Langevin Equation Involving Two Fractional Orders with Boundary Value Conditions

(1)

Boundary Value Problems

Volume 2011, Article ID 516481,17pages doi:10.1155/2011/516481

Research Article

Existence of Solutions to Nonlinear

Langevin Equation Involving Two Fractional Orders with Boundary Value Conditions

Anping Chen

^{1, 2}

and Yi Chen

²

1Department of Mathematics, Xiangnan University, Chenzhou, Hunan 423000, China

2School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411005, China

Correspondence should be addressed to Anping Chen,[email protected]

Received 30 September 2010; Revised 21 January 2011; Accepted 26 February 2011 Academic Editor: Kanishka Perera

Copyrightq2011 A. Chen and Y. Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study a boundary value problem to Langevin equation involving two fractional orders. The Banach fixed point theorem and Krasnoselskii’s fixed point theorem are applied to establish the existence results.

1. Introduction

Recently, the subject of fractional diﬀerential equations has emerged as an important area of investigation. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, electromagnetic, porous media, and so forth. In consequence, the subject of fractional diﬀerential equations is gaining much importance and attention. For some recent developments on the subject, see1–8and the references therein.

Langevin equation is widely used to describe the evolution of physical phenomena in fluctuating environments. However, for systems in complex media, ordinary Langevin equation does not provide the correct description of the dynamics. One of the possible gener- alizations of Langevin equation is to replace the ordinary derivative by a fractional derivative in it. This gives rise to fractional Langevin equation, see for instance9–12and the references therein.

In this paper, we consider the following boundary value problem of Langevin equation with two diﬀerent fractional orders:

CD^β

CD^αλ

ut ft, ut t∈ 0, T, u0 −uT, u0 uT 0,

1.1

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whereT is a positive constant, 1< α≤2, 0 < β≤1,^CD^α, and^CD^βare the Caputo fractional derivatives,f:0, T×R → Ris continuous, andλis a real number.

The organization of this paper is as follows. In Section2, we recall some definitions of fractional integral and derivative and preliminary results which will be used in this paper. In Section3, we will consider the existence results for problem1.1. In Section4, we will give an example to ensure our main results.

2. Preliminaries

In this section, we present some basic notations, definitions, and preliminary results which will be used throughout this paper.

Definition 2.1. The Caputo fractional derivative of orderαof a functionf : 0,∞ → R, is defined as

CD^αft 1

Γn−α _t

0

t−s^n−α−1fⁿsds, n−1< α < n, n α 1, 2.1

whereαdenotes the integer part of the real numberα.

Definition 2.2. The Riemann-Liouville fractional integral of order α > 0 of a function ft, t >0, is defined as

I^αft 1 Γα

_t

0

t−s^α−1fsds, 2.2

provided that the right side is pointwise defined on0,∞.

Definition 2.3. The Riemann-Liouville fractional derivative of orderα > 0 of a continuous functionf:0,∞ → Ris given by

D^αft 1 Γn−α

d dt

_n_t

0

t−s^n−α−1fsds, 2.3

wheren α 1 andαdenotes the integer part of real numberα, provided that the right side is pointwise defined on0,∞.

Lemma 2.4see8. Letα >0, then the fractional diﬀerential equation ^CD^αut 0 has solution ut c₀c₁tc₂t²· · ·c_n−1tⁿ⁻¹, 2.4

wherec_i∈R,i0,1,2, . . . , n−1,n α 1.

Lemma 2.5see8. Letα >0, then

I^αCD^αut ut c0c1tc2t²· · ·c_n−1tⁿ⁻¹, 2.5

for someci∈R,i0,1,2, . . . , n−1,n α 1.

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Lemma 2.6. The unique solution of the following boundary value problem

CD^β

CD^αλ

ut yt, t∈0, T,1< α≤2,0< β≤1,

u0 −uT, u0 uT 0,

2.6

is given by

ut _t

0

t−s^α−1 Γα

_s

0

s−τ^β−1 Γ

β yτdτ−λus

ds

−1 2

_T

0

T−s^α−1 Γα

_s

0

s−τ^β−1 Γ

β yτdτ−λus

ds

T^α−2t^α 2αT^α−1

_T

0

T−s^α−2 Γα−1

_s

0

s−τ^β−1 Γ

ds.

