Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. I

ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS OF LINEAR DIFFERENTIAL/DIFFERENCE EQUATIONS WITHOUT FAVARD’S SEPARATION CONDITION. I TOM´ AS CARABALLO AND DAVID CHEBAN Abstract. The well known Favard’s Theorem states that the linear differential equation (1) x0=A(t)x+f(t) with Bohr almost periodic coefficients admits at least one Bohr almost periodic solution if it has a bounded solution. The main assumption in this theorem is the separation among bounded solutions of homogeneous equations (2) x0=B(t)x, where B∈H(A) := {B|B(t) = lim n→+∞A(t+tn)}.If there are bounded solutions which are non-separated, sometimes almost periodic solutions do not exist (R. Johnson, R. Ortega and M. Tarallo, V. Zhikov and B. Levitan). In this paper we prove that linear differential equation (1) with Levitan almost periodic coefficients has a unique Levitan almost periodic solution, if it has at least one bounded solution, and the bounded solutions of the homogeneous equation (3) x0=A(t)x are homoclinic to zero (i.e. lim |t|→+∞|ϕ(t)|= 0 for all bounded solutions ϕ of (3)). If the coefficients of (1) are Bohr almost periodic and all bounded solutions of all limiting equations (2) are homoclinic to zero, then equation (1) admits a unique almost automorphic solution. The analogue of this result for difference equations is also given. We study the problem of existence of Bohr/Levitan almost periodic solutions of equation (1) in the framework of general non-autonomous dynamical systems (cocycles). Dedicated to Russell Johnson on his 60th birthday 1. Introduction Recall (see, for example, [16, 20]) that a continuous function ϕdefined on the real axis Rwith values in a Banach space Eis called Bohr almost periodic, if for all ε > 0 there exists a positive number l(ε) such that on every interval [a, a+l] (a∈R) Date: April 1, 2008. 1991 Mathematics Subject Classification. 34C27, 34K14, 37B20, 37B55, 39A11. Key words and phrases. almost periodic solution; almost automorphic solutions; nonautonomous dynamical systems; Favard’s condition; cocycle. 1 2 TOM´ AS CARABALLO AND DAVID CHEBAN there exists at least one number τsuch that |ϕ(t+τ)−ϕ(t)|< ε for all t∈R(the number τis called an εalmost period of function ϕ). The following result is well known. Theorem 1.1. (Bochner’s theorem). A continuous function ϕ:R7→ Eis Bohr almost periodic if and only if from each sequence {tn} ⊂ Rthere can be extracted a subsequence {tnk}such that the functional sequence {ϕ(t+tnk)}converges uniformly on the real axis R. A continuous function ϕ:R7→ Eis called [20] Levitan almost periodic, if there exists a Bohr almost periodic function ψ:R7→ F(Fis a Banach space) such that Nψ⊆Nϕ,where Nϕis the family of all sequences {tn} ⊂ Rsuch that the functional sequence {ϕ(t+tn)}converges to ϕuniformly on every compact subset from R. It is evident that every Bohr almost periodic function is Levitan almost periodic. The inverse statement is not true. For example, the function ϕ(t) := (2 + sin t+ sin √2t)−1is Levitan almost periodic, but not Bohr almost periodic [20]. A continuous function ϕ:R7→ Eis called [4] (see also [20, 23, 24]) almost automorphic (or Bohr almost automorphic) if for every sequence {t0 n}there exists a subsequence {tn}for which we have local convergence (i.e. uniform convergence on every compact subset of R) ϕ(t+tn)→˜ϕ(t), the “returning” also holds: ˜ϕ(t−tn)→ϕ(t). It is known (see, for example, [20] and also [9]) that every almost automorphic function is Levitan almost periodic. The inverse, generally speaking, is not true because almost periodic functions are bounded, but a Levitan almost periodic function may be unbounded. Recall also that any Bohr almost periodic function is almost automorphic. This paper is concerned with the study of linear differential (difference) equations with Bohr/Levitan almost periodic and almost automorphic coefficients. This field is called