Computers and Mathematics with Applications 59 (2010) 1318–1325

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Computers and Mathematics with Applications

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On almost automorphic mild solutions for fractional semilinear initial value problems$

Anping Chen a,b, Fulai Chen b,∗, Siqing Deng b a School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411005, China b Department of Mathematics, Xiangnan University, Chenzhou 423000, China article info a b s t r a c t

Keywords: This paper investigates almost automorphic mild solutions of the fractional semilinear Fractional differential equation equation Dαx(t) = Ax(t) + f (t, x(t)), 0 < α < 1, considered in a Banach space X, where Almost automorphic function A is a linear operator of sectorial type ω < 0. Some sufficient conditions are given for the Mild solution existence, uniqueness and uniform stability of almost automorphic mild solutions to this semilinear equation. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Fractional differential equation has received increasing attention during recent years since fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes [1]. Recently, there has been a significant development in some basic theories, including the existence and uniqueness of solutions, for the initial value problems of fractional differential equations involving Riemann–Liouville differential operator (see, for example, [2–6] and the references therein). In this paper, we consider the existence, uniqueness and uniform stability of almost automorphic mild solutions to the following fractional semilinear initial value problem involving Riemann–Liouville differential operator

Dαx(t) = Ax(t) + f (t, x(t)), t ∈ J = [0, T ], 0 < α < 1 α−1 (1) [D x(t)]t=0 = x0, where A : D(A) ⊂ X → X is a linear densely defined operator of sectorial type on a complex Banach space X, f : J × X → X is an almost automorphic function in t for each x ∈ X.

Definition 1.1. A continuous function f : R → X is said to be almost automorphic if for every sequence of real numbers 0 ⊂ 0 := + ∈ (sn)n∈N there exists a subsequence (sn)n∈N (sn)n∈N such that g(t) limn→∞ f (t sn) is well defined for each t R, and f (t) = limn→∞ g(t − sn) for each t ∈ R.

Almost automorphic functions constitute a Banach space AA(X) endowed with the supnorm given by kf k∞ := supt∈J kf (t)k. For more details on almost automorphicity, see [7]. The study of almost automorphic mild solutions of problem (1) in the borderline case α = 1 was well studied in [8,9]. In the case 1 < α < 2, Araya and Lizama [10] proved that the existence and uniqueness of mild solutions of problem

$ This work was supported by the Scientific Research Foundation of Hunan Provincial Education Department (05A057), the Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province, and the Construct Program of the Key Discipline in Hunan Province. ∗ Corresponding author. E-mail addresses: [email protected] (A. Chen), [email protected] (F. Chen), [email protected] (S. Deng).

(1) satisfies a global Lipschitz condition and takes values on X; Cuevas and Lizama [7] discussed almost automorphic mild solutions of the semilinear fractional equation Dαx(t) = Ax(t)+Dα−1f (·, x) in a Banach space X, where A is a linear operator of sectorial type ω < 0. In the case 0 < α < 1, Jaradat et al. [11] investigated the existence of mild solutions for fractional semilinear initial value problems, however, the definition of the mild solution in [11] is inappropriate. The paper has been organized as follows. In Section 2, motivated by [7,10], we study the existence and uniqueness of almost automorphic mild solutions for problem (1) with different conditions of f by means of the Banach contraction principle and the Schauder fixed point theorem. In Section 3, we discuss the uniform stability of almost automorphic mild solutions for problem (1).

2. Existence and uniqueness

We first consider the linear version for problem (1), that is

Dαx(t) = Ax(t) + f (t), t ∈ J, 0 < α < 1, α−1 (2) [D x(t)]t=0 = x0. Similar to Example 4.3 in [1], the solution of the initial value problem (2) can be obtained with the help of the Laplace transform, we have the solution of problem (2) Z t α−1 α α−1 α x(t) = t Eα,α(At )x0 + (t − τ) Eα,α(A(t − τ) )f (τ)dτ, (3) 0 where Eα,β (z) is the Mittag-Leffler function. In fact, taking into account the initial condition of problem (2), the Laplace transform of equation Dαx(t) = Ax(t) + f (t)(0 < α < 1) yields α s Y (s) = AY (s) + F(s) + x0, where Z ∞ − F(s) = L{f (t); s} = e st f (t)dt 0 is called the Laplace transform of the function f (t). Then, from

