Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 483816, 17 pages doi:10.1155/2011/483816 Research Article Nonlocal Cauchy Problem for Nonautonomous Fractional Evolution Equations Fei Xiao Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China Correspondence should be addressed to Fei Xiao, sheaf@mail.ustc.edu.cn Received 28 November 2010; Accepted 29 January 2011 Academic Editor: Toka Diagana Copyright q 2011 Fei Xiao. 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 is properly cited. We investigate the mild solutions of a nonlocal Cauchy problem for nonautonomous fractional evolution equations d q ut/dt q  −Atutft, K 1 ut, K 2 ut, ,K n ut,t∈ I 0,T, u0A −1 0guu 0 , in Banach spaces, where T>0, 0 <q<1. New results are obtained by using Sadovskii’s fixed point theorem and the Banach contraction mapping principle. An example is also given. 1. Introduction During the past decades, the fractional differential equations have been proved to be valuable tools in the investigation of many phenomena in engineering and physics; they attracted many researchers cf., e.g., 1–9. On the other hand, the autonomous and nonautonomous evolution equations and related topics were studied in, for example, 6, 7, 10–20,andthe nonlocal C auchy problem was considered in, for example, 2, 5, 18, 21–26. In this paper, we consider the following nonlocal Cauchy problem for nonautonomous fractional evolution equations d q u  t  dt q  −A  t  u  t   f  t,  K 1 u  t  ,  K 2 u  t  , ,  K n u  t  ,t∈ I   0,T  , u  0   A −1  0  g  u   u 0 , 1.1 in Banach spaces, where 0 <q<1, g : CI; X → X.ThetermsK i ut, i  1, ,n are 2AdvancesinDifference Equations defined by  K i u  t    t 0 k i  t, s  u  s  ds, 1.2 the positive functions k i t, s are continuous on D  {t, s ∈ R 2 :0≤ s ≤ t ≤ T} and K ∗ i  sup t∈0,T  t 0 k i  t, s  ds < ∞. 1.3 Let us assume that u ∈ L0,T; X and At is a family linear closed operator defined in a Banach space X. The fractional order integral of the function u is understood here in the Riemann-Liouville sense, that is, I q u  t   1 Γ  q   t 0  t − s  q−1 u  s  ds. 1.4 In this paper, we denote that C is a positive constant and assume that a family of closed linear {At : t ∈ 0,T} satisfying A1 the domain DA of {At : t ∈ 0 ,T} is dense in the Banach space X and in- dependent of t, A2 the operator Atλ −1 exists in LX for any λ with Re λ ≤ 0and     A  t   λ  −1    ≤ C | λ  1 | ,t∈  0,T  . 1.5 A3 There exists constant γ ∈ 0, 1 and C such that     A  t 1  − A  t 2  A −1  s     ≤ C | t 1 − t 2 | γ ,t 1 ,t 2 ,s∈  0,T  . 