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Existence and continuous dependence for weighted fractional differential equations with infinite delay
Advances in Difference Equations volume 2014, Article number: 190 (2014)
This paper is concerned with weighted fractional differential equations with infinite delay, the Riemann-Liouville derivative, and nonzero initial values. Existence and continuous dependence results of solutions are obtained. An illustrative example is also presented.
Recently, fractional differential equations have attracted a considerable interest in both mathematics and applications, since they have been proved to be valuable tools in modeling many physical phenomena. There has been a significant development in fractional differential equations in the past decades [1–12]. Among these works, some authors studied functional fractional differential equations [5–8, 10]. For example, in , Benchohra et al. studied the fractional order differential equations
with infinite delay
where is the standard Riemann-Liouville fractional derivative, , the phase space defined axiomatically by Hale and Kato, and for . Some existence results are obtained in the special case . Henderson and Ouahab also studied the multivalued version of (1.1) (i.e., fractional differential inclusion) with finite delay in . Zhang  discussed the linear fractional order time-delay system
in a finite dimensional space and got some existence and stability results. In , Lakshmikantham concerned some basic theory for functional fractional differential equations.
However, it is known that the Riemann-Liouville fractional derivative of a function y is unbounded at some neighborhoods of the initial point 0, except that . For this reason, when , the solutions to the functional fractional differential equations given in the mentioned papers may not be well defined.
We investigate some examples. Firstly, we consider fractional differential equations with infinite delay. Let . Then satisfies axioms (A1), (A2) and (B) (specified later, also see ), with and for . Let for in (1.2). Obviously, . We take and . Then equations (1.1)-(1.2) become
Then we get that for . Since for some constant C, we can apply the fractional integral operator to both sides of (1.5). A direct computation gives
for and some constant C, which is unbounded at any right neighborhood of 0. Consequently, equations (1.5)-(1.6) cannot have any continuous solution on .
Similarly, for the case of finite delay, we consider the following fractional differential equation on R with finite delay:
Then for , and equation (1.8) reduces to for . Applying the fractional integral operator to both sides, we get that
for , which is still unbounded in the right neighborhood of 0.
From the above examples we can see that, in the case of nonzero initial value, functional fractional differential equations with Riemann-Liouville derivative and either finite or infinite delay may not have ‘classical’ solutions. Moreover, the functions that appeared in these equations, defined by for , may not be well defined, or not in the phase space. On the other hand, the Riemann-Liouville fractional derivative of the constant function is not zero. In fact, if for and , then . Hence the nonzero initial value problems cannot be transformed to zero initial value problem by the parallel shift method.
where , is the Riemann-Liouville fractional derivative, , is a given function satisfying some assumptions, and ℬ is the phase space that will be specified later. We give the definition of solutions and investigate the existence and continuous dependence of solutions to such equations in the space . An example is presented to illustrate the results.
2 Preliminaries and lemmas
In this section we collect some definitions and results needed in our further investigations.
Let us denote by the space of all continuous real functions defined on and by the space of all real functions defined on which are locally Lebesgue integrable. We also consider the space consisting of all continuous functions such that exists, with the norm .
Definition 2.1 
Let be a fixed number. The Riemann-Liouville fractional integral of order of the function is defined by
provided the right-hand side is pointwisely defined, where denotes the well-known gamma function, i.e., .
Definition 2.2 
Let be fixed and . The Riemann-Liouville fractional derivative of order α of at the point t is defined by
provided the right-hand side is pointwisely defined, where denotes the integer part of the real number α.
When , then
For simplicity, when , we denote and by and , respectively.
Lemma 2.3 
Let . Then the unique solutions to the equation are given by the formula
for , where is a constant, provided . Further, if such that , then
for and some constant .
In the literature devoted to equations with infinite delay, the selection of the state space ℬ plays an important role in the study of both qualitative and quantitative theory. A usual choice is a semi-normed space satisfying suitable axioms, which was introduced by Hale and Kato . For a detailed discussion on the topic, we refer to the book by Hino et al. .
Definition 2.4 
A linear topological space of functions from into X, with seminorm , is called an admissible phase space if ℬ has the following properties.
(A1) There exist a positive constant H and functions , with K continuous and M locally bounded, such that for any and , if , , and is continuous on , then for every , the following conditions hold:
for some ;
(A2) For the function in (A1), is a ℬ-valued continuous function for .
The space ℬ is complete.
3 Existence results
We begin with the definition of solutions to the weighted functional fractional differential equations.
Definition 3.1 A function is said to be a solution to (1.9)-(1.10), if , and satisfies (1.9).
Example 3.1 Consider the weighted version of the previous example
By computation we also get that for ,
Then . Since , we get that . Therefore, the solution to (3.1)-(3.2) is
where is the Mittag-Leffler function, i.e., . Note that
is continuous on , and therefore, is well defined for .
For the existence results on problem (1.9)-(1.10), we need to transform the fractional differential equation into an integral equation. From Lemma 2.3 we can obtain that if and , then the function y solves the fractional differential equation
if and only if y satisfies
for some constant C .
We first give an existence result based on the Banach contraction principle. We list the hypotheses.
(H1) is continuous.
(H2) There exists a constant such that
for and every .
Theorem 3.2 Assume that (H1) and (H2) hold. Then there exists a unique solution to (1.9)-(1.10) on .