Favard’s theory [20, 39], due to the fundamental contributions made by J. Favard [13]. In 1927, J. Favard published his celebrated paper, where he studied the existence of almost periodic solutions of the following equation in RN: (4) x0=A(t)x+f(t), where the matrix A(t) and the vector-function f(t) are almost periodic in the sense of Bohr (see, for example, [16, 20]). Along with equation (4), consider the homogeneous equation (5) x0=A(t)x and the corresponding family of limiting equations (6) x0=B(t)x, ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 3 where B∈H(A),and H(A) denotes the hull of almost periodic matrix A(t) which is composed by those functions B(t) obtained as uniform limits on Rof the type B(t) := lim n→∞ A(t+tn),where {tn}is some sequence in R. From now on, a bounded function on Rwill be simply called a bounded function. Theorem 1.2. (Favard’s theorem [13]) The linear differential equation (4) with Bohr almost periodic coefficients admits at least one Bohr almost periodic solution if it has a bounded solution, and each bounded solution ϕ(t)of every limiting equation (6) (B∈H(A)) is separated from zero, i.e. inf t∈R|ϕ(t)|>0. Remark 1.3. Under the conditions of Favard’s theorem, if ϕ1and ϕ2are two different bounded solutions of the same non-homogeneous limiting equation (7) x0=B(t)x+g(t) ((B, g)∈H(A, g)), then inf t∈R|ϕ1(t)−ϕ2(t)|>0. Remark 1.4. 1. Favard’s theorem is extended to the case of almost automorphic coefficients in the works of L. Faxing [15], and W. Shen and Y. Yi [35]. 2. For equations with Levitan almost periodic coefficients, Favard’s theorem was generalized by B. Levitan [19], B. Levitan and V. Zhikov [20, 39], M. Lyubarskii [21], B. Shcherbakov [33] and M. Shubin [36, 37]. Remark 1.5. In this paper we analyze the existence of Levitan/Bohr almost periodic and almost automorphic solutions of linear differential equations in the framework of a more general problem. Namely, we study the existence of Poisson stable solutions (in particular, periodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkgoff, Levitan almost periodic, almost recurrent in the sense of Bebutov, Poisson stable) of linear differential equations with Poisson stable coefficients. The notions of comparability and uniform comparability of motions by the character of recurrence, introduced by B. Shcherbakov [31]–[34], are powerful tools which we will use to study this problem. Zhikov and Levitan [20, 39] (see also Johnson [18], Ortega and Tarallo [25], Sell [29]) constructed examples of scalar linear differential equations for which all the solutions are bounded, but none of them is almost periodic. In particular, the following result was established in [25]. Theorem 1.6. (Ortega and Tarallo [25]) Let (5) be a linear differential equation with Bohr almost periodic coefficients, and for some B∈H(A)each nontrivial bounded on Rsolution ϕof the equation (6) is homoclinic to zero, i.e. lim |t|→+∞|ϕ(t)|= 0. Then, there exists an almost periodic (in the sense of Bohr) function g:R7→ Rn such that equation (7) has bounded solutions, but none of them is Bohr almost periodic. 4 TOM´ AS CARABALLO AND DAVID CHEBAN From one of our main result (see Section 4, Theorem 4.1 and Corollary 4.4) it follows that, under the conditions of Theorem 1.6, equation (7) has a unique almost automorphic solution, which is a positive statement in contrast to the one in the previous theorem. This paper is organized as follows. In Section 2 we collect some well known facts from the theory of dynamical systems (both autonomous and non-autonomous). Namely, the notions of almost periodic (in the senses of Bohr and Levitan), almost automorphic and recurrent motions; shift dynamical systems and almost periodic and almost automorphic functions; cocycles, skew-product dynamical systems, and general non-autonomous dynamical systems. Section 3 is devoted to the existence of motions comparable (respectively, uniformly comparable) by