2.7

Proof. Similar to the discussion of9, equation1.5, the general solution of

CD^β

CD^αλ

ut yt 2.8

can be written as

ut _t

0

t−s^α−1 Γα

_s

0

s−τ^β−1 Γ

β yτdτ−λus

ds− c0

Γα1t^α−c1t−c2. 2.9

By the boundary conditionsu0 uT 0 andu0 uT 0, we obtain

c₀

Γα1 1 αT^α−1

_T

0

_s

0

s−τ^β−1 Γ

ds,

c10,

c2 1 2

_T

0

T −s^α−1 Γα

_s

0

s−τ^β−1 Γ

β yτdτ−λus

ds

− T 2α

_T

0

T −s^α−2 Γα−1

_s

0

s−τ^β−1 Γ

β yτdτ−λus

ds.

2.10

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Hence,

ut _t

0

t−s^α−1 Γα

_s

0

s−τ^β−1 Γ

β yτdτ−λus

ds

−1 2

_T

0

T−s^α−1 Γα

_s

0

s−τ^β−1 Γ

ds

_T

0

_s

0

s−τ^β−1 Γ

ds.

2.11

Lemma 2.7Krasnoselskii’s fixed point theorem. LetEbe a bounded closed convex subset of a Banach spaceX, and letS,Tbe the operators such that

iSuTv∈Ewheneveru, v∈E, iiSis completely continuous, iiiT is a contraction mapping.

Then there existsz∈Esuch thatzSzTz.

Lemma 2.8H ¨older inequality. Letp >1,1/p 1/q 1,f ∈L^pa, b,g ∈L^qa, b, then the following inequality holds:

_b

a

fxgxdx≤ _b

a

fx^pdx

_1/p_b

a

|gx|^qdx _1/q

. 2.12

3. Main Result

In this section, our aim is to discuss the existence and uniqueness of solutions to the problem 1.1.

LetΩbe a Banach space of all continuous functions from0, T → Rwith the norm usup_t∈0,T{|ut|}.

Theorem 3.1. Assume that

(H1) there exists a real-valued functionμt∈L^1/γ0, T, Rfor someγ∈0,1such that ft, u−ft, v≤μt|u−v|, for almost all t∈0, T, u, v∈R. 3.1 If

Λ

4αβ−γ Γ

β−γ1 μ^∗T^αβ−γ 2αΓ

β Γ

αβ−γ1

1−γ β−γ

1−γ

2|λ|T^α

Γα1<1, 3.2

whereγ ∈ 0,1,β /γ,1 < α ≤ 2, 0 < β ≤ 1, μ^∗ _T

0 μτ^1/γdτ^γ, then problem1.1has a unique solution.

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Proof. Define an operatorF :Ω → Ωby

Fut _t

0

t−s^α−1 Γα

_s

0

s−τ^β−1 Γ

β fτ, uτdτ−λus

ds

−1 2

_T

0

T−s^α−1 Γα

_s

0

s−τ^β−1 Γ

β fτ, uτdτ−λus

ds

_T

0

_s

0

s−τ^β−1 Γ

ds.

3.3

LetMsup_t∈0,T|ft,0|and choose

1 1−δ

4αβ MT^αβ 2αΓ

αβ1

≤r, 3.4

whereδis such thatΛ≤δ <1.

Now we show thatFBr ⊂ Br, whereBr {u ∈ Ω :u ≤ r}. Foru ∈Br, by H ¨older inequality, we have

