مرکزی صفحہ
Abstract and Applied Analysis Nonlinear semigroups and the existence and stability of solutions of semilinear nonautonomous...
Nonlinear semigroups and the existence and stability of solutions of semilinear nonautonomous evolution equations
Aulbach, Bernd, Minh, Nguyen Vanآپ کو یہ کتاب کتنی پسند ہے؟
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جلد:
1
سال:
1996
زبان:
english
رسالہ:
Abstract and Applied Analysis
DOI:
10.1155/s108533759600019x
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NONLINEAR SEMIGROUPS AND THE EXISTENCE AND STABILITY OF SOLUTIONS OF SEMILINEAR NONAUTONOMOUS EVOLUTION EQUATIONS BERND AULBACH AND NGUYEN VAN MINH Abstract. This paper is concerned with the existence and stability of solutions of a class of semilinear nonautonomous evolution equations. A procedure is discussed which associates to each nonautonomous equation the socalled evolution semigroup of (possibly nonlinear) operators. Suﬃcient conditions for the existence and stability of solutions and the existence of periodic oscillations are given in terms of the accretiveness of the corresponding inﬁnitesimal generator. Furthermore, through the existence of integral manifolds for abstract evolutionary processes we obtain a reduction principle for stability questions of mild solutions. The results are applied to a class of partial functional diﬀerential equations. 1. Introduction In the last three decades the theory of semigroups of nonlinear operators has been developed extensively and the achieved results have found many applications in the theory of partial diﬀerential equations (see the survey [11] by M.G. Crandall). Recently, increasing interest has been observed in applications of the methods of dynamical systems to inﬁnite dimensional dynamics (see, e.g., [9], [7], [8], [15], [18], [26], [28] and the references therein). The main idea in this context is to associate a semigroup of nonlinear operators to an evolution equation and then to study the asymptotic behavior of the solutions of this equations in the vicinity of a given stationary solution. Whereas most of those papers deal only with autonomous, i.e. time independent evolution equations, the explicit time dependence of evolution 1991 Mathematics Subject Classiﬁcation. 34G20, 34K30, 47H20. Key words and phrases. Evolutionary process, evolution semigroup, semilinear nonautonomous equation, nonlinear semigroup, stability, periodic solution, accretive operator, integral manifold, instability. The second author was on leave (as a research fellow of the; Alexander von Humboldt Foundation) from the Department of Mathematics, University of Hanoi, Vietnam. Received: July 25, 1996. c 1996 Mancorp Publishing, Inc. 351 352 BERND AULBACH AND NGUYEN VAN MINH equations often arises quite naturally, not only in physics and mechanics, but also in mathematics when one linearizes an autonomous equation along a nonstationary solution. For particular classes of timedependent evolution equations arising from the linearization along a compact invariant subset it has been shown (see e.g. [43]) that one can deﬁne a skewproduct semiﬂow which allows to apply the methods of classical dynamical systems to the underlying nonautonomous equations. To the best of our knowledge, the papers [32, 33] contain the ﬁrst attempt to associate a strongly continuous evolution semigroup to a nonlinear timedependent equation in order to study the asymptotic behavior of solutions. Since the present paper is closely related to those articles we brieﬂy recall some basic results proved in [32, 33]. The right hand sides of the equations considered are deﬁned everywhere and they are supposed to be Lipschitz continuous. To each equation of this kind one associates an evolution semigroup with properties which allow to apply the CrandallLiggett theorem on the generation of nonlinear semigroups. In a recent paper [2] we considered equations with almost periodic coeﬃcients in this semigroup framework. The main obstacles for the application of those results to inﬁnite dimensional systems are apparently due to the assumption that the right hand sides of the equations considered are Lipschitz continuous and that they are deﬁned everywhere. In this paper we are concerned with evolution equations of the form (1) dx = A(t)x + f (t, x) dt where A(t) is a (possibly unbounded) linear operator acting in a real or complex Banach space X and f (·, ·) : R × X → X is a (possibly nonlinear) continuous function. We furthermore assume that the linear part dx/dt = A(t)x of equation (1) is well posed in a sense to be explained. To this kind of equation we manage to associate an evolution semigroup which is strongly continuous and whose generator can be computed explicitly in terms of the generator of the evolution semigroup associated with the linear part of (1) and the nonlinear term f (t, x). Finally, we discuss how to apply this semigroup approach to the study of the asymptotic behavior of mild solutions of equation (1). For the case of a time independent linear part of equation (1) the existence problem for solutions has been investigated by many authors (see e.g. [21], [24], [27], [35], [36], [37], [38], [48] and the references therein). In the present paper we show that the problems arising from the explicit tdependence of A(t) can be overcome by using our evolution semigroup approach. Furthermore, in the study of the asymptotic behavior of mild solutions of equation (1) this approach allows to make use of many results available for dynamical systems. A more detailed outline of our construction is as follows. First we associate to equation (1) with Lipschitz continuous f (t, x) a strongly continuous evolution semigroup whose generator is of the form A + F , where A is the SEMILINEAR NONAUTONOMOUS EVOLUTION EQUATIONS 353 generator of the linear evolution semigroup associated with the linear part of equation (1) and F is an operator acting on a function space induced by f (·, ·). Without any additional assumption on the linear part we do not know any relation between the associated evolution semigroup and the semigroup (if there exists any at all) generated by A + F in the CrandallLiggett sense. Nevertheless, using an appropriate adaptation of a fundamental result due to G.F. Webb [48] for time independent equations we manage to prove that mild solutions of equation (1) exist and that the semigroup generated by A + F in Webb’s sense coincides with the evolution semigroup associated with equation (1). In order to accomplish this we ﬁrst solve the corresponding equation with right hand side A + F in a suitable function space by using Webb’s generation theorem (see [48]), and then we consider equation (1). In doing so we can prove the existence and uniqueness of mild solutions and the coincidence of the semigroup generated by A + F in Webb’sense with the evolution semigroup associated with equation (1). This result is a substantial generalization of a major result on nonlinear equations obtained in [33]. It turns out that for equation (1) with τ periodic coeﬃcients the evolution operator S τ from the evolution semigroup acts like a Poincaré mapping. This analogy provides a suﬃcient condition for the existence of τ periodic mild solutions of equation (1) in terms of the accretiveness of A and F . In order to study the instability of mild solutions of (1) we prove in Section 3 a theorem on the existence of integral manifolds for evolutionary processes by using the Hadamard graph transform. Since this result is derived in a very general setting (without use of any concrete equations) it is applicable to various kinds of equations. An application to partial functional diﬀerential equations is presented in Section 4. 