Open Access

Abstract fractional integro-differential equations involving nonlocal initial conditions in α-norm

Advances in Difference Equations20112011:25

https://doi.org/10.1186/1687-1847-2011-25

Received: 17 December 2010

Accepted: 16 August 2011

Published: 16 August 2011

Abstract

In the present paper, we deal with the Cauchy problems of abstract fractional integro-differential equations involving nonlocal initial conditions in α-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 integro-differential equation with nonlocal initial condition is also considered.

Mathematics subject classification (2000)

26A33, 34G10, 34G20

Keywords

Cauchy problem of abstract fractional evolution equationNonlocal initial conditionFixed point theoremMild 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≥0on a real Banach space (X, ||·||) and 0 ρ(A). Denote by X α , the Banach space D(A α ) endowed with the graph norm ||u|| α = ||A α u|| for u X α . The present paper concerns the study of the Cauchy problem for abstract fractional integro-differential equation involving nonlocal initial condition
c D t β u ( t ) = A u ( t ) + F ( t , u ( t ) , u ( κ 1 ( t ) ) ) + 0 t K ( t - s ) G ( s , u ( s ) , u ( κ 2 ( s ) ) ) d s , t [ 0 , T ] , u ( 0 ) + H ( u ) = u 0
(1.1)

in X α , where c D t β , 0 < β < 1, stands for the Caputo fractional derivative of order β, and K : [0, T] → +, κ 1, κ 2 : [0, T] →[0, T], F, G : [0, T] × X α × X α X, H : C([0, T]; X α ) → X α are given functions to be specified later. As can be seen, H constitutes a nonlocal condition.

The fractional calculus that allows us to consider integration and differentiation of any order, not necessarily integer, has been the object of extensive study for analyzing not only anomalous diffusion on fractals (physical objects of fractional dimension, like some amorphous semiconductors or strongly porous materials; see [13] and references therein), but also fractional phenomena in optimal control (see, e.g., [46]). As indicated in [2, 5, 7] and the related references given there, the advantages of fractional derivatives become apparent in modeling mechanical and electrical properties of real materials, as well as in the description of rheological properties of rocks, and in many other fields. One of the emerging branches of the study is the Cauchy problems of abstract differential equations involving fractional derivatives in time. In recent decades, there has been a lot of interest in this type of problems, its applications and various generalizations (cf. e.g., [811] and references therein). It is significant to study this class of problems, because, in this way, one is more realistic to describe the memory and hereditary properties of various materials and processes (cf. [4, 5, 12, 13]).

In particular, much interest has developed regarding the abstract fractional Cauchy problems involving nonlocal initial conditions. For example, by using the fractional power of operators and some fixed point theorems, the authors studied the existence of mild solutions in [14] for fractional differential equations with nonlocal initial conditions and in [15] for fractional neutral differential equations with nonlocal initial conditions and time delays. The existence of mild solutions for fractional differential equations with nonlocal initial conditions in α-norm using the contraction mapping principle and the Schauder's fixed point theorem have been investigated in [16].

We here mention that the abstract problem with nonlocal initial condition was first considered by Byszewski [17], and the importance of nonlocal initial conditions in different fields has been discussed in [18, 19] and the references therein. Deng [19], especially, gave the following nonlocal initial values: H ( u ) = i = 1 p C i u ( t i ) , where C i (i = 1, ..., p) are given constants and 0 < t 1 < ··· < t p-1< t p < + ∞ (p ), which is used to describe the diffusion phenomenon of a small amount of gas in a transparent tube. In the past several years theorems about existence, uniqueness and stability of Cauchy problem for abstract evolution equations with nonlocal initial conditions have been studied by many authors, see for instance [1928] and references therein.

In this paper, we will study the existence of mild solutions for the fractional Cauchy problem (1.1). New criterions are established. Both Krasnoselkii's fixed point theorem and Schauder's fixed point theorem, and the theory of operator families associated with the function of Wright type and the semigroup generated by A, are employed in our approach. The results obtained are generalizations and continuation of the recent results on this issue.

The paper is organized as follows. In Section 2, some required notations, definitions and lemmas are given. In Section 3, we present our main results and their proofs.

2 Preliminaries

In this section, we introduce some notations, definitions and preliminary facts which are used throughout this work.

We first recall some definitions of fractional calculus (see e.g., [6, 13] for more details).

Definition 2.1 The Riemann-Liouville fractional integral operator of order β > 0 of function f is defined as
I β f ( t ) = 1 Γ ( β ) 0 t ( t - s ) β - 1 f ( s ) d s ,

provided the right-hand side is pointwise defined on [0, ∞), where Γ(·) is the gamma function.

Definition 2.2 The Caputo fractional derivative of order β > 0, m - 1 < β < m, m , is defined as
c D β f ( t ) = I m β D t m f ( t ) = 1 Γ ( m β ) 0 t ( t s ) m β 1 D s m f ( s ) d s ,
where D t m : = d m d t m and f is an abstract function with value in X. If 0 < β < 1, then
c D β f ( t ) = 1 Γ ( 1 β ) 0 t f ( s ) ( t s ) β d s .

Throughout this paper, we let A : D(A) → X be the infinitesimal generator of a compact analytic semigroup of bounded linear operators {T(t)} t≥0on X and 0 ρ(A), which allows us to define the fractional power A α for 0 ≤ α < 1, as a closed linear operator on its domain D(A α ) with inverse A -α .

Let X α denote the Banach space D(A α ) endowed with the graph norm ||u|| α = ||A α u|| for u X α and let C([0, T];X α ) be the Banach space of all continuous functions from [0, T] into X α with the uniform norm topology
| u | α = sup { u ( t ) α , t [ 0 , T ] } .
(X) stands for the Banach space of all linear and bounded operators on X. Let M be a constant such that
M = sup { T ( t ) ( X ) , t [ 0 , ) } .
For k > 0, write
Θ k = { u C ( [ 0 , T ] ; X α ) ; | u | α k } .

