Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 584874, 13 pages doi:10.1155/2011/584874 Research Article Weighted S-Asymptotically ω-Periodic Solutions of a Class of Fractional Differential Equations Claudio Cuevas, 1 Michelle Pierri, 2 and Alex Sepulveda 3 1 Departamento de Matem ´ atica, Universidade Federal de Pernambuco, 50540-740 Recife, PE, Brazil 2 Departamento de F ´ ısica e Matem ´ atica da Faculdade de Filosofia, Ci ˆ encias e Letras de Ribeir ˜ ao Preto, Universidade de S ˜ ao Paulo, 14040-901 Ribeir ˜ ao Preto, SP, Brazil 3 Departamento de Matem ´ atica y Estad ´ ıstica, Universidad de La Frontera, Casilla 54-D, Temuco, Chile Correspondence should be addressed to Claudio Cuevas, cch@dmat.ufpe.br Received 23 September 2010; Accepted 8 December 2010 Academic Editor: J. J. Trujillo Copyright q 2011 Claudio C uevas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work i s properly cited. We study the existence of weighted S-asymptotically ω-periodic mild solutions for a class of abstract fractional differential equations of the form u   ∂ −α1 Au  ft, u, 1 <α<2, where A is a linear sectorial operator of negative type. 1. Introduction S-asymptotically ω-periodic functions have applications to several problems, for example in the theory of functional differential equations, fractional differential equations, integral equations and partial differential equations. The concept of S-asymptotic ω-periodicity was introduced in the literature by Henr ´ ıquez et al. 1, 2. Since then, it attracted the attention of many researchers see 1–10. In Pierri 10 a new S-asymptotically ω-periodic space was introduced. It is called the space of weighted S-asymptotically ω-periodic or Sv- asymptotically ω-periodic functions. In particular, the author has established conditions under which a Sv-asymptotically ω-periodic function is asymptotically ω-periodic and also discusses the existence of Sv-asymptotically ω-periodic solutions for an integral abstract Cauchy problem. The author has applied the results to partial integrodifferential equations. We study in this paper sufficient conditions for the existence and uniqueness of a weighted S-asymptotically ω-periodic mild solution to the following semi-linear integrodifferential equation of fractional order v   t    t 0  t − s  α−2 Γ  α − 1  Av  s  ds  f  t, v  t  ,t≥ 0, 1.1 2 Advances in Difference Equations v  0   u 0 ∈ X, 1.2 where 1 <α<2, A : DA ⊆ X → X is a linear densely defined operator of sectorial type on a complex Banach space X and f : 0, ∞ × X → X is an appropriate function. Note that the convolution integral in 1.1 is known as the Riemann-Liouville fractional integral 11.We remark that there is much interest in developing theoretical analysis and numerical methods for fractional integrodifferential equations because they have recently proved to be valuable in various fields of sciences and engineering. For details, including some applications and recent results, see the monographs of Ahn and MacVinish 12, Gorenflo and Mainardi 13 and Trujillo et al. 14–16 and the papers of Agarwal et al. 17–23,Cuesta11, 24, Cuevas et al. 5, 6, dos Santos and Cuevas 25, Eidelman and Kochubei 26, Lakshmikantham et al. 27–30, Mophou and N’Gu ´ er ´ ekata 31, Ahmed and Nieto 32,andN’Gu ´ er ´ ekata 33.In particular equations of type 1.1 are attracting increasing interest cf. 5, 11, 24, 34. The existence of weighted S-asymptotically ω-periodic mild solutions for integrodif- ferential equation of fractional order of type 1.1 remains an untreated topic in the literature. Anticipating a wide interest in the subject, this paper contributes in filling this important gap. In particular, to illustrate our main results, we examine sufficient conditions for the existence and uniqueness of a weighted S-asymptotically ω-periodic mild solution to a fractional oscillation equation. 