Existence and Uniqueness Analysis for Fractional Differential Equations with Nonlocal Conditions

Chun Wang

Abstract:Fractional differential equations and systems are gradually becoming an essential approach to real world applications in science and engineering technology.The boundary value problems of the fractional differential equations and systems were investigated by using several different methods.Recently,much attention has been paid to investigate fractional differential equations with nonlocal conditions by using the fixed point method.In this paper,by using the Banach fixed point theorem and Krasnoselkii fixed point theorem,the existence and uniqueness of the solutions to the initial value problem of the nonlinear fractional differential equations with nonlocal conditions are investigated,and some sufficient conditions are obtained.We extend some results that already exist.Finally,an example is given to show the usefulness of the theoretical results.

Keywords:initial value problem;fractional differential equation;Krasnoselkii fixed point theorem

1 Introduction

Fractional differential equations are differential equations of any arbitrary real or complex order.The major advantage of the fractional derivatives is that they can describe the property of heredity and memory of many materials.

Fractional differential equations now received increasing attention due to their applications in a variety of disciplines such as physics,chemistry,mechanics,electrical engineering,control theory,heat conduction in materials with memory,etc.(see [1−15]and references therein).Regarding some new real world applications of fractional calculus in science and engineering,we recommend the review article [16]to the readers.

Recently,much attention has been paid to investigate fractional differential equations with nonlocal conditions by using the fixed point method.The reader may consult [17−23].

In [24],by using the monotone iterative method and the fixed point method,Jankowski discussed the initial value problems

In [25],N′Guérékata considered the Cauchy problem for the nonlocal conditions of fractional differential equation

where 0

The above problems with nonlocal conditions can be applied to the description of diffusion phenomenon of a small amount of gas in a transparent tube (see [26]).

Motivated by the above references,we focus our attention on the following problem

with the nonlocal condition

whereDqxdenotes the left-sided Riemann-Liouville fractional derivative ofxwith

gis a bounded linear functional on the space.

whereC1(I,R)=C(I,R) ifq=1.The bounded linear functionalghas many different forms.For example,it could be either of the following:one is that

whereci(i=1,2,···,p) are given constants and 0

whereu(t) is a function such thatu(t)x(t) is integrable on (0,T].

The left-sided Riemann-Liouville fractional derivative ofxwithq∈(0,1) define by

and ifq=1,thenD1x(t)=x′(t),see [27].Here,we also introduce the left-sided fractional integralIqof orderq>0 by

where Γ (q) is the Gamma function,see [27].

The following definition is necessary for our main results.

Definition 1Functionxis called a solution of problem (1)-(2) if

(a)x ∈C(I0,R),t1−qx ∈C(I,R) and its fractional integralI1−qxis continuous onI0;

(b)xsatisfies problem (1)-(2).

2 Main Results

In order to get our main results concerning the existence and uniqueness of the solutions for fractional differential equations with nonlocal conditions,we have to introduce the following norm on the function spaceCq(I,R).

Forx∈Cq(I,R),we define a weighted norm

with a corresponding fixed positive constantλsuch thatλT <1.It is easy to showCq(I,R) is a Banach space with the norm.

The following theorem is our main results.Theorem 1Letq ∈(0,1),f ∈C(I ×R,R).Assume that:

(H1)gis a bounded linear functional onCq(I,R) with

(H2) there exists a nonnegative constantKsuch that

ProofIt is easy to show that problem (1)-(2) is equivalent to the following integral equation

We define an operatorTon the spaceCq(I,R) by

We shall show thatTis a contraction map onCq(I,R) for showing the operatorThas a fixed point.

For anyx,y ∈Cq(I,R),by using the assumptions (H1) and (H2) we get

Corollary 1Whenq=1,assume (H1),(H2)with,then problem (1)-(2) has a unique solution.

ProofIn this case,By Theorem 1,the operatorThas the form

By conditions (H1) and (H2),we can get

According to the Banach fixed point theorem,we show that problem (1)-(2) has a unique solution whenq=1.

The next main result is based on the wellknown Krasnoselkii fixed point Theorem (see[28]).

Theorem 2(Krasnoselkii) LetEbe a closed convex nonempty subset of a Banach space (B,||·||).Suppose that operatorsUandVmapEintoBsuch that

(i) ifx,y ∈E,thenUx+V y ∈E;

(ii)Uis compact and continuous;

(iii)Vis a contraction mapping.

Then there existsz∈Ewithz=Uz+V z.

By using Krasnoselkii fixed point Theorem,we can get the following main result.

Theorem 3Assume thatq∈(0,1),f ∈Cq(I ×R,R),and

(H1)gis a bounded linear functional onCq(I,R) withg(0)=0 and

(H3) |f(s,x(s))|≤µ(s),∀(s,x)∈I ×Cq(I,R),whereµ(s)∈Lα(I,R+),α>1,and

Then problem (1)-(2) has at least one solution onI.

ProofFirstly,it is easy to show that

Choose a real numberrsatisfying

and we consider the ball

Now,we define the operatorsUandVonBrby

By a simple calculations,we get

This shows thatUis uniformly bounded onBr.On the other hand,sincexis continuous,then (Ux)(t) is continuous.

By (H1),for anyy∈Br,we have

By (5),(6) and choice ofr,we obtain

Thus,we show that ifx,y ∈Br,then

By (H1),for anyx1,x2∈Br,we have

it is clear thatVis a contraction mapping forL<1.

Next,we will prove thatUis compact.Firstly,we prove that (Ux)(t) is equicontinuous.For anyt1,t2∈Iandt2

SoU(Br) is relatively compact.By the Arzela-Ascoli Theorem,Uis compact.We now complete the proof of Theorem 3 by using Krasnaselkii’s Theorem above.

3 Example

In this section,we will show an example to illustrate the main results.

Example 1Consider the nonlocal initial problem

By (4) and some simple calculations,we get thatr≥15.So the problem has at least one solution on

by Theorem 3.