The following are basic properties of A α .

Theorem 2.1 ([29], pp. 69-75)).
  1. (a)

    T(t) : XX α for each t > 0, and A α T(t)x = T(t)A α x for each x X α and t ≥ 0.

     
  2. (b)
    A α T(t) is bounded on X for every t > 0 and there exist M α > 0 and δ > 0 such that
    | | A α T ( t ) | | ( X ) M α t α e - δ t .
     
  3. (c)

    A -α is a bounded linear operator in X with D(A α ) = Im(A -α ).

     
  4. (d)

    0 < α 1α 2 , then X α 2 . X α 1

     

Lemma 2.1. [27]The restriction of T(t) to X α is exactly the part of T(t) in X α and is an immediately compact semigroup in X α , and hence it is immediately norm-continuous.

Define two families { S β ( t ) } t 0 and { P β ( t ) } t 0 of linear operators by
S β ( t ) x = 0 Ψ β ( s ) T ( t β s ) x d s , P β ( t ) x = 0 β s Ψ β ( s ) T ( t β s ) x d s
for x X, t ≥ 0, where
Ψ β ( s ) = 1 π β n = 1 ( - s ) n - 1 Γ ( 1 + β n ) n ! sin ( n π β ) , s ( 0 , )
is the function of Wright type defined on (0, ∞) which satisfies
Ψ β ( s ) 0 , s ( 0 , ) , 0 Ψ β ( s ) d s = 1 , a n d 0 s ζ Ψ β ( s ) d s = Γ ( 1 + ζ ) Γ ( 1 + β ζ ) , ζ ( - 1 , ) .
(2.1)

The following lemma follows from the results in [15].

Lemma 2.2. The following properties hold:
  1. (1)
    For every t ≥ 0, S β ( t ) and P β ( t ) are linear and bounded operators on X, i.e.,
    S β ( t ) x M x , P β ( t ) x β M Γ ( 1 + β ) x

    for all x X and 0 ≤ t < ∞.

     
  2. (2)

    For every x X, t S β ( t ) x , t P β ( t ) x are continuous functions from [0, ∞) into X.

     
  3. (3)

    S β ( t ) and P β ( t ) are compact operators on X for t > 0.

     
  4. (4)

    For all x X and t (0, ∞), A α P β ( t ) x C α t - α β x , where C α = M α β Γ ( 2 - α ) Γ ( 1 + β ( 1 - α ) ) .

     

We can also prove the following criterion.

Lemma 2.3. The functions t A α P β ( t ) and t A α S β ( 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 δ 1 , δ 2 > 0 such that
M α r α β 0 δ 1 Ψ β ( s ) s - α d s ε 6 , M α r α β δ 2 Ψ β ( s ) s - α d s ε 6 .
(2.2)
Then, we deduce, in view of the fact tA α T(t) that is continuous in the uniform operator topology on (0, ∞) (see [[30], Lemma 2.1]), that there exists a constant δ > such that
δ 1 δ 2 Ψ β ( s ) A α T t 1 β s - A α T t 2 β s ( X ) d s ε 3 ,
(2.3)

for t 1, t 2r and |t 1 - t 2| < δ.

On the other hand, for any x X, we write
S β ( t 1 ) x - S β ( t 2 ) x = 0 δ 1 Ψ β ( s ) T t 1 β s x - T t 2 β s x d s + δ 1 δ 2 Ψ β ( s ) T t 1 β s x - T t 2 β s x d s + δ 2 Ψ β ( s ) ( T ( t 1 β s ) x - T ( t 2 β s ) x ) d s .
Therefore, using (2.2, 2.3) and Lemma 2.2, we get
A α S β ( t 1 ) x - A α S β ( t 2 ) x 0 δ 1 Ψ β ( s ) A α T t 1 β s ( X ) + A α T t 2 β s ( X ) x d s + δ 1 δ 2 Ψ β ( s ) A α T t 1 β s - A α T t 2 β s ( X ) x d s + δ 2 Ψ β ( s ) A α T t 1 β s ( X ) + A α T t 2 β s ( X ) x d s 2 M α r α β 0 δ 1 Ψ β ( s ) s - α x d s + δ 1 δ 2 Ψ β ( s ) T t 1 β s - T t 2 β s ( X ) x d s + 2 M α r α β δ 2 Ψ β ( s ) s - α x d s ε x ,
that is,
A α S β ( t 1 ) - A α S β ( t 2 ) ε , f o r t 1 , t 2 r a n d | t 1 - t 2 | < δ

which together with the arbitrariness of r > 0 implies that A α P β ( t ) is continuous in the uniform operator topology for t > 0. A similar argument enable us to give the characterization of continuity on A α P β ( t ) . This completes the proof. ■

Lemma 2.4. For every t > 0, the restriction of S β ( t ) to X α and the restriction of P β ( t ) to X α are compact operators in X α .

Proof. First consider the restriction of S β ( t ) to X α . For any r > 0 and t > 0, it is sufficient to show that the set { S β ( t ) u ; u B r } is relatively compact in X α , where B r := {u X α ; ||u|| α r}.

Since by Lemma 2.1, the restriction of T(t) to X α is compact for t > 0 in X α , for each t > 0 and ε (0, t),
ε Ψ β ( s ) T t β s u d s ; u B r = T t β ε ε Ψ β ( s ) T t β s - t β ε u d s ; u B r
is relatively compact in X α . Also, for every u B r , as
ε Ψ β ( s ) T t β s u d s S β ( t ) u , ε 0

in X α , we conclude, using the total boundedness, that the set { S β ( t ) u ; u B r } is relatively compact, which implies that the restriction of S β ( t ) to X α is compact. The same idea can be used to prove that the restriction of P β ( t ) to X α 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 F 1, F 2 be maps of B into E such that F 1 x + F 2 y B for every pair x, y B. If F 1 is a contraction and F 2 is completely continuous, then the equation F 1 x + F 2 x = 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 : BB 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 C([0, T]; X α ) satisfying
u ( t ) = S β ( t ) ( u 0 H ( u ) ) + 0 t ( t s ) β 1 P β ( t s ) ( F ( s , u ( s ) , u ( κ 1 ( s ) ) ) + 0 s K ( s τ ) G ( τ , u ( τ ) , u ( κ 2 ( τ ) ) ) d τ ) d s

for t [0, T].

