Multiple Positive Solutions for Nonlinear

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Volume 2012, Article ID 850871,10pages doi:10.1155/2012/850871

Research Article

Multiple Positive Solutions for Nonlinear

Semipositone Fractional Differential Equations

Wen-Xue Zhou,

^{1, 2}

Ji-Gen Peng,

¹

and Yan-Dong Chu

²

1Department of Mathematics, Xi’an Jiaotong University, Shaanxi, Xi’an 710049, China

2Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, China

Correspondence should be addressed to Wen-Xue Zhou,[email protected] Received 22 September 2011; Accepted 13 December 2011

Academic Editor: Chuanxi Qian

Copyrightq2012 Wen-Xue Zhou et al. 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 present some new multiplicity of positive solutions results for nonlinear semipositone fractional boundary value problemD^α₀ut ptft, ut−qt,0< t <1, u0 u1 u1 0, where 2< α≤3 is a real number andD^α₀is the standard Riemann-Liouville diﬀerentiation. One example is also given to illustrate the main result.

1. Introduction

This paper is mainly concerned with the multiplicity of positive solutions of nonlinear fractional diﬀerential equation boundary value problemBVP for short

D₀^αut ptft, ut−qt, 0< t <1,

u0 u1 u1 0, 1.1

where 2< α≤3 is a real number andD^α₀ is the standard Riemann-Liouville diﬀerentiation, andf, p, qis a given function satisfying some assumptions that will be specified later.

In the last few years, fractional diﬀerential equations in short FDEs have been studied extensively the motivation for those works stems from both the development of the theory of fractional calculus itself and the applications of such constructions in various sciences such as physics, mechanics, chemistry, and engineering. For an extensive collection of such results, we refer the readers to the monographs by Kilbas et al.1, Miller and Ross 2, Oldham and Spanier3, Podlubny4, and Samko et al.5.

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Some basic theory for the initial value problems of FDE involving the Riemann- Liouville diﬀerential operator has been discussed by Lakshmikantham and Vatsala 6–8, Babakhani and Daftardar-Gejji9–11, and Bai12, and others. Also, there are some papers that deal with the existence and multiplicity of solutionsor positive solutionfor nonlinear FDE of BVPs by using techniques of nonlinear analysis fixed point theorems, Leray- Schauders theory, topological degree theory, etc., see13–22and the references therein.

Bai and L ¨u15studied the following two-point boundary value problem of FDEs D^q₀ut ft, ut 0, u0 u1 0, 0< t <1, 1< q≤2, 1.2

whereD^q₀is the standard Riemann-Liouville fractional derivative. They obtained the existence of positive solutions by means of the Guo-Krasnosel’skii fixed point theorem and Leggett- Williams fixed point theorem.

Zhang 22 considered the existence and multiplicity of positive solutions for the nonlinear fractional boundary value problem

cD^q₀ut ft, ut, 0< t <1, u0 u0 0, u1 u1 0, 1.3

where 1 < q ≤ 2 is a real number,f :0,1×0,∞ → 0,∞, and ^cD₀^q is the standard Caputo’s fractional derivative. The author obtained the existence and multiplicity results of positive solutions by means of the Guo-Krasnosel’skii fixed point theorem.

From the above works, we can see the fact that although the fractional boundary value problems have been investigated by some authors to the best of our knowledge, there have been few papers that deal with the boundary value problem1.1for nonlinear fractional diﬀerential equation. Motivated by all the works above, in this paper we discuss the boundary value problem1.1, using the Guo-Krasnosel’skii fixed point theorem, and we give some new existence of multiple positive solutions criteria for boundary value problem 1.1.

The paper is organized as follows. InSection 2, we give some preliminary results that will be used in the proof of the main results. InSection 3, we establish the existence of multiple positive solutions for boundary value problem1.1by the Guo-Krasnosel’skii fixed point theorem. In the end, we illustrate a simple use of the main result.

2. Preliminaries and Lemmas

For the convenience of the reader, we present here the necessary definitions from fractional calculus theory. These definitions can be found in the recent literature such as1,4,15.

