SOME RESULTS FOR TWO KINDS OF FRACTIONAL EQUATIONS WITH BOUNDARY VALUE PROBLEMS

WU Ya-yun，LI Xiao-yan，JIANG Wei

(School of Mathematical Science，Anhui University，Hefei 230601，China）

SOME RESULTS FOR TWO KINDS OF FRACTIONAL EQUATIONS WITH BOUNDARY VALUE PROBLEMS

WU Ya-yun，LI Xiao-yan，JIANG Wei

(School of Mathematical Science，Anhui University，Hefei 230601，China）

In this paper，we study the boundary value problems for two kinds of fractional differential equations，in which the nonlinear term including the derivative of the unknown function. Using the properties of the fractional calculus and the Banach contraction principle，we give the existence results of solutions for these fractional differential equations，which generalize the results of previous literatures.

fractional differential equations；Banach contraction principle；BVPs；monotone positive solution

2010 MR Subject Classification：34A08；34A12；34B15；34B18

Document code：AArticle ID：0255-7797（2016）05-0889-09

1 Introduction

During the last few years the fractional calculus was applied successfully to a variety of applied problems.It drew a great applications in nonlinear oscillations of earthquakes，many physical phenomena such as seepage flow in porous media and in fluid dynamic traffic model，see［1，2］.For more details on fractional calculus theory，one can see the monographs of Kai Diethelm［3］，Kilbas et al.［4］，Lakshmikantham et al.［1］，Podlubny［5］.Fractional differential equations involving the Riemann-Liouville fractional derivative or the Caputo fractional derivative were paid more and more attentions［6-9］.

Recently，the boundary value problems for fractional differential equations provoked a great deal attention and many results were obtained，for example［2，10-12，14，16，17］.

In［2］，Athinson investigated the following boundary value problem（BVP）of integral type

where a（t）：［t0，+∞）→（0，+∞），λ＞0.

As a fractional counterpart of（1.1），some scholars introduced another kind of two-point BVP（see［14］）

where0f（t）stands for the Riemann-Liouville derivative（see Section 2）of order α of some function f，here α∈（0，1），xλ=|x|λsignx and Γ（.）stands for Euler's function Gamma.

Inspired by the work of above papers，the aim of this paper is to solve the BVPs of the following equations

where0stands for the Riemann-Liouville derivative of order β，and M is a constant，β∈（1，2）.And

where M（constant）∈（0，+∞），α∈（0，1），f is continuous functionsis Caputo derivative of order α（see Section 2）andγ（t）=0，β（t）≤γ（t），t≥t0＞0.

2 Preliminaries

In this section，we introduce some definitions about fractional differential equation and theorems that are useful to the proof of our main results.For more details，one can see［3，5］.

Definition 2.1The fractional Riemann-Liouville integral of order α∈（0，1）of a function f：［0，+∞）→R given by

where Γ（.）denotes the Gamma function.

Definition 2.2The Riemann-Liouville and Caputo fractional derivatives are defined respectively as

and

where n is the first integer which is not less than p，D（.）andcD（.）are Riemann-Liouville and Caputo fractional derivatives，respectively.

Definition 2.3For measurable functions m：R→R，define the norm

We give some useful theorems to illustrate the relation between Riemann-Liouvill and Caputo fractional derivative and the operational formula about Riemann-Liouvill derivative.

Theorem 2.1（see［3，p.54］）Assume that η≥0，m=，and f∈Am［a，b］.Then

Theorem 2.2（see［5，p.74］）The composition of two fractional Riemann-Liouville derivative operators:（m-1≤p＜m），and（n-1≤q＜n），m，n are both positive integer，

Theorem 2.3（Banach's fixed point theorem）Assume（U，d）to be a nonempty complete metric space，let 0≤α＜1 and let the mapping A：U→U satisfy the inequality

for every u，v∈U.Then A has a uniquely determined fixed point u?.Furthermore，for any u0∈U，the sequenceconverges to this fixed point u?.

3 Main Results

The following lemma 3.1 we proved will be used to solve（1.3）.

Lemma 3.1In eq.（1.3），let β=1+α，α∈（0，1），and we can have

where v（t）=u＇（t）.

ProofUsing formula（2.5）and the equation+f（t，u（t），u＇（t））=0，we can get

that is

We now prove the following result can hold

Now，we introduce the set X and the metric d（v1，v2）of X.We define

and

for any v1，v2∈X.One can easily prove that（X，d）is a complete metric space by the Lebesgue dominated convergence theorem［13］.