the character of recurrence in the sense of Shcherbakov [31]-[34]. The main results of this section are Theorems 3.6, 3.12 and Corollaries 3.9–3.10, 3.13 (which are, in fact, the main abstract results of the paper) which provide sufficient conditions for the existence of a unique motion comparable (respectively, uniformly comparable) by the character of recurrence for two-sided non-autonomous dynamical systems. In Section 4 we analyze the compatible (respectively, uniformly compatible) solutions of ordinary differential and difference equations in a Banach space. Here we present a test for the existence of Bohr (respectively, Levitan) almost periodic and almost automorphic solutions of non-homogeneous linear differential/difference equations with Bohr (respectively, Levitan) almost periodic and almost automorphic coefficients. 2. Almost Periodic and Almost Automorphic Motions of Dynamical Systems Let us collect in this section some well known concepts and results from the theory of dynamical systems which will be necessary for our analysis in this paper. 2.1. Recurrent, Almost Periodic and Almost Automorphic Motions. Let Xbe a complete metric space, R(Z) be the group of real (integer) numbers. By T we will denote either Ror Z. Let (X, T, π) be a dynamical system on X, i.e. let π:T×X→Xbe a continuous function such that π(0, x) = xfor all x∈X, and π(t1+t2, x) = π(t2, π(t1, x)), for all x∈X, and t1, t2∈T. Given ε > 0,a number τ∈Tis called an ε−shift (respectively, an ε−almost period) of x, if ρ(π(τ, x), x)< ε (respectively, ρ(π(τ+t, x), π(t, x)) < ε for all t∈T). A point x∈Xis called almost recurrent (respectively, Bohr almost periodic), if for any ε > 0 there exists a positive number lsuch that in any segment of length l there is an ε−shift (respectively, an ε−almost period) of the point x∈X. ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 5 If the point x∈Xis almost recurrent, and the set H(x) := {π(t, x)|t∈T}is compact, then xis called recurrent, where the bar denotes the closure in X. Denote by Nx:= {{tn} ⊂ T: such that {π(tn, x)} → xand {tn} → ∞}. A point x∈Xis said to be Levitan almost periodic (see [20]) for the dynamical system (X, T, π) if there exists a dynamical system (Y, T, λ),and a Bohr almost periodic point y∈Ysuch that Ny⊆Nx. Remark 2.1. Let xi∈Xi(i= 1,2, . . . , m) be a Levitan almost periodic point of the dynamical system (Xi,T, πi).Then the point x:= (x1, x2, . . . , xm)) ∈X:= X1×X2×. . . ×Xmis also Levitan almost periodic for the product dynamical system (X, T, π),where π:T×X→Xis defined by the equality π(t, x) := (π1(t, x1), π2(t, x2), . . . , πm(t, xm)) for all t∈Tand x:= (x1, x2, . . . , xm)∈X. A point x∈Xis called stable in the sense of Lagrange (st.L), if its trajectory {π(t, x) : t∈T}is relatively compact. A point x∈Xis called almost automorphic [20, 35] for the dynamical system (X, T, π),if the following conditions hold: (i) xis st.L; (ii) there exists a dynamical system (Y, T, λ),a homomorphism hfrom (X, T, π) onto (Y, T, λ) and an almost periodic (in the sense of Bohr) point y∈Y such that h−1(y) = {x}. Remark 2.2. The following facts hold true. 1. Every almost automorphic point x∈Xis also Levitan almost periodic. 2. A Levitan almost periodic point xwith relatively compact trajectory {π(t, x)t∈ T}is also almost automorphic (see [2]–[5],[10],[22] and [35]). In other words, a Levitan almost periodic point xis almost automorphic, if and only if its trajectory {π(t, x)t∈T}is relatively compact. 3. Let (X, T, π)and (Y, T, λ)be two dynamical systems, x∈X, and assume that the following conditions are fulfilled: (i) there exists a point y∈Ywhich is Levitan almost periodic; (ii) Ny⊆Nx. Then, the point xis also Levitan almost periodic. 4. Let x∈Xbe a st.Lpoint, y∈Yan almost automorphic point, and Ny⊆Nx. Then, the point xis almost automorphic. 