Let us first introduce our basic assumptions.

(H 0) κ 1, κ 2 C([0, T]; [0, T]) and K C([0, T]; +).

(H 1) F, G : [0, T] × X α × X α X are continuous and for each positive number k , there exist a constant γ [0, β(1 - α)) and functions φ k (·) L 1/γ (0, T; +), ϕ k (·) L (0, T; +) such that
sup u α , v α k F ( t , u , v ) φ k ( t ) a n d liminf k + φ k L 1 γ ( 0 , T ) k = σ 1 < , sup u α , v α k G ( t , u , v ) ϕ k ( t ) a n d liminf k + ϕ k L ( 0 , T ) k = σ 2 < .
(H 2) F, G : [0, T] × X α × X α X are continuous and there exist constants L F , L K such that
F ( t , u 1 , v 1 ) - F ( t , u 2 , v 2 ) L F ( u 1 - u 2 α + v 1 - v 2 α ) , G ( t , u 1 , v 1 ) - G ( t , u 2 , v 2 ) L G ( u 1 - u 2 α + v 1 - v 2 α )

for all (t, u 1, v 1), (t, u 2, v 2) [0, T] × X α × X α .

(H 3) H : C([0, T]; X α ) → X α is Lipschitz continuous with Lipschitz constant L H .

(H 4) H : C([0, T]; X α ) → X α is continuous and there is a η (0, T) such that for any u, w C([0, T]; X α ) satisfying u(t) = w(t)(t [η, T]), H(u) = H(w).

(H 5) There exists a nondecreasing continuous function Φ : ++ such that for all u Θ k ,
H ( u ) α Φ ( k ) , a n d liminf k + Φ ( k ) k = μ < .

Remark 3.1. Let us note that (H 4) 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.

In the sequel, we set k ̃ : = 0 T K ( t ) d t . We are now ready to state our main results in this section.

Theorem 3.1. Let the assumptions (H 0), (H 1) and (H 3) be satisfied. Then, for u 0 X α , the fractional Cauchy problem (1.1) has at least one mild solution provided that
M L H + C α σ 1 T ( 1 - α ) β - γ 1 - γ ( 1 - α ) β - γ 1 - γ + C α σ 2 k ̃ T ( 1 - α ) β ( 1 - α ) β < 1 .
(3.1)
Proof. Let v C([0, T]; X α ) be fixed with |v| α ≡ 0. From (3.1) and (H 1), it is easy to see that there exists a k 0 > 0 such that
M ( u 0 α + L H k 0 + H ( ν ) α ) + C α 1 - γ ( 1 - α ) β - γ 1 - γ T ( 1 - α ) β - γ φ k 0 L 1 γ ( 0 , T ) + C α k ̃ T ( 1 - α ) β ( 1 - α ) β ϕ k 0 L ( 0 , T ) k 0 .
Consider a mapping Γ defined on Θ k 0 by
( Γ u ) ( t ) = S β ( t ) ( u 0 - H ( u ) ) + 0 t ( t - s ) β - 1 P β ( t - s ) F ( s , u ( s ) , u ( κ 1 ( s ) ) ) + 0 s K ( s - τ ) G ( τ , u ( τ ) , u ( κ 2 ( τ ) ) ) d τ d s : = ( Γ 1 u ) ( t ) + ( Γ 2 u ) ( t ) , t [ 0 , T ] .
It is easy to verify that (Γu)(·) C([0, T]; X α ) for every u Θ k 0 . Moreover, for every pair v , u Θ k 0 and t [0, T], by (H 1) a direct calculation yields
( Γ 1 v ) ( t ) + ( Γ 2 u ) ( t ) α S β ( t ) ( u 0 - H ( v ) ) α + 0 t ( t - s ) β - 1 A α P β ( t - s ) L ( X ) F ( s , u ( s ) , u ( κ 1 ( s ) ) ) + 0 s K ( s - τ ) G ( τ , u ( τ ) , u ( κ 2 ( τ ) ) ) d τ d s M ( u 0 α + L H k 0 + H ( ν ) α ) + C α 0 t ( t - s ) β ( 1 - α ) - 1 ( φ k 0 ( s ) + k ̃ ϕ k 0 L ( 0 , T ) ) d s M ( u 0 α + L H k 0 + H ( ν ) α ) + C α 1 - γ ( 1 - α ) β - γ 1 - γ T ( 1 - α ) β - γ φ k 0 L 1 γ ( 0 , T ) C α k ̃ T ( 1 - α ) β ( 1 - α ) β ϕ k 0 L ( 0 , T ) k 0 .

That is, Γ 1 v + Γ 2 u Θ k 0 for every pair v , u Θ k 0 . Therefore, the fractional Cauchy problem (1.1) has a mild solution if and only if the operator equation Γ1 u + Γ2 u = u has a solution in Θ k 0 .

In what follows, we will show that Γ1 and Γ2 satisfy the conditions of Lemma 2.5. From (H 3) and (3.1), we infer that Γ1 is a contraction. Next, we show that Γ2 is completely continuous on Ω k 0 .

