Positive Solutions for Boundary Value

Volume 2008, Article ID 437453,15pages doi:10.1155/2008/437453

Research Article

Positive Solutions for Boundary Value

Problems of N-Dimension Nonlinear Fractional Differential System

Aijun Yang and Weigao Ge

Department of Applied Mathematics, Beijing Institute of Technology, Beijing 100081, China

Correspondence should be addressed to Aijun Yang,[email protected] Received 27 October 2008; Accepted 18 December 2008

Recommended by Zhitao Zhang

We study the boundary value problem for a kindN-dimension nonlinear fractional differential system with the nonlinear terms involved in the fractional derivative explicitly. The fractional differential operator here is the standard Riemann-Liouville differentiation. By means of fixed point theorems, the existence and multiplicity results of positive solutions are received.

Furthermore, two examples given here illustrate that the results are almost sharp.

Copyrightq2008 A. Yang and W. Ge. 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.

1. Introduction

We are interested in the followingN-dimension nonlinear fractional differential system:

D^α₀¹x1t f1

t, x2t, D^μ₀¹x2t 0, ...

D^α₀^N−1x_N−1t f_N−1

t, x_Nt, D^μ₀^N−1x_Nt 0, D^α₀^NxNt fN

t, x1t, D^μ₀^Nx1t 0,

0< t <1, 1.1

that is subject to the boundary conditions

x₁0 x₂0 · · ·x_N0 0,

x11 x21 · · ·xN1 0, 1.2

whereD₀^αi is the standard Riemann-Liouville fractional derivative of orderαi,fi ∈C0,1× R×R,R, 1< α_i <2,μ_i>0,i1,2, . . . , N, andα_i−μ_i−1>1,i1,2, . . . , N,μ₀μ_N.

Recently, fractional differential equationsin short FDEs have been studied exten- sively. 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, engineering, and so on. For an extensive collection of such results, we refer the readers to the monographs by Samko et al.1, Podlubny2, Miller and Ross3, and Kilbas et al.4.

Some basic theory for the initial value problems of FDE involving Riemann-Liouville differential operator has been discussed by Lakshmikantham 5–7, El-Sayed et al. 8, 9, Diethelm and Ford10, and Bai11, and so on. Also, there are some papers which deal with the existence and multiplicity of solutions for nonlinear FDE boundary value problems in short BVPs by using techniques of topological degree theory. For example, Su 12 considered the BVP of the coupled system

D^αut f

t, vt, D^μvt

, 0< t <1, D^βvt g

t, ut, D^νut

, 0< t <1,

u0 u1 v0 v1 0.

1.3

By using the Schauder fixed point theorem, one existence result was given.

In13, Bai and L ¨u obtained positive solutions of the two-point BVP of FDE

D^α₀ut f t, ut

, 0< t <1, 1< α≤2,

u0 u1 0 1.4

by means of Krasnosel’skii fixed point theorem and Leggett-Williams fixed point theorem.

D₀^αis the standard Riemann-Liouville fractional derivative.

Zhang discussed the existence of solutions of the nonlinear FDE

cD^α₀ut f t, ut

, 0< t <1, 1< α≤2 1.5

with the boundary conditions

u0 ν /0, u1 ρ /0, 1.6

u0 u0 0, u1 u1 0, 1.7

in14,15, respectively. Since conditions 1.6 and1.7are nonzero boundary values, the Riemann-Liouville fractional derivativeD₀^αis not suitable. Therefore, the author investigated the BVPs1.5-1.6and1.5–1.7by involving in the Caputo fractional derivative^cD^α₀.

From above works, we can see a fact, although the BVPs of nonlinear FDE have been studied by some authors, to the best of our knowledge, higher-dimension fractional equation systems are seldom considered. Su in12studied the two-dimension system, however, the Schauder fixed point theorem cannot ensure the solutions to be positive. Since only positive solutions are useful for many applications, we investigate the existence and multiplicity of positive solutions for BVP 1.1-1.2in this paper. In addition, two examples are given to demonstrate our results.