2. Preliminaries and Basic Results In this section, we introduce notations, definitions and preliminary facts which are used throughout this paper. Let Z, · Z  and Y, · Y  be Banach spaces. The notation BZ, Y  stands for the space of bounded linear operators from Z into Y endowed with the uniform operator topology denoted · BZ,Y  , and we abbreviate to BZ and · BZ whenever Z  Y . In this paper C b 0, ∞,Z denotes the Banach space consisting of all continuous and bounded functions from 0, ∞ into Z with the norm of the uniform convergence. For a closed linear operator B we denote by ρB the resolvent set and by σB the spectrum of B that is, the complement of ρB in the complex plane.SetλI − B −1 the resolvent of B for λ ∈ ρB. 2.1. Sectorial Linear Operators and the Solution Operator for Fractional Equations A closed and linear operator A is said sectorial of type μ if there are 0 <θ<π/2,M>0 and μ ∈ R such that the spectrum of A is contained in the sector μ Σ θ : {μ  λ : λ ∈ C, |arg−λ| <θ} and λ − A −1 ≤M/|λ − μ|, for all λ / ∈μ Σ θ . In order to give an operator theoretical approach for the study of t he abstract system we recall the following definition. Definition 2.1 see 17.LetA be a closed linear operator with domain DA in a Banach space X. One calls A the generator of a solution operator for 1.1-1.2 if there are μ ∈ R and a strongly continuous function S α : R  →BX such that {λ α :Reλ>μ}⊆ρA and λ α−1 λ α − A −1 x   ∞ 0 e −λt S α txdt, for all Re λ>μ,x∈ X. In this case, S α t is called the solution operator generated by A.By35, Proposition 2.6, S α 0I. We observe that the power function λ α is uniquely defined as λ α  |λ| α e i arg λ ,with−π<argλ <π. Advances in Difference Equations 3 We note that if A is a sectorial of type μ with 0 ≤ θ ≤ π1 − α/2, then A is the generator of a solution operator given by S α t :1/2πi  γ e λt λ α−1 λ α −A −1 dλ, t>0, where γ is a suitable path lying outside the sector μ Σ θ cf. 11. Recently, Cuesta 11, Theorem 1 proved that if A is a sectorial operator of type μ<0 for some M>0and0≤ θ ≤ π1 − α/2, then there exists C>0 such that  S α  t   BX ≤ CM 1    μ   t α ,t≥ 0. 2.1 Remark 2.2. In the remainder of this paper, we always assume that A is a a sectorial of type μ<0andM, C, are the constants introduced above. 2.2. Weighted S-Asymptotically ω-Periodic Functions We recall the following definitions. Definition 2.3 see 1.Afunctionf ∈ C b 0, ∞,Z is called S-asymptotically ω-periodic if there exists ω>0 such that lim t →∞ ft  ω − ft  0. In this case, we say that ω is an asymptotic period of f·. Throughout this paper, SAP ω Z represents the space formed for all the Z-valued S- asymptotically ω-periodic functions endowed with the uniform convergence norm denoted · ∞ . It is clear that SAP ω Z is a Banach space see 1, P roposition 3.5. Definition 2.4 see 10.Letv ∈ C b 0, ∞, 0, ∞.Afunctionf ∈ C b 0, ∞,Z is called weighted S-asymptotically ω-periodic or Sv-asymptotically ω-periodic if lim t →∞ ftω− ft/vt0. In this paper, SAP ω Z, v represents the space formed by all the Sv-asymptotically ω-periodic functions endowed with the norm   f   SAP ω Z,v    f   ∞    f   v  sup t≥0   ft   Z  sup t≥0   ft  ω − ft   Z v  t  . 