2.2. Shift Dynamical Systems, Almost Periodic and Almost Automorphic Functions. We show below a general method for the construction of dynamical systems on the space of continuous functions. In this way we will obtain many well known dynamical systems on functional spaces (see, for example, [5, 31]). Let (X, T, π) be a dynamical system on X, Y be a complete pseudo metric space, and Pbe a family of pseudo metrics on Y. We denote by C(X, Y ) the family of all continuous functions f:X→Yequipped with the compact-open topology. This 6 TOM´ AS CARABALLO AND DAVID CHEBAN topology is given by the following family of pseudo metrics {dp K}(p∈P, K ∈ K(X)),where dp K(f, g) := sup x∈K p(f(x), g(x)), and K(X) denotes the family of all compact subsets of X. For every τ∈Twe define a mapping στ:C(X, Y )→C(X, Y ) by the following equality: (στf)(x) := f(π(τ, x)), x∈X. We note that the family of mappings {στ:τ∈T}possesses the next properties: a. σ0=idC(X,Y ); b. στ1◦στ2=στ1+τ2∀τ1, τ2∈T, c. στis continuous, ∀τ∈T. Furthermore, the next lemma ensures that (C(X, Y ),T, σ) is a dynamical system. Lemma 2.3. [8] The mapping σ:T×C(X, Y )→C(X, Y ),defined by the equality σ(τ, f) := στf(f∈C(X, Y ), τ ∈T)is continuous, and, consequently, the triple (C(X, Y ),T, σ)is a dynamical system on C(X, Y ). Consider now some examples of dynamical systems given by the form (C(X, Y ),T, σ) which are useful in applications. Example 2.4. Let X=T, and denote by (X, T, π)a dynamical system on T, where π(t, x) := x+t. The dynamical system (C(T, Y ),T, σ)is called Bebutov’s dynamical system [31] (a dynamical system of translations, or shifts dynamical system). For example, the equality d(f, g) := sup L>0 max{dL(f, g), L−1}, where dL(f, g) := max |t|≤Lρ(f(t), g(t)),defines a complete metric ( Bebutov’s metric) on the space C(T, Y )which is compatible with the compact-open topology on C(T, Y ). Remark 2.5. It is known [31, 34] that d(f, g)< ε (respectively, d(f, g)> ε or d(f, g) = ε) is equivalent to the inequality d1 ε(f, g)< ε (respectively, d1 ε(f, g)> ε or d1 ε(f, g) = ε). It is said that the function ϕ∈C(T, Y )possesses a property (A), if the motion σ(·, ϕ) : T→C(T, Y ) possesses this property in the Bebutov’s dynamical system (C(T, Y ),T, σ), generated by the function ϕ. As property (A) we can consider periodicity, quasi-periodicity, almost periodicity, almost automorphy, recurrence etc. Example 2.6. Let X:= T×W, where Wis a metric space, and let (X, T, π) denote a dynamical system on Xdefined in the following way: π(t, (s, w)) := (s+ t, w). Using the general method proposed above we can define a dynamical system of translations (C(T×W, Y ),T, σ)on C(T×W, Y ). The function f∈C(T×W, Y )is called almost periodic (recurrent, almost automorphic, etc.) with respect to t∈T, uniformly with respect to w∈Won every compact from W, if the motion σ(·, f)is almost periodic (recurrent, almost automorphic, etc.) in the dynamical system (C(T×W, Y ),T, σ). ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 7 Remark 2.7. Let Wbe a compact metric space, then the topology on C(W, Y )is metrizable. For example, the equality d(f, g) := ∞ X k=1 1 2k dk(f, g) 1 + dk(f, g),where dk(f, g) := max |t|≤k, x∈Wρ(f(t, x), g(t, x)), defines a complete metric on the space C(W, X)which is compatible with the compactopen topology on C(W, X).The space C(T×W, Y )is topologically isomorphic to C(T, C(W, Y )) [31], and also the shifts dynamical systems (C(T×W, Y ),T, σ)and (C(T, C(W, Y )),T, σ)are dynamically isomorphic. 