Fractional Differential Equations with Initial Conditions at Inner Points in Banach Spaces

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.


Introduction
Let X be a Banach space.We consider the nonlinear fractional differential equation : , f a b × → X X 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 with 0 1 α < < and the same initial value condition ( ) ( ) ( ) A direct computation deduces that the solutions to the above initial value problems are ( ) ( ) respectively.By a numerical method, we can find that ( ) ( ) y x y x ≠ for 1 x > .This example shows that 0 c D α and 1 c D α are two different "fractional derivatives", and Equations (1.3) and (1.4) are two different equations.
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.

Preliminaries and Lemmas
In this section we collect some definitions and results needed in our further investigations. Let The Caputo fractional derivative of order α of h at the point x is defined by 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 geometry of Banach spaces ([10]- [15]).
where B and convB mean the closure and convex hull of B respectively; (3)

The map
In this paper we denote by β the Hausdorff's measure of noncompactness of X and by c β the Hausdorff's measure of noncompactness of [ ] ( ) , ; C a b X .To discuss the existence we need the following lemmas in this paper.
Lemma 2.4 , where ⊆ is bounded and equicontinuous, then , where

Existence Results
In this section, we study the initial problem for nonlinear fractional differential equations with initial conditions 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 Based on the above analysis (see [1]), we give the definition of mild solutions to the IVP (1.1)-(1.2).Definition 3.1:  Proof.Since ( ) , we can take an 0

Ty x Ty x x t f t y t f t y t t x t f t y t f t y t t x t x t L y t y t t L y t y t t L x a y y
Ty Ty K y y . Thus an application 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 0 x cannot be far away from a.However, the following example shows that we cannot expect that there exists a solution to whose existence interval is [ ) 0, c .However, from the proof of Theorem 3.1 we can see that if the Lipschitz constant L is small enough, then 0 x can be extended to the whole interval.Thus we have the following result.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. ( for a.e.
[ ] Assume that the hypotheses (H 1 )-(H 2 ) hold, and suppose ( ) Further assume that there exists a real number 0 r > solving the inequality Then there exists an 0 h > such that the IVP (1.1)-(1.2) has at least a solution x x a a

Ty x y x t f t y t t x t f t y t t
) which converges to 0 as 2 1 0 x x − → , and the convergence is independent of

D
value condition at an inner point (IVP for short) α is the Caputo fractional derivative, [ ] X be the Banach space of all continuous functions Definition 2.1 ([1]): Let 0 α > be a fixed number.The Riemann-Liouville fractional integral of order 0 α > of the function[ ]: , h a b → R is defined by the Gamma function, i.e., ( ) R , and some other properties of a I α are refered to[1].Definition 2.2 ([1]): Let

[
means the nonsymmetric (or symmetric) Hausdorff distance between B and C in Y; sequence of bounded closed nonempty subsets of Y and compact in Y.

1 )∫
Since the fractional derivative of a function y at an inner point Inserting this into (3.2) we obtain an existence result based on the Banach contraction principle.Let : f G → X be continuous and fulfil a Lipschitz condition with respect to the second variable with a Lipschitz constant L, i.e.

1 :
Considering the differential equation with the Caputo fractional derivative a constant.A direct computation shows that it admits a solution

Theorem 3 . 2 :
Let 0 1 α < < , and [ ] , G a b = × X .Let : f G → X be continuous and fulfil a Lipschitz con-dition with respect to the second variable with a Lipschitz constant L. If the Carathedory condition, i.e. ( ) [ ] continuous for almost every x∈[a,b].
On account of the hypothesis (3.8), we can find constants 0 0 r > large enough and 0 then follows from the hypotheses (H 1 ) − (H 2 ) as well as the Lebesgue dominated convergence theorem that T is well-defined, i.e., Ty is continuous on [ ] bounded subsets into bounded and equi-continuous subsets, and that T is a c β -contraction on 0 r TB .By Darbo-Sadovskii Theorem (Lemma 2.5), we conclude that T has at least a fixed point y in 0 r TB , which is the solution to (1.1)-(1.2) on [ ] 0 , a x h + , and the proof is completed.
One of the most important examples of measure of noncompactness is the Hausdorff's measure of noncompactness Y β , which is defined by inf 0; can be covered with a finite number of balls of radius equal to Y B r B r β = > 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 , B C Y ⊆ be bounded,the following properties are satisfied : (1) B is pre-compact if and only if