2.2 Proposition 2.5. The space SAP v ω X is a Banach space. Proof. Let f n  n∈N be a Cauchy sequence in SAP v ω X. From the definition of · S v ω Z , there exists f ∈ C b 0, ∞,X such that f n → f in C b 0, ∞,X. Next, we prove that f n → f in SAP v ω X. By noting that f n  n is a Cauchy sequence, for ε>0 given there exists N ε ∈ N such that f n − f m  S v ω Z <ε, for all n, m ≥ N ε , which implies    f n − f m   t    <ε, ∀t ≥ 0, ∀n, m ≥ N ε ,    f n − f m   t  ω  −  f n − f m   t    v  t  <ε, ∀t ≥ 0, ∀n, m ≥ N ε . 2.3 4 Advances in Difference Equations Under the above conditions, for t ≥ 0andn ≥ N ε we see that   f n  t  − f  t        f n − f   t  ω  −  f n − f   t    v  t   lim m →∞    f n  t  − f m  t        f n − f m   t  ω  −  f n − f m   t    v  t   ≤ 2, 2.4 which implies that f n − f S v ω Z ≤ 2 for n ≥ N ε and f n − f S v ω Z → 0asn →∞. To conclude the proof we need to show that f ∈ SAP v ω X.LetN ε as above. Since f N ε ∈ SAP v ω X, there exits L ε > 0 such that f N ε t  ω − f N ε t/vt <εfor all t ≥ L ε .Now, by using that f N ε − f S v ω Z ≤ 2,fort ≥ L  we get   f  t  ω  − f  t    v  t  ≤    f  t  ω  − f N ε  t  ω   −  f  t  − f N ε  t     v  t     f N ε  t  ω  − f N ε  t    v  t  < 2ε  ε, 2.5 which implies that lim t →∞ ft  ω − ft/vt  0. This completes the proof. Definition 2.6. A function f ∈ C0, ∞ × Z, Y is called uniformly Sv-asymptotically ω- periodic on bounded sets if for every bounded subset K ⊆ Z,theset{ft, x : t ≥ 0,x ∈ K} is bounded and lim t →∞ ft  ω, x − ft, x Y /vt0, uniformly for x ∈ K.Ifv ≡ 1wesay that f· is uniformly S-asymptotically ω-periodic on bounded sets see 1. To prove some of our results, we need the f ollowing lemma. Lemma 2.7. Let v ∈ C b 0, ∞, 0, ∞. Assume f ∈ C0, ∞ × Z, Y  is uniformly Sv- asymptotically ω-periodic on bounded sets and there is L>0 such that   ft, x − ft, y   Y ≤ L   x − y   Z , ∀t ≥ 0, ∀x, y ∈ Z. 2.6 If u ∈ SAP ω Z, v, then the function t → ft, ut belongs to SAP ω Y, v. Advances in Difference Equations 5 Proof. Using the fact that Ru{ut : t ≥ 0} is bounded, it follows that f·,u· ∈ C b 0, ∞,Y. For >0 be given, we select T  > 0 such that   ft  ω, z − ft, z   Y v  t  ≤  2 ,  ut  ω − ut  Z v  t  ≤  2L , 2.7 for all t ≥ T  and z ∈Ru. Then, for t ≥ T  we see that   ft  ω, ut  ω − ft, ut   Y v  t  ≤   ft  ω, ut  ω − f  t, u  t  ω    Y v  t     ft, ut  ω − f  t, u  t    Y v  t  ≤  2  L  ut  ω − u  t   Z v  t  ≤  2   2  , 2.8 which proves t he assertion. Lemma 2.8. Let v ∈ C b 0, ∞, 0, ∞.Letu ∈ SAP ω X, v and l α : 0, ∞ → X be the f unction defined by l α  t    t 0 S α  t − s  u  s  ds. 2.9 If vtt α−1 →∞as t →∞and Θ : sup t≥0 1/vt  t 0 vt − s/1  |μ|s α ds < ∞,then l α ∈ SAP ω X, v. Proof. From the estimate l α  ∞ ≤ CM|μ| −1/α π/αsinπ/α, it follows that l α ∈ C b 0, ∞,X. For ε>0 be given we select T  > 0 such that  u  t  ω  − u  t   v  t  ≤ ε, CM  1  2 α   u  ∞  α − 1    μ   v  t  t α−1 ≤ ε, 2.10 6 Advances in Difference Equations for all t ≥ T  . Under these conditions, for t ≥ 2T  we have that  l α  t  ω  − l α  t   v  t  ≤ 1 v  t   ω 0  S α t  ω − s  BX  us  X ds  1 v  t   T  0  S α t − s  BX  us  ω − us  X ds  1 v  t   t T   S α t − s  BX  us  ω − us  X ds ≤ CM  u  ∞ v  t    tω t 1 1    μ   s α ds  2  t t−T  1 1    μ   s α ds   CM v  t   t−T  0 v  t − s  1    μ   s α ds ≤ CM  1  2 α   u  ∞  α − 1    μ   1 v  t  t α−1  CMεΘ ≤ ε  1  CMΘ  , 2.11 which completes the proof. 