F(s) x0 Y (s) = + , sα − A sα − A and the inverse Laplace transform of the following transform Z ∞ α−β − − p 1 e pt tβ 1E (atα)dt = ,(Re(p) > |a| α ), α,β α 0 p − a we can obtain Eq. (3). Let α−1 α Sα(t) := t Eα,α(At ), 0 < α < 1, then the solution of problem (2) can be written as Z t x(t) = Sα(t)x0 + Sα(t − τ)f (τ)dτ. 0 ≤ − α For Sα(t), Cuesta’s result [12] proves that if A is a sectorial operator of type ω < 0 for some M > 0 and 0 θ < π(1 2 ), then there exists C > 0 such that CM kSα(t)k ≤ ≤ CM, for t ∈ J. (4) 1 + |ω|tα The above consideration motivates the following definition.

Definition 2.1. A function x : J → X is said to be an almost automorphic mild solution to problem (1) if the function τ → Sαf (τ, x(τ)) is integrable on (0, t) for each t ∈ J and Z t x(t) = Sα(t)x0 + Sα(t − τ)f (τ, x(τ))dτ, (5) 0 for each t ∈ J. 1320 A. Chen et al. / Computers and Mathematics with Applications 59 (2010) 1318–1325

1 R t α−1 Remark 2.1. In [11], a continuous solution x(t) of integral equation x(t) = T (t − t0)x0 + (t − s) T (t − 0(α) t0 s)f (s, x(s), Gx(s), Sx(s))ds is called an almost automorphic mild solution of the initial value problem α D x(t) = Ax(t) + f (t, x(t), Gx(t), Su(t)), t ≥ t0, x(t0) = x0. From the above discussion, we may find that this definition is inappropriate in that it is only a simple extension of the integer-order mild solution.

Theorem 2.1. Assume that A is sectorial of type ω < 0. Let f : J × X → X be almost automorphic in t for each x ∈ X and satisfies the Lipschitz condition kf (t, x) − f (t, y)k ≤ L(t)kx − yk, for all x, y ∈ X, t ∈ J. (6) t Then problem (1) has a unique almost automorphic mild solution, provided that there exists L∗ = sup R L(τ) d such t∈J 0 1+|ω|(t−τ)α τ that CML∗ < 1. Proof. Define the operator P on AA(X) by Z t Px(t) = Sα(t)x0 + Sα(t − τ)f (τ, x(τ))dτ. (7) 0 Then by using the dominated convergence theorem, we can see that P maps AA(X) in AA(X). Next, we shall use the Banach contraction principle to prove that the P defined by (7) has a fixed point. Let x1, x2 be in AA(X), we have Z t

k(Px1)(t) − (Px2)(t)k = Sα(t − τ)[f (τ, x1(τ)) − f (τ, x2(τ))]dτ 0 Z t L(τ) ≤ CM kx (τ) − x (τ)kdτ α 1 2 0 1 + |ω|(t − τ) Z t L(τ) ≤ CMkx − x k∞ dτ 1 2 α 0 1 + |ω|(t − τ) ∗ ≤ CML kx1 − x2k∞. Thus ∗ kPx1 − Px2k∞ ≤ CML kx1 − x2k∞. Since CML∗ < 1, it follows that P is a strict contraction. As a consequence of the Banach fixed point theorem, we deduce that there exists a unique fixed point which is a unique almost automorphic mild solution of problem (1) on J. The proof is complete.