1.6 Under condition A2,eachoperator−As, s ∈ 0,T generates an analytic semigroup exp−tAs, t>0, and there exists a constant C such that   A n  s  exp  −tA  s    ≤ C t n , 1.7 where n  0, 1, t>0, s ∈ 0,T11. We study the existence of mild solution of 1.1 and obtain the existence theorem based on the measures of noncompactness. An example is given to show an application of the abstract results. Advances in Difference Equations 3 2. Preliminaries Throughout this work, we set I 0,T.WedenotebyX a Banach space, LX the space of all linear and bounded operators on X,andCI, X the space of all X-valued continuous functions on I. Lemma 2.1 see 9. 1 I q : L 1 0,T → L 1 0,T. 2 For g ∈ L 1 0,T,wehave  t 0  η 0  t − η  q−1  η − s  γ−1 g  s  ds dη  B  q, γ   t 0  t − s  qγ−1 g  s  ds, 2.1 where Bq, γ is a Beta function. Definition 2.2. Let B be a bounded set of seminormed linear space Y. The Kuratowski’s measure of noncompactness for brevity, α-measure of B is defined as α  B   inf  d>0:B has a finite cover by sets of diameter ≤ d  . 2.2 From the definition, we can get some properties of α-measure immediately, see 27. Lemma 2.3 see 27. Let A and B be bounded sets of X.Then 1 αA ≤ αB,ifA ⊆ B. 2 αAαA cl ,whereA cl denotes the closure of A. 3 αA0 if and only if A is precompact. 4 αλA|λ|αA, λ ∈ R. 5 αA ∪ Bmax{αA,αB}. 6 αA  B ≤ αAαB,whereA  B  {x  y : x ∈ A, y ∈ B}. 7 αA  x 0 αA, for any x 0 ∈ X. For H ⊂ CI, X we define  t 0 H  s  ds    t 0 u  s  ds : u ∈ H  , 2.3 for t ∈ I,whereHs{us ∈ X : u ∈ H}. The following lemma will be needed. Lemma 2.4 see 27. If H ⊂ CI,X is a bounded, equicontinuous set, then 1 αHsup t∈I αHt. 2 α  t 0 Hsds ≤  t 0 αHsds,fort ∈ I. 4AdvancesinDifference Equations Lemma 2.5 see 28. If {u n } ∞ n1 ⊂ L 1 I, X and there exists a m· ∈ L 1 I, R   such that  u n  t  ≤ m  t  , a.e t ∈ I, 2.4 then α{u n t} ∞ n1  is integrable and α   t 0 u n  s  ds  ∞ n1  ≤ 2  t 0 α { u n  s } ∞ n1  ds. 2.5 We need to use the following Sadovskii’s fixed point theorem. Definition 2.6 see 29.LetP be an operator in Banach space X.IfP is continuous and takes bounded, sets into bounded sets, and αPH <αH for every bounded set H of X with αH > 0, then P is said to be a condensing operator on X. Lemma 2.7 Sadovskii’s fixed point theorem 29. Let P be a condensing o perator on Banach space X.IfP B ⊆ B for a convex, closed, and bounded set B of X,thenP has a fixed point in B. According to 4, a mild solution of 1.1 can be defined as follows. Definition 2.8. Afunctionu ∈ CI, X satisfying the equation u  t   A −1  0  g  u   u 0   t 0 ψ  t − η, η  U  η  A  0   A −1  0  g  u   u 0  dη   t 0 ψ  t − η, η  f  η,  K 1 u   η  ,  