|Fut|

_t

0

t−s^α−1 Γα

_s

0

s−τ^β−1 Γ

ds

−1 2

_T

0

T−s^α−1 Γα

_s

0

s−τ^β−1 Γ

ds

_T

0

_s

0

s−τ^β−1 Γ

ds

≤ _t

0

t−s^α−1 Γα

_s

0

s−τ^β−1 Γ

β fτ, uτ−fτ,0 fτ,0 dτ|λus|

ds

1 2

_T

0

T−s^α−1 Γα

_s

0

s−τ^β−1 Γ

ds

T 2α

_T

0

s 0

s−τ^β−1 Γ

ds

≤ _t

0

t−s^α−1 Γα

_s

0

s−τ^β−1 Γ

β

μτ|uτ|fτ,0 dτ|λus|

ds

1 2

_T

0

T−s^α−1 Γα

_s

0

s−τ^β−1 Γ

β

μτ|uτ|fτ,0 dτ|λus|

ds

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T 2α

_T

0

_s

0

s−τ^β−1 Γ

β

μτ|uτ|fτ,0 dτ|λus|

ds

≤ u _t

0

t−s^α−1 Γα

_s

0

s−τ^β−1 Γ

β μτdτ

dsM

_t

0

t−s^α−1 Γα

_s

0

s−τ^β−1 Γ

β dτ ds

|λ|u _t

0

t−s^α−1

Γα ds u 2

_T

0

T−s^α−1 Γα

_s

0

s−τ^β−1 Γ

β μτdτ

ds

M 2

_T

0

T−s^α−1 Γα

_s

0

s−τ^β−1 Γ

β dτ ds |λ|u 2

_T

0

T−s^α−1 Γα ds

Tu 2α

_T

0

_s

0

s−τ^β−1 Γ

β μτdτ

ds

TM 2α

_T

0

_s

0

s−τ^β−1 Γ

β dτ dsT|λ|u 2α

_T

0

T−s^α−2 Γα−1 ds

≤ u

ΓαΓ β

_t

0

t−s^α−1s 0

s−τ^β−1_1/1−γ dτ

_1−γ_s

0

μτ ^1/γdτ γ

ds

M ΓαΓ

β1 _t

0

t−s^α−1s^βds|λ|T^αu Γα1

u 2ΓαΓ

β _T

0

T−s^α−1 s

0

s−τ^β−1_1/1−γ dτ

1−γs 0

μτ ^1/γdτ γ

ds

M 2ΓαΓ

β1 _T

0

T−s^α−1s^βds |λ|T^αu 2Γα1

Tu 2αΓα−1Γ

β _T

0

T−s^α−2 s

0

s−τ^β−1_1/1−γ dτ

1−γs 0

μτ^1/γdτ γ

ds

TM 2αΓα−1Γ

β1 _T

0

T−s^α−2s^βds |λ|T^αu 2Γα1

≤ μ^∗u ΓαΓ

β

1−γ β−γ

_1−γ_t

0

t−s^α−1s^β−γds M ΓαΓ

β1 _t

0

t−s^α−1s^βds

μ^∗u 2ΓαΓ β

1−γ β−γ

_1−γ_T

0

T−s^α−1s^β−γds M 2ΓαΓ

β1 _T

0

T −s^α−1s^βds

Tμ^∗u 2αΓα−1Γ

β

1−γ β−γ

_1−γ_T

0

T−s^α−2s^β−γds

TM 2αΓα−1Γ

β1 _T

0

T−s^α−2s^βds2|λ|T^αu Γα1

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μ^∗ut^αβ−γ ΓαΓ

β

1−γ β−γ

_1−γ₁

0

1−ξ^α−1ξ^β−γdξ Mt^αβ ΓαΓ

β1 ₁

0

1−ξ^α−1ξ^βdξ

μ^∗uT^αβ−γ 2ΓαΓ

β

1−γ β−γ

_1−γ₁

0

1−η ^α−1η^β−γdη MT^αβ 2ΓαΓ

β1 ₁

0

1−η ^α−1η^βdη

μ^∗uT^αβ−γ 2αΓα−1Γ

β

1−γ β−γ

_1−γ₁

0

1−η ^α−2η^β−γdη

MT^αβ 2αΓα−1Γ

β1 ₁

0

1−η ^α−2η^βdη2|λ|T^αu Γα1

≤ rμ^∗T^αβ−γ ΓαΓ

β

1−γ β−γ

_1−γ₁

0

1−ξ^α−1ξ^β−γdξ MT^αβ ΓαΓ

β1 ₁

0

1−ξ^α−1ξ^βdξ

rμ^∗T^αβ−γ 2ΓαΓ

β

1−γ β−γ

_1−γ₁

0

1−η ^α−1η^β−γdη MT^αβ 2ΓαΓ

β1 ₁

0

1−η ^α−1η^βdη

rμ^∗T^αβ−γ 2αΓα−1Γ

β

1−γ β−γ

_1−γ₁

0

1−η ^α−2η^β−γdη

MT^αβ 2αΓα−1Γ

β1 ₁

0

1−η ^α−2η^βdη 2|λ|T^αr Γα1.