2. Evolution semigroups: existence and stability of solutions In this section we consider the evolution semigroups associated with evolutionary processes deﬁned by semilinear equations. One of the main topics to be discussed here is the description of the inﬁnitesimal generators and their use in getting suﬃcient conditions for the existence and stability of solutions of equation (1). We ﬁrst introduce some deﬁnitions and notations which will be used throughout this paper. Without further mention, X will always be a given real or complex Banach space. By Lp (X) , 1 ≤ p < ∞ we denote the space of all(equivalence classes of) Xvalued measurable functions v on R such that R v(t)p dt < ∞ with norm · p . The integral is always to be understood in the Bochner sense (see e.g. [50]). By Cu (R, X) we mean the space of all bounded, uniformly continuous functions from R to X equipped with the supremum norm, while C0 (X) denotes the subspace of functions w ∈ Cu (R, X) with the property limt→∞ w(t) = 0. Various notions of stability and instability will be used in a standard sense (see e.g. [13]). 354 BERND AULBACH AND NGUYEN VAN MINH Deﬁnition 1. A family {X(t, s)  t, s ∈ R , t ≥ s} of (possibly nonlinear) operators acting on X is called an evolutionary process if it satisﬁes the following conditions: (i) X(s, s) x = x for all s ∈ R , x ∈ X; (ii) X(t, s) X(s, r) = X(t, r) for all t ≥ s ≥ r. Such an evolutionary process is called continuous if it satisﬁes the conditions (iii) X(t, s) x − X(t, s) y ≤ Keω(t−s) x − y for all t ≥ s and x, y ∈ X , where K is any positive and ω any real constant, (iv) X(t, s) x is continuous jointly with respect to t, s and x . To every evolutionary process {X(t, s)  t ≥ s} we associate the socalled evolution semigroup {T h  h ≥ 0} deﬁned by the relation (2) (T h v)(t) = X(t, t − h)v(t − h) for all t ∈ R, where v belongs to a suitable space of functions (such as the ones mentioned above). Proposition 1. Assume that {X(t, s)  t ≥ s} is a continuous evolutionary process such that X(t, s) 0 = 0 for all t ≥ s. Then for any of the function spaces Lp (X), 1 ≤ p < ∞, and C0 (X) the associated evolution semigroup {T h  h ≥ 0} is strongly continuous. Proof. We give a proof for the case Lp (X) only because the proof for C0 (X) is essentially the same. We ﬁrst notice that for every v ∈ Lp (X) the function taking t into X(t, t − h)v(t − h) is measurable. Furthermore, we get R (T h v)(t)p dt 1/p = ≤ R R X(t, t − h)v(t − h)p dt Keωh v(t − h)p dt 1/p 1/p ≤ Keωh vp < ∞ . In order to prove that the semigroup {T h  h ≥ 0} is strongly continuous we ﬁrst show that the relation lim (T h v − v) = 0 (3) h→0+ is true for every continuous v with compact support. Indeed, by assumption there exists a positive constant N such that v(t) = 0 for all t ≥ N − 1. Thus (3) is equivalent to lim N h→0+ −N X(t, t − h)v(t − h) − v(t)p dt = 0 . Since {X(t, s)  t ≥ s} and v are continuous, the function X(t, t−h)v(t−h) attains its maximum as (t, h) varies in [−N, N ] × [0, 1]. Thus, the claimed SEMILINEAR NONAUTONOMOUS EVOLUTION EQUATIONS 355 relation (3) is a consequence of the following estimate which uses the FatouLebesgue Lemma: N 0 = lim sup X(t, t − h)v(t − h) − v(t)p dt ≥ −N h→0+ N ≥ lim sup h→0+ −N X(t, t − h)v(t − h) − v(t)p dt ≥ 0 . In order to conclude the proof of the proposition we now consider an arbitrary v ∈ Lp (X) and choose for every positive a continuous function w with compact support and the property w − vp < . Then we get lim sup R h→0+ p X(t, t − h)v(t − h) − v(t) dt ≤ lim sup h→0+ R X(t, t − h)v(t − h) − X(t, t − h)w(t − h)p dt + p R X(t, t − h)w(t − h) − w(t) dt + R p w(t) − v(t) dt h→0+ R 1/p 1/p 1/p ≤ K + lim sup 1/p p X(t, t − h)w(t − h) − w(t) dt 1/p + = (1 + K). Since is arbitrary, this estimate proves (3) for every v ∈ Lp (X) and therefore completes the proof of Proposition 1. Remark. For the linear case the above proposition has been proved in [40]. It is known (see e.g. [12]) that nonlinear semigroups need not have inﬁnitesimal generators even if they are strongly continuous. So in order to get generators of the evolution semigroups associated with continuous evolutionary processes we will consider processes generated by equation (1) under some additional conditions. Deﬁnition 2. The linear equation dx = A(t) x dt is said to be wellposed if there exists a continuous linear evolutionary process {U (t, s)  t ≥ s} such that for every s ∈ R and x ∈ D(A(s)) the function x(t) = U (t, s) x is the uniquely determined solution of equation (4) satisfying x(s) = x. (4) Deﬁnition 3. Suppose the linear equation (4) is wellposed. Then every solution x(t) (deﬁned on some interval [s , s + δ) , δ > 0) of the integral 356 BERND AULBACH AND NGUYEN VAN MINH equation (5) x(t) = U (t, s)x + t s U (t, ξ)f (ξ, x(ξ)) dξ , t≥s is called a mild solution of the semilinear equation (1) starting from x at t = s. Furthermore, equation (1) is said to generate an evolutionary process {X(t, s)  t ≥ s} if for every x ∈ X the function X(t, s) x , t ≥ s is the unique solution of equation (5). Proposition 2. Suppose the following conditions are satisﬁed: (i) The linear equation (4) is wellposed . (ii) The nonlinear function f (t, x) is continuous jointly with respect to t and x and Lipschitz continuous with respect to x uniformly in t ∈ R and f (t, 0) = 0 for all t ∈ R . Then the semilinear equation (1) generates a continuous evolutionary process whose associated evolution semigroup on Lp (X) or C0 (X) is strongly continuous and has an inﬁnitesimal generator of the form A + F , where A is the inﬁnitesimal generator of the linear evolution semigroup associated with the evolutionary process generated by the linear equation (4) in Lp (X) or C0 (X), respectively, and F is the operator taking v from Lp (X) or C0 (X), respectively, into the function t → f (t, v(t)) . Proof. Using standard arguments (see e.g. [45]) one can prove that equation (1) generates an evolutionary process. Furthermore, this evolutionary process is continuous. In fact, from [45] it follows that X(t, s) x is continuous jointly with respect to t, s, x. Now we prove that X(t, s) x also satisﬁes the Lipschitz condition (iii) in Deﬁnition 1. By deﬁnition of the evolutionary process {X(t, s)  t ≥ s} we get X(t, s) x − X(t, s) y ≤ U (t, s) x − U (t, s) y + t ≤ Ke s U (t, ξ) · f (ξ, X(ξ, s)x) − f (ξ, X(ξ, s)y) dξ ω(t−s) x − y + t s Keω(t−ξ) L X(ξ, s)x − X(ξ, s)y dξ, where L is a Lipschitz constant of f (t, x) with respect to x, and K, ω stem from the wellposedness of the linear equation (4). Just for convenience we obviously may choose ω to be positive. Applying Gronwall’s Lemma we get (6) X(t, s) x − X(t, s) y ≤ Ke(ω+KL)(t−s) x − y, t ≥ s, x, y ∈ X. Therefore the evolutionary process {X(t, s)  t ≥ s} is continuous according to Deﬁnition 1. Taking into account that X(t, s) 0 = 0 for all t ≥ s we can apply Proposition 1 to see that the associated evolution semigroup {T h  h ≥ 0} is strongly continuous. Now we are going to compute the inﬁnitesimal generator of this semigroup. To this purpose we ﬁrst prove that for every SEMILINEAR NONAUTONOMOUS EVOLUTION EQUATIONS 357 w ∈ Lp (X) which is continuous and has compact support we get the relation (7) lim h→0+ t f (t, w(t)) − h−1 R t−h U (t, ξ)f ξ, X(ξ, t − h)w(t − h) dξ p dt = 0 . Indeed, by deﬁnition f (·, w(·)) is uniformly continuous. Consequently we get lim sup (8) h→0+ t Furthermore we have (9) f (t, w(t)) − h R t t−h + f (t, w(t)) − h R h R t −1 t−h f (ξ, w(ξ)) − f (t, w(t)) dξ = 0 . t−h −1 ≤ t U (t, ξ)f (ξ, X(ξ, t − h)w(t − h)) dξ t −1 t−h U (t, ξ)f (ξ, w(ξ)) dξ p dt p dt 1/p 1/p p U (t, ξ) [f (ξ, w(ξ)) − f (ξ, X(ξ, t − h)w(t − h))] dξ dt 1/p . Since w has compact support, from the assumptions we observe that f (·, w(·)) has compact support as well. Consequently we get lim h→0+ f (t, w(t)) − h R t −1 t−h U (t, ξ)f (ξ, w(ξ)) dξ p dt 1/p = 0. On the other hand, we have R (10) h ≤ t −1 t−h U (t, ξ)f (ξ, w(ξ)) − f (ξ, X(ξ, t − h)w(t − h)) dξ t R h−1 p t−h Keωh L w(ξ) − X(ξ, t − h)w(t − h) dξ p dt dt 1/p 1/p . Note that the function g(t, ξ, h) := w(ξ)−X(ξ, t−h)w(t−h) is continuous with respect to (t, ξ, h) ∈ {(t, ξ, h)  h ∈ [0 , 1] , −N ≤ t − h ≤ ξ ≤ N }, where supp (w) ⊂ [−N , N ]. Consequently, the function h−1 q(t, h) := t t−h Keωh L w(ξ) − X(ξ, t − h)w(t − h) dξ p is bounded in (t, h) ∈ [−N − 1 , N + 1] × [0, 1]. Now applying the FatouLebesgue Dominant Convergence Lemma we get (11) lim sup h→0+ R t h−1 t−h Keωh L w(ξ) − X(ξ, t − h)w(t − h) dξ p dt 1/p = 0. All of this implies that the claimed relation (7) has been veriﬁed under the assumption that w is continuous and has compact support. 358 BERND AULBACH AND NGUYEN VAN MINH In order to show that (7) is true for all functions from Lp (X), we now choose an arbitrary element v from this space as well as an arbitrary continuous function w with compact support. Then, using (6) and (7) we get lim sup f (t, v(t)) R h→0+ −h−1 t t−h U (t, ξ)f (ξ, X(ξ, t − h)v(t − h)) dξ ≤ lim sup p R h→0+ f (t, v(t)) − f (t, w(t)) dt dt 1/p 1/p + lim sup f (t, w(t)) R h→0+ − h p t −1 t−h U (t, ξ)f (ξ, X(ξ, t − h)w(t − h) dξ + lim sup h R h→0+ −1 t t−h h→0+ p R v(t) − w(t) dt + lim sup h−1 R h→0+ t t−h dt 1/p U (t, ξ) [f (ξ, X(ξ, t − h)w(t − h)) − f (t, X(ξ, t − h)v(t − h))] dξ ≤ lim sup L p p dt 1/p 1/p Keωh L X(ξ, t − h)w(t − h) − X(ξ, t − h)v(t − h)dξ p dt 1/p . Using the estimate (6) we continue this estimate to get ≤ Lv − wp + lim sup R h→0+ t (12) t−h h−1 Keωh (ω+KL)h LKe w(t − h) − v(t − h)dξ = Lv − wp 2 (2ω+KL)h + lim sup LK e R h→0+ 2 ≤ Lv − wp + LK v − wp = (L + LK 2 )w − vp . p dt 1/p p w(t − h) − v(t − h) dt 1/p SEMILINEAR NONAUTONOMOUS EVOLUTION EQUATIONS 359 Since w in (12) is an arbitrary continuous function with compact support, (12) implies for any v ∈ Lp (X) the claimed relation (13) lim h→0+ f (t, v(t)) − h−1 R t t−h U (t, ξ)f ξ, X(ξ, t − h)v(t − h) dξ p dt 1/p = 0. By the deﬁnitions of T h and X(t, s) we obtain T hv − v h = (14) X(t, t − h)v(t − h) − v(t) h U (t, t − h)v(t − h) − v(t) h + h−1 = (t) = t t−h U (t, ξ)f (ξ, X(ξ, t − h)v(t − h)) dξ U (t, t − h)v(t − h) − v(t) + f (t, v(t)) h − f (t, v(t) − h −1 t t−h U (t, ξ)f (ξ, X(ξ, t − h)v(t − h)) dξ . It is clear from (13) and (14) that v belongs to the domain of the inﬁnitesimal generator of {T h  h ≥ 0} if and only if v belongs to the domain of the generator A of the linear evolution semigroup associated with {U (t, s)  t ≥ s}. Since the proof for the C0 (X) case requires no essential changes, the proof of Proposition 2 is complete. Corollary 1. Suppose the assumptions of Proposition 2 are satisﬁed. Then the inﬁnitesimal generator A + F of the evolution semigroup {T h  h ≥ 0} is closed and densely deﬁned in Lp (X) , 1 ≤ p < ∞ or in C0 (R, X), respectively. Proof. Since f (t, x) is Lipschitz continuous with respect to x uniformly in t and f (t, 0) ≡ 0, the operator F is continuous in the function spaces Lp (X) , 1 ≤ p < ∞ and C0 (R, X). Thus, the assertions of the corollary follow from Proposition 2. Now we suppose that all assumptions of Proposition 2 are satisﬁed and that A + F generates a nonlinear semigroup in some sense (e.g. in the CrandallLiggett sense [12]). Then the question arises of how to relate this semigroup to the associated evolution semigroup. In the general Banach space setting we need some additional conditions to see that they indeed coincide. In order to deal with those conditions we recall some notions which turn out to be useful later on. We ﬁrst deﬁne [z , w] = lim z + λw − z /λ λ→0+ and quote some important properties of this bracket [· , ·] from [11, p. 308]. 360 BERND AULBACH AND NGUYEN VAN MINH Proposition 3. For x, y, z ∈ X and α, β ∈ R we get the following properties: i) [· , ·] : X × X → R is uppersemicontinuous, ii) [αx , βy] = β[x , y] if αβ > 0 , iii) [x , αx+ y] = αx + [x , y] , iv) [x , y] ≤ y and [0 , y] = y , v) −[x , −y] ≤ [x , y] , vi) [x , y + z] ≤ [x, y] + [x , z] , vii) [x , y] − [x , z] ≤ y − z . Deﬁnition 4. (see [11]) If A is an operator in X and ω a real number, then A+ωI is called accretive if one (or all) of the following equivalent conditions hold: i) (1 − λω)x − y ≤ x − y + λ(x − y ) for all x ∈ Ax , y ∈ Ay and λ ≥ 0. ii) [x − y , x − y ] ≥ −ω x − y for all x ∈ Ax , y ∈ Ay . iii) If λ > 0 and λω < 1, then (I +λA)−1 is singlevalued and has (1−λω)−1 as a Lipschitz constant. Deﬁnition 5. A continuous function f (t, x) is said to satisfy condition H(Cu ) or H(C0 ), respectively, if the mapping taking v from Cu (R, X) or C0 (X), respectively, into the function f (·, v(·)) is continuous. Corollary 2. Under the assumptions of Proposition 2 the following is true: i) Suppose that −(A + F ) is accretive and R(I − λ(A + F )) equals Lp (X) or C0 (X), where A + F acts in Lp (X) or in C0 (X), respectively, for all suﬃciently small positive λ. Then the zero solution of equation (1) is globally uniformly stable. ii) Suppose that there exists a positive number α such that αI − (A + F ) is accretive and R(I − λ(αI − (A + F )) equals Lp (X) or C0 (X) for all suﬃciently small positive λ. Then the zero solution of equation (1) is globally exponentially stable. Proof. Under the given assumptions the operator A + F generates a semigroup of nonlinear operators on Lp (X) or C0 (X), respectively, in the CrandallLiggett sense [12]. In virtue of Corollary 1 we can apply [6, Corollary 4.3] to see that this semigroup coincides with the evolution semigroup associated with equation (1). Thus the assertion follows. Remark. In Proposition 2 the perturbation f (t, x) is assumed to be Lipschitz continuous (with Lipschitz constant L) with respect to x uniformly in t and f (t, 0) ≡ 0. If ωI − A is accretive, then (L + ω)I − (A + F ) is maccretive (see [48]). Below we will weaken the conditions imposed on f but then we have to restrict our considerations to a smaller class of wellposed equations of the form (4) which generate linear processes {U (t, s)  t ≥ s} such that A is accretive. SEMILINEAR NONAUTONOMOUS EVOLUTION EQUATIONS 361 It turns out that suggested by the above results and by using the operator A + F we can prove a version of Proposition 2 for a larger class of perturbations f as well as the existence of mild solutions of equation (1). As we are concerned with the existence problem for solutions of equations of the form (1) we suppose that equation (1) satisﬁes the Uniqueness Condition for mild solutions, i.e. we suppose that for every ﬁxed s and x, if there exist two mild solutions u1 (t) and u2 (t) deﬁned on some interval [s, s + δ) , δ > 0, then those two solutions coincide on this interval. In the appendix of this paper we describe suﬃcient conditions for this kind of uniqueness. Theorem 1. Let the following conditions be satisﬁed: i) The linear equation (4) is wellposed. ii) Let A denote the inﬁnitesimal generator of the linear evolution semigroup associated with equation (4). Then αI − A is maccretive. iii) f (t, x) satisﬁes condition H(C0 ) and βI − F is accretive. iv) Equation (1) satisﬁes the Uniqueness Condition on mild solutions. Then equation (1) generates a continuous evolutionary process whose associated evolution semigroup is strongly continuous in C0 (R, X) and has A + F as its inﬁnitesimal generator with domain D(A + F ) = D(A) ⊂ C0 (R, X). Furthermore, this evolution semigroup satisﬁes the estimate S h v − S h w ≤ e(α+β)h v − w for all v, w ∈ C0 (R, X) , h ≥ 0 . Proof. Under the assumptions of the theorem the autonomous equation du = (A + F ) u , dt (15) t≥0 generates a strongly continuous semigroup {S t  t ≥ 0} in Webb’s sense (see [48]), i.e. S t u is the unique continuous solution of the integral equation (16) S t u = T (t)u + t 0 T (t − ξ)F S ξ u dξ , where T (t) is generated by the linear operator A . Furthermore, S t u − S t v ≤ e(α+β)t u − v for all t ≥ 0 , u, v ∈ C0 (R, X) . In view of (16) we have (17) t−s t−s S v (t) = T (t − s)v (t) + T (t − s − ξ)F S ξ u (t) dξ 0 t−s = U (t, s)v(s) + = U (t, s)v(s) + 0 t s U (t, ξ + s)(F S ξ v)(s + ξ) dξ U (t, η)f η, (S η−s v)(η) dη . Thus, in view of (17) we observe that for every s ∈ R and x ∈ X equation (5) has at least one continuous solution X(t, s)x = (S t−s v)(t), where v is any element of C0 (R, X) such that v(s) = x. Now we are going to show that 362 BERND AULBACH AND NGUYEN VAN MINH X(t, s)x depends continuously on (t, s, x). Indeed, suppose that x, x ∈ X and (1 − t)x for t ≤ 1, (1 − t)x for t ≤ 1, v(t) = v (t) = 0 for t > 1, 0 for t > 1. Then, since (S t−s v)(t) = X(t, s)v(s) , we get X(t , s )v (s) − X(t, s)x = (S t −s v )(t ) − (S t−s v)(t) ≤ (S t −s v )(t ) − (S t−s v)(t ) + (S t−s v)(t ) − (S t−s v)(t) . If t, s, v are ﬁxed, then limt →t (S t−s v)(t ) − (S t−s v)(t) = 0. On the other hand, (S t −s v )(t ) − (S t−s v)(t ) ≤ sup (S t −s v )(ξ) − (S t−s v)(ξ) ξ = S t −s v −S t−s v ≤ S t −s v − S t −s v + S t −s v − S t−s v . In view of the strong continuity of S t and the property S t −s v − S t −s v ≤ e(α+β)(t −s ) v − v we have (18) lim (t ,s ,v )→(t,s,v) (S t −s v )(t ) − (S t−s v)(t) = 0 . This shows that X(t, s)x depends continuously on (t, s, x). Finally, it is clear that X(t, s) x − X(t, s) y ≤ e(α+β)(t−s) x − y for all t ≥ s , x, y ∈ X . Thus we have proved that {S h  h ≥ 0} is the evolution semigroup associated with the evolutionary process {X(t, s)  t ≥ s}. This completes the proof of the theorem. Now we apply Theorem 1 to investigate the stability of the mild solutions of equation (1). Corollary 3. Let all assumptions of Theorem 1 be satisﬁed with α + β < 0. Then there exists a unique mild solution x : R → C0 (R, X) of equation (1) which is exponentially stable (among mild solutions). Proof. In virtue of Theorem 1 we have S t u − S t v ≤ e(α+β)t u − v for all t ≥ 0 , u, v ∈ C0 (R, X) . Consequently, from the assumptions of the corollary it may be shown that the operators S t , t ≥ 0 have a unique common ﬁxed point v0 ∈ C0 (R, X) which obviously represents a mild solution of equation (1). The stability of this solution follows immediately from the above estimate. Remark. It may be noted that if the evolution semigroup {T h  h ≥ 0} associated with the linear equation (4) is strongly continuous in Cu (R, X) and if F acts on Cu (R, X), then Theorem 1 is still valid for Cu (R, X). Theorem 2. Let the following conditions be satisﬁed: SEMILINEAR NONAUTONOMOUS EVOLUTION EQUATIONS 363 i) The linear equation (4) is well posed. Furthermore, the evolution semigroup associated with the linear process generated by equation (4) is strongly continuous in Cu (R, X) . ii) Let A denote the inﬁnitesimal generator of the above linear evolution semigroup. Then αI − A is maccretive. iii) f satisﬁes condition H(Cu ) and βI − F is accretive. iv) Equation (1) satisﬁes the Uniqueness Condition on mild solutions. Then equation (1) generates a continuous evolutionary process whose associated evolution semigroup is strongly continuous in Cu (R, X) and has A + F as its inﬁnitesimal generator with domain D(A + F ) = D(A) ⊂ Cu (R, X). Furthermore, S h v − S h w ≤ e(α+β)h v − w for all v, w ∈ Cu (R, X) , h ≥ 0 . Proof. The theorem can be proved in the same manner as the previous one. So we omit the details. In particular, if A(t) = 0 for all t, then we get all assertions of Lemmas 1 and 2 of [33]. In this case A = −d/dt with D(A) = Cu1 (R, X). Furthermore, Theorem 2 allows to improve substantially the results for nonlinear equations in [33]. Now we are going to discuss another application of the evolution semigroups {S h  h ≥ 0} acting on Cu (R, X) to investigate the existence of periodic solutions of equation (1). Deﬁnition 6. An evolutionary process {Z(t, s)  t ≥ s} is said to be τ periodic if Z(t + τ, s + τ ) = Z(t, s) for all t ≥ s . Theorem 3. Suppose the following conditions are satisﬁed: i) The linear equation (4) is wellposed and it generates a τ periodic evolutionary process {U (t, s)  t ≥ s} . ii) f (t, x) is τ periodic with respect to t for every ﬁxed x . iii) Equation (1) generates an evolutionary process . iv) x0 (·) is a unique ﬁxed point of S τ in a subset Ω of the space of all bounded functions Cb (R, X) on R which is invariant with respect to the semigroup {S h  h ≥ 0} and the translation Sτ : x(·) → x(· + τ ) . Then x0 (t) is a τ periodic mild solution of equation (1). Proof. We ﬁrst prove that in Cb (R, X) one has S τ Sτ = Sτ S τ . In fact, by deﬁnition X(t + τ, s + τ )x = U (t + τ, s + τ )x + = U (t, s)x + t s t+τ s+τ U (t + τ, ξ)f (ξ, X(ξ, s + τ )x)dξ U (t, ξ)f (ξ, X(ξ + τ, s + τ )x)dξ. Thus from the uniqueness we get X(t, s)x = X(t + τ, s + τ )x for all t ≥ s , x ∈ X . 