2.3. Cocycles, Skew-Product Dynamical Systems and Non-Autonomous Dynamical Systems. Consider now two dynamical systems (X, T, π) and (Y, T, λ). A triplet h(X, T, π),(Y, T, λ), hiis called a non-autonomous dynamical system if h is a homomorphism from (X, T, π) onto (Y, T, λ). Let (Y, T, λ) be a dynamical system on Y,Wa complete metric space, and ϕa continuous mapping from T×W×Yin Wpossessing the following properties: a. ϕ(0, u, y) = u(u∈W, y ∈Y); b. ϕ(t+τ, u, y) = ϕ(τ, ϕ(t, u, y), λ(t, y)) (t, τ ∈T, u ∈W, y ∈Y). Then, the triplet hW, ϕ, (Y, T, λ)i(or shortly ϕ) is called [28] a cocycle on (Y, T, λ) with the fiber W. Given a cocycle hW, ϕ, (Y, T, λ)i, let us set X:= W×Y, and define a mapping π:T×X→Xas follows: π(t, (u, y)) := (ϕ(t, u, y), λ(t, y)) (i.e. π= (ϕ, λ)). Then, (X, T, π) is a dynamical system on X, which is called a skew-product dynamical system [28], and h=pr2:X→Yis a homomorphism from (X, T, π) onto (Y, T, λ) and, hence, h(X, T, π),(Y, T, λ), hiis a non-autonomous dynamical system. Thus, if hW, ϕ, (Y, T, λ)iis a cocycle on the dynamical system (Y, T, λ) with the fiber W, then it generates a non-autonomous dynamical system h(X, T, π),(Y, T, λ), hi (X:= W×Y), called a non-autonomous dynamical system generated by the cocycle hW, ϕ, (Y, T, λ)ion (Y, T, λ). Non-autonomous dynamical systems (cocycles) play a very important role in the study of non-autonomous evolutionary differential equations. Under appropriate assumptions, every non-autonomous differential equation generates a cocycle (a non-autonomous dynamical system). 3. Comparability and Uniform Comparability of Motions by the Character of Recurrence in the Sense of Shcherbakov We will prove now the main abstract results in this paper. First, we start with the following definitions. Let (Ω,T, λ) be a dynamical system. A point ω∈Ω is said to be (see, for example, [34] and [38]) positively (respectively, negatively) stable in the sense of Poisson, if there exists a sequence tn→+∞(respectively, tn→ −∞) such that λ(tn, ω)→ω If the point ωis Poisson stable in both directions, it is called Poisson stable. 8 TOM´ AS CARABALLO AND DAVID CHEBAN Denote by Nω={{tn} ⊂ T|λ(tn, ω)→ω, as n→+∞}. Let (X, h, Ω) be a fiber space, i.e. Xand Ω are two metric spaces and h:X→Ω is a homomorphism from Xonto Ω. The subset M⊆Xis said to be conditionally relatively compact [7, 8] if the pre-image h−1(Ω0)TMof every relatively compact subset Ω0⊆Ω is a relatively compact subset of X, in particular Mω:= h−1(ω)TM is relatively compact for every ω. The set Mis called conditionally compact if it is closed and conditionally relatively compact. Example 3.1. Let Kbe a compact space, X:= K×Ω, and consider h=pr2: X→Ω.Then, the triplet (X, h, Ω) is a fiber space, the space Xis conditionally compact, but it is not compact. The following result characterizes when a closed set is conditionally compact. Lemma 3.2. Let Mbe a closed subset of X. Then, Mis conditionally compact with respect to (X, h, Ω) if and only if the following conditions hold: (i) the set Mω:= h−1(ω) = {x∈M|h(x) = ω}is compact for all ω∈Ω; (ii) the mapping ω7→ Mωis upper semi-continuous. Proof. Necessity. If the closed set Mis conditionally compact, then the set Mω is evidently compact for all ω. Assume now that ωn→ωas n→+∞, and take xn∈Mωn. Since the set Mis conditionally compact, we can assume, without loss of generality, that the sequence {xn}is convergent. Denote by xits limit, then h(x) = lim n→∞ h(xn) = lim n→∞ ωn=ωand, consequently, x∈Mω, i.e. the mapping ω7→ Mωis upper semi-continuous. Sufficiency. Let Ω0⊆Ω be an arbitrary compact set. Since the mapping ω7→ Mωis upper semi-continuous, then the set M∩h−1(Ω0) = ∪{Mω|ω∈Ω0}is compact. ¤ The next lemma provides some examples of conditionally compact sets for a nonautonomous dynamical system. Lemma 3.3. Let hW, ϕ, (Ω,T, λ)ibe a cocycle and h(X, T, π),(Ω,T, λ), hibe a nonautonomous dynamical system associated to the cocycle ϕ. Suppose that x0:= (u0, ω0)∈X:= W×Ω,and that the set Q(u0,ω0):= {ϕ(t, u0, ω0)|t∈T}(respectively, Q+ (u0,ω0):= {ϕ(t, u0, ω0)|t∈T+},where T+:= {t∈T|t≥0}) is compact. Then, the set H(x0) := {π(t, x0)|t∈T}(respectively, {π(t, x0)|t∈T+}:= H+(x0)) is conditionally compact. Proof. Let Ω0be an arbitrary compact subset of Ω,and {xn}an arbitrary sequence in H(x0) (respectively, in H+(x0)). Then, for any n∈Nthere exists tn∈T(respectively, tn∈T+) such that ρ(π(tn, x0), xn)≤1/n (or equivalently, ρ2(λ(tn, ω0), ωn)≤1/n and ρ1(ϕ(tn, u0, ω0), un)≤1/n, where xn:= (un, ωn) and ρ1(respectively, ρ2) denotes the distance on the space W(respectively, Ω). Since the sets Ω0and Q(u0,ω0)are compact, the sequences {un}and {ωn}are relatively compact and, consequently, so is the sequence {xn}.