3.5

Take notice of Beta functions:

B

β−γ1, α ₁

0

1−ξ^α−1ξ^β−γdξ ₁

0

1−η ^α−1η^β−γdη ΓαΓ

β−γ1 Γ

αβ−γ1 ,

B

β1, α ₁

0

1−ξ^α−1ξ^βdξ ₁

0

1−η ^α−1η^βdη ΓαΓ β1 Γ

αβ1 ,

B

β−γ1, α−1 ₁

0

1−η ^α−2η^β−γdη Γα−1Γ

β−γ1 Γ

αβ−γ ,

B

β1, α−1 ₁

0

1−η ^α−2η^βdη Γα−1Γ β1 Γ

αβ .

3.6

We can get

|Fut| ≤ rμ^∗Γ

β−γ1 T^αβ−γ Γ

β Γ

αβ−γ1

1−γ β−γ

_1−γ

MT^αβ Γ

αβ1 rμ^∗Γ

β−γ1 T^αβ−γ 2Γ

β Γ

αβ−γ1

1−γ β−γ

_1−γ

MT^αβ 2Γ

αβ1

(8)

rμ^∗Γ

β−γ1 T^αβ−γ 2αΓ

β Γ

αβ−γ

1−γ β−γ

_1−γ

MT^αβ 2αΓ

αβ 2|λ|T^αr Γα1

4αβ−γ Γ

β Γ

αβ−γ1

1−γ β−γ

_1−γ

2|λ|T^α Γα1

r

4αβ MT^αβ 2αΓ

αβ1

≤Λ 1−δr

≤r.

3.7 Therefore,Fut ≤r.

Foru, v∈Ωand for eacht∈0, T, based on H ¨older inequality, we obtain

|Fut−Fvt|

≤ _t

0

t−s^α−1 Γα

_s

0

s−τ^β−1 Γ

β fτ, uτ−fτ, vτdτ

ds

|λ|

_t

0

t−s^α−1

Γα |us−vs|ds

1 2

_T

0

T−s^α−1 Γα

_s

0

s−τ^β−1 Γ

β fτ, uτ−fτ, vτdτ

ds

|λ|

2 _T

0

T−s^α−1

Γα |us−vs|ds

T 2α

_T

0

_s

0

s−τ^β−1 Γ

ds

|λ|T 2α

_T

0

T−s^α−2

Γα−1 |us−vs|ds

≤ u−v ΓαΓ

β _t

0

t−s^α−1 _s

0

s−τ^β−1μτdτ

ds |λ|T^α

Γα1u−v

u−v 2ΓαΓ

β _T

0

T−s^α−1 _s

0

s−τ^β−1μτdτ

ds |λ|T^α

2Γα1u−v Tu−v

2αΓα−1Γ β

_T

0

T−s^α−2 _s

0

ds |λ|T^α

2Γα1u−v

(9)

≤ u−v ΓαΓ

β _t

0

t−s^α−1s 0

s−τ^β−1_1/1−γ dτ

1−γ_s

0

μτ ^1/γdτ γ

ds

u−v 2ΓαΓ

β _T

0

T−s^α−1s 0

s−τ^β−1_1/1−γ dτ

_1−γ_s

0

μτ ^1/γdτ γ

ds

Tu−v 2αΓα−1Γ

β _T

0

T−s^α−2s 0

s−τ^β−1_1/1−γ dτ

_1−γ_s

0

μτ ^1/γdτ γ

ds

2|λ|T^α

Γα1u−v

≤ μ^∗u−v ΓαΓ

β

1−γ β−γ

_1−γ_t

0

t−s^α−1s^β−γds

μ^∗u−v 2ΓαΓ

β

1−γ β−γ

_1−γ_T

0

μ^∗Tu−v 2αΓα−1Γ β

1−γ β−γ

_1−γ_T

0

T−s^α−2s^β−γds 2|λ|T^α

Γα1u−v

μ^∗u−vt^αβ−γ ΓαΓ

β

1−γ β−γ

_1−γ₁

0

1−ξ^α−1ξ^β−γdξ

μ^∗u−vT^αβ−γ 2ΓαΓ

β

1−γ β−γ

1−γ₁

0

1−η ^α−1η^β−γdη

μ^∗u−vT^αβ−γ 2αΓα−1Γ

β

1−γ β−γ

_1−γ₁

0

1−η ^α−2η^β−γdη 2|λ|T^α

Γα1u−v

≤

4αβ−γ Γ

β Γ

αβ−γ1

1−γ β−γ

1−γ

2|λ|T^α Γα1

u−v

Λu−v.