K 2 u   η  , ,  K n u   η  dη   t 0  η 0 ψ  t − η, η  ϕ  η, s  f  s,  K 1 u  s  ,  K 2 u  s  , ,  K n u  s  ds dη, 2.6 is called a mild solution of 1.1,where ψ  t, s   q  ∞ 0 θt q−1 ξ q  θ  exp  −t q θA  s  dθ, 2.7 and ξ q is a probability density function defined on 0, ∞ such that its Laplace transform is given by  ∞ 0 e −σx ξ q  σ  dσ  ∞  j0  −x  j Γ  1  qj  ,q∈  0, 1  ,x>0, ϕ  t, τ   ∞  k1 ϕ k  t, τ  , 2.8 Advances in Difference Equations 5 where ϕ 1  t, τ    A  t  − A  τ  ψ  t − τ, τ  , ϕ k1  t, τ    t τ ϕ k  t, s  ϕ 1  s, τ  ds, k  1, 2 , U  t   −A  t  A −1  0  −  t 0 ϕ  t, s  A  s  A −1  0  ds. 2.9 To our purpose, the following conclusions will be needed. For the proofs refer to 4. Lemma 2.9 see 4. The operator-valued functions ψt − η, η and Atψt − η, η are continuous in uniform topology in the variables t, η,where0 ≤ η ≤ t − ε, 0 ≤ t ≤ T, for any ε>0. Clearly,   ψ  t − η, η    ≤ C  t − η  q−1 . 2.10 Moreover, we have   ϕ  t, η    ≤ C  t − η  γ−1 . 2.11 Remark 2.10. From the proof of Theorem 2.5 in 4,wecansee 1 Ut≤C  Ct γ . 2 For t ∈ I,  t 0 ψt − η, ηUηdη is uniformly continuous in the norm of LX and       t 0 ψ  t − η, η  U  η  dη      ≤ C 2 t q  1 q  t γ B  q, γ  1   : M  t  . 2.12 3. Existence of Solution Assume that B1 f : I × X × X ×···×X → X satisfies f·,v 1 ,v 2 , ,v n  : I → X is measurable for all v i ∈ X, i  1, 2, ,n and ft, ·, ·, ,· : X × X ×···×X → X is continuous for a.e t ∈ I, and there exist a positive function μ· ∈ L p I, R  p>1/q > 1 and a continuous nondecreasing function ω : 0 , ∞ → 0, ∞ such that   f  t, v 1 ,v 2 , ,v n    ≤ μ  t  ω  n  i1  v i   ,  t, v 1 ,v 2 , ,v n  ∈ I × X × X ×···×X, 3.1 and set T p,q  max{T q−1/p ,T q }. 6AdvancesinDifference Equations B2 For any bounded sets D, D 1 ,D 2 , ,D n ⊂ X,and0≤ τ ≤ s ≤ t ≤ T, α  g  D   ≤ β  t  α  D  , α  ψ  t − s, s  f  s, D 1 ,D 2 , ,D n   ≤ β 1  t, s  α  D 1   β 2  t, s  α  D 2   ··· β n  t, s  α  D n  , α  ψ  t − s, s  ϕ  s, τ  f  τ, D 1 ,D 2 , ,D n   ≤ ζ 1  t, s, τ  α  D 1   ζ 2  t, s, τ  α  D 2   ··· ζ n  t, s, τ  α  D n  , 3.2 where βt is a nonnegative function, and sup t∈I βt : β<∞, sup t∈I  t 0 β i  t, s  ds : β i < ∞,i 1, 2, ,n, sup t∈I  t 0  s 0 ζ j  t, s, τ  dτ ds :  ζ j < ∞,j 1, 2, ,n. 3.3 B3 g : CI; X → X is continuous and there exists 0 <α 1 <  C  M  T   −1 ,α 2 ≥ 0 3.4 such that   g  u    ≤ α 1  u   α 2 . 