Some hypothesis will be introduced here.

［H1］：f meets weak Lipschitz condition with the second and the third variables on X：

especially，|f|≤a（t）d（v，0）=a（t）（‖v‖L1+supt≥0|v|），v∈X.And a（t）can be some nonnegative continuous functions that can make［H2］and［H3］hold.

［H2］：Let ω0（t）=（t-τ）α-1B（τ）dτ+（s-τ）α-1B（τ）dτds，we assume 0＜ω0（t）＜1，where B（t）=a（t）［Γ（α）］-1.

Theorem 3.1Assume that［H1］，［H2］are satisfied.Then the problem（1.3）has a solution u（t）=M-v（s）ds，t＞0，where v∈（C∩L∞∩L1）（［0，+∞），R）.

ProofFrom Lemma 3.1，the operator T can be defined by T：X→（C∩L∞∩L1）（［0，+∞），R）with the formula

From［H1］and（3.5），（Tv）（t）satisfies

so TX?X.

According to（3.4），we get

for any v1，v2∈X.

First，from（3.5）and（3.6），we have

Second，

Then（3.7）and（3.8）yield

By using［H2］and Theorem 2.3，Tv has a fixed point v0.This v0function is the solution of problem（1.3）.The proof is completed.

Remark 3.1In fact，problem（1.2）is the special case of Theorem 3.1.One can easily get the existence result from the procedure of proving Theorem 3.1.

Remark 3.2In the following part，we will prove that some conditions supplied can make 0＜ω0（t）＜1 true.We can choose a simple candidate for a（t）which is provided by the restriction a（t）≤c·t-2，t≥t0＞0，c is undetermined coefficient.Using the restriction，we acquire that

It is easily found that the integral mean value theorem can be used in here to obtain that

In order to obtain existence theorems of equation（1.4），we introduce the following definitions and assumption.

Let

where v（t）=u＇（t）.

We still need to give［H3］to solve（1.4）.

［H3］：a（t）satisfies the following inequalities

and

a（t）is nonnegative continuous function.

Theorem 3.2Assume that［H1］，［H3］hold.Problem（1.4）has a solution on X1.

Similar to the definition about d in Theorem 3.1，we have

One can easily prove the matric space E=（X1，d）is complete by using Lebesgue's dominated convergence theorem.

In fact，we can define T：X1→C（［t0，+∞）；R）given by the formula

where v（t）∈X1.It is easy to see that TX1?X1from the definition about X1.The operator T is contraction in X1，so we have

Considering［H1］，

and

and［H3］，we yield

for all v1，v2∈X1.

Remark 3.3If α=1，we will obtain the equation u＇＇（t）+f（t，u（t），u＇（t））=0，t≥t0＞0.This kind of equation was studied by Octavian and Mustafa（see［13］）with some supplied conditions of u（t）and u＇（t）

where β（t），γ（t）are continuous nonnegative functions satisfying

Remark 3.4From Theorem 2.1，if we have some initial value about u（t）and useinstead ofin problem（1.4），we still can solve that problem by using the similar way in Theorem 3.2.

4 Conclusion

In this paper，we solve two kinds of boundary value problems which include results in ［13，14］.In fact，some equations with some boundary value conditions can also include results in this paper.Our future work is just to solve equation as the following one:f（t，u（t），uα（t））=0，α，β are some fractional numbers.To solve this kind of equation also need some boundary value conditions.

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兩類分數階微分方程的邊值問題

吳亞運，李曉艷，蔣威
（安徽大學數學科學學院，安徽合肥230601）

本文研究了兩類非線性項含有未知函數導數的分數階微分方程的邊值問題.利用分數階微積分的性質及Banach不動點定理，獲得了解的存在唯一性等有關結果，推廣了已有文獻的結論.

分數階微分方程；巴拿赫壓縮定理；邊值問題；單調正解

MR（2010）主題分類號：34A08；34A12；34B15；34B18O175.1

date：2014-06-24Accepted date：2015-01-04

Supported by the National Nature Science Foundation of China（11371027）；Starting Research Fund for Doctors of Anhui University（023033190249）；National Natural Science Foundation of China，Tian Yuan Special Foundation（11326115）；the Special Research Fund for the Doctoral Program of the Ministry of Education of China（20123401120001）.

Biography：Wu Yayun（1990-），male，born at Guzhen，Anhui，postgraduate，major in differential equations.