3.8

SinceΛ<1, consequentlyFis a contraction. As a consequence of Banach fixed point theorem, we deduce thatFhas a fixed point which is a solution of problem1.1.

Corollary 3.2. Assume that

(H1)There exists a constantL >0 such that

ft, u−ft, v≤L|u−v|, ∀t∈0, T, u, v∈R. 3.9

(10)

If

4αβ LT^αβ 2αΓ

αβ1 2|λ|T^α

Γα1 <1, 3.10

then problem1.1has a unique solution.

Theorem 3.3. Suppose that (H1) and the following condition hold:

(H2) There exists a constantl∈0,1and a real-valued functionmt∈L^1/l0, T, Rsuch that

ft, u≤mt, for almost everyt∈0, T, u∈R. 3.11

Then the problem1.1has at least one solution on0, Tif 2αβ−γ Γ

β Γ

αβ−γ1

1−γ β−γ

_1−γ

|λ|T^α

Γα1 <1. 3.12

Proof. Let us fix

4αβ−l Γ

β−l1 m^∗T^αβ−l 2αΓ

β Γ

αβ−l1 1−2|λ|T^α/Γα1 1−l

β−l _1−l

≤r; 3.13

here,m^∗ _T

0 mτ^1/ldτ^l; considerB_r {u∈Ω :^u ≤r}, thenB_r is a closed, bounded, and convex subset of Banach spaceΩ. We define the operatorsSandTonB_r as

Sut _t

0

t−s^α−1 Γα

_s

0

s−τ^β−1 Γ

β fτ, uτdτ−λus

ds,

Tut −1 2

_T

0

T−s^α−1 Γα

_s

0

s−τ^β−1 Γ

ds

_T

0

_s

0

s−τ^β−1 Γ

ds.

3.14

Foru, v∈Br, based on H ¨older inequality, we find that

|SuTv|

≤ _t

0

t−s^α−1 Γα

_s

0

s−τ^β−1 Γ

β fτ, uτdτ|λus|

ds

1 2

_T

0

T−s^α−1 Γα

_s

0

s−τ^β−1 Γ

β fτ, vτdτ|λvs|

ds

(11)