2. Preliminaries

For the convenience of the reader, we first recall some definitions and fundamental facts of fractional calculus theory, which can be found in the recent literatures1–4.

Definition 2.1. The fractional integral of orderτ >0 of a functionf:0,∞ → Ris given by

I₀^τfx 1 Γτ

_x

0

ft

x−t^1−τdt, x >0, 2.1

provided that the integral exists, whereΓτis the Euler gamma function defined by

Γz _∞

0

t^z−1e^−tdt, z >0, 2.2

for which, the reduction formula

Γz1 zΓz, Γ1 1, Γ 1

2

√

π, 2.3

the Dirichlet formula ₁

0

t^z−11−t^ω−1dt ΓzΓω Γzω,

z, ω /∈Z⁻₀

2.4

hold.

Definition 2.2. The fractional derivative of orderτ >0 of a continuous functionf:0,∞ → R can be written as

D₀^τfx 1 Γn−τ

d dx

n_x

0

ft

x−t^τ1−ndt, n τ 1, 2.5

whereτdenotes the integer part ofτ, provided that the right side is pointwise defined on 0,∞.

Remark 2.3. The following properties are useful for our discussion:

I₀^τ I₀^σ ft I₀^τσft, D^τ₀I₀^τ ft ft, τ >0, σ >0, f ∈L0,1, I₀^τ D₀^τft ft c1t^τ−1c2t^τ−2· · ·cnt^τ−n, ci∈R, i1,2, . . . , n, I₀^τ :C0,1−→C0,1, D₀^τ f∈C0,1∩L0,1, τ >0, f ∈C0,1.

2.6

In the following, we present the useful lemmas which are fundamental in the proof of our main results.

Lemma 2.4see16. LetCbe a convex subset of a normed linear spaceEandUbe an open subset ofCwithp^∗∈U. Then every compact continuous mapN:U → Chas at least one of the following two properties:

A1Nhas a fixed point;

A2there is anx∈∂Uwithx 1−λp^∗λNx,for some 0< λ <1.

Definition 2.5. The mapαis said to be a nonnegative continuous concave functional on a cone Pof a real Banach spaceEprovided thatα:P → 0,∞is continuous and

α

tx 1−ty

≥tαx 1−tαy, 2.7

for allx, y∈P,andt∈0,1.

Let α and β be nonnegative continuous convex functionals on the cone P, ψ be a nonnegative continuous concave functional onP. Then for positive real numbersr > aand L, one defines the following convex sets:

Pα, r;β, L {x∈P:αx< r, βx< L}, Pα, r;β, L {x∈P:αx≤r, βx≤L},

Pα, r;β, L;ψ, a {x∈P:αx< r, βx< L, ψx> a}, Pα, r;β, L;ψ, a {x∈P:αx≤r, βx≤L, ψx≥a}.

2.8

The assumptions below about the nonnegative continuous convex functionalsα,βwill be used as follows:

B1there existsM >0 such thatx ≤Mmax{αx, βx},for allx∈P;

B2Pα, r;β, L/∅,for allr >0, L >0.

Lemma 2.6 see 17. Let P be a cone in a real Banach space E,r₂ ≥ d > b > r₁ > 0, and L2 ≥ L1 > 0. Assume thatαandβare nonnegative continuous convex functionals satisfying (B1) and (B2), ψ is a nonnegative continuous concave functional on P such that ψy ≤ αy, for all

y ∈ Pα, r1;β, L1and T : Pα, r2;β, L2 → Pα, r2;β, L2,is a completely continuous operator.

Suppose

C1{y∈Pα, d;β, L₂;ψ, b:ψy> b}/ ∅,ψTy> b, fory∈Pα, d;β, L₂;ψ, b;

C2αTy< r₁,βTy< L₁, for ally∈Pα, r1;β, L₁; C3ψTy> b,for ally∈Pα, d;β, L2;ψ, bwithαTy> d.