We first prove that Γ2 is continuous on Θ k 0 . Let { u n } n = 1 Θ k 0 be a sequence such that u n u as n → ∞ in C([0, T]; X α ). Therefore, it follows from the continuity of F, G, κ 1 and κ 2 that for each t [0, T],
F ( t , u n ( t ) , u n ( κ 1 ( t ) ) ) F ( t , u ( t ) , u ( κ 1 ( t ) ) ) a s n , G ( t , u n ( t ) , u n ( κ 1 ( t ) ) ) G ( t , u ( t ) , u ( κ 2 ( t ) ) ) a s n .
Also, by (H 1), we see
0 t ( t - s ) β - 1 - α β F ( s , u n ( s ) , u n ( κ 1 ( s ) ) ) - F ( s , u ( s ) , u ( κ 1 ( s ) ) ) d s 2 0 t ( t - s ) β - 1 - α β φ k 0 ( s ) d s 2 1 - γ ( 1 - α ) β - γ 1 - γ T ( 1 - α ) β - γ φ k 0 L 1 γ ( 0 , T ) ,
and
0 t ( t - s ) β - 1 - α β 0 s K ( s - τ ) G ( τ , u n ( τ ) , u n ( κ 2 ( τ ) ) ) - G ( τ , u ( τ ) , u ( κ 2 ( τ ) ) ) d τ d s 2 k ̃ ϕ k 0 L ( 0 , T ) 0 t ( t - s ) β - 1 - α β d s 2 k ̃ T ( 1 - α ) β ( 1 - α ) β ϕ k 0 L ( 0 , T ) .
Hence, as
( Γ 2 u n ) ( t ) - ( Γ 2 u ) ( t ) α C α 0 t ( t - s ) β - 1 - α β F ( s , u n ( s ) , u n ( κ 1 ( s ) ) ) - F ( s , u ( s ) , u ( κ 1 ( s ) ) ) d s + C α 0 t ( t - s ) β - 1 - α β 0 s K ( s - τ ) G ( τ , u n ( τ ) , u n ( κ 2 ( τ ) ) ) - G ( τ , u ( τ ) , u ( κ 2 ( τ ) ) ) d τ d s ,
we conclude, using the Lebesgue dominated convergence theorem, that for all t [0, T],
( Γ 2 u n ) ( t ) - ( Γ 2 u ) ( t ) α 0 , a s n ,
which implies that
| Γ 2 u n - Γ 2 u | α 0 , a s n .

This proves that Γ2 is continuous on Θ k 0 .

It suffice to prove that Γ2 is compact on Ω k 0 . For the sake of brevity, we write
N ( t , u ( t ) ) = F ( t , u ( t ) , u ( κ 1 ( t ) ) ) + 0 t K ( t - τ ) G ( τ , u ( τ ) , u ( κ 2 ( τ ) ) ) d τ .
Let t [0, T] be fixed and ε, ε 1 > 0 be small enough. For u Ω k 0 , we define the map Γ ε , ε 1 by
( Γ ε , ε 1 u ) ( t ) = 0 t ε ε 1 β τ Ψ β ( τ ) T ( ( t s ) β τ ) N ( s , u ( s ) ) d τ d s = T ( ε β ε 1 ) 0 t ε ε 1 β τ Ψ β ( τ ) T ( ( t s ) β τ ε β ε 1 ) N ( s , u ( s ) ) d τ d s .
Therefore, from Lemma 2.1 we see that for each t (0, T], the set { Γ ε , ε 1 u ) ( t ) ; u Ω k 0 } is relatively compact in X α . Then, as
( Γ 2 u ) ( t ) ( Γ ε , ε 1 u ) ( t ) α 0 t 0 ε 1 β τ ( t s ) β 1 Ψ β ( τ ) T ( ( t s ) β τ ) N ( s , u ( s ) ) d τ d s α + t ε t ε 1 β τ ( t s ) β 1 Ψ β ( τ ) T ( ( t s ) β τ ) N ( s , u ( s ) ) d τ d s α β M α [ 0 t ( t s ) β ( 1 α ) 1 ( φ k 0 ( s ) + k ˜ ϕ k 0 L ( 0, T ) ) d s 0 ε 1 τ 1 α Ψ β ( τ ) d τ + t ε t ( t s ) β ( 1 α ) 1 ( φ k 0 ( s ) + k ˜ ϕ k 0 L ( 0, T ) ) d s ε 1 τ 1 α Ψ β ( τ ) d τ ] β M α [ ( 1 γ ( 1 α ) β γ ) 1 γ T ( 1 α ) β γ φ k 0 L 1 / γ ( 0, T ) + k ˜ T ( 1 α ) β ( 1 α ) β ϕ k 0 L ( 0, T ) ] 0 ε 1 τ 1 α Ψ β ( τ ) d τ + β M α Γ ( 2 α ) Γ ( 1 + β ( 1 α ) ) [ ( 1 γ ( 1 α ) β γ ) 1 γ φ k 0 L 1 / γ ( 0, T ) ε ( 1 α ) β γ + k ˜ ( 1 α ) β ϕ k 0 L ( 0, T ) ε ( 1 α ) β ] 0 a s ε , ε 1 0 +

in view of (2.1), we conclude, using the total boundedness, that for each t [0, T], the set { Γ 2 u ) ( t ) ; u Ω k 0 } is relatively compact in X α .