T 2α

_T

0

_s

0

s−τ^β−1 Γ

β fτ, vτdτ|λvs|

ds

≤ 1

ΓαΓ β

_t

0

t−s^α−1 _s

0

s−τ^β−1mτdτ

ds|λ|u _t

0

t−s^α−1 Γα ds

1 2ΓαΓ

β _T

0

T−s^α−1 _s

0

s−τ^β−1mτdτ

ds|λ|u 2

_T

0

T−s^α−1 Γα ds

T 2αΓα−1Γ

β _T

0

T−s^α−2 s

0

s−τ^β−1mτdτ

ds |λ|Tu 2α

_T

0

≤ 1

ΓαΓ β

_t

0

t−s^α−1 s

0

s−τ^β−1_1/1−l dτ

1−ls 0

mτ^1/ldτ l

ds

1 2ΓαΓ

β _T

0

T−s^α−1s 0

s−τ^β−1_1/1−l dτ

1−l_s

0

mτ^1/ldτ l

ds

T 2αΓα−1Γ

β _T

0

T−s^α−2s 0

s−τ^β−1_1/1−l dτ

_1−l_s

0

mτ^1/ldτ l

ds

|λ|u _t

0

t−s^α−1

Γα ds|λ|u 2

_T

0

T−s^α−1

Γα ds|λ|Tu 2α

_T

0

≤ m^∗ ΓαΓ

β 1−l

β−l _1−l_t

0

t−s^α−1s^β−lds

m^∗ 2ΓαΓ

β 1−l

β−l _1−l_T

0

T−s^α−1s^β−lds

m^∗T 2αΓα−1Γ

β 1−l

β−l _1−l_T

0

T−s^α−2s^β−lds 2|λ|T^αr Γα1

≤ m^∗T^αβ−l ΓαΓ

β 1−l

β−l _1−l₁

0

1−ξ^α−1ξ^β−ldξ

m^∗T^αβ−l 2ΓαΓ

β 1−l

β−l _1−l₁

0

1−η ^α−1η^β−ldη

m^∗T^αβ−l 2αΓα−1Γ

β 1−l

β−l _1−l₁

0

1−η ^α−2η^β−ldη 2|λ|T^αr Γα1

4αβ−l Γ

β−l1 m^∗T^αβ−l 2αΓ

β Γ

αβ−l1

1−l β−l

_1−l

2|λ|T^αr Γα1

≤r.

3.15 Thus,SuTv ≤r, soSuTv∈Br.

(12)

For u, v ∈ Ω and for each t ∈ 0, T, by the analogous argument to the proof of Theorem3.1, we obtain

|Tut−Tvt|

≤ 1 2

_T

0

T−s^α−1 Γα

_s

0

s−τ^β−1 Γ

ds

|λ|

2 _T

0

T−s^α−1

Γα |us−vs|ds

T 2α

_T

0

_s

0

s−τ^β−1 Γ

ds

|λ|T 2α

_T

0

T−s^α−2

Γα−1 |us−vs|ds

≤ u−v 2ΓαΓ

β _T

0

T−s^α−1 _s

0

ds |λ|T^α

2Γα1u−v

β _T

0

T−s^α−2 _s

0

ds |λ|T^α

2Γα1u−v

≤ u−v 2ΓαΓ

β _T

0

T−s^α−1 s

0

s−τ^β−1^1/1−γdτ

_1−γ_s

0

μτ^1/γdτ γ

ds

β _T

0

T−s^α−2s 0

s−τ^β−1_1/1−γ dτ

_1−γ_s

0

μτ^1/γdτ γ

ds

|λ|T^α

Γα1u−v

≤ μ^∗u−v 2ΓαΓ

β

1−γ β−γ

_1−γ_T

0

μ^∗Tu−v 2αΓα−1Γ β

1−γ β−γ

_1−γ_T

0

T−s^α−2s^β−γds |λ|T^α

Γα1u−v

μ^∗u−vT^αβ−γ 2ΓαΓ

β

1−γ β−γ

_1−γ₁

0

1−η ^α−1η^β−γdη

μ^∗u−vT^αβ−γ 2αΓα−1Γ

β

1−γ β−γ

_1−γ₁

0

1−η ^α−2η^β−γdη |λ|T^α

Γα1u−v

≤

2αβ−γ Γ

β Γ

αβ−γ1

1−γ β−γ

_1−γ

|λ|T^α Γα1

u−v.

3.16

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From the assumption

2αβ−γ Γ

β Γ

αβ−γ1

1−γ β−γ

_1−γ

|λ|T^α

Γα1 <1, 3.17

it follows thatT is a contraction mapping.

The continuity of f implies that the operatorS is continuous. Also,S is uniformly bounded onBr as

