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Abstract fractional integro-differential equations involving nonlocal initial conditions in α-norm
Advances in Difference Equations volume 2011, Article number: 25 (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
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
in X α , where , 0 < β < 1, stands for the Caputo fractional derivative of order β, and K : [0, T] → ℝ+, 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 [1–3] and references therein), but also fractional phenomena in optimal control (see, e.g., [4–6]). 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., [8–11] 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: , where C i (i = 1, ..., p) are given constants and 0 < t 1 < ··· < t p-1< t p < + ∞ (p ∈ ℕ), 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 [19–28] 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
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 ∈ ℕ, is defined as
where and f is an abstract function with value in X. If 0 < β < 1, then
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
ℒ (X) stands for the Banach space of all linear and bounded operators on X. Let M be a constant such that
For k > 0, write
The following are basic properties of A α .
Theorem 2.1 ([29], pp. 69-75)).
-
(a)
T(t) : X → X α for each t > 0, and A α T(t)x = T(t)A α x for each x ∈ X α and t ≥ 0.
-
(b)
A α T(t) is bounded on X for every t > 0 and there exist M α > 0 and δ > 0 such that
-
(c)
A -α is a bounded linear operator in X with D(A α ) = Im(A -α).
-
(d)
0 < α 1 ≤ α 2 , then .↪
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 and of linear operators by
for x ∈ X, t ≥ 0, where
is the function of Wright type defined on (0, ∞) which satisfies
The following lemma follows from the results in [15].
Lemma 2.2. The following properties hold:
-
(1)
For every t ≥ 0, and are linear and bounded operators on X, i.e.,
for all x ∈ X and 0 ≤ t < ∞.
-
(2)
For every x ∈ X, , are continuous functions from [0, ∞) into X.
-
(3)
and are compact operators on X for t > 0.
-
(4)
For all x ∈ X and t ∈ (0, ∞), , where .
We can also prove the following criterion.
Lemma 2.3. The functions and 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
Then, we deduce, in view of the fact t → A 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
for t 1, t 2 ≥ r and |t 1 - t 2| < δ.
On the other hand, for any x ∈ X, we write
Therefore, using (2.2, 2.3) and Lemma 2.2, we get
that is,
which together with the arbitrariness of r > 0 implies that is continuous in the uniform operator topology for t > 0. A similar argument enable us to give the characterization of continuity on . This completes the proof. ■
Lemma 2.4. For every t > 0, the restriction of to X α and the restriction of to X α are compact operators in X α .
Proof. First consider the restriction of to X α . For any r > 0 and t > 0, it is sufficient to show that the set 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),
is relatively compact in X α . Also, for every u ∈ B r , as
in X α , we conclude, using the total boundedness, that the set is relatively compact, which implies that the restriction of to X α is compact. The same idea can be used to prove that the restriction of 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 : 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 ∈ C([0, T]; X α ) satisfying
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]; ℝ+).
(H 1) F, G : [0, T] × X α × X α → X are continuous and for each positive number k ∈ ℕ, there exist a constant γ ∈ [0, β(1 - α)) and functions φ k (·) ∈ L 1/γ(0, T; ℝ+), ϕ k (·) ∈ L ∞(0, T; ℝ+) such that
(H 2) F, G : [0, T] × X α × X α → X are continuous and there exist constants L F , L K such that
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 : ℝ+ → ℝ+ such that for all u ∈ Θ k ,
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 . 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
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
Consider a mapping Γ defined on by
It is easy to verify that (Γu)(·) ∈ C([0, T]; X α ) for every . Moreover, for every pair and t ∈ [0, T], by (H 1) a direct calculation yields
That is, for every pair . 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 .
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 .
We first prove that Γ2 is continuous on . Let 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],
Also, by (H 1), we see
and
Hence, as
we conclude, using the Lebesgue dominated convergence theorem, that for all t ∈ [0, T],
which implies that
This proves that Γ2 is continuous on .
It suffice to prove that Γ2 is compact on . For the sake of brevity, we write
Let t ∈ [0, T] be fixed and ε, ε 1 > 0 be small enough. For , we define the map by
Therefore, from Lemma 2.1 we see that for each t ∈ (0, T], the set is relatively compact in X α . Then, as
in view of (2.1), we conclude, using the total boundedness, that for each t ∈ [0, T], the set is relatively compact in X α .
On the other hand, for 0 < t 1 < t 2 ≤ T and ε' > 0 small enough, we have
where
Therefore, it follows from (H 1), Lemma 2.2, and Lemma 2.3 that
from which it is easy to see that A i (i = 1, 2, 3, 4) tends to zero independently of as t 2 - t 1 → 0 and ε' → 0. Hence, we can conclude that
and the limit is independently of .
For the case when 0 = t 1 < t 2 ≤ T, since
||(Γ2 u)(t 1) - (Γ2 u)(t 2)|| α can be made small when t 2 is small independently of . Consequently, the set is equicontinuous. Now applying the Arzela-Ascoli theorem, it follows that Γ2 is compact on .
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
Proof. The proof is divided into the following two steps.
Step 1. Assume that w ∈ C([η, T]; X α ) is fixed and set
It is clear that w ∈ C([0, T]; X α ). We define a mapping Γ w on C([0, T]; X α ) by
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
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 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
Dividing on both sides by k and taking the lower limit as k → +∞, we get
this contradicts (3.2). Also, for , a direct calculation yields
which together with (3.2) implies that Γ w is a contraction mapping on . Thus, by the Banach contraction mapping principle, Γ w has a unique fixed point , i.e.,
for t ∈ [0, T].
Step 2. Write
It is clear that is a bounded closed convex subset of C([η, T]; X α ).
Based on the argument in Step 1, we consider a mapping on defined by
It follows from (H 5) and (3.2) that maps into itself. Moreover, for , from Step 1, we have
that is,
which yields that is continuous. Next, we prove that has a fixed point in . It will suffice to prove that is a compact operator. Then, the result follows from Lemma 2.6.
Let's decompose the mapping as
Since assumption (H 5) implies that the set is bounded in X α , it follows from Lemma 2.4 that for each t ∈ [η, T], is relatively compact in X α . Also, for η ≤ t 1 ≤ t 2 ≤ T,
independently of . This proves that the set is equicontinuous. Thus, an application of Arzela-Ascoli's theorem yields that is compact.
Observe that the set
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 is compact. Hence, making use of Lemma 2.6 we conclude that has a fixed point . Put . Then,
Since 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
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 be defined by
Then, A has a discrete spectrum and the eigenvalues are -n 2, n ∈ ℕ, with the corresponding normalized eigenvectors . Moreover, A generates a compact, analytic semigroup {T(t)} t≥0. The following results are well also known (see [29] for more details):
-
(1)
.
-
(2)
for each u ∈ X. In particular, .
-
(3)
with the domain
Denote by E ζ, β , the generalized Mittag-Leffler special function (cf., e.g., [4]) defined by
Therefore, we have
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 , then w is absolutely continuous, w' ∈ X, and .
Define
Therefore, it is not difficult to verify that and