On the other hand, for 0 < t 1 < t 2T and ε' > 0 small enough, we have
( Γ 2 u ) ( t 1 ) - ( Γ 2 u ) ( t 2 ) α A 1 + A 2 + A 3 + A 4 ,
where
A 1 = t 1 t 2 ( t 2 - s ) β - 1 - α β N ( s , u ( s ) ) d s , (1)  A 2 = 0 t 1 - ε ( t 1 - s ) β - 1 A α P β ( t 2 - s ) - A α P β ( t 1 - s ) ( X ) N ( s , u ( s ) ) d s , (2)  A 3 = t 1 - ε t 1 ( t 1 - s ) β - 1 ( t 2 - s ) - α β + ( t 1 - s ) - α β N ( s , u ( s ) ) d s , (3)  A 4 = 0 t 1 ( t 2 - s ) β - 1 - ( t 1 - s ) β - 1 ( t 2 - s ) - α β N ( s , u ( s ) ) d s . (4)  (5) 
Therefore, it follows from (H 1), Lemma 2.2, and Lemma 2.3 that
A 1 C α t 1 t 2 ( t 2 s ) β 1 α β ( φ k 0 ( s ) + k ˜ ϕ k 0 L ( 0, T ) ) d s C α ( 1 γ ( 1 α ) β γ ) 1 γ φ k 0 L 1 / γ ( 0, T ) ( t 2 t 1 ) ( 1 α ) β γ + C α k ˜ ϕ k 0 L ( 0, T ) ( 1 α ) β ( t 2 t 1 ) ( 1 α ) β , A 2 sup s [ 0, t 1 ε ] A α α ( t 2 s ) A α α ( t 1 s ) ( X ) × 0 t 1 ε ( t 1 s ) β 1 ( φ k 0 ( s ) + k ˜ ϕ k 0 L ( 0, T ) ) d s [ ( 1 γ β γ ) 1 γ φ k 0 L 1 / γ ( 0, T ) ( t 1 β γ 1 γ ε β γ 1 γ ) 1 γ + k ˜ ϕ k 0 L ( 0, T ) β ( t 1 β ε β ) ] × sup s [ 0, t 1 ε ] A α α ( t 2 s ) A α α ( t 1 s ) ( X ) , A 3 C α t 1 ε t 1 ( t 1 s ) β 1 ( ( t 2 s ) α β + ( t 1 s ) α β ) × ( φ k 0 ( s ) + k ˜ ϕ k 0 L ( 0, T ) ) d s 2 C α t 1 ε t 1 ( t 1 s ) β 1 α β ( φ k 0 ( s ) + k ˜ ϕ k 0 L ( 0, T ) ) d s C α ( 1 γ ( 1 α ) β γ ) 1 γ φ k 0 L 1 / γ ( 0, T ) ε ( 1 α ) β γ + C α k ˜ ϕ k 0 L ( 0, T ) ( 1 α ) β ε ( 1 α ) β , A 4 C α 0 t 1 ( ( t 1 s ) β 1 ( t 2 s ) β 1 ) ( t 2 s ) α β × ( φ k 0 ( s ) + k ˜ ϕ k 0 L ( 0, T ) ) d s C α 0 t 1 ( ( t 1 s ) ( 1 α ) β 1 ( t 2 s ) ( 1 α ) β 1 ) ( φ k 0 ( s ) + k ˜ ϕ k 0 L ( 0, T ) ) d s C α ( 1 γ ( 1 α ) β γ ) 1 γ φ k 0 L 1 / γ ( 0, T ) × [ t 1 ( 1 α ) β γ ( t 2 ( 1 α ) β γ 1 γ ( t 2 t 1 ) ( 1 α ) β γ 1 γ ) 1 γ ] + 2 k ˜ ( 1 α ) β ϕ k 0 L ( 0, T ) ( t 1 ( 1 α ) β t 2 ( 1 α ) β + ( t 2 t 1 ) ( 1 α ) β ) ,
from which it is easy to see that A i (i = 1, 2, 3, 4) tends to zero independently of u Ω k 0 as t 2 - t 1 → 0 and ε' → 0. Hence, we can conclude that
( Γ 2 u ) ( t 1 ) - ( Γ 2 u ) ( t 2 ) α 0 , a s t 2 - t 1 0 ,

and the limit is independently of u Ω k 0 .

For the case when 0 = t 1 < t 2T, since
( Γ 2 u ) ( t 1 ) - ( Γ 2 u ) ( t 2 ) α = 0 t 2 ( t 2 - s ) β - 1 P β ( t 2 - s ) N ( s , u ( s ) ) d s α C α 0 t 2 ( t 2 - s ) β - 1 - α β ( φ k 0 ( s ) + k ̃ ϕ k 0 L ( 0 , T ) ) d s C α 1 - γ ( 1 - α ) β - γ 1 - γ φ k L 1 γ ( 0 , T ) t 2 ( 1 - α ) β - γ + C α k ̃ ϕ k 0 L ( 0 , T ) ( 1 - α ) β t 2 ( 1 - α ) β .

||(Γ2 u)(t 1) - (Γ2 u)(t 2)|| α can be made small when t 2 is small independently of u Ω k 0 . Consequently, the set { ( Γ 2 ) ( ) ; [ 0 , T ] , u Ω k 0 } is equicontinuous. Now applying the Arzela-Ascoli theorem, it follows that Γ2 is compact on Ω k 0 .

Therefore, applying Lemma 2.5, we conclude that Γ has a fixed point, which gives rise to a mild solution of Cauchy problem (1.1). This completes the proof. ■

The second result of this paper is the following theorem.

Theorem 3.2. Let the assumptions (H 0), (H 2), (H 4) and (H 5) be satisfied. Then, for u 0 X α , the fractional Cauchy problem (1.1) has at least one mild solution provided that
M μ + 2 C α T ( 1 - α ) β ( L F + k ̃ L G ) ( 1 - α ) β < 1 .
(3.2)

Proof. The proof is divided into the following two steps.

