Abstract fractional integro-differential equations involving nonlocal

Wang et al. Advances in Difference Equations 2011, 2011:25 http://www.advancesindifferenceequations.com/content/2011/1/25

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Abstract fractional integro-differential equations involving nonlocal initial conditions in a-norm Rong-Nian Wang*, Jun Liu and De-Han Chen * Correspondence: rnwang@mail. ustc.edu.cn Department of Mathematics, NanChang University, NanChang, JiangXi 330031, People’s Republic of China

Abstract In the present paper, we deal with the Cauchy problems of abstract fractional integro-differential equations involving nonlocal initial conditions in a-norm, where the operator A in the linear part is the generator of a compact analytic semigroup. New criterions, ensuring the existence of mild solutions, are established. The results are obtained by using the theory of operator families associated with the function of Wright type and the semigroup generated by A, Krasnoselkii’s fixed point theorem and Schauder’s fixed point theorem. An application to a fractional partial integrodifferential equation with nonlocal initial condition is also considered. Mathematics subject classification (2000) 26A33, 34G10, 34G20 Keywords: Cauchy problem of abstract fractional evolution equation, Nonlocal initial condition, Fixed point theorem, Mild solution, α-norm

1 Introduction Let (A, D(A)) be the infinitesimal generator of a compact analytic semigroup of bounded linear operators {T(t)} t≥0 on a real Banach space (X, ||·||) and 0 Î r(A). Denote by Xa, the Banach space D(Aa) endowed with the graph norm ||u||a = ||Aau|| for u Î Xa. The present paper concerns the study of the Cauchy problem for abstract fractional integro-differential equation involving nonlocal initial condition ⎧c β D u(t) = Au(t) + F(t, u(t), u(κ1 (t))) ⎪ ⎪ ⎨ t t + K(t − s)G(s, u(s), u(κ2 (s)))ds, ⎪ ⎪ 0 ⎩ u(0) + H(u) = u0

t ∈ [0, T],

(1:1)

in Xa, where c Dβt , 0 0 such that ||Aα T(t)||L(X) ≤

Mα −δt e . tα

(c) A-ais a bounded linear operator in X with D(Aa) = Im(A-a). (d) If 0 0. (4) For all x Î X and t Î (0, ∞), Aα Pβ (t)x ≤ Cα t−αβ x , where Cα =

Mα β(2−α) (1+β(1−α)).

We can also prove the following criterion. Lemma 2.3. The functions t → Aα Pβ (t) and t → Aα Sβ (t) are continuous in the uniform operator topology on (0, +∞). Proof. Let ε > 0 be given. For every r > 0, from (2.1), we may choose δ1, δ2 > 0 such that Mα δ1 ε β (s)s−α ds ≤ , αβ r 0 6

ε Mα ∞ β (s)s−α ds ≤ . αβ r δ2 6

(2:2)

Then, we deduce, in view of the fact t ® AaT(t) that is continuous in the uniform operator topology on (0, ∞) (see [[30], Lemma 2.1]), that there exists a constant δ > such that δ2

β β β (s) Aα T t1 s − Aα T t2 s

δ1

for t1, t2 ≥ r and |t1 - t2| 0 implies that Aα Pβ (t) is continuous in the uniform operator topology for t > 0. A similar argument enable us to give the characterization of continuity on Aα Pβ (t). This completes the proof. ■ Lemma 2.4. For every t > 0, the restriction of Sβ (t)to Xa and the restriction of Pβ (t)to Xa are compact operators in Xa. Proof. First consider the restriction of Sβ (t) to Xa. For any r > 0 and t > 0, it is sufficient to show that the set {Sβ (t)u; u ∈ Br }is relatively compact in Xa, where Br := {u Î Xa; ||u||a ≤ r}. Since by Lemma 2.1, the restriction of T(t) to Xa is compact for t > 0 in Xa, for each t > 0 and ε Î (0, t), ∞ β (s)T tβ s uds; u ∈ Br ε ∞ = T tβ ε β (s)T tβ s − tβ ε uds; u ∈ Br ε


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is relatively compact in Xa. Also, for every u Î Br, as ∞