Definition 2.1see1,4. The Riemann-Liouville fractional integral of orderαα > 0of a functionf:0,∞ → Ris given by

I₀^αft _t

0

t−s^α−1

Γα fsds, 2.1

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

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Definition 2.2see1,4. The Riemann-Liouville fractional derivative of orderαα >0of a continuous functionf:0,∞ → Ris given by

D^α₀f

t 1 Γn−α

d dt

n_t

0

t−s^n−α1fsds, 2.2

provided that the right side is pointwise defined on0,∞, wheren α1 andαdenotes the integer part ofα.

Lemma 2.3see15. Letα >0. If one assumesu ∈C0,1∩L0,1, then fractional diﬀerential equation

D^αut 0 2.3

has

ut C1t^α−1C2t^α−2· · ·CNt^α−N, Ci∈R, i1,2, . . . , N, 2.4

as unique solutions, whereNis the smallest integer greater than or equal toα.

Lemma 2.4see15. Assume thath∈C0,1∩L0,1with a fractional derivative of orderα >0 that belongs toC0,1∩L0,1. Then

I^αD^αht ht C₁t^α−1C₂t^α−2· · ·C_Nt^α−N, 2.5 for someC_i∈R,i1,2, . . . , N, whereNis the smallest integer greater than or equal toα.

In the following, we present Green’s function of the fractional diﬀerential equation boundary value problem.

Lemma 2.5. Leth∈C0,1and 2< α≤3, then the unique solution of D^αut ht 0, 0< t <1,

u0 u1 u1 0 2.6

is given by

ut ₁

0

Gt, shsds, 2.7

whereGt, sis Green’s function given by

Gt, s 1 Γα

⎧⎨

⎩

1−s^α−1t^α−1−t−s^α−1, if 0≤s≤t≤1,

1−s^α−1t^α−1, if 0≤t≤s≤1. 2.8

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The following properties of Green’s function form the basis of our main work in this paper.

Lemma 2.6. The functionGt, sdefined by2.8possesses the following properties:

iGt, s G1−s,1−tfort, s∈0,1;

iit^α−11−ts1−s^α−1≤Gt, sΓα≤α−1s1−s^α−1fort,s∈0,1;

iiit^α−11−ts1−s^α−1≤Gt, sΓα≤α−1t^α−11−tfort,s∈0,1;

ivGt, s>0 fort, s∈0,1.

The following Krasnosel’skii’s fixed point theorem will play a major role in our next analysis.

Lemma 2.7see23. LetXbe a Banach space, and letP ⊂Xbe a cone inX. AssumeΩ1,Ω2are open subsets ofXwith 0∈Ω1 ⊂Ω1⊂Ω2, and letA :P → P be a completely continuous operator such that either

i Au ≤ u , u∈P∩∂Ω1, Au ≥ u , u∈P∩∂Ω2, or ii Au ≥ u , u∈P∩∂Ω1, Au ≤ u , u∈P∩∂Ω2. ThenAhas a fixed point inP∩Ω2\Ω1.

3. Main Results

In this section, we establish some new existence results for the fractional diﬀerential equation 1.1. Givena∈L¹0,1, we write aa 0, ifa≥0 fort∈0,1, and it is positive in a set of positive measure.

Let us list the following assumptions:

H1f :0,1×0,∞ → 0,∞is continuous,p, q0;

H2there existsθ∈0,1/2, such that _1−θ

θ

pss1−s^α−1ds >0. 3.1

In view of Lemmas2.5and2.6, we obtain the following.

Lemma 3.1. Letq∈L¹0,1withq >0 on (0,1), andγtis the unique solution of D^α₀ut qt, 0< t <1,

u0 u1 u1 0, 3.2

Then

0≤γt≤ α−1

Γα q ₁t^α−11−t:Ct^α−11−t, for t∈0,1, 3.3

whereC α−1/Γα q ₁, q ₁₁

0|qt|dt.

