ta-original="http://html.scirp.org/file/1-5300920x105.png" /> is said to be a mild solution to (1.1)-(1.2) if it satisfies
We first give an existence result based on the Banach contraction principle.
Theorem 3.1: Let, and. Let be continuous and fulfil a Lipschitz con- dition with respect to the second variable with a Lipschitz constant L, i.e.
Then for with, there exist an with and a unique
mild solution to the IVP (1.1)-(1.2).
Proof. Since, we can take an with such that
We define a mapping by
for and. Then for any and, we have
It then follows that
with. Since, we get that. Thus an appli-
cation of Banach’s fixed point theorem yields the existence and uniqueness of solution to our integral equation (3.3).
Remark 3.1: The condition means that the point cannot be far away from a. How-
ever, the following example shows that we cannot expect that there exists a solution to (1.1)-(1.2) for each.
Example 3.1: Considering the differential equation with the Caputo fractional derivative
where is a constant. A direct computation shows that it admits a solution
whose existence interval is.
However, from the proof of Theorem 3.1 we can see that if the Lipschitz constant L is small enough, then can be extended to the whole interval. Thus we have the following result.
Theorem 3.2: Let, and. Let be continuous and fulfil a Lipschitz con-
dition with respect to the second variable with a Lipschitz constant L. If, then for every
, there exists an with and a unique mild solution to the IVP (1.1)-(1.2).
Next we want to study the case that f satisfies the Carathedory condition. For simplicity, we limit to the case that f is locally bounded. We list the hypotheses.
(H1): satisfies the Carathedory condition, i.e. is measurable for every and is continuous for almost every xÎ[a,b].
(H2): For every, there is a constant, such that for a.e. and with.
(H3): There exists with such that
for a.e. and any bounded subset.
Theorem 3.3: Let and. Assume that the hypotheses (H1)-(H2) hold, and suppose satisfying
Further assume that there exists a real number solving the inequality
Then there exists an such that the IVP (1.1)-(1.2) has at least a solution.
Proof. On account of the hypothesis (3.8), we can find constants large enough and with
Due to the hypothesis (3.6), we can take small enough such that
Define an operator by
for and. It then follows from the hypotheses (H1) − (H2) as well as the Lebesgue dominated convergence theorem that T is well-defined, i.e., Ty is continuous on for every, and that T is continuous. Further, let. Then is a bounded closed subset of. For every and, we have
due to (H2) and (3.8) which implie that.
Below we show that T satisfies the hypotheses of Darbo-Sadovskii Theorem (Lemma 2.5). We first prove that T maps bounded subsets in into bounded subsets. For this purpose we show that is bounded for every with fixed. Let. Then by (H2), for every, we have
It follows that which is independent of. Hence is bounded.
Next we prove that T maps bounded subsets into equi-continuous subsets. Let be arbitrary and with. Then we have
which converges to 0 as, and the convergence is independent of. Thus is equi- continuous.
Now we verify that T is a -contraction. Take any bounded subset, then W is equi-continuous. So we get from Lemma 2.4, 2.6 and 2.8 that
The assumption implies that, which shows that the function with for every. Hence an employment of Hölder inequality yields
From the inequality (3.9), we deduce that, which means that T is a -con- traction on.
We have now shown that that T maps bounded subsets into bounded and equi-continuous subsets, and that T is a -contraction on. By Darbo-Sadovskii Theorem (Lemma 2.5), we conclude that T has at least a fixed point y in, which is the solution to (1.1)-(1.2) on, and the proof is completed.
This research was supported by the National Natural Science Foundation of China (11271316, 11571300 and 11201410) and the Natural Science Foundation of Jiangsu Province (BK2012260).
Cite this paper
Xiaoping Xu,Guangxian Wu,Qixiang Dong, (2015) Fractional Differential Equations with Initial Conditions at Inner Points in Banach Spaces. Advances in Pure Mathematics,05,809-816. doi: 10.4236/apm.2015.514075