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This paper is concerned with nonlinear fractional differential equations with the Caputo fractional derivatives in Banach spaces. Local existence results are obtained for initial value problems with initial conditions at inner points for the cases that the nonlinear parts are Lipschitz and non-Lipschitz, respectively. Hausdorff measure of non-compactness and Darbo-Sadovskii fixed point theorem are employed to deal with the non-Lipschitz case. The results obtained in this paper extend the classical Peano’s existence theorem for first order differential equations partly to fractional cases.

Let

with the initial value condition at an inner point (IVP for short)

where

Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, biology, economics, control theory, signal and image processing, etc. which involve fractional order derivatives. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes. Consequently, the subject of fractional differential equations is gaining much importance and attention (see [

In the most of the mentioned works above, the initial value problems for fractional differential equations were studied with the initial conditions at the endpoints of the definition interval, recalling that the classical existence and uniqueness theorem are for first order differential equations, where the initial conditions are at any inner points of the considered interval. On the other hand, classical integer order derivatives at a point are determined by some neighbourhoods of this point, while the fractional derivatives are determined by intervals from the endpoints up to this point. Fractional derivatives at the same point with different endpoints of the definition intervals are in fact different derivatives. Let us investigate the fractional differential equations

and

with

A direct computation deduces that the solutions to the above initial value problems are

and

respectively. By a numerical method, we can find that

Motivated by the above comment, in this paper, we study the existence of solutions to the nonlinear Caputo fractional differential equation modeled as (1.1), with the initial conditions at inner points of the definition interval of the fractional derivative. In this case, the equivalent integral equation is a Volterra-Fredholm equation. Local existence results are obtained for the cases that the function f on the righthand side of the equation is Lipschitz and Caratheodory type, respectively. The theory of measure of non-compactness is employed to deal with the non-Lipschitz case. In this sense, the classical Peano’s theorem is extended to fractional cases.

In this section we collect some definitions and results needed in our further investigations.

Let

such that

Definition 2.1 ( [

where

It has been shown that the fractional integral operator

Definition 2.2 ( [

Lemma 2.1 ( [

for

In recent decades measures of noncompactness play very important role in nonlinear analysis [

for bounded set B in a Banach space Y.

The following properties of Hausdorff’s measure of noncompactness are well known.

Lemma 2.2 ( [

(1) B is pre-compact if and only if

(2)

(3)

(4)

(5)

(6)

(7) If the map

(8)

(9) If

The map

Lemma 2.3 ( [

In this paper we denote by

Lemma 2.4 ( [

for all

Lemma 2.5 ( [

Lemma 2.6 ( [

for all

In this section, we study the initial value problem for nonlinear fractional differential equations with initial con- ditions at inner points. More precisely, we will prove a Peano type theorem of the fractional version. We begin with the definition of the solutions to this problem. Consider initial value problem

Since the fractional derivative of a function y at an inner point

The initial condition then implies that

Inserting this into (3.2) we obtain

Based on the above analysis (see [

Definition 3.1: A contionuous function

where

We first give an existence result based on the Banach contraction principle.

Theorem 3.1: Let

Then for

mild solution

Proof. Since

We define a mapping

for

It then follows that

with

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

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

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

Theorem 3.2: Let

dition with respect to the second variable with a Lipschitz constant L. If

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.

(H_{1}):

(H_{2}): For every

(H_{3}): There exists

for a.e.

Theorem 3.3: Let _{1})-(H_{2}) hold, and suppose

Further assume that there exists a real number

Then there exists an

Proof. On account of the hypothesis (3.8), we can find constants

Due to the hypothesis (3.6), we can take

Define an operator

for _{1}) − (H_{2}) as well as the Lebesgue dominated convergence theorem that T is well-defined, i.e., Ty is continuous on

due to (H_{2}) 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 _{2}), for every

It follows that

Next we prove that T maps bounded subsets into equi-continuous subsets. Let

which converges to 0 as

Now we verify that T is a

The assumption

From the inequality (3.9), we deduce that

We have now shown that

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).

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