Step 1. Assume that w C([η, T]; X α ) is fixed and set
w ̃ ( t ) = w ( t ) , t [ η , T ] , w ( η ) , t [ 0 , η ] .
It is clear that w C([0, T]; X α ). We define a mapping Γ w on C([0, T]; X α ) by
( Γ w u ) ( t ) = S β ( t ) ( u 0 H ( w ˜ ) ) + 0 t ( t s ) β 1 P β ( t s ) ( F ( s , u ( s ) , u ( κ 1 ( s ) ) ) + 0 s K ( s τ ) G ( τ , u ( τ ) , u ( κ 2 ( τ ) ) ) d τ ) d s , t [ 0, T ] .
Clearly, (Γ w u)(·) C([0, T]; X α ) for every u C([0, T]; X α ). Moreover, for u Θ k , from (H 2), it follows that
( Γ w u ) ( t ) α S β ( t ) ( u 0 H ( w ˜ ) ) α + 0 t ( t s ) β 1 P β ( t s ) ( F ( s , u ( s ) , u ( κ 1 ( s ) ) ) + 0 s K ( s τ ) G ( τ , u ( τ ) , u ( κ 2 ( τ ) ) ) d τ ) α d s M ( u 0 α + H ( w ˜ ) α ) + C α 0 t ( t s ) β ( 1 α ) 1 [ L F ( u ( s ) α + u ( κ 1 ( s ) ) α ) + F ( s , ν , ν ) + L G 0 s K ( s τ ) ( u ( τ ) α + u ( κ 2 ( τ ) ) α + G ( s , ν , ν ) ) d τ ] d s M ( u 0 α + H ( w ˜ ) α ) + 2 k C α ( L F + k ˜ L G ) t ( 1 α ) β ( 1 α ) β + C α ( max 0 s T F ( s , ν , ν ) + k ˜ max 0 s T G ( s , ν , ν ) ) T ( 1 α ) β ( 1 α ) β ,
where v C([0, T]; X α ) is fixed with |v| α ≡ 0, which implies that there exists a integer k 0 > 0 such that Γ w maps Θ k 0 into itself. In fact, if this is not the case, then for each k > 0, there would exist u k Θ k and t k [0, T] such that ||(Γ w u k )(t k )|| α > k. Thus, we have
k < ( Γ w u k ) ( t k ) α M ( u 0 α + H ( u ̃ ) α ) + 2 k C α ( L F + k ̃ L G ) T ( 1 - α ) β ( 1 - α ) β + C α max 0 s T F ( s , ν , ν ) + k ̃ max 0 s T G ( s , ν , ν ) T ( 1 - α ) β ( 1 - α ) β
Dividing on both sides by k and taking the lower limit as k → +∞, we get
1 2 C α ( L F + k ̃ L G ) T ( 1 - α ) β ( 1 - α ) β ,
this contradicts (3.2). Also, for u , v Θ k 0 , a direct calculation yields
( Γ w u ) ( t ) - ( Γ w v ) ( t ) α = 0 t ( t - s ) β - 1 P β ( t - s ) F ( s , u ( s ) , u ( κ 1 ( s ) ) ) - F ( s , v ( s ) , v ( κ 1 ( s ) ) ) + 0 s K ( s - τ ) ( G ( τ , u ( τ ) , u ( κ 2 ( τ ) ) ) - G ( τ , v ( τ ) , v ( κ 2 ( τ ) ) ) ) d τ d s α C α 0 t ( t - s ) β - 1 - α β F ( s , u ( s ) , u ( κ 1 ( s ) ) ) - F ( s , v ( s ) , v ( κ 1 ( s ) ) ) d s + 0 s K ( s - τ ) G ( τ , u ( τ ) , u ( κ 2 ( τ ) ) ) - G ( τ , v ( τ ) , v ( κ 2 ( τ ) ) ) d τ d s C α 0 t ( t - s ) β - 1 - α β L F ( u ( s ) - v ( s ) α + u ( κ 1 ( s ) ) - v ( κ 1 ( s ) ) α ) + L G 0 s K ( s - τ ) ( u ( τ ) - v ( τ ) α + u ( κ 2 ( τ ) ) - v ( κ 2 ( τ ) ) α ) d τ d s 2 C α T ( 1 - α ) β ( L F + k ̃ L G ) ( 1 - α ) β | u - v | α ,
which together with (3.2) implies that Γ w is a contraction mapping on Θ k 0 . Thus, by the Banach contraction mapping principle, Γ w has a unique fixed point u w Θ k 0 , i.e.,
u w = S β ( t ) ( u 0 H ( w ˜ ) ) + 0 t ( t s ) β 1 P β ( t s ) ( F ( s , u w ( s ) , u w ( κ 1 ( s ) ) ) + 0 s K ( s τ ) G ( τ , u w ( τ ) , u w ( κ 2 ( τ ) ) ) d τ ) d s

for t [0, T].

Step 2. Write
Θ k 0 η = { u C ( [ η , T ] ; X α ) ; u ( t ) α k 0 f o r a l l t [ η , T ] } .

It is clear that Θ k 0 η is a bounded closed convex subset of C([η, T]; X α ).

Based on the argument in Step 1, we consider a mapping on Θ k 0 η defined by
( w ) ( t ) = u w , t [ η , T ] .
It follows from (H 5) and (3.2) that maps Θ k 0 η into itself. Moreover, for w 1 , w 2 Θ k 0 η , from Step 1, we have
1 - 2 C α T ( 1 - α ) β ( L F + k ̃ L G ) ( 1 - α ) β | u w 1 - u w 2 | α M H ( w ¯ 1 ) - H ( w ¯ 2 ) α ,
that is,
sup t [ η , T ] w 1 ( t ) - w 2 ( t ) α 0 a s w 1 w 2 i n Θ k 0 η ,

which yields that is continuous. Next, we prove that has a fixed point in Θ k 0 η . It will suffice to prove that is a compact operator. Then, the result follows from Lemma 2.6.

Let's decompose the mapping = 1 + 2 as
( 1 w ) ( t ) = S β ( t ) ( u 0 H ( w ˜ ) ) , ( 2 w ) ( t ) = 0 t ( t s ) β 1 P β ( t s ) ( F ( s , u w ( s ) , u w ( κ 1 ( s ) ) ) + 0 s K ( s τ ) G ( τ , u w ( τ ) , u w ( κ 2 ( τ ) ) ) d τ ) d s .
Since assumption (H 5) implies that the set H ( w ̃ ) ; w Θ k 0 η is bounded in X α , it follows from Lemma 2.4 that for each t [η, T], ( 1 w ) ( t ) ; w Θ k 0 η is relatively compact in X α . Also, for ηt 1t 2T,
( S β ( t 2 ) - S β ( t 1 ) ) ( u 0 - H ( w ̃ ) ) α 0 a s t 2 - t 1 0

independently of w Θ k 0 η . This proves that the set ( 1 w ) ( ) ; w Θ k 0 η is equicontinuous. Thus, an application of Arzela-Ascoli's theorem yields that 1 is compact.