Corollary 2.1. Assume that A is sectorial of type ω < 0. Let f : J × X → X be almost automorphic in t for each x ∈ X and satisfies the Lipschitz condition kf (t, x) − f (t, y)k ≤ Lkx − yk, for all x, y ∈ X, t ∈ J. (8) Then problem (1) has a unique almost automorphic mild solution, provided that the constant L satisfies that CMLT < 1. t Proof. Let L t ≡ L in 6 , L∗ = sup R L d ≤ LT , then CMLT 1 implies that CML∗ 1. According to ( ) ( ) t∈J 0 1+|ω|(t−τ)α τ < < Theorem 2.1, there exists a unique mild solution of problem (1) on J. The proof is complete.

Theorem 2.2. Assume that A is sectorial of type ω < 0. Suppose that f : J × X → X is Lebesgue measurable and almost + automorphic with respect to t and is continuous with respect to x. For each k ∈ N, there exists a function hk : J → R and a constant γ ≥ 0 such that

(i) supkxk≤k kf (t, x)k ≤ hk(t); (ii) the function → 1 h t − belongs to L1 J R+ such that τ 1+|ω|τ α k( τ) ( , ) 1 Z t 1 lim inf hk(τ)dτ = γ < +∞. →+∞ α k k 0 1 + |ω|(tk − τ) Then problem (1) at least has an almost automorphic mild solution on J whenever CMγ < 1. Proof. Let P be the function defined by (7). We shall use the Schauder fixed point theorem to prove that P has a fixed point. The proof will be given in several steps. A. Chen et al. / Computers and Mathematics with Applications 59 (2010) 1318–1325 1321

Step 1. P is continuous.

Let {xn} be a sequence such that xn → x on J. This, together with the continuity of f (t, x) with respect to x, implies that f (τ, xn(τ)) → f (τ, x(τ)) on J as n → ∞. For each t ∈ J Z t

k(Pxn)(t) − (Px)(t)k = Sα(t − τ)[f (τ, xn(τ)) − f (τ, x(τ))]dτ 0 Z t CM ≤ kf (·, x (·)) − f (·, x(·))k dτ n α 0 1 + |ω|(t − τ) Z t ≤ kf (·, xn(·)) − f (·, x(·))k CMdτ 0

≤ CMT kf (·, xn(·)) − f (·, x(·))k.

Since f (·, xn(·)) is convergent to f (·, x(·)) as n → ∞, we have

kPxn − Pxk → 0 as n → ∞, this follows that P is continuous. Step 2. P maps bounded sets into bounded sets on J. For each k ∈ N, let

Bk = {x ∈ AA(X) : kx(t)k ≤ k, t ∈ J}. ∗ We show that there exists a k ∈ N such that PBk∗ ⊂ Bk∗ . If it is not true, then for each k ∈ N, there would exist xk ∈ Bk and tk ∈ R such that k(PBk)(t)k > k. This, together with the assumption of f , yields Z t

k < k(PBk)(t)k = Sα(t)x0 + Sα(t − τ)f (τ, xk(τ))dτ 0 Z t

≤ kSα(t)x0k + Sα(t − τ)f (τ, xk(τ))dτ 0 Z t 1 ≤ CMkx k + CM kf (τ, x (τ))kdτ 0 α k 0 1 + |ω|(t − τ) Z t 1 ≤ CMkx k + CM h (τ)dτ. (9) 0 α k 0 1 + |ω|(t − τ) Dividing both sides of (9) by k, we obtain

1 1 Z t 1 1 < CMkx k + CM h (τ)dτ. (10) 0 α k k k 0 1 + |ω|(t − τ) Taking the lower limit on both sides of (10) as k → ∞, assumption (ii) follows that

1 ≤ CMγ . ∗ This is a contradiction with CMγ < 1. Then PBk∗ ⊂ Bk∗ for some k ∈ N. Step 3. P maps bounded sets into equicontinuous sets on J.