¤ ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 9 Let h(X, T, π),(Ω,T, λ), hibe a two-sided (a group) non-autonomous dynamical system, and ω∈Ω be a positively Poisson stable point. Denote by Eω:= {ξ| ∃{tn} ∈ Nωsuch that π(tn,·)|Xω→ξ}, where Xω:= {x∈X|h(x) = ω}and →means the pointwise convergence. Lemma 3.4. [7, 8] Let ω∈Ωbe a Poisson stable point, h(X, T, π),(Ω,T, λ), hia non-autonomous dynamical system, and Xa conditionally compact space. Then, Eω is a nonempty compact sub-semigroup of the semigroup XXω ω(w.r.t. the composition of mappings). Recall that if Xis a compact metric space, then XXdenotes the collection of all maps from Xto itself, provided with the product topology, or, in other words, the topology of pointwise convergence. By Tychonoff’s theorem, XXis compact. XXpossesses a semi-group structure defined by the composition of maps. Let Ebe a semi-group. A right ideal in Eis a non-empty subset Isuch that IE ⊂ I, where IE:= {ξ◦η:ξ∈I, η ∈ E}, and ξ◦ηis the composition of ξand ηdefined in the following way: (ξ◦η)(x) := η(ξ(x)),for x∈X. It is worth noticing that we are using the original notation for the composition which was used in the works [1], [5] and [11]. Needless to say that this notation can be misunderstanding, and it would be possible to use the standard definition, but then we should change all the terminology about right and left ideals, and the results already proved in the literature concerning these sets. For this reason, we prefer to keep the original notation and recommend the reader to be careful with this notation. A right ideal is said to be a minimal right ideal if it does not contain any proper right ideal. An idempotent in a semigroup Eis an element u∈ E such that u2=u. Remark 3.5. 1. Every compact semigroup admits at least one minimal right ideal [1, 5, 11]. 2. Every compact semigroup contains at least one idempotent element [1, 5, 11]. Now, we can prove our first main abstract result. Theorem 3.6. Let Xbe a conditionally compact metric space, and h(X, T, π) (Ω,T, λ), hia non-autonomous dynamical system. Suppose that the following conditions are fulfilled: (i) There exists a Poisson stable point ω∈Ω; (ii) lim |t|→+∞ρ(π(t, x1), π(t, x2)) = 0 for all x1, x2∈Xω:= h−1(ω) = {x∈X: h(x) = ω}. Then there exists a unique point xω∈Xωsuch that ξ(xω) = xωfor all ξ∈ Eω. 16 TOM´ AS CARABALLO AND DAVID CHEBAN Example 4.9. Consider the following two-dimensional system of linear differential equations x0=A(t)x, where (17) A(t) = µ−a(t)−b(t) b(t)−a(t)¶, the function a(t)is defined by equality (13) and b(t) := (2 + sin t+ sin √2t)−1for all t∈R. It is easy to check that the matrix A(t)is Levitan almost periodic, but not almost automorphic because it is unbounded on R. Let µx1(t) x2(t)¶be a solution of the system (17), then we have (18) d dt(x2 1(t) + x2 2(t)) = −2a(t)(x2 1(t) + x2 2(t)) for all t∈R. It follows from (18) that (19) |ϕ(t, x, A)|=e−h(t)|x| for all x∈R2and t∈R,where ϕ(t, x, A)is a solution of equation (17) passing through the point x∈R2at the initial moment, and h(t) := Rt 0a(s)ds. Taking into account the results in Example 4.8 we conclude that lim |t|→+∞|ϕ(t, x, A)|= 0. Thus, if f∈C(R,R2)is a Levitan almost periodic function and the equation y0=A(t)y+f(t) admits a bounded solution then, according to Theorem 4.1 and Corollary 4.2, it has a unique Levitan almost periodic solution. 4.2. Linear Difference Equations. As our second class of applications, consider the following difference equation (20) u(t+ 1) = A(t)u(t) + f(t) and its corresponding homogeneous equation (21) u(t+ 1) = A(t)u(t), where (A, f)∈C(Z,[E]) ×C(Z, E).Along with equations (20) and (21), we also consider the H-class of equation (20) (respectively, (21)), that is the family of equations (22) v(t+ 1) = B(t)v(t) + g(t), (respectively, (23) v(t+ 1) = B(t)v(t) ) with (B, g)∈H(A, f) := {(Aτ, fτ)|τ∈Z}(respectively, B∈H(A)), Aτ(t) = A(t+τ), fτ(t) := f(t+τ) and t∈Z,where the bar denotes closure in C(Z,[E]) × C(Z, E) (respectively, C(Z,[E])). Let ϕ(t, v, (B, g)) (respectively, ϕ(t, v, B)) be the solution of equation (22) (respectively, (23)) that satisfies the condition ϕ(0, v, (B, g)) = v(respectively, ϕ(0, v, B) = v). ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 17 Now, in order to have a two-sided non-autonomous dynamical system, we need to impose the following condition on the rest of the paper: Example 4.10. Condition (C): the operator B(n)is invertible for all B∈H(A) and n∈Z. Remark 4.11. Assuming that condition (C) is fulfilled from now on, we can ensure that the solution ϕ(n, v, (B, g)) of equation (22) is defined on the whole Z. We set now Y:= H(A, f),and denote the dynamical system of shifts on H(A, f) by (Y, Z, σ).Consider X:= E×Y, and define a dynamical system on Xby setting π(τ, (v, B, g)) := (ϕ(τ, v, (B, g)), Bτ, gτ) for all (v, (B, g)) ∈E×Yand τ∈Z. Then h(X, Z, π),(Y, Z, σ), hiis a group non-autonomous dynamical system, where h:= pr2:X→Y. Our aim now is to apply the results of Sections 2–3 to this system, and obtain some results concerning the difference equation (22). As before, a solution ϕ∈C(Z, E) of equation (20) is called [34] compatible by the character of recurrence if N(A,f)⊆Nϕ,where N(A,f):= {{tn} ⊂ Z|(Atn, ftn)→ (A, f)}, and, respectively, Nϕ:= {{tn} ⊂ Z|ϕtn→ϕ}. Following a scheme similar to the one used in the first application (and which is motivated by the structure of the general theory developed in Section 3) we can prove now similar results for the discrete non-autonomous dynamical system generated by equation (20). Although the proofs may seem a repetition of the previous ones, they are necessary to justify every statement, and this is the reason why we prefer not to omit them. Theorem 4.12. Let (A, f)∈C(Z,[E]) ×C(Z, E)be Poisson stable. Suppose that the following conditions hold: (i) equation (20) admits a relatively compact solution ϕ(t, u0,(A, f)),i.e. the set Q(u0,(A,f)) := ϕ(Z, u0,(A, f)) is compact in E; (ii) all the relatively compact solutions of equation (21) tend to zero as the time ttends to ∞,i.e. lim |t|→+∞|ϕ(t, u, A)|= 0 if ϕ(n, u, A)is relatively compact (this means that the set ϕ(Z, u, A)) is relatively compact in E). Then, equation (20) has a unique compatible solution ϕ(n, ¯u, f)with values from the compact Q(u0,(A,f)). Proof. Denote by h(X, Z, π),(Y, Z, σ), hithe group non-autonomous dynamical system generated by equation (20) (see construction above). By Lemma 3.3, under the conditions of the theorem, the invariant set H(x0)⊂X(where x0:= (u0,(A, f)) ∈Xand H(x0) := {π(t, x0)|t∈R}) is conditionally compact. Let now x1, x2∈H(x0)∩X(A,f), where X(A,f):= E× {(A, f)}(i.e. xi= (ui,(A, f)) and ui∈E(i=1,2)), then lim |t|→+∞ρ(π(t, x1), π(t, x2)) = lim |t|→+∞|ϕ(t, u1,(A, f)) −ϕ(t, u2,(A, f))|= 0. Now to finish the proof it is sufficient to refer to Theorem 3.6, Corollary 3.7 and Corollary 3.9. ¤ 18 TOM´ AS CARABALLO AND DAVID CHEBAN Corollary 4.13. Under the conditions of Theorem 4.12 if (A, f)∈C(Z,[E]) × C(Z, E)is τ-periodic (respectively, Levitan almost periodic, almost recurrent, Poisson stable), then equation (20) admits a unique τ-periodic (respectively, Levitan almost periodic, almost recurrent, Poisson stable) solution. Proof. This statement follows from Theorem 4.12 and Corollary 3.10. ¤ Corollary 4.14. Under the conditions of Theorem 4.12 if (A, f)∈C(Z,[E]) × C(Z, E)is almost automorphic, then equation (9) admits a unique almost automorphic solution. Proof. Since the function ϕ(t, ¯u, (A, f)) is relatively compact, it easily follows that ¯ϕ:= ϕ(·,¯u, (A, f)) ∈C(Z, E) is a Lagrange stable point of the dynamical system (C(Z, E),Z, σ).On the other hand, by Corollary 4.13 the function ¯ϕis Levitan almost periodic and, consequently, almost automorphic. ¤ Corollary 