TWO DEFINITIONS OF EXPONENTIAL DICHOTOMY FOR SKEW-PRODUCT SEMIFLOW IN BANACH SPACES
|
|
- Frederica Wilson
- 6 years ago
- Views:
Transcription
1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 4, April 1996 TWO DEFINITIONS OF EXPONENTIAL DICHOTOMY FOR SKEW-PRODUCT SEMIFLOW IN BANACH SPACES SHUI-NEE CHOW AND HUGO LEIVA (Communicated by Hal L. Smith) Abstract. In this paper we introduce a concept of exponential dichotomy for linear skew-product semiflows (LSPS) in infinite dimensional Banach spaces, which is an extension of the classical concept of exponential dichotomy for time dependent linear differential equations in Banach spaces. We prove that the concept of exponential dichotomy used by Sacker-Sell and Magalhães in recent years is stronger than this one, but they are equivalent under suitable conditions. Using this concept we where able to find a formula for all the bounded negative continuations. After that, we characterize the stable and unstable subbundles in terms of the boundedness of the corresponding projector along (forward/backward) the LSPS and in terms of the exponential decay of the semiflow. The linear theory presented here provides a foundation for studying the nonlinear theory. Also, this concept can be used to study the existence of exponential dichotomy and the roughness property for LSPS. 1. Introduction The concept of exponential dichotomy of linear differential equations was introduced by Perron [14], which is concerned with the problem of conditional stability of a system ẋ = A(t)x and its connection with the existence of bounded solutions of the equation ẋ = A(t)x + f(x, t), where the state space is a Banach space X and t A(t) : R L(X) is bounded, continuous in the strong operator topology. An important contribution to these problems is the work done by Massera-Schäffer [12], Daleckii-Krein [5], Levinson [8], Coppel [4], Sacker-Sell [15] and Palmer [13]. The need for a new approach arose from the fact that for a time dependent linear differential equation with unbounded operator A(t), the solutions, generally speaking, either cannot be extended in the direction of the negative times, or can be extended, but not uniquely. For example, for parabolic partial differential equations many authors have studied these problems, including Henry [7], Xiao-Biao Lin [10] and J. Hale [6]. For the case of functional differential equations we can see the work done by X.B. Lin [9]. All the problems above can be treated in the unified setting of a linear skewproduct semiflow (LSPS). In [16] Sacker-Sell use a concept of exponential dichotomy for skew-product semiflow with the restriction that the unstable subspace has finite Received by the editors April 14, Mathematics Subject Classification. Primary 34G10; Secondary 35B40. Key words and phrases. Skew-product semiflow, exponential dichotomy, stable and unstable manifolds. This research was partially supported by NSF grant DMS c 1996 American Mathematical Society 1071
2 1072 SHUI-NEE CHOW AND HUGO LEIVA dimension, and they give a sufficient condition for the existence of exponential dichotomy for skew-product semiflow. This concept is also used by Magalhães in [11]. In this work we introduce a concept of exponential dichotomy for skewproduct semiflow weaker than the concept used by Sacker-Sell and Magalhães; here we allow the unstable subspace to have infinite dimension. We prove that the concept of exponential dichotomy used by Sacker-Sell and Magalhães implies this one, and that they are equivalent, if we suppose that the unstable subspace has finite dimension (or infinite dimension) in both definitions. Using this concept, we will find a formula for all the bounded negative continuations. After that, we will characterize the stable and unstable subbundles in terms of the boundedness of the corresponding projector along (forward/backward) the LSPS and in terms of the exponential decay of the semiflow. The linear theory presented here provides a foundation for studying the nonlinear theory. Also this concept can be used to study the existence of exponential dichotomy and the roughness property for LSPS. 2. Preliminaries In this section we shall present some definitions, notations and results about skew-product semiflow in infinite dimensional Banach spaces Linear skew product semiflow. We begin with the notion of skew-product semiflow on the trivial Banach bundle E = X Θ, where X is a fixed a Banach space (the state space) and Θ is a compact Hausdorff space. Definition 2.1. Suppose that σ(θ, t) =θ tis a flow on Θ, i.e., the mapping (θ, t) θ t is continuous, θ 0=θand θ (s + t) =(θ s) t, for all s, t R. Alinear skew-product semiflow π =(Φ,σ)onE= X Θ is a mapping π(x, θ, t) =(Φ(θ, t)x, θ t) fort 0, with the following properties: (1) Φ(θ, 0) = I, the identity operator on X, for all θ Θ. (2) lim t 0 + Φ(θ, t)x = x, uniformly in θ. This means that for every x X and every ɛ>0thereisaδ=δ(x, ɛ) > 0 such that Φ(θ, t)x x ɛ, for all θ Θ and 0 t δ. (3) Φ(θ, t) is a bounded linear operator from X into X that satisfies the cocycle identity: (2.1) Φ(θ, t + s) =Φ(θ t, s)φ(θ, t), θ Θ, 0 s, t. (4) For all t 0 the mapping from E into X given by (x, θ) Φ(θ, t)x is continuous. The properties (2) and (3) imply that for each (x, θ) Ethe solution operator t Φ(θ, t)x is right continuous for t 0. In fact : Φ(θ, t + h)x Φ(θ, t)x = [Φ(θ t, h) I]Φ(θ, t)x, whichgoesto0ashgoes to 0 +. Since E = X Θ is a trivial Banach bundle, then for any subset F Ewedefine the fiber (2.2) F(θ) :={x X:(x, θ) F}, θ Θ. So E(θ) =X, θ Θ.
3 DICHOTOMY FOR SKEW-PRODUCT SEMIFLOW Projectors and subbundles. A mapping P : E Eis said to be a projector if P is continuous and has the form P(x, θ) =(P(θ)x, θ), where P (θ) is a bounded linear projection on the fiber E(θ). For any projector P we define the range and null space by R = R(P) ={(x, θ) E:P(θ)x=x}, N =N(P)={(x, θ) E:P(θ)x=0} The continuity of P implies that the fibers R(θ) andn(θ) vary continuously in θ. This also means that P (θ) is strongly continuous as a function of θ. The following result can be found in Sacker-Sell [16]. Lemma 2.1. Let P be a projector on E. Then R and N are closed subsets in E and we have R(θ) N(θ)={0}, R(θ)+N(θ)=E(θ) for all θ Θ. Definition 2.2. A subset V is said to be a subbundle of E, if there is a projectorp on E with the property that R(P) =V;inthiscaseW=N(P) is a complementary subbundle, i.e., E = V + W as a Whitney sum of subbundles The stable, unstable and the initial bounded sets. Definition 2.3. Apoint(x, θ) Eis said to have a negative continuation with respect to π if there exists a continuous function φ = φ(x, θ), φ :(, 0] E, satisfying the following properties: (1) φ(t) =(φ x (t),θ t) where φ x :(, 0] X, (2) φ(0) = (x, θ), (3) π(φ(s),t) =φ(s+t) for each s 0 and 0 t s, (4) π(φ(s),t)=π(x, θ, t + s), for each 0 s t. In this case the function φ is said to be a negative continuation of the point (x, θ). Now we shall define the following sets: M := {(x, θ) E:(x, θ) has a negative continuation φ}, X u := {(x, θ) M: there is a negative continuation φ of (x, θ) satisfying φ x (t) 0 as t }, B + := {(x, θ) E:sup t 0 Φ(θ, t)x < }, Bu := {(x, θ) M:(x, θ) has a unique bounded negative continuation φ}, B := {(x, θ) M: there is a bounded negative continuation φ of (x, θ)}, X s := {(x, θ) E: Φ(θ, t)x 0 as t }, B:= B + B. The set X u is called the unstable set, X s is the stable set and B is the initial bounded set. Definition 2.4. For θ Θ we shall call the fibers X s (θ) and X u (θ) the stable and unstable linear space of π =(Φ,σ) respectively. Proposition 2.1. Let φ and ψ be negative continuations of (x, θ) and (y, θ) respectively. Then (a) h(t) =(h x+y (t),θ t)=(φ x (t)+ψ y (t),θ t), t 0, is a negative continuation of (x+y, θ). (b) For all λ R, h λ (t)=(λφ x (t),θ t), t 0, is a negative continuation of (λx, θ).