ThenT has at least three fixed pointsy1, y2, y3∈Pα, r2;β, L2with y₁∈P

α, r₁;β, L₁ , y₂∈

y∈P

α, r₂;β, L₂;ψ, b

:ψy> b , y3∈P

α, r2;β, L2

\ P

α, r2;β, L2;ψ, b

∪P

α, r1;β, L1

.

2.9

3. Related lemmas

LetXX1×X2× · · · ×XNwith the norm xmax x_i

Xi :i1,2, . . . , N

, forx

x₁, x₂, . . . , x_N

∈X, 3.1

whereXi{xi ∈C0,1:D^μ₀ⁱ⁻¹xi∈C0,1},i1,2, . . . , Nwith x_i

Xi x_i

∞ D^μi−1x_i

∞, 3.2

where · _∞is the standard sup norm of the spaceC0,1. Throughout, we denoteμ₀ μ_N andx_N1 x₁. ThenXis a Banach spacesee12.

Define the coneP ⊂Xby

P

x

x₁, x₂, . . . , x_N

∈X:x_it≥0, x_i0 0, t∈0,1, i1,2, . . . , N

. 3.3

Lemma 3.1. Ifx∈P, thenxi_∞≤1/Γ1μ_i−1D^μi−1x_i∞,i1,2, . . . , N.

Proof. Forx x1, x₂, . . . , x_N∈P, we have

x_it I₀^μi−1D₀^μi−1x_it

≤ 1 Γ

μ_{i−1 t}

0

D^μi−1x_is t−s^1−μi−1ds

≤ 1

Γ

1μ_i−1 D^μi−1x_i

∞, i1,2, . . . , N.

3.4

That is,xi_∞≤1/Γ1μ_i−1D^μi−1x_i∞,i1,2, . . . , N.

It is well known that the solution for the system BVP1.1-1.2is equivalent to the fixed point of the following integral system:

T₁x₂t ₁

0

G₁t, sf1

s, x₂s, D^μ₀¹x₂s ds, ...

T_N−1xNt ₁

0

G_N−1t, sf_N−1

s, xNs, D^μ₀^N−1xNs ds,

T_Nx₁t ₁

0

G_Nt, sfN

s, x₁s, D^μ₀^Nx₁s ds,

0< t <1, 3.5

forx∈X, where

Git, s 1 Γ

α_i

⎧⎨

⎩

t1−s_αi−1

−t−s^αi−1, 0≤s≤t≤1,

t1−sαi−1, 0≤t≤s≤1. 3.6

DenoteTx: T1x₂, . . . , T_N−1x_N, T_Nx₁, we can see

Tix_i1t t^αi−1I₀^αifi

1, x_i11, D^μix_i11

−I₀^αifi

t, x_i1t, D^μix_i1t

, 3.7

i1,2, . . . , N. For the Green functionsG_it, s,i1,2, . . . , N, we can obtain

iG_it, s≥0,fort, s ∈0,1,γ_isGis, s≤G_it, s≤G_is, s,fort, s∈θ,1−θ× 0,1,θ∈0,1/2, where

γis

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

1−θ1−sαi−1−1−θ−s^αi−1

s1−sαi−1 , 0< s≤ri, θ^αi−1

s^αi−1, ri≤s <1,

3.8

here,ri∈θ,1−θis the unique solution of the equation 1−θ1−sαi−1−1−θ−s^αi−1

θ1−sαi−1; 3.9

iimax_t∈0,11

0Git, sds αi−1^αi−1/α^α_iⁱΓαi1 :ρi1and min_{t∈θ,1−θ1}

0Git, sds θ1−θ^αi−1/Γαi1 :ρi2.

Lemma 3.2. T:P → Pis completely continuous.

Proof. We divide the proof into three steps.

Step 1. T :P → P. In fact, for anyx ∈P, sincef_it, x_i1t, D^μ₀ⁱx_i1t ≥0 fort ∈0,1and G_it, s ≥ 0,fort, s ∈ 0,1,T_ix_i1t ≥ 0,fort ∈ 0,1. Moreover,G0, s 0 implies that Tix_i10 0.