Su ≤ Γ

β−l1 m^∗T^αβ−l Γ

β Γ

αβ−l1

1−l β−l

_1−l

|λ|T^αr

Γα1. 3.18

On the other hand, letNmax_t,u∈0,T_×B_r|ft, ut|1, for allε >0, setting

σmin

⎧⎨

⎩ 1 2

εΓ αβ 2N

_1/αβ , 1

2

εΓα 2|λ|r

1/α⎫

⎬

⎭. 3.19

For eachu∈B_r, we will prove that ift₁, t₂∈0, Tand 0< t₂−t₁< σ, then

|Sut2−Sut1|< ε. 3.20

In fact, we have

|Sut2−Sut1|

_t₂

0

t2−s^α−1 Γα

_s

0

s−τ^β−1 Γ

ds

− _t₁

0

t1−s^α−1 Γα

_s

0

s−τ^β−1 Γ

β fτ, uτdτ−λus

ds

_t₁

0

t2−s^α−1 Γα

_s

0

s−τ^β−1 Γ

ds

_t₂

t1

t2−s^α−1 Γα

_s

0

s−τ^β−1 Γ

ds

− _t₁

0

t1−s^α−1 Γα

_s

0

s−τ^β−1 Γ

ds

_t₁

0

t2−s^α−1−t1−s^α−1 Γα

_s

0

s−τ^β−1 Γ

ds

_t₂

t1

t2−s^α−1 Γα

_s

0

s−τ^β−1 Γ

ds

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≤ _t₁

0

t2−s^α−1−t1−s^α−1 Γα

_s

0

s−τ^β−1 Γ

β fτ, uτdτ

ds

|λ|u _t₁

0

t2−s^α−1−t1−s^α−1 Γα ds

_t₂

t1

t2−s^α−1 Γα

_s

0

s−τ^β−1 Γ

β fτ, uτdτ

ds|λ|u _t₂

t1

t2−s^α−1 Γα ds

≤ N

Γ

αβ1

t^αβ₂ −t^αβ₁

|λ|r Γα1

t^α₂−t^α₁ .

3.21

In the following, the proof is divided into two cases.

Case 1. Forσ≤t₁< t₂< T, we have

|Sut2−Sut1| ≤ N Γ

αβ1

|λ|r Γα1

t^α₂−t^α₁

≤ N

Γ

αβ1

αβ σ^αβ−1t2−t₁ |λ|r

Γα1ασ^α−1t2−t₁

< N Γ

αβ σ^αβ |λ|r Γασ^α

<

1 2

_αβ ε 2

1 2

_α ε 2

< ε.

3.22

Case 2. for 0≤t₁< σ,t₂<2σ, we have.

|Sut2−Sut1| ≤ N Γ

αβ1

|λ|r Γα1

t^α₂−t^α₁

≤ N

Γ

αβ1 t^αβ₂ |λ|r Γα1t^α₂

< N Γ

αβ1 2σ^αβ |λ|r

Γα12σ^α

< ε 2 ε

2 ε.

3.23

Therefore,Sis equicontinuous and the Arzela-Ascoli theorem implies thatSis compact on Br, so the operatorSis completely continuous.

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Thus, all the assumptions of Lemma2.7are satisfied and the conclusion of Lemma2.7 implies that the boundary value problem1.1has at least one solution on0, T.

Corollary 3.4. Suppose that the condition (H1)hold and, assume that 2αβ LT^αβ

2αΓ

αβ1 |λ|T^α

Γα1 <1. 3.24

Further assume that

(H2)there exists a constantK >0 such that

ft, u≤K, ∀t∈0, T, u∈R, 3.25

then problem1.1has at least one solution on0, T.

4. Example

Letα2,β1,λ1/8,Tπ/2. We consider the following boundary value problem

CD¹

CD²1 8

ut ft, ut, 0≤t≤ π 2, u0 uπ

2

0, u0 uπ 2

0,

4.1

where

ft, u 1 t2²

u

1u, t, u∈0, T×0,∞. 4.2

Because of|ft, u−ft, v| ≤1/4|u−v|, letμt≡ 1/4, thenμt ∈L²0, π/2, we have γ1/2 andμ^∗ _T

0 μτ^1/γdτ^γ _π/2

0 1/4²dτ^1/2 √ π/4√

2. Further, 4αβ−γ Γ

β Γ

αβ−γ1

1−γ β−γ

_1−γ

2|λ|T^α Γα1

17/2Γ3/2μ^∗T^5/2

4Γ1Γ7/2 2|λ|T² Γ3 17π³

15×64 π² 32

≈0.86<1.

4.3

Then BVP4.1has a unique solution on0, π/2according to Theorem3.1.

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On the other hand, we find that 2αβ−γ Γ

β Γ

αβ−γ1

1−γ β−γ

_1−γ

|λ|T^α Γα1

9/2Γ3/2μ^∗T^5/2

4Γ7/2 |λ|T² Γ3 9π³

64×15 π² 64

≈0.44<1.

4.4

Then BVP4.1has at least one solution on0, π/2according to Theorem3.3.

Acknowledgments

This work was supported by the Natural Science Foundation of China 10971173, the Natural Science Foundation of Hunan Province10JJ3096, the Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province, and the Construct Program of the Key Discipline in Hunan Province.

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