Observe that the set
F ( t , u ( t ) , u ( κ 1 ( t ) ) ) + 0 t K ( t - τ ) G ( τ , u ( τ ) , u ( κ 2 ( τ ) ) ) d τ ; t [ 0 , T ] , w Θ k 0 η
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 2 is compact. Hence, making use of Lemma 2.6 we conclude that has a fixed point w * Θ k 0 η . Put q = u w * . Then,
q ( t ) = S β ( t ) ( u 0 H ( w ˜ ) ) + 0 t ( t s ) β 1 P β ( t s ) ( F ( s , q ( s ) , q ( κ 1 ( s ) ) ) + 0 s K ( s τ ) G ( τ , q ( τ ) , q ( κ 2 ( τ ) ) ) d τ ) ) d s , t [ 0, T ] .

Since u w * = 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
c t 1 2 u ( t , x ) = 2 u ( t , x ) x 2 + q 1 ( t ) | u ( sin ( t ) , x ) | 1 + | u ( sin ( t ) , x ) | + 0 t q 2 ( s ) 1 + ( t - s ) 2 | u ( s , x ) | 1 + | u ( s , x ) | d s , 0 t 1 , 0 x π , u ( t , 0 ) = u ( t , π ) = 0 , 0 t 1 , u ( 0 , x ) = η 1 ln [ e u ( s , x ) ( | u ( s , x ) | + 1 ) ] d s + u 0 ( x ) , 0 x π ,
(4.1)

where the functions q 1, q 2 are continuous on [0, 1] and 0 < η < 1.

Let X = L 2[0, π] and the operators A = 2 x 2 : D ( A ) X X be defined by
D ( A ) = { u X ; u , u are absolutely continuous , u X , and u ( 0 ) = u ( π ) = 0 } .
Then, A has a discrete spectrum and the eigenvalues are -n 2, n , with the corresponding normalized eigenvectors y n ( x ) = 2 π sin ( n x ) . Moreover, A generates a compact, analytic semigroup {T(t)} t≥0. The following results are well also known (see [29] for more details):
  1. (1)

    T ( t ) u = n = 1 e - n 2 t ( u , y n ) y n , T ( t ) ( X ) e - t f o r a l l t 0 .

     
  2. (2)

    A - 1 2 u = n = 1 1 n ( u , y n ) y n for each u X. In particular, A - 1 2 ( X ) = 1 .

     
  3. (3)
    A 1 2 u = n = 1 n ( u , y n ) y n with the domain
    D A 1 2 = u X ; n = 1 n ( u , y n ) y n X .
     
Denote by E ζ, β , the generalized Mittag-Leffler special function (cf., e.g., [4]) defined by
E ζ , β ( t ) = k = 0 t k Γ ( ζ k + β ) ζ , β > 0 , t .
Therefore, we have
S β ( t ) u = n = 1 E β ( - n 2 t β ) ( u , y n ) y n , u X ; S β ( t ) ( X ) 1 , P β ( t ) u = n = 1 e β ( - n 2 t β ) ( u , y n ) y n , u X ; P β ( t ) ( X ) β Γ ( 1 + β )

for all t ≥ 0, where E β (t) := E β,1(t) and e β (t) := E β, β (t).

The consideration of this section also needs the following result.

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

Define
F ( t , u ( t ) , u ( κ 1 ( t ) ) ) ( x ) = q 1 ( t ) | u ( sin ( t ) , x ) | 1 + | u ( sin ( t ) , x ) | , K ( t ) = 1 t 2 + 1 , κ 1 ( t ) = sin ( t ) , κ 2 ( t ) = t , G ( t , u ( t ) , u ( κ 2 ( t ) ) ) ( x ) = q 2 ( t ) | u ( t , x ) | 1 + | u ( t , x ) | , H ( u ) ( x ) = η 1 ln [ e u ( s , x ) ( | u ( s , x ) | + 1 ) ] d s .
Therefore, it is not difficult to verify that F , G : [ 0 , 1 ] × X 1 2 × X 1 2 X and H : C ( [ 0 , 1 ] ; X 1 2 ) X 1 2 are continuous,
F ( t , u ( t ) , u ( κ 1 ( t ) ) ) - F ( t , v ( t ) , v ( κ 1 ( t ) ) ) μ 1 A - 1 2 ( X ) u ( κ 1 ( t ) ) - v ( κ 1 ( t ) ) 1 2 , G ( t , u ( t ) , u ( κ 1 ( t ) ) ) - G ( t , v ( t ) , v ( κ 1 ( t ) ) ) μ 2 A - 1 2 ( X ) u ( t ) - v ( t ) 1 2 ,
where μ i := sup t [0, 1]|q i (t)|, and for any u satisfying | u | 1 2 k ,
H ( u ) 1 2 = A 1 2 H ( u ) ( ) = H ( u ) ( ) 2 ( 1 - η ) k

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 (H 0), (H 2), (H 4) and (H 5) hold with
α = 1 2 , T = 1 , L F = μ 1 , L G = μ 2 , Φ ( k ) = 2 ( 1 - η ) k , μ = 2 ( 1 - η ) .

Thus, when 1 - η + 4 M 1 2 ( μ 1 + μ 2 ) < 1 2 such that condition (3.2) is verified, (4.1) has at least one mild solution due to Theorem 3.2.

