4. L < ν 2N , where N and ν there are the numbers figuring in the Definition. Then there exists a positive number ε0 such that for any 0 < ε ≤ ε0 1. equation (4) has a unique solution ψε ∈ Cb(R+,B) with Pψε(0) = 0;2. if the function f ∈ Cb(R+, B) is asymptotically stationary (respectively, τ -periodic, quasi-periodic, Bohr almost periodic, Bohr almost automorphic, Birkhoff recurrent, positively Lagrange stable), then equation (1) has a unique solution φε ∈ Cb(R+,B) with Pφε(0) = 0 which is asymptotically stationary (respectively, τ -periodic, quasi-periodic, Bohr almost periodic, Bohr almost automorphic, Birkhoff recurrent, positively Lagrange stable); 3. lim ε→0 sup t∈R+ |ψε(t) − ¯ ψ| = 0, where ¯ ψ is a unique stationary solution of equation (5).