β (s)T tβ s uds → Sβ (t)u,

ε→0

ε

in X a , we conclude, using the total boundedness, that the set {Sβ (t)u; u ∈ Br } is relatively compact, which implies that the restriction of Sβ (t) to X a is compact. The same idea can be used to prove that the restriction of Pβ (t) to Xa is also compact. ■ The following fixed point theorems play a key role in the proofs of our main results, which can be found in many books. Lemma 2.5 (Krasnoselskii’s Fixed Point Theorem). Let E be a Banach space and B be a bounded closed and convex subset of E, and let F1, F2 be maps of B into E such that F1x + F2y Î B for every pair x, y Î B. If F1 is a contraction and F2 is completely continuous, then the equation F1x + F2x = x has a solution on B. Lemma 2.6 (Schauder Fixed Point Theorem). If B is a closed bounded and convex subset of a Banach space E and F : B ® B is completely continuous, then F has a fixed point in B.

3 Main results Based on the work in [[15], Lemma 3.1 and Definition 3.1], in this paper, we adopt the following definition of mild solution of Cauchy problem (1.1). Definition 3.1. By a mild solution of Cauchy problem (1.1), we mean a function u Î C([0, T]; Xa) satisfying u(t) = Sβ (t)(u0 − H(u)) +

t 0

(t − s)β−1 Pβ (t − s)(F(s, u(s), u(κ1 (s))) s + K(s − τ )G(τ , u(τ ), u(κ2 (τ )))dτ )ds 0

for t Î [0, T]. Let us first introduce our basic assumptions. (H0) 1, 2 Î C([0, T]; [0, T]) and K Î C([0, T]; ℝ+). (H1) F, G : [0, T] × Xa × Xa ® X are continuous and for each positive number k Î N, there exist a constant g Î [0, b(1 - a)) and functions k(·) Î L1/g(0, T; ℝ+), jk(·) Î L∞(0, T; ℝ+) such that sup

F(t, u, v) ≤ ϕk (t)

and

sup

G(t, u, v) ≤ φk (t)

and

uα , vα ≤k uα , vα ≤k

ϕk L1/γ (0,T)

= σ1 < ∞, k φk L∞ (0,T) lim inf = σ2 < ∞. k→+∞ k

lim inf k→+∞

(H2) F, G : [0, T] × Xa × Xa ® X are continuous and there exist constants LF, LK such that F(t, u1 , v1 ) − F(t, u2 , v2 ) ≤ LF ( u1 − u2 α + v1 − v2 α ), G(t, u1 , v1 ) − G(t, u2 , v2 ) ≤ LG ( u1 − u2 α + v1 − v2 α )


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for all (t, u1, v1), (t, u2, v2) Î [0, T] × Xa × Xa. (H3) H : C([0, T]; Xa) ® Xa is Lipschitz continuous with Lipschitz constant LH. (H4) H : C([0, T]; Xa) ® Xa is continuous and there is a h Î (0, T) such that for any u, w Î C([0, T]; Xa) satisfying u(t) = w(t)(t Î[h, T]), H(u) = H(w). (H5) There exists a nondecreasing continuous function F : ℝ+ ® ℝ+ such that for all u Î Θk, H(u)α ≤ (k),

and

lim inf k→+∞

(k) = μ < ∞. k

Remark 3.1. Let us note that (H4) is the case when the values of the solution u(t) for t near zero do not affect H(u). We refer to [19]for a case in point. T In the sequel, we set k := K(t)dt. We are now ready to state our main results in 0

this section. Theorem 3.1. Let the assumptions (H0), (H1) and (H3) be satisfied. Then, for u0 Î Xa, the fractional Cauchy problem (1.1) has at least one mild solution provided that MLH + Cα σ1 T (1−α)β−γ

1−γ (1 − α)β − γ

1−γ +

kT (1−α)β Cα σ2 < 1. (1 − α)β

(3:1)

Proof. Let v Î C([0, T]; Xa) be fixed with |v|a ≡ 0. From (3.1) and (H1), it is easy to see that there exists a k0 > 0 such that M( u0 α + LH k0 + H(ν)α ) + Cα +

1−γ (1 − α)β − γ

1−γ T (1−α)β−γ ϕk0 L1/γ (0,T)

kT (1−α)β Cα φk0 L∞ (0,T) ≤ k0 . (1 − α)β

Consider a mapping Γ defined on k0 by (u)(t) = Sβ (t)(u0 − H(u)) + +

s

t

(t − s)β−1 Pβ (t − s) F(s, u(s), u(κ1 (s)))

0

K(s − τ )G(τ , u(τ ), u(κ2 (τ )))dτ ds

0

:= (1 u)(t) + (2 u)(t),

t ∈ [0, T].

