Fractional Differential Equations with Initial Conditions at Inner Points in Banach Spaces ()
Received 12 June 2015; accepted 4 December 2015; published 7 December 2015
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1. Introduction
Let
be a Banach space. We consider the nonlinear fractional differential equation
(1.1)
with the initial value condition at an inner point (IVP for short)
(1.2)
where
,
is the Caputo fractional derivative,
is a given function satisfying some assumptions that will be specified later.
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 [1] - [5] ). There are a large number of papers dealing with the existence or properties of solutions to fractional differential equations. For an extensive collection of such results, we refer the reader to the monograph [1] and [3] and references therein.
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
(1.3)
and
(1.4)
with
and the same initial value condition
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A direct computation deduces that the solutions to the above initial value problems are
![](//html.scirp.org/file/1-5300920x16.png)
and
![](//html.scirp.org/file/1-5300920x17.png)
respectively. By a numerical method, we can find that
for
. This example shows that
and
are two different “fractional derivatives”, and Equations (1.3) and (1.4) are two different equ- ations.
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.
2. Preliminaries and Lemmas
In this section we collect some definitions and results needed in our further investigations.
Let
be the Banach space of all continuous functions
with the norm
, and
the Banach space of all measurable functions ![]()
such that
are Lebesgue integrable, equipped with the norm
with
.
Definition 2.1 ( [1] ): Let
be a fixed number. The Riemann-Liouville fractional integral of order
of the function
is defined by
![]()
where
denotes the Gamma function, i.e.,
.
It has been shown that the fractional integral operator
transforms the space
into
, and some other properties of
are refered to [1] .
Definition 2.2 ( [1] ): Let
, and
. The Caputo fractional derivative of order
of h at the point x is defined by
![]()
is also called the Caputo fractional differential operator.
Lemma 2.1 ( [1] ): Let
and
. Then
![]()
for
.
In recent decades measures of noncompactness play very important role in nonlinear analysis [6] - [9] . They are often applied to the theories of differential and integral equations as well as to the operator theory and geo- metry of Banach spaces ( [10] - [15] ). One of the most important examples of measure of noncompactness is the Hausdorff’s measure of noncompactness
, which is defined by
![]()
for bounded set B in a Banach space Y.
The following properties of Hausdorff’s measure of noncompactness are well known.
Lemma 2.2 ( [8] ): Let Y be a real Banach space and
be bounded,the following properties are satisfied :
(1) B is pre-compact if and only if
;
(2)
where
and
mean the closure and convex hull of B respec- tively;
(3)
when
;
(4)
where
;
(5)
;
(6)
for any
;
(7) If the map
is Lipschitz continuous with constant k then
for any bounded subset
, where Z be a Banach space;
(8)
, where
means the nonsymmetric (or symmetric) Hausdorff distance between B and C in Y;
(9) If
is a decreasing sequence of bounded closed nonempty subsets of Y and
, then
is nonempty and compact in Y.
The map
is said to be a
if there exists a positive constant
such that
for any bounded closed subset
, where Y is a Banach space.
Lemma 2.3 ( [8] ): (Darbo-Sadovskii) If
is bounded closed and convex, the continuous map
is a
-contraction, then the map Q has at least one fixed point in W.
In this paper we denote by
the Hausdorff’s measure of noncompactness of X and by
the Hausdorff’s measure of noncompactness of
. To discuss the existence we need the following lemmas in this paper.
Lemma 2.4 ( [8] ): If
is bounded, then
![]()
for all
, where
. Furthermore if W is equicontinuous on [a,b], then
is continuous on
and
![]()
Lemma 2.5 ( [14] [15] ): If
is uniformly integrable, then
is measurable and
(2.1)
Lemma 2.6 ( [8] ): If
is bounded and equicontinuous, then
is continuous and
(2.2)
for all
, where
.
3. Existence Results
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
(3.1)
Since the fractional derivative of a function y at an inner point
is determined by the values of y on the interval
, for
and
, we get from Lemma 2.3 that
(3.2)
The initial condition then implies that
![]()
Inserting this into (3.2) we obtain
![]()
Based on the above analysis (see [1] ), we give the definition of mild solutions to the IVP (1.1)-(1.2).
Definition 3.1: A contionuous function
is said to be a mild solution to (1.1)-(1.2) if it satisfies
(3.3)
where
and
.
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
(3.4)
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
(3.5)
for a.e.
and any bounded subset
.
Theorem 3.3: Let
and
. Assume that the hypotheses (H1)-(H2) hold, and suppose
satisfying
(3.6)
Further assume that there exists a real number
solving the inequality
(3.7)
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
(3.8)
Due to the hypothesis (3.6), we can take
small enough such that
(3.9)
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
(3.10)
The assumption
implies that
, which shows that the function
with
for every
. Hence an employment of Hölder inequality yields
(3.11)
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.
Acknowledgements
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).