Declarations

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’ Affiliations

(1)
Department of Mathematics, NanChang University

References

  1. Anh Vo V, Leonenko NN: Spectral analysis of fractional kinetic equations with randomdata. J Stat Phys 2001, 104: 1349-1387. 10.1023/A:1010474332598View ArticleMATHGoogle Scholar
  2. Eidelman SD, Kochubei AN: Cauchy problem for fractional diffusion equations. J Differ Equ 2004,199(2):211-255. 10.1016/j.jde.2003.12.002MathSciNetView ArticleMATHGoogle Scholar
  3. Metzler R, Klafter J: The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 2000, 339: 1-77. 10.1016/S0370-1573(00)00070-3MathSciNetView ArticleMATHGoogle Scholar
  4. Miller KS, Ross B: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York; 1993.Google Scholar
  5. Podlubny I, Petraš I, Vinagre BM, O'Leary P, Dorčak L: Analogue realizations of fractional-order controllers: fractional order calculus and its applications. Nonlinear Dyn 2002, 29: 281-296. 10.1023/A:1016556604320View ArticleMATHGoogle Scholar
  6. Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, Yverdon; 1993.Google Scholar
  7. Mainardi F: Fractional calculus: some basic problems in continuum and statistical mechanics. In Fractals and Fractional Calculus in Continuum Mechanics. Edited by: Carpinteri A, Mainardi F. Springer, New York; 1997:291-348.View ArticleGoogle Scholar
  8. Agarwal RP, Lakshmikantham V, Nieto JJ: On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal 2010, 72: 2859-2862. 10.1016/j.na.2009.11.029MathSciNetView ArticleMATHGoogle Scholar
  9. Agarwal RP, Zhou Y, He YY: Existence of fractional neutral functional differential equations. Comput Math Appl 2010, 59: 1095-1100.MathSciNetView ArticleMATHGoogle Scholar
  10. Cuevas C, de Souza JC: S -asymptotically ω -periodic solutions of semilinear fractional integro-differential equations. Appl Math Lett 2009, 22: 865-870. 10.1016/j.aml.2008.07.013MathSciNetView ArticleMATHGoogle Scholar
  11. Hernández E, O'Regan D, Balachandran K: On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Anal 2010, 73: 3462-3471. 10.1016/j.na.2010.07.035MathSciNetView ArticleMATHGoogle Scholar
  12. Hilfer H: Applications of Fractional Calculus in Physics. World Scientific Publ. Co., Singapure; 2000.Google Scholar
  13. Kilbas AA, Srivastava HM, Juan Trujillo J: Theory and applications of fractional differential equations. In North-Holland Mathematics Studies. Volume 204. Elsevier, Amsterdam; 2006.Google Scholar
  14. Zhou Y, Jiao F: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal Real World Appl 2010, 11: 4465-4475. 10.1016/j.nonrwa.2010.05.029MathSciNetView ArticleMATHGoogle Scholar
  15. Zhou Y, Jiao F: Existence of mild solutions for fractional neutral evolution equations. Comput Math Appl 2010, 59: 1063-1077.MathSciNetView ArticleMATHGoogle Scholar
  16. Diagana T, Mophou GM, N'Guérékata GM: On the existence of mild solutions to some semilinear fractional integro-differential equations. E J Qual Theory Diff Equ 2010, 58: 1-17.MATHGoogle Scholar
  17. Byszewski L: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J Math Anal Appl 1991, 162: 494-505. 10.1016/0022-247X(91)90164-UMathSciNetView ArticleMATHGoogle Scholar
  18. Byszewski L, Lakshmikantham V: Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Appl Anal 1991, 40: 11-19. 10.1080/00036819008839989MathSciNetView ArticleMATHGoogle Scholar
  19. Deng K: Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions. J Math Anal Appl 1993, 179: 630-637. 10.1006/jmaa.1993.1373MathSciNetView ArticleMATHGoogle Scholar
  20. Aizicovici S, Staicu V: Multivalued evolution equations with nonlocal initial conditions in Banach spaces. NoDEA Nonlinear Differ Equ Appl 2007, 14: 361-376. 10.1007/s00030-007-5049-5MathSciNetView ArticleMATHGoogle Scholar
  21. Ezzinbi K, Fu X, Hilal K: Existence and regularity in the -norm for some neutral partial differential equations with nonlocal conditions. Nonlinear Anal 2007, 67: 1613-1622. 10.1016/j.na.2006.08.003MathSciNetView ArticleMATHGoogle Scholar
  22. Fan Z: Impulsive problems for semilinear differential equations with nonlocal conditions. Nonlinear Anal 2010, 72: 1104-1109. 10.1016/j.na.2009.07.049MathSciNetView ArticleMATHGoogle Scholar
  23. Liang J, van Casteren J, Xiao TJ: Nonlocal Cauchy problems for semilinear evolution equations. Nonlinear Anal 2002, 50: 173-189. 10.1016/S0362-546X(01)00743-XMathSciNetView ArticleMATHGoogle Scholar
  24. Liang J, Liu JH, Xiao TJ: Nonlocal Cauchy problems governed by compact operator families. Nonlinear Anal 2004, 57: 183-189. 10.1016/j.na.2004.02.007MathSciNetView ArticleMATHGoogle Scholar
  25. Liang J, Liu JH, Xiao TJ: Nonlocal impulsive problems for nonlinear differential equations in Banach spaces. Math Comput Model 2009, 49: 798-804. 10.1016/j.mcm.2008.05.046MathSciNetView ArticleMATHGoogle Scholar
  26. Liang J, Xiao TJ: Semilinear integrodifferential equations with nonlocal initial conditions. Comput Math Appl 2004, 47: 863-875. 10.1016/S0898-1221(04)90071-5MathSciNetView ArticleMATHGoogle Scholar
  27. Liu H, Chang JC: Existence for a class of partial differential equations with nonlocal conditions. Nonlinear Anal 2009, 70: 3076-3083. 10.1016/j.na.2008.04.009MathSciNetView ArticleMATHGoogle Scholar
  28. N'Guérékata GM: A Cauchy problem for some fractional abstract differential equation with nonlocal conditions. Nonlinear Anal 2009, 70: 1873-1876. 10.1016/j.na.2008.02.087MathSciNetView ArticleMATHGoogle Scholar
  29. Pazy A: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York; 1983.