It is easy to verify that (Γu)(·) Î C([0, T]; Xa) for every u ∈ k0. Moreover, for every pair v, u ∈ k0 and t Î [0, T], by (H1) a direct calculation yields (1 v)(t) + (2 u)(t)α ≤ Sβ (t)(u0 − H(v))α +

t

(t − s)β−1 Aα Pβ (t − s)L(X) F(s, u(s), u(κ1 (s)))

0

+ K(s − τ ) G(τ , u(τ ), u(κ2 (τ )))dτ ds s 0

≤ M( u0 α + LH k0 + H(ν)α ) t k φk0 L∞ (0,T) )ds +Cα (t − s)β(1−α)−1 (ϕk0 (s) + 0

≤ M( u0 α + LH k0 + H(ν)α )

1−γ 1−γ Cα kT (1−α)β T (1−α)β−γ ϕk0 L1/γ (0,T) +Cα φk0 L∞ (0,T) (1 − α)β − γ (1 − α)β ≤ k0 .


Page 8 of 16

That is, 1 v + 2 u ∈ k0 for every pair v, u ∈ k0. Therefore, the fractional Cauchy problem (1.1) has a mild solution if and only if the operator equation Γ1u + Γ2u = u has a solution in k0. In what follows, we will show that Γ1 and Γ2 satisfy the conditions of Lemma 2.5. From (H3) and (3.1), we infer that Γ1 is a contraction. Next, we show that Γ2 is completely continuous on k0. We first prove that Γ2 is continuous on k0. Let {un }∞ n=1 ⊂ k0 be a sequence such that un ® u as n ® ∞ in C([0, T]; Xa). Therefore, it follows from the continuity of F, G, 1 and 2 that for each t Î [0, T], F(t, un (t), un (κ1 (t))) → F(t, u(t), u(κ1 (t))) as n → ∞, G(t, un (t), un (κ1 (t))) → G(t, u(t), u(κ2 (t))) as n → ∞.

Also, by (H1), we see t

(t − s)β−1−αβ F(s, un (s), un (κ1 (s))) − F(s, u(s), u(κ1 (s))) ds

0

t ≤ 2 (t − s)β−1−αβ ϕk0 (s)ds 0

1−γ 1−γ T (1−α)β−γ ϕk0 L1/γ (0,T) , ≤2 (1 − α)β − γ

and t

(t − s)β−1−αβ

0

s

K(s − τ ) G(τ , un (τ ), un (κ2 (τ )))

0

−G(τ , u(τ ), u(κ2 (τ ))) dτ ds t ≤ 2 k φk0 L∞ (0,T) (t − s)β−1−αβ ds 0

2 kT (1−α)β ≤ φk0 L∞ (0,T) . (1 − α)β

Hence, as (2 un )(t) − (2 u)(t)α t ≤ Cα (t − s)β−1−αβ F(s, un (s), un (κ1 (s))) − F(s, u(s), u(κ1 (s))) ds 0

+Cα

t

(t − s)β−1−αβ

0

s

K(s − τ ) G(τ , un (τ ), un (κ2 (τ )))

0

−G(τ , u(τ ), u(κ2 (τ ))) dτ ds,

we conclude, using the Lebesgue dominated convergence theorem, that for all t Î [0, T], (2 un )(t) − (2 u)(t)α → 0,

as n → ∞,

which implies that |2 un − 2 u|α → 0,

as n → ∞.