364 BERND AULBACH AND NGUYEN VAN MINH This proves that S τ Sτ = Sτ S τ . Now since S τ commutes with all other operators of the evolution semigroup x0 has to be a common ﬁxed point of S h and Sτ . This implies that x0 is a mild solution of equation (1) which is τ periodic. The proof of the theorem is complete. Theorem 4. Let all assumptions of Theorem 2 be satisﬁed with α + β < 0. Furthermore, let the following conditions be fulﬁlled: i) The linear equation (4) generates a τ periodic process and the evolution semigroup {T h  h ≥ 0} associated with (4) is strongly continuous in Cu (R, X) . ii) f (t, x) is τ periodic with respect to t for every x . Then equation (1) has a unique τ periodic mild solution which is globally exponentially stable. Proof. This theorem is an immediate consequence of Theorems 2 and 3. Another application of Theorem 3 is related to the concept of exponential dichotomy whose deﬁnition due to Henry [18] we recall next. Deﬁnition 7. A linear evolutionary process {U (t, s)  t, s ∈ R , t ≥ s} is said to have an exponential dichotomy if there exist positive constants N, α and projections P (t) , t ∈ R, bounded uniformly in t, i.e. sup P (t) < ∞ , t∈R such that the following three conditions hold: i) U (t, s)P (s) = P (t)U (t, s) for all t ≥ s . ii) For t ≥ s the restriction U (t, s)Ker P (s) is an isomorphism from Ker P (s) onto Ker P (t) and we deﬁne U (s, t) as the inverse mapping from Ker P (t) onto Ker P (s) . iii) The inequalities U (t, s)P (s)x ≤ N e−α(t−s) P (s)x for all t ≥ s , x ∈ X , U (t, s)Q(t)x ≤ N e−α(s−t) Q(s)x for all s ≥ t , x ∈ X , hold true where X(t, s) for s ≥ t is deﬁned in ii) and Q(s) := I − P (s) . By abuse of terminology we say that a semigroup {T (t)  t ≥ 0} has an exponential dichotomy (or that it is hyperbolic) if the process {U (t, s)  t ≥ s} deﬁned by U (t, s) = T (t − s) for all t, s ∈ R , t ≥ s has an exponential dichotomy. Proposition 4. Suppose that the following conditions are satisﬁed: i) A(t) is constant (with value a) for t ≥ 0 and the semigroup {T (t)  t ≥ 0} has an exponential dichotomy, ii) f (t, x) is τ periodic with respect to t, continuous with respect to (t, x) and Lipschitz continuous with respect to x with Lipschitz constant δ . SEMILINEAR NONAUTONOMOUS EVOLUTION EQUATIONS 365 Then for suﬃciently small δ the equation dx = a + f (t, x) dt has a unique τ periodic mild solution. (19) Proof. First notice that equation (19) generates an evolutionary process. Furthermore, observe that X(t, s)x − X(t, s)y ≤ Keω(t−s) x − y + t s Keω(t−ξ) δX(ξ, s)x − X(ξ, s)y dξ. Hence, using Gronwall’s inequality we have X(t, s)x − X(t, s)y ≤ Ke(ω+δK)(t−s) x − y . Consequently, the evolution semigroup {S h  h ≥ 0} associated with equation (19) acts on the space Cb (R, X) of bounded functions on R . Now observe that X(t, s)x − U (t, s)x − X(t, s)y − U (t, s)y ≤ δ ≤ δ t s t s ≤ Keω(t−ξ) X(ξ, s)x − X(ξ, s)y dξ ≤ Keω(t−ξ) Ke(ω+KL)(ξ−s) dξ x − y , where U (t, s) = T (t − s) for all t ≥ s. Thus, if δ is suﬃciently small we can apply the Inverse Function Theorem for Lipschitz mappings (see e.g. [27], [34]) to conclude that S τ has a unique ﬁxed point. Now we are in a position to apply Theorem 3 to see that equation (19) has a unique τ periodic mild solution. Remark. The results derived in this section can be generalized to hold for equations which are deﬁned in closed subsets of the extended phase space. The corresponding proofs are based on the above approach applied to the results available for the autonomous case (see e.g. [24], [27]). 3. Evolution semigroups: unstable integral manifolds and instability of solutions In this section we discuss the application of evolution semigroups to study the instability of solutions. To this end we prove the existence of unstable manifolds for semilinear equations whose linear parts have an exponential dichotomy. Since we deal with evolutionary processes rather than with concrete equations our results can be applied to a large class of evolution equations such as partial functional diﬀerential equations. In this section we consider the (possibly nonlinear) perturbation {X(t, s)  t ≥ s} of a given linear evolutionary process {U (t, s)  t ≥ s}. By abuse of 366 BERND AULBACH AND NGUYEN VAN MINH terminology we say that a function x : R → X is a solution of a given process {X(t, s)  t ≥ s} if X(t, s)x(s) = x(t) for all t ≥ s . We then put (20) φ(t, s)x = X(t, s)x − U (t, s)x for all t ≥ s , x ∈ X . For convenience, in this section we always assume that all the evolutionary processes {Z(t, s)  t ≥ s} under consideration have the property (21) Z(t, s)0 = 0 for all t ≥ s . We say that the process {Z(t, s)  t ≥ s} has bounded growth if (22) Z(t, s)x ≤ M eω(t−s) x for all t ≥ s , x ∈ X for some positive constants ω and M . Below we suppose that all evolutionary processes in consideration have bounded growth. Deﬁnition 8. A set M ⊂ R × X is said to be an integral manifold of the evolutionary process {X(t, s)  t ≥ s} if for every t ∈ R the phase space X splits into a direct sum X = Xt1 ⊕ Xt2 such that (23) def inf Sn(Xt1 , Xt2 ) = inf t∈R inf t∈R xi ∈Xti ,xi =1,i=1,2 x1 + x2 > 0 and if there exists a family of Lipschitz continuous mappings gt : Xt1 → Xt2 , t ∈ R, with Lipschitz constants independent of t such that M = {(t, x, gt (x)) ∈ R × (Xt1 ⊕ Xt2 )  t ∈ R , x ∈ Xt1 } and X(t, s)(gr(gs )) = gr(gt ) for all t ≥ s , where gr(gs ) denotes the graph {(x, y) ∈ Xs1 ⊕ Xs2  y = gs (x)} of the mapping gs . An integral manifold M is said to be proper if the set {(t, 0, 0) ∈ R × (Xt1 ⊕ Xt2 )  t ∈ R} is contained in M . We are going to show that every nonlinear process {Z(t, s)  t ≥ s} which is close enough to a linear process having an exponential dichotomy has an unstable integral manifold. The method of proof we use is the socalled graph transform (see e.g. [20], [34]). Suppose that the linear process {U (t, s)  t ≥ s} has an exponential dichotomy with positive constants K, α and projections P (t) , t ∈ R. Since P(t) is bounded uniformly in t, i.e. supt∈R P (t) < ∞, from the wellknown fact [13] that 1 2 ≤ Sn Im P (t), Ker P (t) ≤ , P (t) P (t) it follows that (24) inf Sn Im P (t), Ker P (t) t∈R = γ > 0. SEMILINEAR NONAUTONOMOUS EVOLUTION EQUATIONS 367 From now on we use the notation Xt1 = Im P (t) , Xt2 = Ker P (t). Furthermore, for every ﬁxed r > 0 we denote by Bt1 (r) , Bt2 (r) and B(r) the open balls of radius r in the Banach spaces Xt1 , Xt2 and X, respectively. We have x ≤ P (t)x + Q(t)x ≤ (2 sup P (t) + 1)x , t where Q(t) = I − P (t). Thus we get for all t ∈ R and x ∈ X (25) 1 x ≤ max P (t)x, Q(t)x ≤ 1 + sup P (t) x. 2 t Below we shall assume that (26) φ(t, s)x − φ(t, s)y ≤ eµ(t−s) x − y for all t ≥ s , x, y ∈ X, for some positive constants and µ. Putting Oδ = {gt : Xt2 → Xt1  gt (0) = 0 , Lip(gt ) ≤ δ , t ∈ R} we deﬁne in Oδ a distance d(g, h) = ∞ 1 k=1 2k sup t∈R,x≤k gt (x) − ht (x) . It is easily checked that (Oδ , d) is a complete metric space. Proposition 5. Assume that the linear process {U (t, s)  t ≥ s} has an exponential dichotomy with constants K, α and projections P (t) , t ∈ R and suppose h0 is a given positive number. Then there exists a positive constant δ0 (depending only on {U (t, s)  t ≥ s} and h0 ) such that for any 0 < δ < δ0 the mapping Q(t)U (t, s)(gs (x), x) is a homeomorphism with respect to x from Xs2 onto Xt2 for all 0 ≤ t − s ≤ h0 . Similarly, for δ < δ0 /2 and < δ0 e−µh0 /2 the mapping Q(t)X(t, s) ◦(gs (x), x) is a homeomorphism with respect to x from Xs2 onto Xt2 . Proof. Consider the inclusion i : x −→ (gs (x), x) and the mapping Q(t)U (t, s)i . Evidently, Q(t)U (t, s)i is a linear homeomorphism from Xs2 onto Xt2 . Let us deﬁne Γgs as Γgs x = (gs (x), x). Applying the Inverse Function Theorem for Lipschitz continuous mappings (see e.g. [27], [34]) and putting ψ(t, s)x = Q(t)U (t, s)x − Q(t)U (t, s)Γgs x , we see that Lip(ψ) ≤ δ, and if −1 1 ≤ Q(t)Y (t, s)i K then Q(t)U (t, s)Γgs is a homeomorphism. Thus δ ≤ δ0 = −1 , 1 . K Similarly, if δ < δ0 /2 and < δ0 e−µh0 /2, then Q(t)U (t, s)Γgs is a homeomorphism. 368 BERND AULBACH AND NGUYEN VAN MINH Proposition 6. Under the assumptions and notations of the previous proposition, if (27) δ Q(s)(x − y) ≥ P (s)(x − y), then δ Q(t) X(t, s)x − X(t, s)y (28) ≥ P (t) X(t, s)x − X(t, s)y , where δ = (29) δKe−α(t−s) + 2 eµ(t−s) . (1/K)eα(t−s) − 2 eµ(t−s) Proof. Putting f = X(t, s) , S = U (t, s) and φ = φ(t, s) for simplicity we get Q(t)f (x) − Q(t)f (y) ≥ (Q(t)Sx − Q(t)Sx) + (φ(x) − φ(y)) . Since Q(t)U (t, s) = Q(t)U (t, s)Q(s) we have Q(t)f (x) − Q(t)f (y) ≥ (1/K)eα(t−s) Q(s)(x − y) − eµ(t−s) x − y . Taking into account (25) and (27), for suﬃciently small δ (δ < δ0 ) we get (30) Q(t)f (x) − Q(t)f (y) ≥ (1/K)eα(t−s) − 2 eµ(t−s) Q(s)(x − y) . On the other hand, we have P (t)f (x) − P (t)f (y) = P (t)Sx − P (t)Sy + φ(x) − φ(y) ≤ Ke−α(t−s) P (s)(x − y) + eµ(t−s) x − y . According to (25) and (27) we have (31) P (t)f (x) − P (t)f (y) ≤ δKe−α(t−s) + 2 eµ(t−s) Q(s)(x − y) . Thus, from (30) and (31) it follows that P (t)f (x) − P (t)f (y) ≤ δ Q(t)f (x) − Q(t)f (y) . This completes the proof of the proposition. From Propositions 5 and 6 we see that for a given positive h0 if δ < δ0 /2 and < (δ0 e−µh0 )/2, then S h is well deﬁned as a mapping from Oδ to Oδ for 0 < h < h0 . Now we choose k ∈ N such that 1 Ke−αk = q < 2 and then h0 = 2k. Thus, for δ < δ0 /2 = 1/(2K) and (32) 0 < < min e−2µk δ(q −1 − q) , e−2µk 2K 2(1 + δ) S k maps Oδ into itself by the formula gr (S k g)t = X(t, t − k) gr(gt−k ) for all g ∈ Oδ . SEMILINEAR NONAUTONOMOUS EVOLUTION EQUATIONS 369 Proposition 7. Under the above assumptions on {U (t, s)  t ≥ s} and {X(t, s)  t ≥ s}, for suﬃciently small , S k is a contraction mapping in Oδ . Proof. It is suﬃcient to show that for some 0 < q < 1 the estimate P (t)X(t, t − k)x − (S k g)t Q(t)X(t, t − k)x (33) ≤ q P (t − k)x − gt−k Q(t − k)x) is true for every g ∈ Oδ and x ∈ X. In fact, suppose that h ∈ Oδ , substituting x by (ht−k (Q(t − k)x), Q(t − k)x) into (33) we get (34) (S k h)t (Q(t)X(t, t − k)x) − (S k g)t (Q(t)X(t, t − k)x) ≤ q ht−k (Q(t − k)x) − gt−k (Q(t − k)x) for all x ∈ X and t ∈ R. Put y = Q(t − k)x. Then [Q(t)X(t, t − k)Q(t − k)]−1 ({z ≤ r}) is contained in {y ≤ r}. Thus, for every n ∈ N sup t∈R , y≤n (S k h)t (y) − (S k g)t (y) ≤ q sup t∈R , y≤n ht (y) − gt (y) . Hence, for suﬃciently small (such that q < 1), S k is a contraction mapping. Now we prove that (33) holds. For simplicity of notation put f = X(t, t − k) , S = U (t, t − k) and φ = X(t, t − k) − U (t, t − k). We then have Q(t)f (x) − Q(t)f (gt−k (Q(t − k)x) + Q(t − k)x) (35) ≤ Q(t)φ(x) − Q(t)φ(gt−k (Q(t − k)x) + Q(t − k)x) + Q(t)S(x) − Q(t)S(gt−k (Q(t − k)x) + Q(t − k)x) ≤ sup Q(t)eµk P (t − k)x − gt−k (Q(t − k)x). t On the other hand, we have P (t)f (x) − P (t)f (gt−k (Q(t − k)x) + Q(t − k)x) ≤ P (t)(φ(x) − φ(gt−k (Q(t − k)x) + Q(t − k)x)) + P (t)(S(x) − S(gt−k (Q(t − k)x) + Q(t − k)x)) (36) ≤ sup P (t)eµk P (t − k)x − gt−k (Q(t − k)x) t + Ke−αk P (t − k)x − gt−k (Q(t − k)x) = (q + sup P (t)eµk )P (t − k)x − gt−k (Q(t − k)x). t 370 BERND AULBACH AND NGUYEN VAN MINH Note that (S k g)t (Q(t)Z(t, t − k)x) = P (t)f (gt−k (Q(t − k)x) + Q(t − k)x). Now, combining (35) and (36), we get P (t)X(t, t − k)x − (S k g)t (Q(t)X(t, t − k)x) ≤ P (t)f (x) − (S k g)t (Q(t)X(t, t − k)(gt−k (Q(t − k)x) + Q(t − k)x)) + (S k g)t (Q(t)X(t, t − k)(gt−k (Q(t − k)x) + Q(t − k)x)) − (S k g)t (Q(t)X(t, t − k)x) (37) ≤ P (t)f (x) − P (t)f (gt−k (Q(t − k)x) + Q(t − k)x) + P (t)f (gt−k (Q(t − k)x) + Q(t − k)x) − (S k g)t (Q(t)Z(t, t − k)x ≤ (q + sup P (t)eµk + δ)P (t − k)x − gt−k (Q(t − k)x). t q Thus = q + δ + supt P (t)eµk is less than 1 if and δ are suﬃciently small. This proves the assertion of the proposition. Supposing that g is the ﬁxed point of S k in Oδ , we next prove that g is the ﬁxed point of S h for all h ≥ 0 in some sense. Theorem 5. Under the assumptions of Propositions 6 and 7 there exists a socalled unstable integral manifold (which is proper and Lipschitz continuous) for the nonlinear evolutionary process {X(t, s)  t ≥ s} . Proof. We only need to prove that M = {gr(g t )  t ∈ R} is left invariant by the process {X(t, s)  t ≥ s}, i.e. that gr(g t ) = X(t, s)(gr(g s )) for all t ≥ s . To this end we consider the action of S h , 0 ≤ h ≤ 2k on Oδ for suﬃciently small δ and , by the formula (38) gr(gt ) = X(t, t − h)(gr(gt−h )), where g = {gt  t ∈ R} ∈ Oδ . According to Proposition 6 we can choose δ and suﬃciently small so that sup 0 ≤ t−s ≤ 2k δ < δ0 = δ1 4 where δ is deﬁned by (29), δ0 = Ke−2αk . Thus S h is a mapping from Oδ to Oδ . Suppose that and δ are chosen such that (32) holds. Then for any ξ ∈ [0, k) we consider the mappings S k+ξ : Oδ → Oδ1 and S h : Oθ → Oθ , δ ≤ θ ≤ δ1 . We then get S k+ξ = S ξ · S k (: Oδ → Oδ1 ) = S k · S ξ (Oδ → Oδ1 ) . From Proposition 7 we have S ξ · S k g = S ξ g and S ξ · S k g = S k · S ξ g = S ξ g . SEMILINEAR NONAUTONOMOUS EVOLUTION EQUATIONS 371 From the uniqueness of the ﬁxed point of S k it follows that S ξ g = g where g is the ﬁxed point of S k acting on Oδ . This proves the theorem. Remark. It is apparent that Lip(g t ) → 0 as → 0, where g t , t ∈ R is determined in the proof of Theorem 5. Combining the proofs of Propositions 6 and 7 with Theorem 5 we can deduce the following local version of Theorem 5: Theorem 6. Assume that {U (t, s)  t ≥ s} is as in Theorem 5 and that {X(t, s)  t ≥ s} is deﬁned in the open ball {x ∈ X  x < 2r}. Furthermore, suppose that (26) holds for all x and y in this ball. Then for suﬃciently small there exists a ”local” unstable integral manifold which is represented by g = {gt : Bt2 (rt ) → Bt1 (rt )} , Lip(gt ) < δ = δ() , inf t rt > 0 such that gr(gt ) = X(t, s)(gr(gs )) ∩ B(rs ) for all t ≥ s where B(r) denotes the ball {x ∈ X  max{P (t)x , Q(t)x} < r} . Proof. We can deﬁne a function ρ : X → [0, 1] with the property 1 for x ≤ r 0 for x ≥ 1.5r with Lipschitz constant L. We then deﬁne ρ(x) = X (t, s)x = ρ(x)X(t, s)x for all x ∈ X . Now, in order to complete the proof it suﬃces to apply Theorem 5 to {X (t, s)  t ≥ s}. Next we are going to apply the above results to investigate the asymptotic behaviour of the process {X(t, s)  t ≥ s} around the ”zero solution”. Proposition 8. Under the assumptions of Theorem 5 we get the limiting relation lim d(Z(t, s)x , Mt ) = 0 (39) t→∞ where Mt = gr(gt ) and d(y , Mt ) = inf z∈Mt y − z. Proof. From (33) it follows that (40) d X(t, t − k)x , Mt Thus we have ≤ q d(x , Mt−k ) for all t ∈ R , x ∈ X. lim d Z(s + nk, s)x , Ms+k ) = 0 . n→∞ From the bounded growth of {X(t, s)  t ≥ s} we get the claimed relation (39). Below we shall consider the case where the linear process {U (t, s)  t ≥ s} satisﬁes a condition more general than that of an exponential dichotomy. Deﬁnition 9. A linear process {U (t, s)  t ≥ s} with bounded growth is said to satisfy condition H if there exist positive constants K, α, β with α > β and nontrivial projections P (t) , t ∈ R which are bounded uniformly in t such that the following three conditions are met: 372 BERND AULBACH AND NGUYEN VAN MINH i) P (t)U (t, s) = U (t, s)P (s) for all t ≥ s , ii) The restriction U (t, s)Ker P (s) is an isomorphism from Ker P (s) onto Ker P (t) (whose inverse is denoted by U (s, t) for s ≤ t) . iii) With Q(s) = I − P (s) we have U (t, s)P (s)x ≤ Ke−α(t−s) P (s)x for all t ≥ s , x ∈ X , U (t, s)Q(s)x ≥ K −1 e−β(t−s) Q(s)x for all t ≥ s , x ∈ X . Examples. It is apparent that every linear process with an exponential dichotomy satisﬁes condition H. More generally, one can show that a linear process {U (t, s)  t ≥ s} having bounded growth satisﬁes condition H if and only if (see e.g. [25]) there exists an r ∈ (0 , 1) such that the circle with radius r belongs to the resolvent set ρ(T (1)) and that σ(T (1)) ∩ {z ∈ C  z < r} = ∅, where {T (t)  t ≥ 0} is the evolution semigroup associated with {U (t, s)  t ≥ s} in Lp . It is easy to see that if {U (t, s)  t ≥ s} satisﬁes condition H, then {U ∗ (t, s)  t ≥ s} deﬁned as (41) U ∗ (t, s)x = eγ(t−s) U (t, s)x for all x ∈ X where γ = (α − β)/2 has an exponential dichotomy with constants K, (α − β)/2 and the same projection P (t) , t ∈ R as {U (t, s)  t ≥ s}. Consider the ”change of variables” for the nonlinear process {X(t, s)  t ≥ s} as follows: instead of {X(t, s)  t ≥ s} we consider the process {X ∗ (t, s)  t ≥ s} deﬁned as (42) X ∗ (t, s)x = eγt X(t, s)(e−γs x) for all t ≥ s , x ∈ X . {X ∗ (t, s)  t ≥ s} is nonlinear as well. Furthermore, Observe that the process if the process φ(t, s)x = X(t, s)x − U (t, s)x satisﬁes (26), then denoting φ∗ (t, s) = X ∗ (t, s)x − U ∗ (t, s)x we have (43) φ∗ (t, s)x − φ∗ (t, s)y ≤ eγt φ(t, s)(eγs x) − φ∗ (t, s)(eγs y) ≤ eγt eµ(t−s) eγs x − y = e(γ+µ)(t−s) x − y. Now we are in a position to apply Theorem 5 to the processes {U ∗ (t, s)  t ≥ s} and {X ∗ (t, s)  t ≥ s}. It follows that for suﬃciently small there exists a g ∈ Oδ (where δ = δ() and lim#→0 δ() = 0) such that M = {Mt = gr(gt )  t ∈ R} is an integral manifold of {Z ∗ (t, s)  t ≥ s}. Let us deﬁne gt∗ (x) = e−γt gt (eγ tx). Obviously, we then get gr(gt∗ ) = e−γt gr(gt ). Since g = {gt  t ∈ R} satisﬁes gr(gt ) = X ∗ (t, s)(gr(gs )) for all t ≥ s SEMILINEAR NONAUTONOMOUS EVOLUTION EQUATIONS 373 we get eγt gr(gt∗ ) = eγt Z(t, s)(e−γs eγs gr(gs∗ )) . This shows that the set N = {Nt  t ∈ R}, where Nt = gr(g ∗ ) , t ∈ R is an integral manifold of the process {X(t, s)  t ≥ s}. Now we are going to apply the above result to investigate the instability of solutions. Theorem 7. (Reduction Principle for Stability of Evolutionary Processes.) Assume that the linear process {U (t, s)  t ≥ s} satisﬁes condition H. In addition assume that for the nonlinear process {X(t, s)  t ≥ s} the condition Lip X(t, s)x − U (t, s)x ≤ eµ(t−s) for all t ≥ s holds for some positive µ and . Then for suﬃciently small > 0 there exists an integral manifold M = {Mt  t ∈ R} of {X(t, s)  t ≥ s} such that the zero solution of {X(t, s)  t ≥ s} is stable if and only if for every ∗ > 0 and s ∈ R there exists a δ = δ(∗ , s) > 0 such that X(t, s)x < ∗ for all t ≥ s if x ∈ Ms and x ≤ δ. Similarly, the asymptotic, uniform and exponential stability of the zero solution of {X(t, s)  t ≥ s} are equivalent to the respective stability type of {X(t, s)Ms  t ≥ s}. Proof. From (33) we can easily prove the assertions of the theorem. Thus the proof is similar to that of Proposition 8. Remark. In the case where for every pair (t, s) the operator U (t, s) is compact we observe that codim Im P (t) < ∞. Thus the integral manifold M in Theorem 6 is of ﬁnite dimension, i.e. dim D(gt ) < ∞ for all t ∈ R. Theorem 8. (Linearized Instability Theorem) Under the assumptions of Theorem 5, if the projections P (t) , t ∈ R are nontrivial, i.e. P (t) = I and P (t) = 0 for all t, then for suﬃciently small the zero solution of {X(t, s)  t ≥ s} is unstable. Proof . It is suﬃcient to prove that X(s + kn , s)x tends to ∞ as kn → ∞ for every x = 0 in Ms , where M = {Mt  t ∈ R} is the integral manifold provided by Theorem 5. For suﬃciently small we have Mt = gr(gt ) , lim δ() = 0 , #→0 374 BERND AULBACH AND NGUYEN VAN MINH where Lip(gt ) ≤ δ = δ(). Thus we can assume that δ < 1/2. We have X(t,s)x = P (t)X(t, s)x + Q(t)X(t, s)x = gt (Q(t)X(t, s)x) + Q(t)X(t, s)x (44) ≥ (1 − δ)Q(t)X(t, s)x 1 ≥ Q(t)X(t, s)(gs (Q(s)x) + Q(s)x) 2 1−δ Q(t)X(t, s)Q(s)x = 2 1 ≥ Q(t)X(t, s)Q(s)x 4 1 1 ≥ Q(t)U (t, s)Q(s)x − sup Q(t)Lip(φ(t, s))Q(s)x 4 t 4 1 1 α(t−s) e ≥ − sup Q(t) eµ(t−s) Q(s)x. 4 K t Hence, if we ﬁx t − s = k0 , where k0 is chosen such that (1/K)eα(t−s) > 8, then for suﬃciently small we have (45) Q(k0 + s)X(k0 + s, s)Q(s)x ≥ pQ(s)x, where p > 1. Since x ∈ gr(gs ) we can apply (44) repeatedly to get (46) Q(nk0 + s)X(nk0 + s, s)Q(s)x ≥ pn Q(s)x . 0. Now from (43) we observe Since x = 0, x ∈ gr(gs ) we have Q(s)x = that lim X(nk0 + s, s)x = ∞ . n→∞ This completes the proof of the theorem. 