Let t1, t2 ∈ J, t1 < t2, Bk∗ be a bounded set on J as in Step 2, and x ∈ Bk∗ . Then

Z t2 Z t1

k(Px)(t2) − (Px)(t1)k ≤ kSα(t2)x0 − Sα(t1)x0k + Sα(t2 − τ)f (τ, x(τ))dτ − Sα(t1 − τ)f (τ, x(τ))dτ 0 0 Z t2 ≤ k α−1 α − α−1 α kk k + − − t2 Eα,α(At2 ) t1 Eα,α(At1 ) x0 Sα(τ)f (t2 τ, x(t2 τ))dτ 0 Z t1

− Sα(τ)f (t1 − τ, x(t1 − τ))dτ 0 − α 1 Z t1 α−1 α t1 α ≤ t E (At ) − E (At ) kx k + S (τ)[f (t − τ, x(t − τ)) 2 α,α 2 α,α 1 0 α 2 2 t2 0 Z t2

− f (t1 − τ, x(t1 − τ))]dτ + Sα(τ)f (t2 − τ, x(t2 − τ))dτ t1 1322 A. Chen et al. / Computers and Mathematics with Applications 59 (2010) 1318–1325

Z t1 ≤ α−1k α − α kk k + k − − t2 Eα,α(At2 ) Eα,α(At1 ) x0 CM f (t2 τ, x(t2 τ)) 0 Z t2 1 − f (t − τ, x(t − τ))kdτ + CM h ∗ (t − τ)dτ 1 1 + | | α k 2 t1 1 ω τ A Z t1 ≤ α−1 0 α k k α − α + k − − t2 Eα,α(At2 ) x0 (t2 t1 ) CM f (t2 τ, x(t2 τ)) α 0 Z t2 1 − f (t − τ, x(t − τ))kdτ + CM h ∗ (t − τ)dτ 1 1 + | | α k 2 t1 1 ω τ

:= A1 + A2 + A3. → α → α As t2 t1, t2 t1 follows that A1 tends to 0, the Lebesgue measurability of f with respect to t implies that A2 tends to 0, the integrability of 1 h ∗ t − with respect to implies that A tends to 0. Hence P is equicontinuous. 1+|ω|τ α k ( 2 τ) τ 3 It remains to prove that V (t) = {(Px)(t) : x ∈ Bk∗ } is relatively compact in X. Sα(·) is compact in X since it is generated by the dense operator A. Then V (0) = Sα(0)x0 is relatively compact in X. Fix t ∈ (0, T ], for each ∈ (0, t) and x ∈ Bk∗ , define (Px)(t) as follows Z t− (Px)(t) = Sα(t)x0 + Sα(t − τ)f (τ)dτ. (11) 0

Then the sets V(t) = {(Px)(t) : x ∈ Bk∗ } are relatively compact in X. Moreover, for each x ∈ Bk∗ , we have Z t

k(Px)(t) − (Px)(t)k ≤ Sα(t − τ)f (τ)dτ t− Z t ≤ kSα(t − τ)f (τ)kdτ t− Z t hk∗ ≤ CM dτ. (12) α t− 1 + |ω|(t − τ) which, combining condition (ii), follows that there are relatively compact sets arbitrarily close to V (t) and hence V (t) is also relatively compact in X. Thus, the Arzela–Ascoli theorem implies that PBk∗ is relatively compact, P is completely continuous on Bk∗ . As a consequence of Steps 1 to 3 together with the Schauder fixed point theorem, we deduce that P has a fixed point in Bk∗ which is an almost automorphic mild solution of problem (1). The proof is complete.

Theorem 2.3. Assume that A is sectorial of type ω < 0. Suppose that f : J × X → X is Lebesgue measurable and almost automorphic with respect to t and is continuous with respect to x. There exists a function h : J → R+ such that (i)0 sup kf (t, x)k ≤ h(t); t ii 0 the integral R 1 h t − d exists for all t ∈ J. ( ) 0 1+|ω|τ α ( τ) τ Then problem (1) at least has an almost automorphic mild solution on J. Proof. Let P be the function defined by (7). We shall also apply the Schauder fixed point theorem to prove this theorem. The proof of Step 1 in this theorem is the same as the proof of Step 1 in Theorem 2.2 and so is omitted. In the following, we start our proof from Step 2. t Step 2. Let B = {x ∈ AA X : kx t k ≤ d t ∈ J}, where d = CM kx k + h∗ , and h∗ = sup R 1 h t − d . ( ) ( ) , ( 0 ) t∈J 0 1+|ω|τ α ( τ) τ For each x ∈ B, we have Z t

