Abstract
Of concern is the existence of solutions for a class of boundary value problems for impulsive fractional differential equations involving the Caputo fractional derivative in a Banach space. Our approach is based upon the techniques of noncompactness measures and the fixed point theory. Two examples are presented to illustrate the results.
MSC: 34A08, 34A37, 34K30, 47H08.
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1 Introduction
The fractional differential equations have received increasing attention during recent years and have been studied extensively (see, e.g., [1–8] and references therein). This is mostly due to the fact that fractional calculus provides an efficient and excellent instrument to describe many practical dynamical phenomena which arise in viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc.
A strong motivation for studying impulsive fractional differential equations comes from the fact they can be used to model phenomena that cannot be modeled by traditional initial value problems, such as the dynamics of populations subject to abrupt changes (harvesting, diseases, etc.) and mechanics, electrical engineering, medicine biology, ecology, and so on. We refer the readers to [9–15] for the general theory and applications of impulsive differential equations. Recently, Ahmad et al. [14] applied the measure of noncompactness to a class of impulsive integrodifferential equations in a Banach space. Liu and Ahmad [15] discussed the existence and uniqueness of solutions for initial value problems of nonlinear singular multiterm impulsive Caputo type fractional differential equations on the half line.
Moreover, some researchers (see [16–20] and the references therein) have addressed the theory of boundary value problems for impulsive fractional differential equations. However, to the best of our knowledge, few papers can be found in the literature for the impulsive fractional differential equations with boundary value conditions in abstract spaces.
In this paper, we study the following fractional impulsive differential equations with boundary value conditions in a Banach space X:
where , . The fractional derivative is understood here in the Caputo sense. Here , (), are appropriate functions to be specified later. The impulsive moments are given such that , represents the jump of function u at , which is defined by , where , represent the right and left limits of at , respectively.
Observe that problem (1.1)-(1.4) reduces to an anti-periodic boundary value problem in [21] for and . If or , then it reduces to an integral boundary problem.
The rest of this paper is organized as follows. In Section 2, we state some basic concepts, notations and preliminary results about fractional calculus and measure of noncompactness. In Section 3, we discuss a new existence result (Theorem 3.1) for solutions of problem (1.1)-(1.4) on X, and obtain the corresponding result (Theorem 3.2) in R immediately. We shall give two illustrative examples for our results in Section 4.
2 Preliminaries
Throughout this paper, we denote by X a separable Banach space with norm . For measurable functions , define the norm
Let be the Banach space of all Lebesgue measurable functions with .
Let , , , := {; , , and the right limit exists for } with the norm and := {; , , and the right limits , exist for } with the norm . Obviously, and are Banach spaces.
Let us recall the following known definitions. For more details see [4].
Let , the set of all integrable functions from to R.
Definition 2.1 ([4])
The fractional integral of order q with the lower limit a for a function is defined as
provided the right side is point-wise defined on , where is the gamma function.
Definition 2.2 ([4])
Riemann-Liouville derivative of order q with the lower limit a for a function can be written as
Definition 2.3 ([4])
The Caputo derivative of order q for the function can be written as
Remark 2.4
-
(1)
If , then
-
(2)
The Caputo derivative of a constant is equal to zero.
Moreover, from the definition of Caputo derivative, we can obtain the following auxiliary results.
Lemma 2.5 For , the general solution of fractional differential equation is given by
where , () and denotes the integer part of the real number q.
In view of Lemma 2.5, it follows that the result holds.
Lemma 2.6 Let , then
where , , .
Definition 2.7 A function is said to be a solution of (1.1)-(1.4) if u satisfies the equation on , and conditions , , and , .
Lemma 2.8 Let be continuous. A function is a solution of the following fractional integral equation:
if and only if is a solution of the problem
Proof Let u be the solution of (2.2)-(2.5). If , then Lemma 2.6 implies that
for some . Obviously, , .
If , then Lemma 2.6 implies that
for some . Thus, we have
Applying the impulsive condition (2.3), we derive
Hence, for ,
If , then Lemma 2.6 implies that
for some . Thus, we have
Applying the impulsive condition (2.3), we obtain
Hence, for ,
By repeating the process, for , we have
Now, applying the boundary conditions (2.4), (2.5) to (2.2), we get
Now, it is clear that (2.6)-(2.9) imply (2.1).
Conversely, if we assume that u satisfies (2.1), by a direct computation, it follows that the solution given by (2.1) satisfies (2.2)-(2.5). This completes the proof. □
Next, we recall that the Hausdorff measure of noncompactness on each bounded subset Ω of Banach space Y is defined by
This measure of noncompactness satisfies some basic properties as follows.
Lemma 2.9 ([22])
Let Y be a Banach space and let be bounded. Then
-
(1)
if and only if U is precompact;
-
(2)
, where and convU mean the closure and convex hull of U, respectively;
-
(3)
if ;
-
(4)
;
-
(5)
, where ;
-
(6)
, for any ;
-
(7)
if the map is Lipschitz continuous with constant k, then for any bounded subset , where Z is a Banach space.
Definition 2.10 A continuous map is said to be a χ-contraction if there exists a positive constant such that for any bounded closed subset .
Theorem 2.11 (Darbo-Sadovskii [22])
If is bounded closed and convex, the continuous map is a χ-contraction, then the map ℱ has at least one fixed point in U.
In Section 3, we will use the above fixed point theorem to obtain main result. To this end, we present the following assertion about χ-estimates for a multivalued integral (Theorem 4.2.3 of [23]).