4 1074 SHUI-NEE CHOW AND HUGO LEIVA Proof. It follows directly from the Definition 2.2. Proposition 2.2. If Bu then Bu = B. Proof. Clearly Bu B. It easy to see that 0 Bu(θ) for all θ Θ. Now, suppose that φ(t) =(φ x (t),θ t)andψ(t)=(ψ x (t),θ t) are two bounded negative continuations of the point (x, θ) B.Thenh(t)=(φ x (t) ψ x (t),θ t), t 0, is a bounded negative continuation of (0,θ). Therefore, φ x (t) =ψ x (t), t 0 φ = ψ. This means that each point of B has only one bounded negative continuation. Hence, B Bu. 3. Exponential dichotomy for linear skew-product semiflow Now we shall introduce two concepts of exponential dichotomy for skew-product semiflow in infinite dimensional Banach spaces. The first one is used by Sacker and Sell in [16] and by Magalhães in [11]. The second one is an extension of the concept of exponential dichotomy for evolution operator given in Henry [7]. Definition 3.1. A projector P on E is say to be invariant if it satisfies the following property: (3.1) i.e., P (θ t)φ(θ, t) =Φ(θ, t)p(θ), t 0, θ Θ, P π(,t)=π(,t) P, t 0. Proposition 3.1. (a) For all θ Θ, Bu (θ) is a linear subspace of X. (b) For all invariant projectors P and (x, θ) Bu with the corresponding negative bounded continuation φ(t) =(φ x (t),θ t),iffort 0we define Φ(θ, t)x := φ x (t), then we have that: Φ(θ, t) is linear mapping from Bu (θ) to Bu (θ t) and (3.2) Φ(θ, t + s)x =Φ(θ t, s)φ(θ, t)x, s, t R, (3.3) P (θ t)φ(θ, t)x =Φ(θ, t)p(θ)x, t R. Definition 3.2 (Sacker-Sell). We shall say that a linear skew-product semiflow π on E has an exponential dichotomy over Θ,ifdimRange(I P (θ)) < and Range(I P (θ)) Bu (θ)foreachθ Θ, and there are constants k 1, β > 0 such that the following inequalities hold : Φ(θ, t)p(θ) ke βt, t 0, θ Θ, Φ(θ, t)(i P (θ) ke βt, t 0, θ Θ. Remark 3.1. It is easy to see that if Φ(θ, t) is one-to-one for all t>0, then every negative continuation is unique. Uniqueness of negative continuations is a common feature in the study of partial differential equations, see, for example, Hale [6]. The following definition of exponential dichotomy for a skew-product semiflow is weaker than Definition 3.2. Basically, the unstable subspace is not required to be finite dimensional. But, they are equivalent if the unstable subspace is finite in both definitions (or if the unstable subspace is infinite in both definitions). Both definitions do allow for the possibility that the linear operator Φ(θ, t) neednotbe one-to-one for some t>0, i.e., Φ(θ, t) may has a nontrivial null space. Because of this, it maybe possible for a point (x, θ) Eto have more than one negative continuation.
5 DICHOTOMY FOR SKEW-PRODUCT SEMIFLOW 1075 Definition 3.3. We shall say that a linear skew-product semiflow π on E has an exponential dichotomy over Θ, if there are constants k 1, β > 0and invariant projector P such that for all θ Θwehavethefollowing: (1) Φ(θ, t) :N(P(θ)) N(P(θ t)), t 0, is an isomorphism with inverse: Φ(θ t, t) :N(P(θ t)) N(P(θ)), t 0. (2) Φ(θ, t)p(θ) ke βt, t 0. (3) Φ(θ, t)(i P (θ) ke βt, t 0. From N (P (θ)) = R(I P (θ)) and the Open Mapping Theorem we have that Φ(θ, t)(i P (θ)) is well defined and is a linear bounded operator for t 0. Proposition 3.2. Definition 3.2 (Sacker-Sell) implies Definition 3.3. Proof. We only have to prove that Φ(θ, t) :N(P(θ)) N(P(θ t)), t 0, is an isomorphism. In fact, since Range(I P (θ)) = N (P (θ)) Bu(θ), θ Θ, then for all x N(P(θ)) the point (x, θ) has a unique bounded negative continuation φ(t) =(φ x (t),θ t). Then for t 0 we shall define Φ(θ, t)x := φ x (t). Moreover, from Definition 3.1 we get Φ(θ, t + s)x =Φ(θ t, s)φ(θ, t)x, s, t R. Hence (3.4) x =Φ(θ t, t)φ(θ, t)x, t R. So, if Φ(θ, t)x =0, then x= 0. On the other hand, from Definition 3.1 we have that P (θ t)φ(θ, t)x =Φ(θ, t)p(θ)x t R. Therefore, Φ(θ, t)x N(P(θ t). Finally, if y N(P(θ t)), then So, if we put x =Φ(θ t, t)y, thenwegety=φ(θ, t)x. y Bu(θ t). Lemma 3.1. If π =(Φ,σ) is a linear skew-product semiflow on E = X Θ which admits an exponential dichotomy over Θ according to Definition 3.3 with an invariant projector P, then for all θ Θ we have that: B(θ) ={0}, X s (θ)=r(p(θ)) = B + (θ) and X u (θ) =N(P(θ)) = B (θ). Moreover, X = R(P (θ)) + N (P (θ) =X s (θ)+x u (θ) Proof. Consider x B(θ)andφ(t) = (φ x (t),θ t) the corresponding bounded negative continuation of the point (x, θ). Set y = P (θ)x and z =(I P(θ))x. Then x=y+z. From the Definition 2.3 of negative continuation we get that Φ(θ, t + s)x =Φ(θ t, s)φ x (t), 0 t s. So, if we put s = t, thenx=φ(θ t, t)φ x (t), for t 0. Therefore, for t 0we have the following: y = P (θ)x = P (θ t ( t))φ(θ t, t)φ x (t) =Φ(θ t, t)p (θ t)φ x (t). Then y ke βt φ x (t), t 0. Since, φ x (t) is bounded, then y =0. From the Definition 3.3 of exponential dichotomy, we know that Φ(θ, t) :N(P(θ)) N(P(θ t)), t 0,
6 1076 SHUI-NEE CHOW AND HUGO LEIVA is an isomorphism with inverse: Φ(θ t, t) :N(P(θ t)) N(P(θ)), t 0. Since z =(I P(θ))x N(P(θ)) = R(I P (θ)), we get that z =Φ(θ t, t)φ(θ, t)z =Φ(θ t, t)φ(θ, t)(i P (θ))x =Φ(θ t, t)(i P (θ t))φ(θ, t)x, t 0. Hence, z ke βt Φ(θ, t)x, t 0. Since Φ(θ, t)x is bounded, then z =0. Therefore, x =0. SoB(θ) ={0}. Clearly, N (P (θ)) X u (θ) B + (θ)andr(p(θ)) X s (θ) B (θ). The proof follows from X = R(P (θ)) + N (P (θ). Remark 3.2. From Proposition 3.1 and Lemma 3.1 we get that in Definition 3.2 the condition R(I P (θ)) Bu (θ), θ Θ, is equivalent to R(I P (θ)) = (θ), θ Θ. From now on, we will work with Definition 3.3. B u Proposition 3.3. If the skew-product semiflow π =(Φ,σ) has an exponential dichotomy over Θ according to Definition 3.3, then for all θ Θ and t, s R we have that Φ(θ, t + s)(i P (θ)) = Φ(θ t, s)φ(θ, t)(i P (θ)). Proof. (i) If t, s 0, then it follows from the cocycle property (2.1). (ii) If t<0ands<0, then Φ(θ t, s)φ(θ, t)(i P (θ)) =(Φ(θ (t+s), s) N(P(θ t)) ) 1 (Φ(θ t, t) N (P (θ)) ) 1 (I P (θ)) =[Φ(θ t, t)φ(θ (t + s), s) N (P (θ)) ] 1 (I P (θ)). Now, using the cocycle property (2.1), we get that Φ(θ t, s)φ(θ, t)(i P (θ)) = (Φ(θ (t + s), (t + s)) N (P (θ)) ) 1 (I P (θ)) =Φ(θ, t + s)(i P (θ)). (iii) If t>0, s < 0andt+s<0, then Φ(θ t, s)φ(θ, t)(i P (θ)) =(Φ(θ (t+s), s) N(P(θ t)) ) 1 (Φ(θ t, t) N (P (θ)) ) 1 (I P (θ)) =[Φ(θ t, t)φ(θ (t + s), s) N (P (θ)) ] 1 (I P (θ)) =[Φ(θ t, t)φ(θ (t + s), (t + s)+t) N(P(θ)) ] 1 (I P (θ)). Since (t + s) > 0 and t>0, we can apply the cocycle property (2.1) to get Φ(θ t, s)φ(θ, t)(i P (θ)) =[Φ(θ t, t)φ(θ, t)φ(θ (t + s), (t + s)) N (P (θ)) ] 1 (I P (θ)) =(Φ(θ (t+s), (t+s)) N (P (θ)) ) 1 (I P (θ)) =Φ(θ, t + s)(i P (θ)). The case (iv) t>0, s < 0 and t+s>0 is similar.