3.5 B4 The functions μ and ω satisfy the following c ondition: C  1  CB  q, γ  T γ p,q Ω p,q  n  i1 K ∗ i    μ   L p lim inf τ →∞ ω  τ  τ < 1 − α 1  C  M  T   , 3.6 where Ω p,q p − 1/pq − 1 p−1/p ,andT γ p,q  max{T p,q ,T p,qγ }. Theorem 3.1. Suppose that (B1)–(B4) are satisfied, and if C  MTβ  4Σ n i1 β i  2ζ i K ∗ i  < 1, then 1.1 has a mild solution on 0,T. Advances in Difference Equations 7 Proof. Define the operator F : CI; X → CI; X by F  u  t   A −1  0  g  u   u 0   t 0 ψ  t − η, η  U  η  A  0   A −1  0  g  u   u 0  dη   t 0 ψ  t − η, η  f  η,  K 1 u   η  ,  K 2 u   η  , ,  K n u   η  dη   t 0  η 0 ψ  t − η, η  ϕ  η, s  f  s,  K 1 u  s  ,  K 2 u  s  , ,  K n u  s  ds dη, t ∈ I. 3.7 Then we proceed in five steps. Step 1. We show that F is continuous. Let u i be a sequence that u i → u as i →∞.Sincef satisfies B1,wehave f  t,  K 1 u i  t  ,  K 2 u i  t  , ,  K n u i  t  −→ f  t,  K 1 u  t  ,  K 2 u  t  , ,  K n u  t  , as i −→ ∞ . 3.8 Then  F  u i  t  − F  u  t  ≤    A −1  0       g  u i  − g  u      t 0   ψ  t − η, η  U  η      g  u i  − g  u    dη   t 0   ψ  t − η, η  f  η,  K 1 u i   η  ,  K 2 u i   η  , ,  K n u i   η  −f  η,  K 1 u   η  ,  K 2 u   η  , ,  K n u   η    dη   t 0  η 0   ψ  t − η, η  ϕ  η, s  f  s,  K 1 u i  s  ,  K 2 u i  s  , ,  K n u i  s  −f  s,  K 1 u  s  ,  K 2 u  s  , ,  K n u  s     ds dη. 3.9 According to the condition A2, 2.12, and the continuity of g,wehave    A −1  0       g  u i  − g  u    −→ 0, as i −→ ∞ ;  t 0   ψ  t − η, η  U  η      g  u i  − g  u    dη −→ 0, as i −→ ∞ . 3.10 8AdvancesinDifference Equations Noting that u i → u in CI,X,thereexistsε>0suchthatu i − u≤ε for i sufficiently large. Therefore, we have    f  t,  K 1 u i  t  ,  K 2 u i  t  , ,  K n u i  t  − f  t,  K 1 u  t  ,  K 2 u  t  , ,  K n u  t     ≤ μ  t  ⎡ ⎣ ω ⎛ ⎝ n  j1    K j u i   t    ⎞ ⎠  ω n  j1    K j u   t    ⎤ ⎦ ≤ μ  t  ⎡ ⎣ ω ⎛ ⎝ n  j1 K ∗ j  u   ε  ⎞ ⎠  ω ⎛ ⎝ n  j1 K ∗ j  u  ⎞ ⎠ ⎤ ⎦ . 3.11 Using 2.10 and by means of the Lebesgue dominated convergence theorem, we obtain  t 0   ψ  t − η, η  f  η,  K 1 u i   η  ,  K 2 u i   η  , ,  K n u i   η  −f  η,  K 1 u   η  ,  K 2 u   η  , ,  K n u   η    dη ≤ C  t 0  t − η  q−1    f  η,  K 1 u i   η  ,  K 2 u i   η  , ,  K n u i   η  −f  η,  K 1 u   η  ,  K 2 u   η  , ,  K n u   η    dη, −→ 0, as i −→ ∞ . 3.12 Similarly, by 2.10 and 2.11,wehave  t 0  η 0   ψ  t − η, η  ϕ  η, s  ×  f  s,  K 1 u i  t  ,  K 2 u i  t  , ,  K n u i  t  −f  s,  K 1 u  s  ,  K 2 u  s  , ,  K n u  s     ds dη ≤ C 2  t 0  η 0  t − η  q−1  η − s  γ−1 ×   f  s,  K 1 u i  t  ,  K 2 u i  t  , ,  K n u i  t  −f  s,  K 1 u  s  ,  K 2 u  s  , ,  K n u  s    ds dη −→ 0, as i −→ ∞ . 3.13 Therefore, we deduce that lim i →∞  F  u i  − F  u   0. 