Step 2. T is continuous onP, which is valid due to the continuity of the functionf.

Step 3. We will show thatT is relatively compact. For any given bounded setU ⊂ P, there existsM >0 such thatx ≤M,for allx∈U. We takeκimax{|fit, u, v|:t∈0,1, |u| ≤ M, |v| ≤M}.Forx∈U, lett₁, t₂∈0,1be such thatt₁< t₂, we have

Tix_i1 t1

−Tix_i1

t2t^α₁ⁱ⁻¹−t^α₂ⁱ⁻¹ I₀^αifi

1, x_i11, D₀^μix_i11

− I₀^αif_i

t₁, x_i1 t₁

, D₀^μix_i1 t₁

−I₀^αif_i t₂, x_i1

t₂

, D₀^μix_i1 t₂

≤t^α₁ⁱ⁻¹−t^α₂ⁱ⁻¹ 1 Γ

αi

₁

0

1−s^αi−1f_i

s, x_i1s, D₀^μix_i1s ds

1

Γ αi

_t2

0

t₂−s_αi−1 f_i

s, x_i1s, D₀^μix_i1s ds

− 1 Γ

α_{i t1}

0

t1−s_αi−1 fi

s, x_i1s, D^μ₀ⁱx_i1s ds

≤ κi

Γ

αi1t^α₁ⁱ⁻¹−t^α₂ⁱ⁻¹ κi

Γ α_i ^t2

t1

t2−s_αi−1 ds

_t1

0

t2−s_αi−1

−

t1−s_αi−1ds

κ_i Γ

αi1

t^α₂ⁱ⁻¹−t^α₁ⁱ⁻¹t^α₂ⁱ−t^α₁ⁱ

−→0, ast₂−t₁−→0.

Notice that 3.10

D^μ₀ⁱ⁻¹T_ix_i1t I^α₀ⁱf_i

1, x_i11, D^μix_i11

·D^μ₀ⁱ⁻¹t^αi−1−I₀^{αi−μi−1}f_i

t, x_i1t, D^μi−1x_i1t , 3.11 one gets

D₀^μi−1Tix_i1 t1

−D^μ₀ⁱ⁻¹Tix_i1 t2 I₀^αif_i

1, x_i11, D^μix_i11

D₀^μi−1t^α₁ⁱ⁻¹−D₀^μi−1t^α₂ⁱ⁻¹

−

I₀^{αi−μi−1}fi

t1, x_i1 t1

, D^μix_i1 t1

−I₀^{αi−μi−1}fi

t2, x_i1 t2

, D^μix_i1 t2

≤ κi

α_iΓ

α_i−μ_i−1t^α₁^{i−μi−1−1}−t^α₂^{i−μi−1−1} κ_i

Γ

αi−μ_i−1 ^t2

t1

t₂−sαi−μi−1−1ds _t1

0

t₂−sαi−μi−1−1−

t₁−sαi−μi−1−1ds

κi

α_iΓ

α_i−μ_i−1

t^α₂^{i−μi−1−1}−t^α₁^{i−μi−1−1} κi

Γ

α_i−μ_i−11

t^α₂^i−μi−1−t^α₁^i−μi−1

−→0, ast2−t1−→0,

3.12 wherei1,2, . . . , N, we can see thatTUis an equicontinuous set. Now, we proof thatT is uniformly bounded. For anyx∈U,