Proof By Lemma 2.3 and the above remark, a function y is a solution to (1.9)-(1.10) if and only if y satisfies
For given which belongs to ℬ, let be a function defined by
Then we have . For , where is the Banach space consisting of all continuous functions such that exists, endowed with the norm , we extend to , also denoted by , defined by
It is easily seen that if satisfies the integral equation
we can decompose as , which implies that for , and the function satisfies
Set . For , define , then becomes a Banach space. Define an operator by
We can see that if is a fixed point of P, then is a solution of (1.9)-(1.10).
Let , where is the function that appeared in Definition 2.4. Let , and . Then and
We first focus on the interval . Let and define for . Then is a Banach space. Define the operator by
For and , we have
From (3.4) and the Banach contraction principle we know that there exists a unique satisfying
for , which is the unique solution to the integral equation (3.3) on the interval .
Next we consider the interval . Restrict the functions on the interval to construct and define for . Then is a Banach space. For , rewrite equation (3.3) as
Since the function z is uniquely defined on , the second integral can be considered as a known function. Using the same arguments as above, we can obtain that there exists a unique function satisfying
for , which is the unique solution to the integral equation (3.3) on the interval . Taking the next interval , repeating this process, we conclude that there exists a unique solution to the integral equation (3.3) on the interval . Set , then y is the unique solution to the fractional differential equation (1.9)-(1.10).
Below we consider the existence result which is based on the Schauder fixed point theorem. We need the following hypothesis.
(H3) There exist an with and a continuously non-decreasing function such that
for and every .
Theorem 3.3 Assume that hypotheses (H1) and (H3) hold. If
then there exists at least a solution to (1.9)-(1.10) on .
Proof As in the proof of Theorem 3.2, we define the operator . The continuity of P can be derived by hypothesis (H3) and the Lebesgue dominated convergence theorem. We will verify that P is completely continuous.
We first show that P maps bounded subsets in W into bounded subsets. Let . Then, for any and , we have
where , it follows from (H3) and Holder’s inequality that
where and with . Therefore, for every , which implies that P maps bounded subsets into bounded subsets in W.
Next, we prove that P maps bounded subsets into equicontinuous subsets in W. Let and with , we have
where , and . It follows that as , and the convergence is independent of , which implies that the set is equicontinuous.
Now we have proved that P maps bounded subsets into bounded and equicontinuous subsets in W. By the Arzelá-Ascoli theorem, we conclude that P is a completely continuous operator.
To apply Schauder’s fixed point theorem, we need to verify that there exists a closed convex bounded subset in such that . To this end, we derive from inequality (3.5) that there exists a constant such that
Define , then B is closed, convex and bounded in W. Then, for every and , we have
It follows that for all , and hence .
An application of Schauder’s fixed point theorem shows that there exists at least a fixed point z of P in W. Then is the solution to (1.9)-(1.10), and the proof is completed. □
4 Dependence on initial data
In this section, we investigate the influence of a perturbation of initial data to the solutions. We first look at the dependence of solutions on initial values. For this purpose, we denote by the solution to equation (1.9) with initial condition (1.10), and by the solution to (1.9) with the initial condition
We will prove that the solution mapping is Lipschitz continuous.
Theorem 4.1 Let the assumptions of Theorem 3.2 hold. If , then there exists a constant such that
for every .
Proof From Theorem 3.2 we know that for every , equation (1.9) has solutions and on , respectively. Further, there are such that and , satisfying
and , for . Then, by (H2) and axiom (A1) (in Definition 2.4), for we have
Since , let . Then we obtain
as desired, which completes the proof. □
Next we investigate the influence of changes in the given function on the right-hand side of the differential equation. Now we denote by the solution to the differential equation (1.9) with initial condition (1.10) and by the solution to the differential equation
with initial condition (1.10).
Theorem 4.2 Let f and fulfill hypotheses (H1) and (H2). If , then there exists a constant such that
Proof The existence of solutions can be ensured by Theorem 3.2. Let be such that and . Then and satisfy
For , we have
Take , we get the required result. □
5 An example
In this section we give an example to illustrate our main results. Let , where is a constant. Then satisfies axioms (A1), (A2) and (B) in Definition 2.4 with and . For any , consider the weighted fractional functional differential equation
with infinite delay
where , and are constants. Let . If , then it is easily seen that f satisfies the Lipschitz condition with respect to x on any bounded interval. So, by Theorem 3.2, problem (5.1)-(5.2) has a unique solution on . If , then f does not satisfy the Lipschitz condition with respect to x in some neighborhoods of 0. However, in this case, we can define , for , and for and constant ρ with . Then hypothesis (H3) holds. Also, since , condition (3.5) is satisfied. Hence, by Theorem 3.3, problem (5.1)-(5.2) has at least a solution on .
Remark 5.1 In , the authors studied the weighted fractional Cauchy problem
where , and with initial conditions
The nonlinear case () is discussed in Example 3.4 on p.181, and the linear case () in Examples 4.2-4.3 on pp.225-227. For the homogeneous case, the explicit solutions are constructed by using generalized Mittag-Leffler functions. However, in general, we cannot expect to find explicit solutions for delayed fractional differential equations, even for linear and homogeneous cases.
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This research was done while the author visited the University of Southern Mississippi, which was supported financially by Jiangsu Government Scholarship for Overseas Studies (JS-2011-159). This work is supported by the National Natural Science Foundation of China (11271316 and 11201410) and the Natural Science Foundation of Jiangsu Province (BK2012260 and BK20141271).
The author declares that he has no competing interests.