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Next, we consider

D^α₀ut ptg

t, ut−γt

, 0< t <1,

u0 u1 u1 0, 3.4

where

gt, u

⎧⎨

⎩

ft, u, ifu≥0

ft,0, ifu <0, 3.5

Then3.4is equivalent to the following integral equation:

ut ₁

0

Gt, spsg

s, us−γs

ds. 3.6

Lemma 3.2. Letut ≥ γtfort ∈ 0,1, andutis positive solution of the problem3.4. Then ut−γtis positive solution of the problem1.1.

Proof. In fact, letxt ut−γt. Thenxt≥0 andut xt γt. Sinceutis positive solution of the problem3.4, we have

D^α₀

xt γt

ptgt, xt, 0< t <1, xγ

0 xγ

1 xγ

1 0. 3.7

So

D₀^αxt ptgt, xt−qt, 0< t <1,

x0 x1 x1 0. 3.8

For our constructions, we will consider the Banach spaceE C0,1equipped with standard norm u max_0≤t≤1|ut|, u∈E.

Define a coneKby

K

u∈E:ut≥ t^α−11−t

α−1 u , ∀t∈0,1, α∈2,3

. 3.9

Let the operatorA:K → Ebe defined by the formula

Axt: ₁

0

Gt, spsg

s, xs−γs

ds, 0≤t≤1, u∈K. 3.10 Lemma 3.3. Assume that (H1) holds. ThenAK⊂K.

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Proof. Notice from3.10andLemma 2.6that, forx∈K,Axt≥0 on0,1and

Ax ≤ 1 Γα

₁

0

α−1s1−s^α−1psg

s, xs−γs

ds. 3.11

On the other hand, we have

Axt≥ 1 Γα

₁

0

t^α−11−ts1−s^α−1psg

s, xs−γs ds

≥ t^α−11−t α−1Γα Ax .

3.12

Thus we haveAK⊂K. The proof is finished.

It is standard thatA:K → Kis continuous and completely continuous.

For convenience, we introduce the following notations: N α − 1 p ₁/Γα max_0≤s≤1s1−s^α−1,Nσ_1−θ

θ s1−s^α−1/Γαpsds,σmin_{θ≤t≤1−θ}1−tt^α−1.

Theorem 3.4. Assume that (H1) and (H2) are satisfied. Also suppose the following conditions are satisfied:

A1there exists a constantR₁>α−1Csuch thatNf t, u≤R₁for allt, u∈0,1×0, R1; A2there exists a constantR2>2R1such thatNft, u> R2for allt, u∈0,1×σR2, R2; A3lim_u_→_∞max_0≤t≤1ft, u/u 0.

Then the problem1.1has at least two positive solutions.

Proof. To show that1.1has at least two positive solutions, we will assume the problem3.4 has at least two positive solutionsx1andx2withR1≤ x1 < R2< x2 ≤R3.

We now show

Ax ≤ x , forx∈K∩∂Ω1, 3.13

To see this, letΩ1{x∈K| x < R₁}, then forx∈K∩∂Ω1,t∈0,1, byLemma 3.1 andA1, we have

xt−γt≤xt≤ x R1, xt−γt≥ t^α−11−t

α−1 R₁− Ct^α−11−t≥ R₁

α−1− C

t^α−11−t≥0.

3.14

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Thus, we see, fromLemma 2.6andA1, that

Ax max

0≤t≤1Aut max

0≤t≤1

₁

0

Gt, spsg

s, xs−γs ds

≤ α−1 Γα

₁

0

s1−s^α−1psf

s, xs−γs ds

≤ α−1 Γα

₁

0

R₁

Ns1−s^α−1psds

≤R₁,

3.15

from which we see that Ax ≤ x , forx∈K∩∂Ω1. Next we now show

Ax ≥ x , forx∈K∩∂Ω2. 3.16

To see this, letΩ2{x∈K| x < R₂}; then, forx∈K∩∂Ω2,t∈0,1, byR₂>2R₁, we have xt−γt≥ t^α−11−t