This proves that Γ2 is continuous on k0. It suffice to prove that Γ2 is compact on k0. For the sake of brevity, we write t

N (t, u(t)) = F(t, u(t), u(κ1 (t))) +

K(t − τ )G(τ , u(τ ), u(κ2 (τ )))dτ . 0


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Let t Î [0, T] be fixed and ε, ε1 > 0 be small enough. For u ∈ k0, we define the map by

ε,ε1

( ε,ε1 u)(t) =

t−ε∞

βτ β (τ )T((t − s)β τ )N (s, u(s))dτ ds

0 ε1 β

t−ε∞

= T(ε ε1 )

βτ β (τ )T((t − s)β τ − εβ ε1 )N (s, u(s))dτ ds.

0 ε1

Therefore, from Lemma 2.1 we see that for each t Î (0, T], the set { ε,ε1 u)(t); u ∈ k0 } is relatively compact in Xa. Then, as ( − ( ε,ε1 u)(t)α 2tu)(t) ε1 β−1 β τ )N (s, u(s))dτ ds ≤ βτ (t − s) (τ )T((t − s) β 0 0 α t ∞ β−1 β + βτ (t − s) (τ )T((t − s) τ ) N (s, u(s))dτ ds β t−ε ε1 α t ε 1 1−α β(1−α)−1 ≤ βMα (t − s) (ϕk0 (s) + k φk0 L∞ (0,T) )ds τ β (τ )dτ 0 0 t ∞ 1−α β(1−α)−1 (ϕk (s) + k φk L∞ (0,T) )ds τ β (τ )dτ + (t − s)

0

t−ε

1−γ (1 − α)β − γ

0

1−γ

ε1

T (1−α)β−γ ϕk0 L1/γ (0,T) ε1 1−α kT (1−α)β + τ β (τ )dτ φk0 L∞ (0,T) (1 − α)β 0

1−γ 1−γ βMα (2 − α) ϕk0 L1/γ (0,T) ε(1−α)β−γ + (1 + β(1 − α)) (1 − α)β − γ k + φk0 L∞ (0,T) ε(1−α)β (1 − α)β → 0 as ε, ε1 → 0+

≤ βMα

in view of (2.1), we conclude, using the total boundedness, that for each t Î [0, T], the set {2 u)(t); u ∈ k0 } is relatively compact in Xa. On the other hand, for 0 0, there would exist uk Î Θ k and t k Î [0, T] such that ||(Γ w uk )(t k)|| a >k. Thus, we have 2kCα (LF + kLG )T (1−α)β k < (w uk )(tk )α ≤ M( u0 α + H( u)α ) +

(1 − α)β Cα max F(s, ν, ν) +k max G(s, ν, ν) T (1−α)β 0≤s≤T 0≤s≤T + (1 − α)β


Page 12 of 16

Dividing on both sides by k and taking the lower limit as k ® +∞, we get 1≤

2Cα (LF + kLG )T (1−α)β , (1 − α)β

this contradicts (3.2). Also, for u, v ∈ k0, a direct calculation yields ( wt u)(t) − (w v)(t)α β−1 = (t − s) Pβ (t − s) F(s, u(s), u(κ1 (s))) − F(s, v(s), v(κ1 (s))) 0 s + K (s − τ )(G(τ , u(τ ), u(κ2 (τ ))) − G(τ , v(τ ), v(κ2 (τ ))))dτ ds 0

≤ Cα

t

β−1−αβ

(t − s)

0

+

s

α

F(s, u(s), u(κ1 (s))) − F(s, v(s), v(κ1 (s))) ds

K(s − τ ) G(τ , u(τ ), u(κ2 (τ ))) − G(τ , v(τ ), v(κ2 (τ ))) dτ ds

0

≤ Cα

t

(t − s)β−1−αβ LF ( u(s) − v(s)α + u(κ1 (s)) − v(κ1 (s))α )

0

+LG

s

K(s − τ )( u(τ ) − v(τ )α + u(κ2 (τ )) − v(κ2 (τ ))α )dτ ds

0

kLG ) 2Cα T (1−α)β (LF + |u − v|α , ≤ (1 − α)β

which together with (3.2) implies that Γw is a contraction mapping on k0. Thus, by the Banach contraction mapping principle, Γw has a unique fixed point uw ∈ k0, i.e., t uw = Sβ (t)(u0 − H( w)) +

(t − s)β−1 Pβ (t − s)(F(s, uw (s), uw (κ1 (s)))

0

s K(s − τ )G(τ , uw (τ ), uw (κ2 (τ )))dτ )ds

+ 0

for t Î [0, T]. Step 2. Write η

k0 = {u ∈ C([η, T]; Xα ); u(t)α ≤ k0

for all t ∈ [η, T]}.