k(Px)(t)k ≤ kSα(t)x0k + Sα(t − τ)f (τ, x(τ))dτ 0 Z t 1 ≤ CMkx k + CM kf (τ, x(τ))kdτ 0 α 0 1 + |ω|(t − τ) Z t 1 ≤ CMkx k + CM kf (t − τ, x(t − τ))kdτ 0 α 0 1 + |ω|τ Z t 1 ≤ CMkx k + CM h(t − τ)dτ 0 α 0 1 + |ω|τ ∗ ≤ CM(kx0k + h ) = d. Therefore, P : B → B. A. Chen et al. / Computers and Mathematics with Applications 59 (2010) 1318–1325 1323

Step 3. Let t1, t2 ∈ J, t1 < t2, B be a bounded set on J as in Step 2, and x ∈ B. Then, similar to the proof of Step 3 in Theorem 2.2, we have

Z t2 Z t1

− Sα(τ)f (t1 − τ, x(t1 − τ))dτ 0 α−1 t Z t1 ≤ α−1k α − 1 α kk k + [ − − t2 Eα,α(At2 ) Eα,α(At1 ) x0 Sα(τ) f (t2 τ, x(t2 τ)) t2 0 Z t2

− f (t1 − τ, x(t1 − τ))]dτ + Sα(τ)f (t2 − τ, x(t2 − τ))dτ t1 Z t1 ≤ α−1k α − α kk k + k − − t2 Eα,α(At2 ) Eα,α(At1 ) x0 CM f (t2 τ, x(t2 τ)) 0 Z t2 1 − f (t − τ, x(t − τ))kdτ + CM h(t − τ)dτ 1 1 + | | α 2 t1 1 ω τ A Z t1 ≤ α−1 0 α k k α − α + k − − t2 Eα,α(At2 ) x0 (t2 t1 ) CM f (t2 τ, x(t2 τ)) α 0 Z t2 1 − f (t − τ, x(t − τ))kdτ + CM h(t − τ)dτ 1 1 + | | α 2 t1 1 ω τ

:= A1 + A2 + A4. → α → α As t2 t1, t2 t1 follows that A1 tends to 0, the Lebesgue measurability of f with respect to t implies that A2 tends to 0, the integrability of 1 h t − with respect to implies that A tends to 0. We can also obtain that P is equicontinuous. 1+|ω|τ α ( 2 τ) τ 4 Set V (t) = {(Px)(t) : x ∈ B}. Fix t ∈ (0, T ], for each ∈ (0, t) and x ∈ B, let P be the function defined by (11), then the sets V(t) = {(Px)(t) : x ∈ B} are relatively compact in X. Meanwhile, (12) implies that V(t) arbitrarily close to V (t) and V (t) is also relatively compact in X. Thus, the Arzela–Ascoli theorem implies that PB is relatively compact, P is completely continuous on B. According to the conclusions derived in Steps 1 to 3, we can conclude that P : B → B is continuous and completely continuous. Thus, P has a fixed point in B by using the Schauder fixed point theorem. This implies that problem (1) has at least an almost automorphic mild solution on J. The proof is complete.

3. Stability

In this section, we study the uniform stability of the almost automorphic mild solution of problem (1).

Theorem 3.1. Assume that A is sectorial of type ω < 0. Let f : J × X → X be almost automorphic in t for each x ∈ X and satisfies the Lipschitz condition

kf (t, x) − f (t, y)k ≤ L(t)kx − yk, for all x, y ∈ X, t ∈ J. ∗ Then the almost automorphic mild solution of problem (1) is uniformly stable, provided that there exists L = supt∈J t R L(τ) d such that CML∗ 1. 0 1+|ω|(t−τ)α τ <