7 DICHOTOMY FOR SKEW-PRODUCT SEMIFLOW 1077 Proposition 3.4. If the skew-product semiflow π =(Φ,σ) has an exponential dichotomy over Θ according to Definition 3.3 with an invariant projector P on E, then for all θ Θ and t, s R we have that Φ(θ, t)(i P (θ)) = (I P (θ t))φ(θ, t), on N (P (θ)). Proof. If t 0 there is nothing to prove. Suppose t<0. Then, from (3.1) we get that Φ(θ t, t)(i P (θ t)) = (I P (θ))φ(θ t, t). Therefore, (Φ(θ, t) N(P(θ)) ) 1 (I P (θ t)) = (I P (θ))(φ(θ, t) N(P(θ)) ) 1. Then, (I P (θ t)) = Φ(θ, t)(i P (θ))(φ(θ, t) N(P(θ)) ) 1. So, (I P (θ t))φ(θ, t) =Φ(θ, t)(i P (θ)). Proposition 3.5. If the skew-product semiflow π =(Φ,σ) has an exponential dichotomy over Θ according to Definition 3.3, then for all x X fixed, the mapping t φ x (t) :=Φ(θ, t)(i P (θ))x is continuous in R. Moreover, the mapping φ(t) := (φ x (t),θ t) is a negative continuation of the point ((I P (θ))x, θ). Proof. First, we shall prove the continuity at t = 0, which is enough to prove that lim Φ(θ, t)(i P (θ))x =(I P(θ))x. t 0 In fact, taking ɛ>0 and using Proposition 3.3, we get that lim t 0 φx (t) = lim Φ(θ, t)(i P (θ))x t 0 = lim Φ(θ, ɛ + t + ɛ)(i P (θ))x t 0 = lim Φ(θ ( ɛ),t+ɛ)φ(θ, ɛ)(i P (θ))x, t + ɛ>0. t 0 From Definition 2.1 we get that for all z X the mapping s Φ(θ, s)z is continuous for s 0 uniformly on Θ. Therefore, lim t 0 φx (t) =Φ(θ ( ɛ),ɛ)φ(θ, ɛ)(i P (θ))x =(I P(θ))x. Hence lim Φ(θ, t)(i P (θ))x =(I P(θ))x. t 0 Now, consider t<0andh Rsmall enough. Then from Propositions 3.3 and 3.4 we get: lim Φ(θ, t + h)(i P (θ))x = lim Φ(θ t, h)φ(θ, t)(i P (θ))x h 0 h 0 = lim Φ(θ t, h)(i P (θ t))φ(θ, t)x. h 0
8 1078 SHUI-NEE CHOW AND HUGO LEIVA If we put z =Φ(θ, t)x, then lim Φ(θ, t + h)(i P (θ))x = lim Φ(θ t, h)(i P (θ t))z h 0 h 0 =(I P(θ t))z =Φ(θ, t)(i P (θ))x. Corollary 3.1. If the skew-product semiflow π =(Φ,σ) has an exponential dichotomy over Θ according to Definition 3.3 with projector P, then each (x, θ) N (P) has a bounded negative continuous (3.5) φ(t) =(φ x (t),θ t):=(φ(θ, t)(i P (θ))x, θ t), t 0. Moreover, φ x (t) ke tβ x, t 0. Corollary 3.2. If π =(Φ,σ) is a linear skew-product semiflow on E = X Θ which admits an exponential dichotomy over Θ according to Definition 3.3 with an invariant projector P and Bu, then for all θ Θ we have that: X u (θ) =N(P(θ)) = B (θ) =Bu (θ). Moreover, all the bounded negative continuations all given by formula (3.5). Proof. From Proposition 2.2 and Lemma 3.1 we get that N (P (θ)) = B (θ) = Bu (θ), θ Θ. Then from Corollary 3.1 we get that all bounded negative continuations are given by the formula (3.5). Theorem 3.1. If in both Definitions 3.3 and 3.2 of exponential dichotomy we assume that dimr(i P (θ)) <, θ Θ (or dimr(i P (θ)) =, θ Θ ), then Definitions 3.3 and 3.2 are equivalent Characterization of the stable and unstable manifolds. We begin this section with the following definition: Definition 3.4. Given a point (x, θ) E=X Θ, we shall say that Φ(θ, t)x is well defined on R if it is a continuous function on t R and satisfies (a) Φ(θ, t + s)x =Φ(θ t, s)φ(θ, t)x, t, s R, (b) P (θ t)φ(θ, t)x =Φ(θ, t)p(θ)x, t R. Also, we define the set M w := {(x, θ) E:Φ(θ, t)x is well defined}. Remark 3.3. Clearly B u M w. Also, if π =(Φ,σ) has an exponential dichotomy according to Definition 3.3, then N (P) M w. Lemma 3.2. If π =(Φ,σ) has an exponential dichotomy over Θ, then (3.6) X s (θ)=r(p(θ)) = {x X :sup (I P(θ t))φ(θ, t)x < } =: Z u (θ), t 0 (3.7) X u (θ) =N(P(θ)) = {x M w (θ):sup P(θ t)φ(θ, t)x < } =: Z s (θ) t 0 for all θ Θ.
