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Initial value problem for fractional evolution equations
Advances in Difference Equations volume 2012, Article number: 49 (2012)
Abstract
This article is concerned with the existence of mild solutions to the initial value problem for a class of semilinear evolution equations with fractional order. New existence theorems are obtained by means of fixed point theorem for condensing maps. The results extend some related results in this direction.
Mathematics Subject Classification (2000): 34A12; 35F25.
1 Introduction
This article deal with the existence of mild solutions to the initial value problem (IVP) for a class of semilinear evolution equations with fractional order of the form
where Dβis the standard Caputo's derivative of order 0 < β < 1, J = [0, 1], is a linear closed densely defined operator, -A is the infinitesimal generator of a C0-semigroup T(t)(t ≥ 0) of operators on is continuous and u0 is an element of the Banach space .
Differential equations of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetism, etc. (see [1–5]). There has been a significant development in the study of fractional differential equations and inclusions in recent years; see the monograph of Kilbas [6], Lakshmikantham [7], Podlubny [4], and the survey by Agarwal [8]. For some recent contributions on fractional differential equations, see [9–15] and the references therein.
Very recently some basic theory for the initial value problems of fractional differential equations involving Riemann-Liouville differential operator of order 0 < q ≤ 1 has been discussed by Lakshmikantham and Vatsala [16–18].
Among the previous research, only a few be concerned with evolution equations of fractional order under noncompactness conditions. for some recent and deeper results on fractional differential equations under noncompactness conditions, see [19, 20]. In this article, we prove the existence of mild solutions for the IVP (1.1) under noncompactness measure condition of nonlinear term f. For the details of the definition and properties of the measure of noncompactness, see [21].
The rest of this article is organized as follows: In Section 2, we recall briefly some basic definitions, lemmas and preliminary facts which are used throughout this article. The existence theorems of mild solutions for the IVP (1.1) and their proofs are arranged in Section 3.
2 Preliminaries
In this section, we introduce preliminary facts which are used in what follows.
Let be the Banach space of all linear bounded operators on . Throughout this article, let -A be the infinitesimal generators of a C0-semigroup T(t)(t ≥ 0) of bounded linear operators on . Clearly
Let P be a cone in which define a partial ordering in by x ≤ y if and only if y - x ∈ P. If x ≤ y and x ≠ y, we write x < y.
P is said to be normal if there exists a positive constant N such that θ ≤ x ≤ y implies ∥x∥ ≤ N∥y∥, where θ denotes the zero element of .
Denote by the Banach space of all continuous functions with norm . Set , then P C is a cone in space , and so, is partially ordered by P C : u ≤ v if and only if v - u ∈ P C , i.e., u(t) ≤ v(t) for t ∈ J.
Now let Φ β be the Mainardi function:
then
For the details we refer to [20–22].
We set
Then we have the following result.
Lemma 2.1 [23, 24]. Let and be the operators defined, respectively, by (2.6) and (2.7). Then
(i) for all and t ≥ 0.
(ii) The operators and are strongly continuous.
Definition 2.2. A C0-semigroup R(t)(t ≥ 0) in is said to be positive, if order inequality R(t)x ≥ θ holds for each x ≥ θ, and t ≥ 0.
Remark 2.3. According to (2.6), (2.7) and Definition 2.2, if T(t)(t ≥ 0) is positive, then and are also positive.
Definition 2.4 [25, 26]. Let and be operators defined, respectively, by (2.6) and (2.7).
Then a continuous function satisfying for any t ∈ [0, 1] the equation
is called a mild solution of the problem (1.1).
Lemma 2.5. Let T(t)(t ≥ 0) is positive, and be the operators defined, respectively, by (2.6) and (2.7), v, , and
-
(1)
-
(2)
, 0 ≤ t ≤ 1,
one of the foregoing inequalities being strict. Suppose further that f(t, x) is nondecreasing in x for each t and
Then we have
Proof. Suppose that the conclusion (2.10) is not true. Then, because of the continuity of the functions involved and (2.9), it follows that there exists a t1 such that 0 < t1 ≤ 1 and
Since v(0) < w(0) and is positive, so
Similarly, using the nondecreasing nature of f and (2.11), we obtain
Without loss of generality, let us suppose that the inequality (2) is strict, according to (2.12) and (2.13) we get
which is a contradiction in view of (2.11). Hence the conclusion (2.10) is valid and the proof is complete.
Let α(·) denotes the Kuratowski measure of noncompactness of the bounded set. For any and t ∈ J, set . If B is bounded in , then B(t) is bounded in , and α(B(t)) ≤ α(B).
Lemma 2.6 [27]. Let be bounded. Then there exists a countable set D0 ⊂ D, such that α(D) ≤ 2α(D0).