3. Existence of Weighted S-Asymptotically ω-Periodic Solutions In this section we discuss the existence of weighted S-asymptotically ω-periodic solutions for the abstract system 1.1-1.2. To begin, we recall the definition of mild solution for 1.1- 1.2. Definition 3.1 see 5.Afunctionu ∈ C b 0, ∞,X is called a mild solution of the abstract Cauchy problem 1.1-1.2  if u  t   S α  t  u 0   t 0 S α  t − s  f  s, u  s  ds, ∀t ∈ R  . 3.1 Now, we can establish our first existence result. Theorem 3.2. Assume f : 0, ∞ × X → X is a uniformly S-asymptotically ω-periodic on bounded sets function and there is a mesurable bounded function L f : 0, ∞ → R  such that   f  t, x  − f  t, y    ≤ L f  t    x − y   , ∀t ∈ R, ∀x, y ∈ X. 3.2 If Λ : CMsup t≥0  t 0 L f s/1  |μ|t − s α ds < 1, then there exits a unique S-asymptotically ω-periodic mild solution u· of 1.1-1.2. Suppose, there is a function L u : 0, ∞ → R  such that 1  |μ|· α L u · ∈ L 1 0, ∞ and ft  ω, x − ft, x≤L u t, for every x ∈Ru{us : s ≥ 0} and all t ≥ 0.Ifv ∈ C b 0, ∞, 0, ∞ is such that 1/vt1  |μ|t α e 2 α CM  t 0 L f sds → 0 as t →∞,thenu· is weighted S-asymptotically ω-periodic. Advances in Difference Equations 7 Proof. Let F α :SAP ω X → C b 0, ∞,X be the operator defined by F α u  t   S α  t  u 0   t 0 S α  t − s  f  s, u  s  ds : S α  t  u 0  F 1 α u  t  . 3.3 We show initially that F α is SAP ω X-valued. Since S α tu 0 → 0, as t →∞,itis sufficient to show that the function F 1 α is SAP ω X-valued. Let u ∈ SAP ω X. Using the fact that f·,u· is a bounded function, it follows that F 1 α u ∈ C b 0, ∞,X. For ε>0 be given, we select a constant T  > 0 such that sup t≥T  ,s≥0    f  t  ω, u  s  − f  t, u  s      u  t  ω  − u  t    < ε 2 , 2CM   f  ·,u  ·    ∞  ∞ T  1 1    μ   s α ds < ε 2 . 3.4 Then, for t ≥ 2L  we see that    F 1 α u  t  ω  − F 1 α u  t     ≤  ω 0   S α  t  ω − s  f  s, u  s    ds   T  0   S α  t − s   f  s  ω, u  s  ω  − f  s, u  s  ω     ds   T  0   S α  t − s   f  s, u  s  ω  − f  s, u  s     ds   t T    S α  t − s   f  s  ω, u  s  ω  − f  s, u  s  ω     ds   t T    S α  t − s   f  s, u  s  ω  − f  s, u  s     ds ≤ CM   f  ·,u  ·    ∞   ∞ t 1 1    μ   s α ds   ∞ t/2 1 1    μ   s α ds   ε 2 CMsup τ≥0  τ 0 L f  τ −s  1    μ   s α ds < ε 2  ε 2  ε, 3.5 which implies that F 1 α ut  ω − F 1 α ut → 0ast →∞, F 1 α u ∈ SAP ω X and hence F α SAP ω X ⊂ SAP ω X. Moreover, from the above estimate it is easy to infer that F α u − F α v≤Λu − v, for all u, v ∈ SAP ω X, F α is a contraction and there exists a unique S-asymptotically ω-periodic mild solution u· of 1.1-1.2. 8 Advances in Difference Equations Next, we prove that last assertion. Let ξ : 0, ∞ → R  be the function defined by ξtut  ω − ut/vt. For t ≥ 0, we get ξ  t  ≤  S α  t  ω  u 0 − S α  t  u 0  v  t     F 1 α u  t  ω  − F 1 α u  t    v  t  ≤ 2CM  u 0  v  t   1    μ   t α   1 v  t   ω 0  S α  t  ω − s   BX   f  s, u  s    ds  1 v  t   t 0  S α t − s  BX   f  s  ω, u  s  ω  − f  s, u  s    ds  2CM  u 0  v  t   1    μ   t α   I 1  I 2 . 