α−1 R₂− Ct^α−11−t≥ t^α−11−t

2α−1 R₂. 3.17

Forx∈∂Ω2;t∈θ,1−θσ, then, it follows from3.17that

R₂≤ t^α−11−t

2α−1 R₂≤xt−γt≤R₂. 3.18

In view ofA2,3.17andLemma 2.6, we have that for allx∈∂Ω2,t∈θ,1−θσ Au ≥

₁

0

Gt, spsg

s, xs−γs ds

≥t^α−11−t ₁

0

s1−s^α−1 Γα psf

s, xs−γs ds

> t^α−11−t _1−θ

θ

s1−s^α−1R₂ ΓαN psds

≥σ _1−θ

θ

s1−s^α−1R2

ΓαN psds R2,

3.19

from which we see that Ax > x , forx∈K∩∂Ω2. On the other hand, letε >0, where

εα−1 Γα max

0≤t≤1t1−t^α−1p

1≤1. 3.20

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Supposing thatA3holds, one can findN > R2>0, so that

ft, u≤εu, ∀t∈0,1, u≥N. 3.21

Setting

R₃ α−1max0≤t≤1t1−t^α−1 p 1maxt,u∈0,1×0,Nft, u Γα−εα−1max0≤t≤1t1−t^α−1 p 1

N, 3.22

thenR₃> N > R₂, and so

Au max

0≤t≤1

₁

0

Gt, spsg

s, xs−γs ds

≤ ₁

0

α−1s1−s^α−1

Γα ps max

s,u∈0,1×0,Nfs, uds

₁

0

α−1s1−s^α−1 Γα psε

xs−γs ds

≤R3,

3.23

from which we see that Ax ≤ x , forx∈K∩∂Ω3.

In view ofLemma 2.7, the problem3.4has at least two positive solutionsx1andx2

withR₁≤ x1 < R₂< x2 ≤R₃. SinceR₂ > R₁>α−1C, we have

x₁t−γt≥ t^α−11−t

α−1 R₁− Ct^α−11−t≥ R₁

α−1− C

t^α−11−t≥0,

x₂t−γt≥ t^α−11−t

α−1 R₂− Ct^α−11−t≥ R₂

α−1− C

t^α−11−t≥0.

3.24

Thereforex₁, x₂are solutions of the problem1.1. This completes the proof.

Theorem 3.5. Suppose that (H1), (H2) are satisfied. Furthermore assume that

A4there exists a constantR₁ > 2α−1Csuch thatNft, u ≥ R₁ for allt, u ∈0,1× σR1, R1;

A5there exists a constantR₂> max{R1,R₁/NN} such thatNf t, u< R₂for allt, u∈ 0,1×0, R2;

A6limu→∞min_{θ≤t≤1−θ}ft, u/u ∞.

Then the problem1.1has at least two positive solutions.

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4. An Example

As an application of the main results, we consider

D^5/2yt f y

− 1

√t, 0< t <1, y0 y0 y1 0,

4.1

Set

f y

⎧⎪

⎪⎨

⎪⎪

⎩

−2y−7²1100, if 0≤y≤7,

−2 y−7

1100, if 7≤y≤450, y−4502214, if y≥450,

4.2

Then we haveC α−1/Γα q ₁≈2.25676,N α−1 q 1/Γαmax0≤s≤1s1−s^α−1≈0.4, letting,θ 1/4, thenσ min_{θ≤t≤1−θ}1−tt^α−1 ≈ 0.09375, N σ_3/4

1/4s1−s^α−1/Γαds ≈ 0.008, choosing R1 7,R2 450, thenR1 > 2α−1C 6.77,R2 > max{R1,R₁/NN}

max{7,350} 350; therefore, we have Nfu 0.008−2y−7² 1100 ≥ 8.156 >

R₁, t, u → 1/4,3/4×0.65625,7,Nf u 0.4−2y−7²1100≤ 440 < R₂, t, u → 0,1×0,7, Nf u 0.4−2y −7 1100 ≤ 440 < R2, t, u → 0,1×7,450, and lim_y_→_∞fy/y lim_y_→∞y−450²214/y ∞.

It is clear thatf : 0,1×0,∞ → 0,∞is continuous. Since all the conditions of Theorem 3.5are satisfied, the problem4.1has at least two positive solutions.

Acknowledgments

This work supported by the Nature Science Foundation of China under the Contact no.

10901075 and the Key Project of Chinese Ministry of Education210226.

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