9 DICHOTOMY FOR SKEW-PRODUCT SEMIFLOW 1079 Proof. Suppose x R(P(θ)) = X s (θ). Then P (θ)x = x. Soweget (I P(θ t))φ(θ, t)x = Φ(θ, t)(i P (θ))x =0. Therefore, x Z s (θ). So, X s (θ) Z s (θ). Suppose x Z s (θ) Then there exists a constant C>0such that (I P (θ t))φ(θ, t)x C<, for t 0. Then Φ(θ, t)x =Φ(θ, t)(i P (θ))x +Φ(θ, t)p(θ)x =(I P(θ t))φ(θ, t)x +Φ(θ, t)p(θ)x. So Φ(θ, t)x C+ke βt x, t 0. Hence, x B + =R(P(θ)) = X s (θ). So, Z s (θ) =X s (θ). Now, suppose that x N(P(θ)) = X u (θ). Then P (θ)x = 0. Hence, using Proposition 2.1, we get the following: P (θ t)φ(θ, t)x =Φ(θ, t)p(θ)x =0, t 0. Therefore, X u (θ) Z u (θ). Suppose that x Z u (θ). Then there exists C>0 such that P (θ t)φ(θ, t)x <C, t 0. On the other hand, from Definition 3.4 we get Φ(θ, t)x =(I P(θ t))φ(θ, t)x + P(θ )Φ(θ, t)x =Φ(θ, t)(i P (θ))x + P (θ t)φ(θ, t)x. Then Φ(θ, t)x C+ke βt x, t 0. So, Φ(θ, t)x is bounded for t 0. Therefore x B (θ)=n(p(θ)) = X u (θ). Hence, X u (θ) =Z u (θ). Lemma 3.3. If π =(Φ,σ) has an exponential dichotomy over Θ, then for all η (0,β) we have (3.8) X s (θ) ={x X:supe Φ(θ, t)x < } t 0 (3.9) X u (θ) ={x M w (θ):supe Φ(θ, t)x < } t 0 for all θ Θ. Proof. Denote the right side of (3.8) by Z s (θ). Then clearly X s (θ) Z s (θ). Assume x Zs(θ). Then Φ(θ, t)x Ce ηt, t 0, and x = P (θ)x +(I P(θ))x. It is enough to prove that (I P (θ))x =0. Infact,for t 0 we get (I P(θ))x = Φ(θ ( t),t)φ(θ, t)(i P (θ))x = Φ(θ ( t),t)(i P (θ ( t)))φ(θ, t)x Φ(θ ( t),t)(i P (θ ( t))) Φ(θ, t)x ke βt Ce ηt = kce (β η)t 0, as t. Hence Z s (θ) X s (θ).
10 1080 SHUI-NEE CHOW AND HUGO LEIVA Denote the right side of (3.9) by Z u (θ). Then clearly X u (θ) Z u (θ). Suppose x Z u (θ). Then x M w (θ), Φ(θ, t)x Ce ηt, t 0, and x = P (θ)x +(I P(θ))x. It is enough to prove that P (θ)x =0. Infact,for t 0, from Definition 3.4 we get that P (θ)x = Φ(θ ( t),t)φ(θ, t)p(θ)x = Φ(θ ( t),t)p(θ ( t))φ(θ, t)x Φ(θ ( t),t)p(θ ( t)) Φ(θ, t)x ke βt Ce ηt = kce (η β)t 0, as t +. Hence Z u (θ) X u (θ). In conclusion we have the following theorem: Theorem 3.2. If π =(Φ,σ) has an exponential dichotomy over Θ, then we have the following: (a) X s, X u are invariant subbundles of E under the flow π and E = X s + X u and X s = B +, X u = B. (the Whitney sum of two subbundles). (b) We get the following characterization of X s and X u : (3.10) X s = {(x, θ) E:sup (I P(θ t))φ(θ, t)x < }, t 0 (3.11) X u = {(x, θ) M w :sup P(θ t)φ(θ, t)x < }. t 0 (c) For η (0,β) we get (3.12) X s = {(x, θ) E:supe ηt Φ(θ, t)x < }, t 0 (3.13) X u = {(x, θ) M w :supe ηt Φ(θ, t)x < }. t 0 References [1] S.N.ChowandH.Leiva,Dynamical spectrum for time dependent linear systems in banach spaces, Japan J. Indust. Appl. Math. 11 (1994), MR 95i:34106 [2] S. N. Chow and H. Leiva, Dynamical spectrum for skew-product flow in banach spaces, Boundary Problems for Functional Differential Equations, World Sci. Publ., Singapore, 1995, pp [3] S. N. Chow and H. Leiva, Existence and roughness of the exponential dichotomy for skewproduct semiflow in banach spaces, J. Differential Equations 120 (1995), CMP 95:17 [4] W. A. Coppel, Dichotomies in stability theory, Lect. Notes in Math, vol. 629, Springer-Verlag, New York, MR 58:1332 [5] J. L. Daleckii and M. G. Krein, Stability of solutions of differential equations in Banach space, Transl. Math. Monographs, vol. 43, Amer. Math. Soc., Providence, RI, MR 50:5126 [6]J.K.Hale,Asymptotic behavior of dissipative systems, Math. Surveys and Monographs, vol. 25, Amer. Soc., Providence, R.I., MR 89g:58059 [7] D. Henry, Geometric theory of semilinear parabolic equations, Springer-Verlag, New York, MR 83j:35084 [8] N. Levinson, The asymptotic behavior of system of linear differential equations, Amer. J. Math. vol. 68, pp. 1 6, MR 7:381f [9] X.B.Lin,Exponential dichotomies and homoclinic orbits in functional-differential equations, J. Differential Equations 63 (1986), MR 87j:34138
11 DICHOTOMY FOR SKEW-PRODUCT SEMIFLOW 1081 [10] X. B. Lin, Exponential dichotomies in intermediate spaces with applications to a diffusively perturbed predator-prey model, J. Differential Equations 108 (1994), MR 95c:35139 [11] L. T. Magalhães, The spectrum of invariant sets for dissipative semiflows, in Dynamics Of Infinite Dimensional Systems, NATO Adv. Sci. Inst. Ser. F: Comput. Systems Sci., vol. 37, Springer Verlag, New York, 1987, pp CMP 20:06 [12] J. L. Massera and J. J. Schäffer, Linear differential equations and function spaces, Academic Press, New York, MR 35:3197 [13] K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations, vol. 55, pp , MR 86d:58088 [14] O. Perron, Die stabilit atsfrage bei differentialgleichungen, Math. Z vol. 32, pp , [15] R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splitting for linear differential systems I, II, III J. Differential Equations. 15 (1974), , 22 (1976), , MR 49:6209 [16] R. J. Sacker and G. R. Sell, Dichotomies for linear evolutionary equations in Banach spaces, J. Differential Equations 113 (1994), CMP 95:01 CDSNS Georgia Tech, Atlanta, Georgia address: chow@math.gatech.edu CDSNS Georgia Tech, Atlanta, Georgia and ULA-Venezuela address: leiva@math.gatech.edu
DISCRETE METHODS AND EXPONENTIAL DICHOTOMY OF SEMIGROUPS. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXIII, 2(2004), pp. 97 205 97 DISCRETE METHODS AND EXPONENTIAL DICHOTOMY OF SEMIGROUPS A. L. SASU Abstract. The aim of this paper is to characterize the uniform exponential
More informationOn the dynamics of strongly tridiagonal competitive-cooperative system
On the dynamics of strongly tridiagonal competitive-cooperative system Chun Fang University of Helsinki International Conference on Advances on Fractals and Related Topics, Hong Kong December 14, 2012
More informationOn (h, k) trichotomy for skew-evolution semiflows in Banach spaces
Stud. Univ. Babeş-Bolyai Math. 56(2011), No. 4, 147 156 On (h, k) trichotomy for skew-evolution semiflows in Banach spaces Codruţa Stoica and Mihail Megan Abstract. In this paper we define the notion of
More informationSome Concepts of Uniform Exponential Dichotomy for Skew-Evolution Semiflows in Banach Spaces
Theory and Applications of Mathematics & Computer Science 6 (1) (2016) 69 76 Some Concepts of Uniform Exponential Dichotomy for Skew-Evolution Semiflows in Banach Spaces Diana Borlea a, a Department of
More informationEVANESCENT SOLUTIONS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS
EVANESCENT SOLUTIONS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS Cezar Avramescu Abstract The problem of existence of the solutions for ordinary differential equations vanishing at ± is considered. AMS
More informationA. L. Sasu EXPONENTIAL INSTABILITY AND COMPLETE ADMISSIBILITY FOR SEMIGROUPS IN BANACH SPACES
Rend. Sem. Mat. Univ. Pol. Torino - Vol. 63, 2 (2005) A. L. Sasu EXPONENTIAL INSTABILITY AND COMPLETE ADMISSIBILITY FOR SEMIGROUPS IN BANACH SPACES Abstract. We associate a discrete-time equation to an
More informationExponential stability of families of linear delay systems