3.14 Advances in Difference Equations 9 Step 2. We show that F maps bounded sets of CI, X into bounded sets in CI, X. For any r>0, we set B r  {u ∈ CI, X : u≤r}. Now, for u ∈ B r ,byB1,wecansee   f  t,  K 1 u  t  ,  K 2 u  t  , ,  K n u  t    ≤ μ  t  ω ⎛ ⎝ n  j1 K ∗ j r ⎞ ⎠ . 3.15 Based on 2.12,wedenotethatSt :  t 0 ψt − η, ηUηdη,wehave  S  t  A  0  u 0  ≤ C 2 t q  1 q  t γ B  q, γ  1    A  0  u 0   M  t  A  0  u 0  . 3.16 Then for any u ∈ B r ,byA2, 2.10, 2.11,andLemma 2.1,wehave  Fu  t  ≤    A −1  0  g  u       u 0     S  t  g  u      S  t  A  0  u 0    t 0   ψ  t − η, η  f  η,  K 1 u   η  ,  K 2 u   η  , ,  K n u   η    dη   t 0  η 0   ψ  t − η, η  ϕ  η, s  f  s,  K 1 u  s  ,  K 2 u  s  , ,  K n u  s    ds dη ≤  C  Mt    g  u      u 0   Mt  A  0  u 0   C  t 0  t − η  q−1 μ  η  ω ⎛ ⎝ n  j1 K ∗ j r ⎞ ⎠ dη  C 2  t 0  η 0  t − η  q−1  η − s  γ−1 μ  s  ω ⎛ ⎝ n  j1 K ∗ j r ⎞ ⎠ ds dη ≤ α 1  C  Mt   u   α 2  C  M  t     u 0   Mt  A  0  u 0   M 1  C  t 0  t − η  q−1 μ  η  dη  C 2 B  q, γ   t 0  t − η  qγ−1 μ  η  dη  , 3.17 where M 1  ω  n j1 K ∗ j r. By means of the H ¨ older inequality, we have  t 0  t − η  q−1 μ  η  dη  t pq−1/p M p,q   μ   L p ≤ T p,q Ω p,q   μ   L p ,  t 0  t − η  γq−1 μ  η  dη ≤ T p,qγ Ω p,qγ   μ   L p . 3.18 10 Advances in Difference Equations Thus  Fu  t  ≤ α 1  C  M  T   r  α 2  C  M  T     u 0   M  T  A  0  u 0   M 1 Ω p,q T γ p,q  C  C 2 B  q, γ     μ   L p : r. 3.19 This means FB r  ⊂ B r . Step 3. We show that there exists m ∈ N such that FB m  ⊂ B m . Suppose the contrary, that for every m ∈ N,thereexistsu m ∈ B m and t m ∈ I,suchthat Fu m t m  >m. However, on the other hand   f  t,  K 1 u m  t  ,  K 2 u m  t  , ,  K n u m  t    ≤ μ  t  ω ⎛ ⎝ n  j1 K ∗ j m ⎞ ⎠ , 3.20 we have m<  Fu m  t m  ≤ α 1  C  M  T    u m   α 2  C  M  T     u 0   M  T  A  0  u 0   M 1  C  t m 0  t m − η  q−1 μ  η  dη  C 2 B  q, γ   t m 0  t m − η  qγ−1 μ  η  dη  ≤ α 1  C  M  T    u m   α 2  C  M  T     u 0   M  T  A  0  u 0   M 1 Ω p,q T γ p,q  C  C 2 B  q, γ     μ   L p ≤ α 1  C  M  T   m  α 2  C  M  T     u 0   M  T  A  0  u 0   M 1 Ω p,q T γ p,q  C  C 2 B  q, γ     μ   L p . 3.21 Dividing both sides by m and taking the lower limit as m →∞,weobtain C  1  CB  q, γ  T γ p,q Ω p,q n  j1 K ∗ j   μ   L p lim inf m →∞ w  m  m ≥ 1 − α 1  C  M  T   3.22 which contradicts B4. Step 4. 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We investigate the mild solutions of a nonlocal Cauchy problem for nonautonomous fractional evolution equations d q ut/dt q  −Atutft, K 1 ut, K 2 ut,. example, 2, 5, 18, 21–26. In this paper, we consider the following nonlocal Cauchy problem for nonautonomous fractional evolution equations d q u  t  dt q  −A  t  u  t   f  t,  K 1 u  t  ,  K 2 u  t  ,