4. An application to partial functional differential equations Since the coeﬃcientoperators A(t) in the equations considered above are not assumed to be bounded the results of the previous sections have applications in the theory of partial diﬀerential equations. For a standard procedure of such an application we refer to [3], [37], [38]. On the other hand, taking into account that in Section 3 we deal with evolutionary processes rather than with concrete evolution equations, we will consider an application of the results of Section 3 to study the asymptotic behavior of solutions of a class of partial functional diﬀerential equations (for a standard procedure see [47]). In a forthcoming paper we shall deal with the evolution semigroups associated with this kind of equations in the context of the theory of strongly continuous semigroups of operators. In the sequel we will use the following terminology. By C = C([−r, 0], X), r > 0, we denote the Banach space of continuous Xvalued functions on [−r, 0] equipped with the supremum norm. If u is a continuous function SEMILINEAR NONAUTONOMOUS EVOLUTION EQUATIONS 375 from [a − r , b] to X and t ∈ [a, b], then ut denotes the element of C given by ut (θ) = u(t + θ) for −r ≤ θ ≤ 0. For the reader’s convenience the following result is quoted from [47]: Proposition 9. Suppose F : [a, b] × C → X is continuous and satisﬁes F (t, φ) − F (t, ψ)X ≤ L ψ − φC for all t ∈ [a , b] , φ, ψ ∈ C , where L is a positive constant. Furthermore let {T (t)  t ≥ 0} be a strongly continuous semigroup of linear operators acting on X. Then for every φ ∈ C there exists a unique continuous function u : [a − r , b] → X which solves the initial value problem t a u(t) = T (t − a)φ(0) + (47) uα = φ . T (t − s)F (s, us ) ds for all t ∈ [a , b], Proposition 10. Let the assumptions of Proposition 9 be satisﬁed for all t ∈ [a , b] and suppose L is independent of a and b. Then equation (47) provides a (nonlinear) evolutionary process {X(t, s)  t ≥ s} on C. If in addition F (t, 0) = 0 and T (t) ≤ eµt for all t ∈ R as well as L < 1, then for {X(t, s)  t ≥ s} the following holds: ∆φ − ∆ψC ≤ eω (t−s) φ − ψC for all t ≥ s , φ, ψ ∈ C , where = Le2µr , ω = 1 + µ + eµr , ∆φ = X(t, s)φ − U (t, s)φ , (U (t, s)φ)(θ) = T (t + θ − s)φ(0) for all t ≥ s , −r ≤ θ ≤ 0 . Proof. Suppose that u(t) is the solution of equation (47). Then we put X(t, a)φ = ut . Now we show that {X(t, s)  t ≥ s} is an evolutionary process. To this end, it is suﬃcient to prove that X(t, s) · X(s, τ ) = X(t, τ ) for all t ≥ s ≥ τ . In virtue of Proposition 9, if u(t) denotes the solution of the equation u(t) = T (t − s)[X(s, τ )φ](0) + us = X(s, τ )φ , then we have t s T (t − ξ)F (ξ, uξ ) dξ u∗ (t) = T (t − s) T (s − τ )φ(0) + + t s s τ T (s − ξ)F (ξ, X(ξ, τ )φ) dξ T (t − ξ)F (ξ, uξ ) dξ = T (t − τ )φ(0) + where u∗ (ξ) = t τ for all t ≥ s , T (t − ξ)F (ξ, u∗ξ ) dξ , u(ξ) for ξ ≥ s X(ξ, τ )φ for τ ≤ ξ ≤ s . 376 BERND AULBACH AND NGUYEN VAN MINH From the uniqueness of solutions it follows that u∗ (ξ) is the solution of the equation u∗ (t) = T (t − τ )φ(0) + u∗τ = φ . t τ T (t − ξ)F (ξ, u∗ (ξ) dξ , Thus, by deﬁnition we have X(t, τ )φ = X(t, s)[X(s, τ )φ] for all φ ∈ C . This shows that {X(t, s)  t ≥ s} is indeed a (nonlinear) evolutionary process. By assumptions there exists a positive constant µ such that T (t) ≤ eµt for all t ≥ 0 . Hence, since F (t, 0) ≡ 0, for all t ≥ s and φ ∈ C we have (see [47]) X(t, s)φ ≤ e(µ+L)(t−s) φ −µr e−µr e(µ+Le )(t−s) φ for µ ≥ 0 , for µ < 0 . By deﬁnition we have ∆φ − ∆ψ (48) ≤ sup t+θ −r ≤ θ ≤ 0 s L T (t + θ − ξ)X(ξ, s)φ − X(ξ, s)ψ dξ. On the other hand, we get X(t, s)φ − X(t, s)ψC ≤ eµr eµ(t−s) φ − ψC + + Putting t s eµr eµ(t−ξ) L X(ξ, s)φ − X(ξ, s)ψC dξ . g(t) = e−µt X(t, s)φ − X(t, s)ψC we have g(t) ≤ M + N t s g(ξ) dξ for all t ≥ s for all t ≥ s, where M = eµ(r−s) φ − ψC and N = Leµr . Now by applying Gronwall’s inequality (or precisely, a generalized version of it) we get the estimate (49) X(t, s)φ − X(t, s)ψC ≤ Keω(t−s) φ − ψC , where K = eµr and ω = µ + Leµr . Now substituting (48) into (47) we get ∆φ − ∆ψC ≤ sup t+θ eµ(t+r−ξ) LKeω(ξ−s) φ − ψC dξ −r ≤ θ ≤ 0 s µr (t−s) ≤ eµr 1 − e−Le (µ+Leµr )(t−s) e . Using the elementary estimates 1 − ex  ≤ xex and t − s ≤ et−s SEMILINEAR NONAUTONOMOUS EVOLUTION EQUATIONS 377 as well as L < 1 we get ∆φ − ∆ψC ≤ eω (t−s) φ − ψC for all t ≥ s , φ, ψ ∈ C , where = Le2µr and ω = 1 + µ + eµr . The proof of the proposition is complete. We are now in a position to apply the results achieved in Section 3 to study the instability of solutions of the evolution equation with delay (47). Proposition 11. Let all assumptions of Proposition 10 be fulﬁlled. In addition let {T (t)  t ≥ 0} have an exponential dichotomy with nontrivial projections. Then for suﬃciently small L there exists an unstable integral manifold for equation (47), and consequently, the zero solution of equation (47) is unstable. Proof. First note that the exponential dichotomy of {T (t)  t ≥ 0} provides an exponential dichotomy for the linear process {U (t, s)  t ≥ s} deﬁned in Proposition 10. Now in view of this proposition it is suﬃcient to apply the results of the previous section to get the claimed assertion. 5. Appendix. uniqueness of mild solutions For the reader’s convenience in this appendix we present a suﬃcient condition for the uniqueness of mild solutions of equation (1). Since this result is primarily a minor adaptation of a result proved in [21] we only sketch the details. We ﬁrst describe some assumptions on the function f (t, x) in the right hand side of equation (1). Deﬁnition 10. A function g : R × R :→ R is said to satisfy Condition G if it satisﬁes the following conditions: (1) g(t, w) is continuous in w for each ﬁxed t and Lebesgue measurable in t for each ﬁxed w and for each r > 0 there exists a locally integrable function Lr (t) deﬁned on R such that g(t, w) ≤ Lr (t) for all t ∈ R and w ∈ [−r, r]; (2) g(t, 0) = 0 and w(t) = 0 is the maximal solution of the initialvalue problem w (t) = g(t, w(t)) , w(a) = 0 , for a < t < b , where a and b are arbitrary real numbers such that a < b. Proposition 12. Let the following conditions be fulﬁlled: i) Equation (4) is well posed. ii) g(t, w) deﬁned as a function on R × R such that x − y , −(f (t, x) − f (t, y)) ≥ g(t, x − y) for all (t, x), (t, y) ∈ D(f ) satisﬁes condition G. 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Bernd Aulbach Department of Mathematics University of Augsburg D86135 Augsburg GERMANY Email address: aulbach@math.uniaugsburg.de Nguyen Van Minh Department of Mathematics University of Augsburg D86135 Augsburg, GERMANY and Department of Mathematics University of Tübingen D72076 Tübingen, GERMANY Email address: ming@michelangelo.mathematik.unituebingen.de Advances in Operations Research Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Advances in Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Applied Mathematics Algebra Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Probability and Statistics Volume 2014 The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Differential Equations Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 Submit your manuscripts at http://www.hindawi.com International Journal of Advances in Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Mathematics and Mathematical Sciences Mathematical Problems in Engineering Journal of Mathematics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Discrete Mathematics Journal of Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Discrete Dynamics in Nature and Society Journal of Function Spaces Hindawi Publishing Corporation http://www.hindawi.com Abstract and Applied Analysis Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Journal of Stochastic Analysis Optimization Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014