Exponential stability of families of linear delay systems F. Wirth Zentrum für Technomathematik Universität Bremen 28334 Bremen, Germany fabian@math.uni-bremen.de Keywords: Abstract Stability, delay systems,
More informationA REMARK ON THE GLOBAL DYNAMICS OF COMPETITIVE SYSTEMS ON ORDERED BANACH SPACES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 A REMARK ON THE GLOBAL DYNAMICS OF COMPETITIVE SYSTEMS ON ORDERED BANACH SPACES KING-YEUNG LAM
More informationTHE MULTIPLICATIVE ERGODIC THEOREM OF OSELEDETS
THE MULTIPLICATIVE ERGODIC THEOREM OF OSELEDETS. STATEMENT Let (X, µ, A) be a probability space, and let T : X X be an ergodic measure-preserving transformation. Given a measurable map A : X GL(d, R),
More informationComputing Sacker-Sell spectra in discrete time dynamical systems
Computing Sacker-Sell spectra in discrete time dynamical systems Thorsten Hüls Fakultät für Mathematik, Universität Bielefeld Postfach 100131, 33501 Bielefeld, Germany huels@math.uni-bielefeld.de March
More informationSpectrum for compact operators on Banach spaces
Submitted to Journal of the Mathematical Society of Japan Spectrum for compact operators on Banach spaces By Luis Barreira, Davor Dragičević Claudia Valls (Received Nov. 22, 203) (Revised Jan. 23, 204)
More informationProblem Set 6: Solutions Math 201A: Fall a n x n,
Problem Set 6: Solutions Math 201A: Fall 2016 Problem 1. Is (x n ) n=0 a Schauder basis of C([0, 1])? No. If f(x) = a n x n, n=0 where the series converges uniformly on [0, 1], then f has a power series
More informationMIHAIL MEGAN and LARISA BIRIŞ
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA DIN IAŞI (S.N.) MATEMATICĂ, Tomul LIV, 2008, f.2 POINTWISE EXPONENTIAL TRICHOTOMY OF LINEAR SKEW-PRODUCT SEMIFLOWS BY MIHAIL MEGAN and LARISA BIRIŞ Abstract.
More informationConvergence in Almost Periodic Fisher and Kolmogorov Models
Convergence in Almost Periodic Fisher and Kolmogorov Models Wenxian Shen Department of Mathematics Auburn University Auburn, AL 36849 and Yingfei Yi School of Mathematics Georgia Institute of Technology
More informationA FIXED POINT THEOREM FOR GENERALIZED NONEXPANSIVE MULTIVALUED MAPPINGS
Fixed Point Theory, (0), No., 4-46 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html A FIXED POINT THEOREM FOR GENERALIZED NONEXPANSIVE MULTIVALUED MAPPINGS A. ABKAR AND M. ESLAMIAN Department of Mathematics,
More informationTHE PERRON PROBLEM FOR C-SEMIGROUPS
Math. J. Okayama Univ. 46 (24), 141 151 THE PERRON PROBLEM FOR C-SEMIGROUPS Petre PREDA, Alin POGAN and Ciprian PREDA Abstract. Characterizations of Perron-type for the exponential stability of exponentially
More informationHOMOLOGICAL LOCAL LINKING
HOMOLOGICAL LOCAL LINKING KANISHKA PERERA Abstract. We generalize the notion of local linking to include certain cases where the functional does not have a local splitting near the origin. Applications
More informationSOME GENERALIZATION OF MINTY S LEMMA. Doo-Young Jung
J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math. 6(1999), no 1. 33 37 SOME GENERALIZATION OF MINTY S LEMMA Doo-Young Jung Abstract. We obtain a generalization of Behera and Panda s result on nonlinear
More informationA Finite-time Condition for Exponential Trichotomy in Infinite Dynamical Systems
Canad. J. Math. Vol. 67 (5), 015 pp. 1065 1090 http://dx.doi.org/10.4153/cjm-014-03-3 c Canadian Mathematical Society 014 A Finite-te Condition for Exponential Trichotomy in Infinite Dynamical Systems
More informationCenter manifold and exponentially-bounded solutions of a forced Newtonian system with dissipation
Nonlinear Differential Equations, Electron. J. Diff. Eqns., Conf. 5, 2, pp. 69 8 http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp) Center manifold
More informationOn the simplest expression of the perturbed Moore Penrose metric generalized inverse
Annals of the University of Bucharest (mathematical series) 4 (LXII) (2013), 433 446 On the simplest expression of the perturbed Moore Penrose metric generalized inverse Jianbing Cao and Yifeng Xue Communicated
More informationTakens embedding theorem for infinite-dimensional dynamical systems
Takens embedding theorem for infinite-dimensional dynamical systems James C. Robinson Mathematics Institute, University of Warwick, Coventry, CV4 7AL, U.K. E-mail: jcr@maths.warwick.ac.uk Abstract. Takens
More informationViscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces
Viscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces YUAN-HENG WANG Zhejiang Normal University Department of Mathematics Yingbing Road 688, 321004 Jinhua
More informationCompetitive Exclusion in a Discrete-time, Size-structured Chemostat Model
Competitive Exclusion in a Discrete-time, Size-structured Chemostat Model Hal L. Smith Department of Mathematics Arizona State University Tempe, AZ 85287 1804, USA E-mail: halsmith@asu.edu Xiao-Qiang Zhao
More informationThe best generalised inverse of the linear operator in normed linear space
Linear Algebra and its Applications 420 (2007) 9 19 www.elsevier.com/locate/laa The best generalised inverse of the linear operator in normed linear space Ping Liu, Yu-wen Wang School of Mathematics and
More informationTOPOLOGICAL ENTROPY FOR DIFFERENTIABLE MAPS OF INTERVALS
Chung, Y-.M. Osaka J. Math. 38 (200), 2 TOPOLOGICAL ENTROPY FOR DIFFERENTIABLE MAPS OF INTERVALS YONG MOO CHUNG (Received February 9, 998) Let Á be a compact interval of the real line. For a continuous
More informationGLOBAL ATTRACTIVITY IN A CLASS OF NONMONOTONE REACTION-DIFFUSION EQUATIONS WITH TIME DELAY
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 17, Number 1, Spring 2009 GLOBAL ATTRACTIVITY IN A CLASS OF NONMONOTONE REACTION-DIFFUSION EQUATIONS WITH TIME DELAY XIAO-QIANG ZHAO ABSTRACT. The global attractivity
More informationFIXED POINT THEOREMS AND CHARACTERIZATIONS OF METRIC COMPLETENESS. Tomonari Suzuki Wataru Takahashi. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 8, 1996, 371 382 FIXED POINT THEOREMS AND CHARACTERIZATIONS OF METRIC COMPLETENESS Tomonari Suzuki Wataru Takahashi
More informationA NONLOCAL REACTION-DIFFUSION POPULATION MODEL WITH STAGE STRUCTURE
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 11, Number 3, Fall 23 A NONLOCAL REACTION-DIFFUSION POPULATION MODEL WITH STAGE STRUCTURE Dedicated to Professor Paul Waltman on the occasion of his retirement
More informationSENSITIVITY AND REGIONALLY PROXIMAL RELATION IN MINIMAL SYSTEMS
SENSITIVITY AND REGIONALLY PROXIMAL RELATION IN MINIMAL SYSTEMS SONG SHAO, XIANGDONG YE AND RUIFENG ZHANG Abstract. A topological dynamical system is n-sensitive, if there is a positive constant such that
More informationLipschitz shadowing implies structural stability
Lipschitz shadowing implies structural stability Sergei Yu. Pilyugin Sergei B. Tihomirov Abstract We show that the Lipschitz shadowing property of a diffeomorphism is equivalent to structural stability.