Lemma 2.7 [28]. Let is bounded and equicontinuous. Then
Lemma 2.8 [28]. Let H be a countable set of strongly measurable function such that there exists an g ∈ L(J, [0, +∞)) such that ∥x(t)∥ ≤ g(t) a.e. t ∈ J for all x ∈ H. Then α(H(t)) ∈ L(J,[0,+∞)) and
Lemma 2.9 [29]. Suppose b ≥ 0, q > 0 and a(t) is a nonnegative function locally integrable on 0 ≤ t < T (some T ≤ ∞), and suppose x(t) is nonnegative and locally integrable on 0 ≤ t < T with
on this interval; then
Lemma 2.10 [21]. Let be a Banach space and Ω is a bounded convex closed set in , Q : Ω → Ω be condensing. Then Q has a fixed point in Ω.
3 Main Results
Theorem 3.1. Let be an ordered Banach space, whose positive cone P is normal, . Suppose that the following conditions are satisfied:
(H1) T(t)(t ≥ 0) is equicontinuous, i.e., T(t) is continuous in the uniform operator topology for t > 0.
(H2) There exists a constant such that
where .
(H3) There exists a function μ(t) ∈ L∞(J,ℝ+) such that
Then the IVP (1.1) has a mild solution in .
Proof. Let
where
Define the operator by
It is obvious that the mild solution of the IVP (1.1) is equivalent to the fixed point of Q. Then we proceed in two steps.
Step 1. Q : Ω → Ω.
In view of (2.1), (H3) and Lemma 2.1, we have for u ∈ Ω and t ∈ J,
that is
So Q: Ω → Ω.
Step 2. Q : Ω → Ω is condensing.
First, by using analog argument performed in [30], one can prove Q(Ω) is equicontinuous, we omit it here.
For any B ⊂ Ω, by Lemma 2.6, there exists a countable set B1 = {u n } ⊂ B, such that
Since Q(B1) ⊂ Q(Ω) is equicontinuous, in view of Lemma 2.7
For t ∈ J, according to Lemma 2.1, Lemma 2.8 and (H2), we have
So, we can conclude that
Thus, a combination of (3.2), (3.3), and (3.4) gives that
From (H2), Q : Ω → Ω is condensing.
Finally, Lemma 2.10 guarantees that Q has a fixed point in Ω.
Now we discuss the existence of minimal and maximal mild solutions for IVP (1.1).
Theorem 3.2. Let be an ordered Banach space, whose positive cone P is normal with normal constant N, T(t)x > θ holds for each x > θ, and t ≥ 0, . If conditions (H1)-(H3) and the following condition are satisfied:
(H4) t ∈ J, u1 ≤ u2 implies f(t,u1) ≤ f(t,u2).
Then the IVP (1.1) has minimal and maximal mild solutions in .
Proof. Let
where
Set
where
We consider the following fractional evolution equation
by Lemma 2.4, if u(t) is a mild solution of IVP (3.7), then
It follows from (3.6), (3.8), (H3) and Lemma 2.1 that
Thus
From the proof of Theorem 3.1, we know that the IVP (3.7) has a mild solution u(t,ε n ) in Ω1.
By (3.8), we know that
where
This yields
Combining (H4) with Lemma 2.5, we have
Hence
Let
From (3.9), (H2), Lemmas 2.1 and 2.8, we have
This together with Lemma 2.9, we obtain that φ(t) ≡ 0 on J. This means that V(t) is precompact in . On the other hand, from the proof of Theorem 3.1 we know that Q(Ω1) is equicontinuous, consequently, V is also equicontinuous. By Ascoli-Arzela theorem, we can obtain that V is relatively compact in , and so, there exists a subsequence of {u(t,ε n )} which converges uniformly on J to some . In view of (3.10), we see that {u(t,ε n )} is non-increasing. Let converge to u*, for i > j, we have , which implies that u* ≤ u(t,ε n ). For any ϵ > 0, there exists k such that
Thus, for n ≥ n k , we get . This, together with the normality of P, yields that
which implies that {u(t,ε n )} itself converges to u* uniformly on J. So, we have
On the other hand, we know that
It follows from (3.9), (3.11), (3.12) and the Lebesgue dominated convergence theorem that
Consequently, u* is a mild solution of the IVP (1.1).
Let u(t) be any solution of the IVP (1.1). It is obvious that
By (H4) and Lemma 2.5, we deduce that
Let n → ∞, we have
Thus u* is a maximal mild solution of the IVP (1.1).
Similar to the above proof, one can prove that the IVP (1.1) has a minimal mild solution in Ω1, we omit it here.
The proof is complete.
Remark 3.3. In Theorem 3.2, we do not assume f(t,B r ) = {f(t,u) : u ϵ B r } is relatively compact in for any t ∈ J and r > 0, therefore, Theorem 3.2 in this article is the extension of the main result in [15, Theorem 2.1].
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Acknowledgements
This study was supported by the NSFC (11101335, 11126296).
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H-XF carried out the main part of this manuscript. JM participated discussion and corrected the main theorem. All authors read and approved the final manuscript.
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Fan, H., Mu, J. Initial value problem for fractional evolution equations. Adv Differ Equ 2012, 49 (2012). https://doi.org/10.1186/1687-1847-2012-49
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DOI: https://doi.org/10.1186/1687-1847-2012-49
Keywords
- fractional evolution equations
- mild solutions
- initial value problem
- condensing maps
- measure of noncompactness