Existence and continuous dependence for weighted fractional differential equations with infinite delay
© Dong; licensee Springer. 2014
Received: 2 May 2014
Accepted: 1 July 2014
Published: 22 July 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.
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.
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 .
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 
provided the right-hand side is pointwisely defined, where denotes the well-known gamma function, i.e., .
Definition 2.2 
provided the right-hand side is pointwisely defined, where denotes the integer part of the real number α.
For simplicity, when , we denote and by and , respectively.
Lemma 2.3 
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.
for some ;
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).
is continuous on , and therefore, is well defined for .
for some constant C .
We first give an existence result based on the Banach contraction principle. We list the hypotheses.
(H1) is continuous.
for and every .
Theorem 3.2 Assume that (H1) and (H2) hold. Then there exists a unique solution to (1.9)-(1.10) on .
We can see that if is a fixed point of P, then is a solution of (1.9)-(1.10).
for , which is the unique solution to the integral equation (3.3) on the interval .
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.
for and every .
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.
where and with . Therefore, for every , which implies that P maps bounded subsets into bounded subsets in W.
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.
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
We will prove that the solution mapping is Lipschitz continuous.
for every .
as desired, which completes the proof. □
with initial condition (1.10).
Take , we get the required result. □
5 An example
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 .
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.
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).
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