More informationRenormings of c 0 and the minimal displacement problem
doi: 0.55/umcsmath-205-0008 ANNALES UNIVERSITATIS MARIAE CURIE-SKŁODOWSKA LUBLIN POLONIA VOL. LXVIII, NO. 2, 204 SECTIO A 85 9 ŁUKASZ PIASECKI Renormings of c 0 and the minimal displacement problem Abstract.
More informationA note on the monotonicity of matrix Riccati equations
DIMACS Technical Report 2004-36 July 2004 A note on the monotonicity of matrix Riccati equations by Patrick De Leenheer 1,2 Eduardo D. Sontag 3,4 1 DIMACS Postdoctoral Fellow, email: leenheer@math.rutgers.edu
More informationALMOST PERIODIC SOLUTIONS OF NON-AUTONOMOUS BEVERTON-HOLT DIFFERENCE EQUATION. 1. Introduction
ALMOST PERIODIC SOLUTIONS OF NON-AUTONOMOUS BEVERTON-HOLT DIFFERENCE EQUATION DAVID CHEBAN AND CRISTIANA MAMMANA Abstract. The article is devoted to the study of almost periodic solutions of difference
More informationTHE IMPLICIT FUNCTION THEOREM FOR CONTINUOUS FUNCTIONS. Carlos Biasi Carlos Gutierrez Edivaldo L. dos Santos. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 32, 2008, 177 185 THE IMPLICIT FUNCTION THEOREM FOR CONTINUOUS FUNCTIONS Carlos Biasi Carlos Gutierrez Edivaldo L.
More informationFURTHER STUDIES OF STRONGLY AMENABLE -REPRESENTATIONS OF LAU -ALGEBRAS
FURTHER STUDIES OF STRONGLY AMENABLE -REPRESENTATIONS OF LAU -ALGEBRAS FATEMEH AKHTARI and RASOUL NASR-ISFAHANI Communicated by Dan Timotin The new notion of strong amenability for a -representation of
More informationPermanence Implies the Existence of Interior Periodic Solutions for FDEs
International Journal of Qualitative Theory of Differential Equations and Applications Vol. 2, No. 1 (2008), pp. 125 137 Permanence Implies the Existence of Interior Periodic Solutions for FDEs Xiao-Qiang
More informationMathematical Journal of Okayama University
Mathematical Journal of Okayama University Volume 46, Issue 1 24 Article 29 JANUARY 24 The Perron Problem for C-Semigroups Petre Prada Alin Pogan Ciprian Preda West University of Timisoara University of
More informationWeakly Compact Composition Operators on Hardy Spaces of the Upper Half-Plane 1
Int. Journal of Math. Analysis, Vol. 4, 2010, no. 37, 1851-1856 Weakly Compact Composition Operators on Hardy Spaces of the Upper Half-Plane 1 Hong Bin Bai School of Science Sichuan University of Science
More informationATTRACTORS FOR SEMILINEAR PARABOLIC PROBLEMS WITH DIRICHLET BOUNDARY CONDITIONS IN VARYING DOMAINS. Emerson A. M. de Abreu Alexandre N.
ATTRACTORS FOR SEMILINEAR PARABOLIC PROBLEMS WITH DIRICHLET BOUNDARY CONDITIONS IN VARYING DOMAINS Emerson A. M. de Abreu Alexandre N. Carvalho Abstract Under fairly general conditions one can prove that
More informationHomoclinic trajectories of non-autonomous maps
Homoclinic trajectories of non-autonomous maps Thorsten Hüls Department of Mathematics Bielefeld University Germany huels@math.uni-bielefeld.de www.math.uni-bielefeld.de:~/huels Non-autonomous dynamical
More informationON RIGIDITY OF ALGEBRAIC DYNAMICAL SYSTEMS
ON RIGIDITY OF ALGEBRAIC DYNAMICAL SYSTEMS ALEX CLARK AND ROBBERT FOKKINK Abstract. We study topological rigidity of algebraic dynamical systems. In the first part of this paper we give an algebraic condition
More informationSYNCHRONIZATION OF NONAUTONOMOUS DYNAMICAL SYSTEMS
Electronic Journal of Differential Equations, Vol. 003003, No. 39, pp. 1 10. ISSN: 107-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu login: ftp SYNCHRONIZATION OF
More informationTHE DAUGAVETIAN INDEX OF A BANACH SPACE 1. INTRODUCTION
THE DAUGAVETIAN INDEX OF A BANACH SPACE MIGUEL MARTÍN ABSTRACT. Given an infinite-dimensional Banach space X, we introduce the daugavetian index of X, daug(x), as the greatest constant m 0 such that Id
More informationHalf of Final Exam Name: Practice Problems October 28, 2014
Math 54. Treibergs Half of Final Exam Name: Practice Problems October 28, 24 Half of the final will be over material since the last midterm exam, such as the practice problems given here. The other half
More informationThe Mild Modification of the (DL)-Condition and Fixed Point Theorems for Some Generalized Nonexpansive Mappings in Banach Spaces
Int. Journal of Math. Analysis, Vol. 6, 2012, no. 19, 933-940 The Mild Modification of the (DL)-Condition and Fixed Point Theorems for Some Generalized Nonexpansive Mappings in Banach Spaces Kanok Chuikamwong
More informationEXISTENCE OF CONJUGACIES AND STABLE MANIFOLDS VIA SUSPENSIONS
Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 172, pp. 1 11. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF CONJUGACIES AND STABLE MANIFOLDS
More informationPropagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R
Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R P. Poláčik School of Mathematics, University of Minnesota Minneapolis, MN 55455 Abstract We consider semilinear
More informationA Criterion for Flatness of Sections of Adjoint Bundle of a Holomorphic Principal Bundle over a Riemann Surface
Documenta Math. 111 A Criterion for Flatness of Sections of Adjoint Bundle of a Holomorphic Principal Bundle over a Riemann Surface Indranil Biswas Received: August 8, 2013 Communicated by Edward Frenkel
More informationCOMPOSITION SEMIGROUPS ON BMOA AND H AUSTIN ANDERSON, MIRJANA JOVOVIC, AND WAYNE SMITH
COMPOSITION SEMIGROUPS ON BMOA AND H AUSTIN ANDERSON, MIRJANA JOVOVIC, AND WAYNE SMITH Abstract. We study [ϕ t, X], the maximal space of strong continuity for a semigroup of composition operators induced
More informationTwo-Step Iteration Scheme for Nonexpansive Mappings in Banach Space
Mathematica Moravica Vol. 19-1 (2015), 95 105 Two-Step Iteration Scheme for Nonexpansive Mappings in Banach Space M.R. Yadav Abstract. In this paper, we introduce a new two-step iteration process to approximate
More informationAN EFFECTIVE METRIC ON C(H, K) WITH NORMAL STRUCTURE. Mona Nabiei (Received 23 June, 2015)
NEW ZEALAND JOURNAL OF MATHEMATICS Volume 46 (2016), 53-64 AN EFFECTIVE METRIC ON C(H, K) WITH NORMAL STRUCTURE Mona Nabiei (Received 23 June, 2015) Abstract. This study first defines a new metric with
More informationFUNCTION BASES FOR TOPOLOGICAL VECTOR SPACES. Yılmaz Yılmaz
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 33, 2009, 335 353 FUNCTION BASES FOR TOPOLOGICAL VECTOR SPACES Yılmaz Yılmaz Abstract. Our main interest in this
More informationBanach Spaces V: A Closer Look at the w- and the w -Topologies
BS V c Gabriel Nagy Banach Spaces V: A Closer Look at the w- and the w -Topologies Notes from the Functional Analysis Course (Fall 07 - Spring 08) In this section we discuss two important, but highly non-trivial,
More informationComputations of Critical Groups at a Degenerate Critical Point for Strongly Indefinite Functionals
Journal of Mathematical Analysis and Applications 256, 462 477 (2001) doi:10.1006/jmaa.2000.7292, available online at http://www.idealibrary.com on Computations of Critical Groups at a Degenerate Critical
More informationBEST APPROXIMATIONS AND ORTHOGONALITIES IN 2k-INNER PRODUCT SPACES. Seong Sik Kim* and Mircea Crâşmăreanu. 1. Introduction
Bull Korean Math Soc 43 (2006), No 2, pp 377 387 BEST APPROXIMATIONS AND ORTHOGONALITIES IN -INNER PRODUCT SPACES Seong Sik Kim* and Mircea Crâşmăreanu Abstract In this paper, some characterizations of
More informationTHE NEARLY ADDITIVE MAPS
Bull. Korean Math. Soc. 46 (009), No., pp. 199 07 DOI 10.4134/BKMS.009.46..199 THE NEARLY ADDITIVE MAPS Esmaeeil Ansari-Piri and Nasrin Eghbali Abstract. This note is a verification on the relations between
More informationON CONTINUITY OF MEASURABLE COCYCLES
Journal of Applied Analysis Vol. 6, No. 2 (2000), pp. 295 302 ON CONTINUITY OF MEASURABLE COCYCLES G. GUZIK Received January 18, 2000 and, in revised form, July 27, 2000 Abstract. It is proved that every
More informationESTIMATES ON THE NUMBER OF LIMIT CYCLES OF A GENERALIZED ABEL EQUATION
Manuscript submitted to Website: http://aimsciences.org AIMS Journals Volume 00, Number 0, Xxxx XXXX pp. 000 000 ESTIMATES ON THE NUMBER OF LIMIT CYCLES OF A GENERALIZED ABEL EQUATION NAEEM M.H. ALKOUMI
More informationON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 7, July 1996 ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES M. I. OSTROVSKII (Communicated by Dale Alspach) Abstract.