Tix_i1tt^αi−1I₀^αifi

1, x_i11, D₀^μix_i11

−I^α₀ⁱfi

t, x_i1t, D^μ₀ⁱx_i1t

≤ 1 Γ

α_{i 1}

0

1−s^αi−1fi

s, x_i1s, D^μ₀ⁱx_i1s ds

1 Γ

αi

_t

0

t−s^αi−1f_i

s, x_i1s, D₀^μix_i1s ds

≤ 2κi

Γ

α_i1 <∞, D^μ₀ⁱ⁻¹T_ix_i1tI₀^αif_i

1, x_i11, D^μix_i11

D^μ₀ⁱ⁻¹t^αi−1−I₀^{αi−μi−1}f_i

t, x_i1t, D^μix_i1t

≤ κ_i αiΓ αi

Γαi Γ

αi−μ_i−1 κ_i Γ

αi−μ_{i−1 t}

0

t−s^{αi−μi−1−1}ds

≤ κi

2αi−μ_i−1 αiΓ

αi−μ_i−11 <∞,

3.13 wherei 1,2, . . . , N. That is,TUis uniformly bounded. Thus, T is relatively compact. By means of the Arzela-Ascoli theorem,T :P → Pis completely continuous.

4. The existence of one positive solution

Theorem 4.1. If there exista_i, b_i, c_i∈C0,1,R,i1,2, . . . , Nsatisfying bi

∞ ci

∞<min α^α_iⁱΓ

αi1 α_i−1αi−1,α_iΓ

α_i−μ_i−11 2αi−μ_i−1

, 4.1

such that

f_it, x, y≤a_it b_itxc_ity. 4.2 Then the BVP1.1-1.2has at least one positive solution.

Proof. Lemma 3.2indicates thatT :P → Pis completely continuous.

Fori1,2, . . . , N, let

Q_i >max

α_i−1αi−1 a_i

∞

α^α_iⁱΓ α_i1

−

α_i−1_αi−1 b_{i ∞} c_{i ∞}, 2αi−μ_i−1 ai

∞

α_iΓ

α_i−μ_i−11

−

2α_i−μ_i−1 b_i

∞ c_i

∞

,

Qmax

Qi:i1,2, . . . , N .

4.3

DefineΩ {x x1, x₂, . . . , x_N∈P :xiXi < Q_i, i1,2, . . . , N}, thenx< Q. For∀x∈∂Ω, xiXi Qi. Thus,xi_∞≤QiandD₀^μi−1xi_∞≤Qi:

Tix_i1t ₁

0

Git, sfi

s, x_i1s, D^μ₀ⁱx_i1s ds

≤ ₁

0

G_it, s

a_is b_isx_i1s c_isD^μ₀ⁱx_i1s ds

≤ ai

∞ bi

∞ ci

∞

Qi

α_i−1αi−1

α^α_iⁱΓ

α_i1 < Qi, D^μ₀ⁱ⁻¹T_ix_i1tI₀^αif_i

1, x_i11, D^μix_i11

D^μ₀ⁱ⁻¹t^αi−1−I₀^{αi−μi−1}f_i

t, x_i1t, D^μix_i1t

≤ 1 Γ

αi

₁

0

1−s^αi−1f_i

s, x_i1s, D₀^μix_i1s

ds· Γ αi

Γ

αi−μ_i−1t^{αi−μi−1−1} 1

Γ

α_i−μ_{i−1 t}

0

t−s^{αi−μi−1−1}fi

s, x_i1s, D₀^μix_i1s ds

≤ ai

∞ bi

∞ ci

∞

Qi

αiΓ

αi−μ_i−1 ai

∞ bi

∞ ci

∞

Qi

Γ

αi−μ_i−11 ai

∞ bi

∞ ci

∞

Qi

2α_i−μ_i−1 α_iΓ

α_i−μ_i−11 < Qi

4.4 indicate thatTix_i1Xi < Qi, and thenTx max{Tix_i1Xi : i 1,2, . . . , N} < Q. Take p^∗ 0 inLemma 2.4, for anyx∈∂Ω,xλTx0< λ <1does not hold. Hence, the operator Thas at least a fixed point, then the BVP1.1-1.2has at least one positive solution.