4.15. Under the conditions of Theorem 4.12 if (A, f)∈C(Z,[E]) × C(Z, E)is Bohr almost periodic, then equation (20) admits a unique almost automorphic solution. Proof. This statement follows from Corollary 4.14 because every Bohr almost periodic function is almost automorphic. ¤ A solution ϕ∈C(Z, E) of equation (20) is called (see [31],[34]) uniformly compatible by the character of recurrence, if M(A,f)⊆Mϕ,where M(A,f):= {{tn} ⊂ Z|such that the sequence {(Atn, ftn)}is convergent }(respectively, Mϕ:= {{tn} ⊂ Z| such that the sequence {ϕtn}is convergent }). Theorem 4.16. Let (A, f)∈C(Z,[E]) ×C(Z, E)be recurrent. Suppose that the following conditions hold: (i) the equation (20) admits a relatively compact on Zsolution ϕ(t, u0,(A, f)); (ii) for all B∈H(A)the relatively compact on Zsolutions of equation (23) tend to zero as the time tends to ∞,i.e. lim |t|→+∞|ϕ(t, u, B)|= 0, if ϕ(t, u, B)is relatively compact on Z. Then, equation (20) has a unique uniformly compatible solution ϕ(t, ¯u, f)with values from the compact Q(u0,(A,f)). Proof. Denote by h(X, Z, π),(Y, Z, σ), hithe group non-autonomous dynamical system generated by equation (20). Under the conditions of the theorem, the invariant set H(x0)⊂X(where x0:= (u0,(A, f)) ∈Xand H(x0) := {π(t, x0)|t∈Z}) is compact. Let now x1, x2∈H(x0)∩X(B,g), where (B, g)∈H(A, f) and X(B,g):= E×{(B, g)}(i.e. xi= (ui,(B, g)) and ui∈E(i=1,2)). Then lim |t|→+∞ρ(π(t, x1), π(t, x2)) = lim |t|→+∞|ϕ(t, u1,(B, g)) −ϕ(y, u2,(B, g))|= 0. Now to finish the proof it is sufficient to refer to Theorem 3.12. ¤ ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 19 Corollary 4.17. Under the conditions of Theorem 4.16, if (A, f)∈C(Z,[E]) × C(Z, E)is τ-periodic (respectively, Bohr almost periodic, almost automorphic, recurrent), then equation (20) admits a unique τ-periodic (respectively, Bohr almost periodic, almost automorphic, recurrent) solution. Proof. This statement follows from Theorem 4.16 and Corollary 3.13. ¤ Remark 4.18. Note that the results of this subsection can also hold even without assuming condition (C), but the proofs in this case may need of some new ideas and abstract results. Briefly, the main difference is as follows: as we have seen, equation (22) with condition (C) generates a group non-autonomous dynamical system, but without condition (C) the non-autonomous dynamical system generated by (22) is only one-sided (i.e. a semi-group system). We plan to develop this situation in our next paper [6]. Acknowledgments 1. The authors would like to thank the anonymous referee for his/her comments and suggestions on a preliminary version of this article. 2. This paper was written while the second author was visiting the Department of Differential Equations and Numerical Analysis, University of Seville (Spain), in the winter of 2007/2008. He would like to thank people of this university for their very kind hospitality. 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[34] B.A. Shcherbakov, Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations, S¸tiint¸a, Chi¸sin˘au, 1985. (In Russian) [35] W. Shen, Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc. 136 (1998), no. 647. [36] M. A. Subin, Almost periodic functions and partial differential operators, Russ. Math. Surv. 33 (1978), no. 2, 1-52. [37] Shubin M., Local Favard theory, Mosc. Univ. Math. Bul., 34 (1978), no.2, 32–37 . [38] K.S. Sibirsky, Introduction to Topological Dynamics, Noordhoff, Leyden, 1975. [39] V.V. Zhikov, B.M. Levitan, Favard theory, Russian Math. Survey 32 (1977), 129-180. E-mail address, T. Caraballo: [email protected] E-mail address, D. Cheban: [email protected] ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS ... 21 (T. Caraballo) Departamento de Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. Correos 1160, 41080-Sevilla (Spain) (D. Cheban) State University of Moldova, Department of Mathematics and Informatics, A. Mateevich Street 60, MD–2009 Chis¸in˘ au, Moldova