η

It is clear that k0 is a bounded closed convex subset of C([h, T]; Xa). η

Based on the argument in Step 1, we consider a mapping F on k0 defined by (F w)(t) = uw ,

t ∈ [η, T]. η

η

It follows from (H5) and (3.2) that F maps k0 into itself. Moreover, for w1 , w2 ∈ k0, from Step 1, we have 2Cα T (1−α)β (LF + kLG ) 1− |uw1 − uw2 |α ≤ M H(w1 ) − H(w2 )α , (1 − α)β that is, sup F w1 (t) − F w2 (t)α → 0 t∈[η,T]

η

as w1 → w2 in k0 ,


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η

which yields that F is continuous. Next, we prove that F has a fixed point in k0. It will suffice to prove that F is a compact operator. Then, the result follows from Lemma 2.6. Let’s decompose the mapping F = F1 + F2 as (F1 w)(t) = Sβ (t)(u0 − H( w)), t β−1 (F2 w)(t) = (t − s) Pβ (t − s)(F(s, uw (s), uw (κ1 (s))) 0

s + K(s − τ )G(τ , uw (τ ), uw (κ2 (τ )))dτ )ds. 0

η w); w ∈ k0 is bounded in Xa, it folSince assumption (H5) implies that the set H( η lows from Lemma 2.4 that for each t Î [h, T], (F1 w)(t); w ∈ k0 is relatively com-

pact in Xa. Also, for h ≤ t1 ≤ t2 ≤ T, (Sβ (t2 ) − Sβ (t1 ))(u0 − H( w)) α → 0

as t2 − t1 → 0

η η independently of w ∈ k0. This proves that the set (F1 w)(·); w ∈ k0 is equicontin-

uous. Thus, an application of Arzela-Ascoli’s theorem yields that F1 is compact. Observe that the set ⎧ ⎫ t ⎬ ⎨ η F(t, u(t), u(κ1 (t))) + K(t − τ )G(τ , u(τ ), u(κ2 (τ )))dτ ; t ∈ [0, T], w ∈ k0 ⎭ ⎩ 0

is bounded in X. Therefore, using Lemma 2.1, Lemma 2.2 and Lemma 2.3, it is not difficult to prove, similar to the argument with Γ2 in Theorem 3.1, that F2 is compact. η

Hence, making use of Lemma 2.6 we conclude that F has a fixed point w∗ ∈ k0. Put q = uw∗. Then, $∗ + q(t) = Sβ (t) u0 − H w

t

(t − s)β−1 Pβ (t − s) F(s, q(s), q(κ1 (s)))

0

s +

K (s − τ )G(τ , q(τ ), q(κ2 (τ )))dτ ) ds,

t ∈ [0, T].

0

Since uw∗ = F w∗ = w∗ (t ∈ [η, T]), H(w∗ ) = H(q) and hence q is a mild solution of the fractional Cauchy problem (1.1). This completes the proof. ■

4 Example In this section, we present an example to our abstract results, which do not aim at generality but indicate how our theorem can be applied to concrete problem. We consider the partial differential equation with Dirichlet boundary condition and nonlocal initial condition in the form


Page 14 of 16

⎧ 1 ∂ 2 u(t, x) q1 (t)|u(sin(t), x)| ⎪ c 2 ⎪ ⎪ ∂ u(t, x) = + t ⎪ 2 ⎪ ∂x 1 + |u(sin(t), x)| ⎪ ⎪ ⎪ t ⎪ q2 (s) |u(s, x)| ⎨ + ds, 0 ≤ t ≤ 1, 0 ≤ x ≤ π , 2 1 + |u(s, x)| 0 1 + (t − s) ⎪ ⎪ u(t, 0) = u(t, π ) = 0, 0 ≤ t ≤ 1, ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ u(s,x) (|u(s, x)| + 1)]ds + u (x), ⎪ 0 ≤ x ≤ π, 0 ⎩ u(0, x) = ln[e

(4:1)

η

where the functions q1, q2 are continuous on [0, 1] and 0 0,

t ∈ R.