3.6 Concerning the quantities I 1 and I 2 ,wenotethat I 1 ≤ CM   f  ·,u  ·    ∞ v  t    tω t 1 1    μ   s α ds  ≤ CMω   f  ·,u  ·    ∞ v  t   1    μ   t α  , I 2 ≤ 1 v  t   t 0  S α t − s  BX   f  s  ω, u  s  ω  − f  s, u  s  ω    ds  1 v  t   t 0  S α t − s  BX   f  s, u  s  ω  − f  s, u  s    ds ≤ CM v  t   t 0 L u  s  1    μ    t − s  α ds  CM v  t   t 0 L f  s  v  s  ξ  s  1    μ    t − s  α ds. 3.7 Using the estimates 3.7 in 3.6,weseethat v  t   1    μ   t α  ξ  t  ≤ CM  2  u 0     f  ·,u  ·    ∞   CM  t 0 1    μ   t α 1    μ    t − s  α L u  s  ds  CM  t 0 1    μ   t α 1    μ    t − s  α L  s  v  s  ξ  s  ds ≤ CM   u 0     f  ·,u  ·    ∞  2 α  t 0  1    μ   s α  L u  s  ds   2 α−1 CM  t 0 L f  s  v  s   1    μ   s α  ξ  s  ds ≤ P  2 α−1 CM  t 0 L f  s  v  s   1    μ   s α  ξ  s  ds, 3.8 Advances in Difference Equations 9 where P is a positive constant independent of t. Finally, by using the Gronwall-Bellman inequality we infer that lim t →∞  u  t  ω  − u  t   v  t   0, 3.9 which shows that u ∈ SAP ω X, v. This completes the proof. Example 3.3. We set X  L 2 0,π, A  −ρ α I with 0 <ρ<1. Let g : R → R be a function such that |gx − gy|≤L g x − y, for all x, y ∈ R and let f : 0, ∞ × X → X be defined by ft, xξe −t α gxξ, ξ ∈ 0,π. We observe that   ft  ω, x − f  t, x    L 2 ≤ √ 2  e −tω α − e −t α   L g  x  L 2    g  0    √ π  , 3.10 whence f is S-asymptotically ω-periodic on bounded sets. By Theorem 3.2 we conclude that if L g <αsinπ/α/πρ −1 , then there is a unique S-asymptotically ω-periodic mild solution u· of 1.1-1.2. Moreover u ∈ SAP ω X, 1/1  ρ α t. Theorem 3.4. Let v ∈ C b 0, ∞, 0, ∞. Assume G ∈ SAP ω BX,v, 1/vtt α−1 → 0 as t → ∞ and Λ : CM  G  SAP ω BX,v    μ   −1/α π α sin  π/α   ω sup t≥0  1 v  t   1    μ   t α    2Θ  < 1, 3.11 where Θ is the constant introduced in Lemma 2.8.Then there is a unique weighted S-asymptotically ω-periodic mild solution of u   t    t 0  t − s  α−2 Γ  α − 1  Au  s  ds  G  t  u  t  ,t≥ 0, u  0   u 0 ∈ X. 3.12 Proof. The proof is based in Lemmas 2.7 and 2.8.LetΓ :SAP ω X, v → C b 0, ∞,X be the map defined by Γu  t   S α  t  u 0   t 0 S α  t − s  G  s  u  s  ds  S α  t  u 0 Γ 1 u  t  ,t≥ 0. 3.13 We show initially that Γ is SAP ω X, v-valued. From the estimate  S α  t  ω  u 0 − S α  t  u 0  v  t  ≤ 2CM  u 0    μ   1 v  t  t α 3.14 we have that S α ·u 0 ∈ SAP ω X, v. 10 Advances in Difference Equations Let u ∈ SAP ω X, v.FromLemma 2.7, we have that s → Gsus is a weighted S- asymptotically ω-periodic function and by Lemma 2.8 we obtain that Γu ∈ SAP ω X, v.Thus, the map Γ is SAP ω X, v-valued. In order to prove that Γ is a contraction, we note that for u ∈ SAP ω X, v and t ≥ 0,  Γ 1 u  t   ≤ CM  t 0 1 1    μ    t − s  α  G  s   u  s   ds ≤ CM   t 0 1 1    μ   s α ds   G  ∞  u  ∞ ≤ CM   μ   −1/α π α sin  π/α   G  ∞  u  ∞ , 3.15 so that,  Γ 1 u  ∞ ≤ CM   μ   −1/α π α sin  π/α   G  SAP ω BX,v  u  SAP ω X,v . 3.16 On the another hand, for t ≥ 0weseethat  Γ 1 u  t  ω  − Γ 1 u  t   v  t  ≤ 1 v  t    ω 0  S α t  ω − s  BX ds   G  ∞  u  ∞  1 v  t   t 0  S α t − s  BX  G  s  ω  u  s  ω  − G  s  u  s   ds ≤ CMω v  t   1    μ   t α   G  ∞  u  ∞  CM v  t   t 0 1 1    μ    t − s  α  Gs  ω − Gs  BX  u  s  ω   ds  CM v  t   t 0 1 1    μ    t − s  α  Gs  BX  u  s  ω  − u  s   ds ≤ CMω v  t   1    μ   t α   G  ∞  u  ∞  CM  1 v  t   t 0 v  t − s  1    μ   s α ds   G  v  u  ∞  CM  1 v  t   t 0 v  t − s  1    μ   s α ds   G  ∞  u  v , 3.17 [...]... 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Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 584874, 13 pages doi:10.1155/2011/584874 Research Article Weighted S-Asymptotically. untreated topic in the literature. Anticipating a wide interest in the subject, this paper contributes in filling this important gap. In particular, to illustrate our main results, we examine sufficient.    μ   s α  ξ  s  ds, 3.8 Advances in Difference Equations 9 where P is a positive constant independent of t. Finally, by using the Gronwall-Bellman inequality we infer that lim t →∞  u  t