Example 4.2. Consider the problem D^5/3₀ x₁t f₁

t, x₂t, D₀^1/4x₂t

0, 0< t <1, D^3/2₀ x2t f2

t, x1t, D₀^1/3x1t

0, 0< t <1, x10 x11 x20 x21 0,

4.5

where

f1t, u, v 10 9 Γ

2 3

−1 9Γ

1 3

1 2√ π Γ1/4

t1

9Γ 1

3

1 12√ π 5Γ1/4

t²

1 9Γ

1 3

√ tu1

9Γ 1

3

t^3/4v,

f₂t, u, v 3 4

√π−1 2

1Γ2/3 Γ1/3

t 1

4

25Γ2/3 2Γ1/3

t²1

2t^1/3u 1 4t^2/3v, α₁ 5

3, α₂ 3

2, μ₁ 1

4, μ₂ 1 3.

4.6

Choose

a₁t 10 9 Γ

2 3

1

9Γ 1

3

1 12√ π 5Γ1/4

t², b₁t 1 9Γ

1 3

√

t, c₁t 1 9Γ

1 3

t^3/4,

a2t 3 4

√π1 4

25Γ2/3 2Γ1/3

t², b2t 1

2t^1/3, c2t 1 4t^2/3.

4.7 It is easy to check that4.1holds. Thus, byTheorem 4.1, the BVP4.5has at least one positive solution. In fact,xt t^3/21−t, t^1/21−tis such a solution.

5. The existence of triple positive solutions

Let the nonnegative continuous convex functionals α, β and the nonnegative continuous concave functionalψbe defined on the coneP by

αx max x_i

∞:i1,2, . . . , N , βx max D^μ₀ⁱ⁻¹xi

∞:i1,2, . . . , N ,

ψx min

θ≤t≤1−θmin x_it:i1,2, . . . , N .

5.1

Obviously,αandβsatisfyB1andB2,ψx≤αx,for allx∈P.

For simplicity, we denote

ρi3: _1−θ

θ

γisGis, sds, ρi4: 2α_i−μ_i−1 α_iΓ

α_i−μ_i−11, σ:max

1 Γ

1μi

:i1,2, . . . , N

.

5.2

Theorem 5.1. Assume that there exist constantsσL≥b/θ > b > σl >0 such thatbΓμi1≤θL, fori1,2, . . . , N. Suppose

H1fit, u, v≤min{σL/ρi1, L/ρi4},t, u, v∈0,1×0, σL×−L, L;

H2f_it, u, v> b/ρ_i2,t, u, v∈0,1×b, b/θ×−L, L;

H3fit, u, v<min{σl/ρi1, l/ρi4},t, u, v∈0,1×0, σl×−l, l;

H4f_it, u, v> b/ρ_i3,t, u, v∈θ,1−θ×b, σL×−L, L.

Then the BVP1.1-1.2has at least three positive solutionsx x1, x₂, . . . , x_N,y y1, y₂, . . . , yN,andz z1, z2, . . . , zNsuch that

0≤xit≤σl, 0≤yit≤σL, σl≤zit≤σL, t∈0,1, D^μ₀ⁱ⁻¹x_i

∞≤l, D^μ₀ⁱ⁻¹y_i

∞≤L, −l≤D^μ₀ⁱ⁻¹z_it≤L, t∈0,1, yit> b, zit≤b, t∈θ,1−θ, fori1,2, . . . , N.

5.3

Proof. Lemma 3.2has showed thatT :P → Pis completely continuous. Now, we will verify that all the conditions ofLemma 2.6are satisfied. The proof is based on the following steps.

Step 1. We will show thatH1impliesT :Pα, σL;β, L → Pα, σL;β, L.

In fact, forx∈Pα, σL;β, L,αx≤σL,βx≤L, and thenxi_∞≤σL,D₀^μi−1xi_∞≤L, i1,2, . . . , N. In view ofH1, we have

T_ix_i1

∞max

0≤t≤1

₁

0

G_it, sfi

s, x_i1s, D₀^μix_i1s ds

≤ max

t,u,v∈0,1×0,σL×−L,Lfit, u, v·max

0≤t≤1

₁

0

Git, sds

≤ σL

ρ_i1 ·ρi1σL, D^μi−1Tix_i1

∞max

0≤t≤1I₀^αifi

1, x_i11, D^μix_i11

·D^μ₀ⁱ⁻¹t^αi−1−I₀^{αi−μi−1}f_i

t, x_i1t, D^μix_i1t

≤ max

t,u,v∈0,1×0,σL×−L,Lft, u, v

·max

0≤t≤1

1 Γ

αi

₁

0

1−s^αi−1ds Γ α_i Γ

αi−μ_i−1t^{αi−μi−1−1} 1

Γ

α_i−μ_{i−1 t}

0

t−s^{αi−μi−1−1}ds

≤ L

ρ_i4 ·ρ_i4 L.