More informationFUNCTIONAL COMPRESSION-EXPANSION FIXED POINT THEOREM
Electronic Journal of Differential Equations, Vol. 28(28), No. 22, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) FUNCTIONAL
More informationEVOLUTIONARY SEMIGROUPS AND DICHOTOMY OF LINEAR SKEW-PRODUCT FLOWS ON LOCALLY COMPACT SPACES WITH BANACH FIBERS
EVOLUTIONARY SEMIGROUPS AND DICHOTOMY OF LINEAR SKEW-PRODUCT FLOWS ON LOCALLY COMPACT SPACES WITH BANACH FIBERS Y. LATUSHKIN, S. MONTGOMERY-SMITH, AND T. RANDOLPH Abstract. We study evolutionary semigroups
More informationSOME REMARKS ON KRASNOSELSKII S FIXED POINT THEOREM
Fixed Point Theory, Volume 4, No. 1, 2003, 3-13 http://www.math.ubbcluj.ro/ nodeacj/journal.htm SOME REMARKS ON KRASNOSELSKII S FIXED POINT THEOREM CEZAR AVRAMESCU AND CRISTIAN VLADIMIRESCU Department
More informationarxiv: v1 [math.fa] 2 Jan 2017
Methods of Functional Analysis and Topology Vol. 22 (2016), no. 4, pp. 387 392 L-DUNFORD-PETTIS PROPERTY IN BANACH SPACES A. RETBI AND B. EL WAHBI arxiv:1701.00552v1 [math.fa] 2 Jan 2017 Abstract. In this
More informationNONLINEAR DIFFERENTIAL INEQUALITY. 1. Introduction. In this paper the following nonlinear differential inequality
M athematical Inequalities & Applications [2407] First Galley Proofs NONLINEAR DIFFERENTIAL INEQUALITY N. S. HOANG AND A. G. RAMM Abstract. A nonlinear differential inequality is formulated in the paper.
More informationEXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC SCHRÖDINGER EQUATIONS
Electronic Journal of Differential Equations, Vol. 017 (017), No. 15, pp. 1 7. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC
More informationSome Characterizations for the Uniform Exponential Expansiveness of Linear Skew-evolution Semiflows
Ξ45ffΞ3 μ fl Ω $ Vol.45, No.3 206 5" ADVANCES IN MATHEMATICS CHINA May, 206 doi: 0.845/sxjz.20473b Some Characterizations for the Uniform Exponential Expansiveness of Linear Skew-evolution Semiflows YUE
More informationEXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM
EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM JENICĂ CRÎNGANU We derive existence results for operator equations having the form J ϕu = N f u, by using
More informationCOMMON COMPLEMENTS OF TWO SUBSPACES OF A HILBERT SPACE
COMMON COMPLEMENTS OF TWO SUBSPACES OF A HILBERT SPACE MICHAEL LAUZON AND SERGEI TREIL Abstract. In this paper we find a necessary and sufficient condition for two closed subspaces, X and Y, of a Hilbert
More information1. Classifying Spaces. Classifying Spaces
Classifying Spaces 1. Classifying Spaces. To make our lives much easier, all topological spaces from now on will be homeomorphic to CW complexes. Fact: All smooth manifolds are homeomorphic to CW complexes.
More informationMultiplication Operators with Closed Range in Operator Algebras
J. Ana. Num. Theor. 1, No. 1, 1-5 (2013) 1 Journal of Analysis & Number Theory An International Journal Multiplication Operators with Closed Range in Operator Algebras P. Sam Johnson Department of Mathematical
More informationApplied Mathematics Letters
Applied Mathematics Letters 25 (2012) 545 549 Contents lists available at SciVerse ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml On the equivalence of four chaotic
More informationON CHARACTERISTIC 0 AND WEAKLY ALMOST PERIODIC FLOWS. Hyungsoo Song
Kangweon-Kyungki Math. Jour. 11 (2003), No. 2, pp. 161 167 ON CHARACTERISTIC 0 AND WEAKLY ALMOST PERIODIC FLOWS Hyungsoo Song Abstract. The purpose of this paper is to study and characterize the notions
More informationarxiv:math/ v1 [math.fa] 26 Oct 1993
arxiv:math/9310217v1 [math.fa] 26 Oct 1993 ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES M.I.Ostrovskii Abstract. It is proved that there exist complemented subspaces of countable topological
More informationSung-Wook Park*, Hyuk Han**, and Se Won Park***
JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 16, No. 1, June 2003 CONTINUITY OF LINEAR OPERATOR INTERTWINING WITH DECOMPOSABLE OPERATORS AND PURE HYPONORMAL OPERATORS Sung-Wook Park*, Hyuk Han**,
More informationTWO MAPPINGS RELATED TO SEMI-INNER PRODUCTS AND THEIR APPLICATIONS IN GEOMETRY OF NORMED LINEAR SPACES. S.S. Dragomir and J.J.