Let be the family of all nonempty subset of Y, be a multifunction. It is called:
-
(i)
integrable, if it admits a Bochner integrable selection , for a.e. ;
-
(ii)
integrably bounded, if there exists a function such that
Proposition 2.12 For an integrable, integrably bounded multifunction where X is a separable Banach space, let
where . Then for all .
To end this section, we introduce the following -type Arzela-Ascoli theorem ([24], Theorem 2.1) which will be used in Section 3.
Theorem 2.13 Let Y be a Banach space and . If the following conditions are satisfied:
-
(i)
is a uniformly bounded subset of ;
-
(ii)
is equicontinuous in , , where , ;
-
(iii)
, and are relatively compact subsets of Y,
then is a relatively compact subset of .
3 Main results
In this section, we will discuss the existence of solutions to (1.1)-(1.4). For this end, we consider the following assumptions.
(H1)
-
(i)
satisfies is measurable for all and is continuous for a.e. , and there exists a function () such that
-
(ii)
For any bounded set , there exists a function () such that
(H2) The functions () are continuous and there exist positive constants , such that
(H3) There exist constants such that
Theorem 3.1 Assume that (H1)-(H3) are satisfied, then there exists a solution of (1.1)-(1.4) on J provided that
Proof Consider the operator defined by
Clearly, ℱ is well defined and the fixed point of ℱ is the solution of problem (1.1)-(1.4) by Lemma 2.8.
The operator ℱ can be rewritten in the form , for (), where
Then
Let be a sequence such that in as . Since f satisfies (H1)(i), for almost every , we get as . Moreover, .
It follows from Lebesgue’s dominated convergence theorem that
Moreover, noting that (H2) and (H3), we have as . Now we can see that ℱ is continuous.
Let , where and
For , , using (H1)(i) and the Hölder inequality, we have
Therefore
then
Moreover, by (H2) and (H3), we obtain
and
therefore
From (3.2) and (3.3), we have . This shows that .
Let , for any , we have
Therefore,
Moreover, from the Hölder inequality, we have
where is a constant and . Similarly,
Thus,
This shows that the set is equicontinuous.
If is bounded and the elements of are equicontinuous on each (), we can consider the measure of noncompactness on the space in the following way:
Let be a nonempty, bounded set. Clearly, we can see
Moreover, for , we set
Consider the multifunction
Obviously, is integrable and from (H1)(i) it follows that is integrably bounded. Moreover, noting that (H1)(ii), we have the following estimate for a.e. ,
Applying Proposition 2.12, we obtain
Therefore, combining with (3.4), (3.5) and the Hölder inequality, we have
Similarly,
Since and are equicontinuous on every , by Proposition 7.3 of [25], we find that
Then, according to inequalities (3.6) and (3.7), we have
Moreover, by (H2) and (H3), for any ,
This means that is Lipschitz continuous with Lipschitz constant . It follows from (7) in Lemma 2.9 that
Therefore, from (3.8) and (3.9), we get
Hence, ℱ is a χ-contraction on by Definition 2.10. According to Theorem 2.11, the operator ℱ has at least one fixed point u in , which is a solution of problem (1.1)-(1.4). This ends the proof. □
When , we rely on Schauder’s fixed point theorem, which gives an existence result for solutions of problem (1.1)-(1.4) under the following assumptions.
(H1′) is jointly continuous, and there exists a function () such that .
(H2′) The functions are continuous and there exist positive constants , such that and , , .
(H3′) There exist constants such that , , .
Theorem 3.2 Assume that the assumptions (H1′)-(H3′) hold. Then problem (1.1)-(1.4) has at least one solution on J provided that
Proof Let be defined as in the proof of Theorem 3.1. In the proof of Theorem 3.1, we can see that ℱ is continuous, ℱ maps bounded sets into bounded sets and equicontinuous sets. Then we can deduce that ℱ is compact on as a result of the -type Arzela-Ascoli theorem (see Theorem 2.13 in the case of ).
As a consequence of Schauder’s fixed point theorem, we conclude that ℱ has a fixed point, that is, problem (1.1)-(1.4) has at least one solution on R. The proof is complete. □
4 Applications
In this section, we give two examples to illustrate the usefulness of our main results.
Example 4.1 Let . Consider the following impulsive integral boundary problem:
where , , , , , are bounded functions on R such that , , the functions , () are measurable and there exists a constant such that .
For , , , we set
Then we can rewrite (4.1) in the abstract form (1.1)-(1.4).
Obviously, we have for . For any ,
which implies that for any bounded set , , .
Moreover, for any ,
Suppose further that
then problem (4.1) has at least a solution by Theorem 3.1.
Example 4.2 Let . Consider the following impulsive anti-periodic boundary problem of fractional order:
Set , and . Obviously, . Moreover, , . It can be found that . Therefore, due to the fact that all the assumptions of Theorem 3.2 hold, problem (4.2) has a solution.
5 Conclusion
In this paper, a generalized boundary value problem for impulsive fractional differential equations involving Caputo fractional derivative in abstract space has been studied. A reasonable formula and definition of solutions for such problem is introduced. The new existence results are obtained based upon the techniques of measures of noncompactness and the fixed point theory.
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Acknowledgements
The authors are grateful to the referee for his/her valuable suggestion. This work was supported partly by the NSF of China (11201413), the NSF of Yunnan Province (2013FB034) and the Educational Commission of Yunnan Province (2012Z010).
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Each of the authors, FL and HW, contributed to each part of this study equally and read and approved the final version of the manuscript.
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Li, F., Wang, H. Solvability of boundary value problems for impulsive fractional differential equations in Banach spaces. Adv Differ Equ 2014, 202 (2014). https://doi.org/10.1186/1687-1847-2014-202
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DOI: https://doi.org/10.1186/1687-1847-2014-202