Therefore, we have

Sβ (t)u = Pβ (t)u =

∞ n=1 ∞ n=1

Eβ (−n2 tβ )(u, yn )yn ,

u ∈ X;

Sβ (t)L(X) ≤ 1,

eβ (−n2 tβ )(u, yn )yn ,

u ∈ X;

Pβ (t)L(X) ≤

β (1 + β)

for all t ≥ 0, where Eb(t) := Eb,1(t) and eb(t) := Eb, b(t). The consideration of this section also needs the following result. 1

Lemma 4.1. [31]If w ∈ D(A 2 ), then w is absolutely continuous, w’ Î X, and 1 2 w = A w .


Page 15 of 16

Define F(t, u(t), u(κ1 (t)))(x) =

q1 (t)|u(sin(t), x)| , 1 + |u(sin(t), x)|

1 , κ1 (t) = sin(t), κ2 (t) = t, t2 + 1 q2 (t)|u(t, x)| G(t, u(t), u(κ2 (t)))(x) = , 1 + |u(t, x)| 1 H(u)(x) = ln[eu(s,x) (|u(s, x)| + 1)]ds. K(t) =

η

Therefore, it is not difficult to verify that F, G : [0, 1] × X 1 × X 1 → X and 2

2

H : C([0, 1]; X 1 ) → X 1 are continuous, 2

2

F(t, u(t), u(κ 1 (t))) − F(t, v(t), v(κ1 (t))) −1 u(κ1 (t)) − v(κ1 (t)) 1 , ≤ μ1 A 2 L(X)

2

G(t, u(t), u(κ1 (t))) − G(t, v(t), v(κ1 (t))) −1 ≤ μ2 u(t) − v(t) 1 , A 2 L(X)

where μi := suptÎ H(u) 1 2

2

[0, 1]|qi(t)|,

and for any u satisfying |u| 1 ≤ k, 2

1 = H(u) (·) ≤ 2(1 − η)k 2 = A H(u)(·)

in view of Lemma 4.1. Now, we note that the problem (4.1) can be reformulated as the abstract problem (1.1) and the assumptions (H0), (H2), (H4) and (H5) hold with α=

1 , 2

T = 1,

LF = μ 1 ,

LG = μ2 ,

Thus, when 1 − η + 4M 1 (μ1 + μ2 ) < 2

1 2

(k) = 2(1 − η)k,

μ = 2(1 − η).

such that condition (3.2) is verified, (4.1)

has at least one mild solution due to Theorem 3.2. Acknowledgements We would like to thank the referees for their valuable comments and suggestions. This research was supported in part by NNSF of China (11101202), NSF of JiangXi Province of China (2009GQS0018), and Youth Foundation of JiangXi Provincial Education Department of China (GJJ10051). Authors’ contributions All authors contributed equally to the manuscript and read and approved the final manuscript. Competing interests The authors declare that they have no competing interests. Received: 17 December 2010 Accepted: 16 August 2011 Published: 16 August 2011 References 1. Anh Vo, V, Leonenko, NN: Spectral analysis of fractional kinetic equations with randomdata. J Stat Phys. 104, 1349–1387 (2001). doi:10.1023/A:1010474332598 2. Eidelman, SD, Kochubei, AN: Cauchy problem for fractional diffusion equations. J Differ Equ. 199(2), 211–255 (2004). doi:10.1016/j.jde.2003.12.002 3. Metzler, R, Klafter, J: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep. 339, 1–77 (2000). doi:10.1016/S0370-1573(00)00070-3 4. Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993)


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doi:10.1186/1687-1847-2011-25 Cite this article as: Wang et al.: Abstract fractional integro-differential equations involving nonlocal initial conditions in a-norm. Advances in Difference Equations 2011 2011:25.

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