RGMIA Research Report Collection, Vol. 2, No. 1, 1999 http://sci.vu.edu.au/ rgmia TWO MAPPINGS RELATED TO SEMI-INNER PRODUCTS AND THEIR APPLICATIONS IN GEOMETRY OF NORMED LINEAR SPACES S.S. Dragomir and
More informationTwo perturbation results for semi-linear dynamic equations on measure chains
Two perturbation results for semi-linear dynamic equations on measure chains CHRISTIAN PÖTZSCHE1 Department of Mathematics, University of Augsburg D-86135 Augsburg, Germany E-mail: christian.poetzsche@math.uni-augsburg.de
More informationDEGREE OF EQUIVARIANT MAPS BETWEEN GENERALIZED G-MANIFOLDS
DEGREE OF EQUIVARIANT MAPS BETWEEN GENERALIZED G-MANIFOLDS NORBIL CORDOVA, DENISE DE MATTOS, AND EDIVALDO L. DOS SANTOS Abstract. Yasuhiro Hara in [10] and Jan Jaworowski in [11] studied, under certain
More informationDIFFERENTIABLE CONJUGACY NEAR COMPACT INVARIANT MANIFOLDS
DIFFERENTIABLE CONJUGACY NEAR COMPACT INVARIANT MANIFOLDS CLARK ROBINSON 0. Introduction In this paper 1, we show how the differentiable linearization of a diffeomorphism near a hyperbolic fixed point
More informationCONVERGENCE OF APPROXIMATING FIXED POINTS FOR MULTIVALUED NONSELF-MAPPINGS IN BANACH SPACES. Jong Soo Jung. 1. Introduction
Korean J. Math. 16 (2008), No. 2, pp. 215 231 CONVERGENCE OF APPROXIMATING FIXED POINTS FOR MULTIVALUED NONSELF-MAPPINGS IN BANACH SPACES Jong Soo Jung Abstract. Let E be a uniformly convex Banach space
More informationPROBLEMS. (b) (Polarization Identity) Show that in any inner product space
1 Professor Carl Cowen Math 54600 Fall 09 PROBLEMS 1. (Geometry in Inner Product Spaces) (a) (Parallelogram Law) Show that in any inner product space x + y 2 + x y 2 = 2( x 2 + y 2 ). (b) (Polarization
More informationON THE REGULARITY OF ONE PARAMETER TRANSFORMATION GROUPS IN BARRELED LOCALLY CONVEX TOPOLOGICAL VECTOR SPACES
Communications on Stochastic Analysis Vol. 9, No. 3 (2015) 413-418 Serials Publications www.serialspublications.com ON THE REGULARITY OF ONE PARAMETER TRANSFORMATION GROUPS IN BARRELED LOCALLY CONVEX TOPOLOGICAL
More informationPolarization constant K(n, X) = 1 for entire functions of exponential type
Int. J. Nonlinear Anal. Appl. 6 (2015) No. 2, 35-45 ISSN: 2008-6822 (electronic) http://dx.doi.org/10.22075/ijnaa.2015.252 Polarization constant K(n, X) = 1 for entire functions of exponential type A.
More informationTANGENT BUNDLE EMBEDDINGS OF MANIFOLDS IN EUCLIDEAN SPACE
TANGENT BUNDLE EMBEDDINGS OF MANIFOLDS IN EUCLIDEAN SPACE MOHAMMAD GHOMI Abstract. For a given n-dimensional manifold M n we study the problem of finding the smallest integer N(M n ) such that M n admits
More informationStrictly convex functions on complete Finsler manifolds
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 126, No. 4, November 2016, pp. 623 627. DOI 10.1007/s12044-016-0307-2 Strictly convex functions on complete Finsler manifolds YOE ITOKAWA 1, KATSUHIRO SHIOHAMA
More informationSome sequence spaces in 2-normed spaces defined by Musielak-Orlicz function
Acta Univ. Sapientiae, Mathematica, 3, 20) 97 09 Some sequence spaces in 2-normed spaces defined by Musielak-Orlicz function Kuldip Raj School of Mathematics Shri Mata Vaishno Devi University Katra-82320,
More informationPERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 207 222 PERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS Fumi-Yuki Maeda and Takayori Ono Hiroshima Institute of Technology, Miyake,
More informationBoundedly complete weak-cauchy basic sequences in Banach spaces with the PCP
Journal of Functional Analysis 253 (2007) 772 781 www.elsevier.com/locate/jfa Note Boundedly complete weak-cauchy basic sequences in Banach spaces with the PCP Haskell Rosenthal Department of Mathematics,
More informationMAT 578 FUNCTIONAL ANALYSIS EXERCISES
MAT 578 FUNCTIONAL ANALYSIS EXERCISES JOHN QUIGG Exercise 1. Prove that if A is bounded in a topological vector space, then for every neighborhood V of 0 there exists c > 0 such that tv A for all t > c.
More informationRESEARCH ANNOUNCEMENTS OPERATORS ON FUNCTION SPACES
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 78, Number 5, September 1972 RESEARCH ANNOUNCEMENTS The purpose of this department is to provide early announcement of significant new results, with
More informationAdapted metrics for dominated splittings
Adapted metrics for dominated splittings Nikolaz Gourmelon January 15, 27 Abstract A Riemannian metric is adapted to an hyperbolic set of a diffeomorphism if, in this metric, the expansion/contraction
More informationThe Morozov-Jacobson Theorem on 3-dimensional Simple Lie Subalgebras
The Morozov-Jacobson Theorem on 3-dimensional Simple Lie Subalgebras Klaus Pommerening July 1979 english version April 2012 The Morozov-Jacobson theorem says that every nilpotent element of a semisimple
More informationRepresentations and Derivations of Modules
Irish Math. Soc. Bulletin 47 (2001), 27 39 27 Representations and Derivations of Modules JANKO BRAČIČ Abstract. In this article we define and study derivations between bimodules. In particular, we define
More informationITERATIVE SCHEMES FOR APPROXIMATING SOLUTIONS OF ACCRETIVE OPERATORS IN BANACH SPACES SHOJI KAMIMURA AND WATARU TAKAHASHI. Received December 14, 1999
Scientiae Mathematicae Vol. 3, No. 1(2000), 107 115 107 ITERATIVE SCHEMES FOR APPROXIMATING SOLUTIONS OF ACCRETIVE OPERATORS IN BANACH SPACES SHOJI KAMIMURA AND WATARU TAKAHASHI Received December 14, 1999
More informationA PROOF OF A CONVEX-VALUED SELECTION THEOREM WITH THE CODOMAIN OF A FRÉCHET SPACE. Myung-Hyun Cho and Jun-Hui Kim. 1. Introduction
Comm. Korean Math. Soc. 16 (2001), No. 2, pp. 277 285 A PROOF OF A CONVEX-VALUED SELECTION THEOREM WITH THE CODOMAIN OF A FRÉCHET SPACE Myung-Hyun Cho and Jun-Hui Kim Abstract. The purpose of this paper
More informationAW -Convergence and Well-Posedness of Non Convex Functions
Journal of Convex Analysis Volume 10 (2003), No. 2, 351 364 AW -Convergence Well-Posedness of Non Convex Functions Silvia Villa DIMA, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy villa@dima.unige.it
More informationP.S. Gevorgyan and S.D. Iliadis. 1. Introduction
MATEMATIČKI VESNIK MATEMATIQKI VESNIK 70, 2 (208), 0 9 June 208 research paper originalni nauqni rad GROUPS OF GENERALIZED ISOTOPIES AND GENERALIZED G-SPACES P.S. Gevorgyan and S.D. Iliadis Abstract. The
More informationAvailable online at J. Nonlinear Sci. Appl., 10 (2017), Research Article
Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 (2017), 2719 2726 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa An affirmative answer to
More informationGinés López 1, Miguel Martín 1 2, and Javier Merí 1
NUMERICAL INDEX OF BANACH SPACES OF WEAKLY OR WEAKLY-STAR CONTINUOUS FUNCTIONS Ginés López 1, Miguel Martín 1 2, and Javier Merí 1 Departamento de Análisis Matemático Facultad de Ciencias Universidad de
More informationALGEBRAIC HYPERBOLICITY OF THE VERY GENERAL QUINTIC SURFACE IN P 3
ALGEBRAIC HYPERBOLICITY OF THE VERY GENERAL QUINTIC SURFACE IN P 3 IZZET COSKUN AND ERIC RIEDL Abstract. We prove that a curve of degree dk on a very general surface of degree d 5 in P 3 has geometric
More information