Proof. Let x(t) be a solution of

Z t x(t) = Sα(t)x0 + Sα(t − τ)f (τ, x(τ))dτ, (13) 0 and let x˜(t) be a solution of (13) such that x˜(0) = x˜0, where x0, x˜0 ∈ X. Then Z t ˜ x(t) − x˜(t) = Sα(t)(x0 − x˜0) + Sα(t − τ)[f (τ, x(τ)) − f (τ, x(τ))]dτ 0 Z t ˜ kx(t) − x˜(t)k ≤ kSα(t)kkx0 − x˜0 k∞ + kSα(t − τ)kkf (τ, x(τ)) − f (τ, x(τ))kdτ 0 1324 A. Chen et al. / Computers and Mathematics with Applications 59 (2010) 1318–1325

Z t L(τ) ˜ ≤ CMkx − x˜ k∞ + CM kx(τ) − x(τ)kdτ 0 0 α 0 1 + |ω|(t − τ) Z t L(τ) ≤ CMkx − x˜ k∞ + CMkx − x˜k∞ dτ 0 0 α 0 1 + |ω|(t − τ) ∗ ≤ CMkx0 − x˜0k∞ + CML kx − x˜k∞. Thus ∗ kx − x˜k∞ ≤ CMkx0 − x˜0k∞ + CML kx − x˜k∞, CML∗ < 1 yields CM kx − x˜k∞ ≤ kx0 − x˜0k∞. 1 − CML∗

Therefore, if kx0 − x˜0k∞ < δ(ε), then kx − x˜k∞ < ε, which implies that the almost automorphic mild solution of problem (1) is uniformly stable. The proof is complete.

Corollary 3.1. Assume that A is sectorial of type ω < 0. Let f : J × X → X be almost automorphic in t for each x ∈ X and satisfies the Lipschitz condition kf (t, x) − f (t, y)k ≤ Lkx − yk, for all x, y ∈ X, t ∈ J. Then the almost automorphic mild solution of problem (1) is uniformly stable, provided that the constant L satisfies that CMLT < 1.

4. Example

As the application of our main results, we consider the following example.

Example 4.1. Let X = L2[0, π] and the operator A of problem (1) defined on X by 00 Ax = x − µx, (µ > 0) with domain D(A) = {x ∈ L2[0, π]: x00 ∈ L2[0, π], x(0) = x(π) = 0}. Then µI − A is sectorial of type ω = −µ < 0 because that ∆x = x00 is the generator of an analytic semigroup on L2[0, π] [7]. Let f (t, x)(s) = βb(t) sin(x(s)) for all x ∈ X and s ∈ [0, π], t ∈ R with b ∈ AA(X), β ∈ R, which implies that t → f (t, x) is almost automorphic in t for each x ∈ X and Z π k − k2 ≤ 2| |2| − |2 ≤ 2| |2k − k2 f (t, x) f (t, y) 2 β b(t) sin(x(s)) sin(y(s)) ds β b(t) (x y) 2. 0 ∗ Therefore, problem (1) has a unique almost automorphic mild solution, provided that there exists L = supt∈[0,π] t 2| |2 R β b(τ) d such that CML∗ 1 according to Theorem 2.1. 0 1+µ(t−τ)α τ < 2 2 Let hk(t) ≡ β |b(t)| for k ∈ N, problem (1) at least has an almost automorphic mild solution on J whenever CMγ < 1 t according to Theorem 2.2, where = lim inf 1 R 1 h d +∞. γ k→+∞ k 0 1+µ(t−τ)α k(τ) τ < Let h(t) = β2|b(t)|2 for k ∈ N, problem (1) at least has an almost automorphic mild solution on J if the integral t R 1 h t − d exists for all t ∈ J according to Theorem 2.3. 0 1+µτ α ( τ) τ The almost automorphic mild solution of problem (1) is uniformly stable, provided that there exists L∗ = t 2| |2 sup R β b(τ) d such that CML∗ 1 according to Theorem 3.1. t∈[0,π] 0 1+µ(t−τ)α τ <

Acknowledgments

The authors thank the referees and the editor for their valuable suggestions and comments.

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