5.4

ThenαTx≤σLandβTx≤L, that is,Tx∈Pα, σL;β, L.

Step 2. To check the conditionC1inLemma 2.6, we choosex^∗t b/θt^μN,b/θt^μ1, . . . , b/θt^μN−1,t∈0,1. It is easy to see that

α x^∗

max

t∈0,1max b

θt^μi

:i1,2, . . . , N

b θ, β

x^∗ max

t∈0,1max b

θD^μ₀ⁱt^μi

:i1,2, . . . , N

max b

θΓ 1μ_i

:i1,2, . . . , N

≤L,

ψ x^∗

min

t∈θ,1−θmin b

θt^μi

:i1,2, . . . , N

min b

θθ^μi :i1,2, . . . , N

> b.

5.5 Consequently,{x∈Pα, b/θ;β, L;ψ, b:ψx> b}/∅. For anyx∈Pα, b/θ;β, L;ψ, b, from H2, one gets

t∈θ,1−θmin T_ix_i1t min

t∈θ,1−θ

₁

0

G_it, sfi

s, x_i1s, D₀^μixs ds

≥ min

t,u,v∈0,1×b,b/θ×−L,Lfit, u, v· min

t∈θ,1−θ

₁

0

Git, sds

> b

ρ_i2 ·ρ_i2b,

5.6

then we can obtainψTx> b.

Step 3. It is similar toStep 1that we can proveT :Pα, σl;β, l → Pα, σl;β, lby condition H3, that is,C2inLemma 2.6holds.

Step 4. We verify thatC3inLemma 2.6is satisfied. Forx∈Pα, σL;β, L;ψ, bwithαTx>

b/θ, we have

t∈θ,1−θmin Tix_i1t≥ ₁

0

γisGis, sfi

s, x_i1s, D^μ₀ⁱx_i1s ds

≥ _1−θ

θ

γisGis, sds· min

t,u,v∈θ,1−θ×b,σL×−L,Lfit, u, v

> ρ_i3· b ρ_i3 b.

5.7

Thus,ψTx> b,C3inLemma 2.6is satisfied.

Therefore, the operatorThas three pointsx, y, z∈Pα, σL;β, Lwith x∈Pα, σl;β, l, y∈Pα, σL;β, L;ψ, b,

z∈Pα, σL;β, L\Pα, σL;β, L;ψ, b∪Pα, σl;β, l. 5.8

Then the BVP1.1-1.2has three positive solutionsx, y, z∈Pα, σL;β, Lsuch that

0≤x_it≤σl, 0≤y_it≤σL, σl≤z_it≤σL, t∈0,1, D^μ₀ⁱ⁻¹x_i

∞≤l, D^μ₀ⁱ⁻¹y_i

∞≤L, −l≤D^μ₀ⁱ⁻¹z_it≤L, t∈0,1, yit> b, zit≤b, t∈θ,1−θ, fori1,2, . . . , N.

5.9

Example 5.2. Consider the problem

D^3/2₀ x₁t f₁

t, x₂t, D₀^1/2x₂t

0, 0< t <1, D^7/4₀ x₂t f₂

t, x₁t, D₀^1/4x₁t

0, 0< t <1, x₁0 x₁1 x₂0 x₂1 0,

5.10

where

f₁t, u, v

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩ 1

2 _t

u² 10² |v|

10⁶, u∈

0,56

25

, 1

2 t

213749

24890 u² |v|

10⁶ 3136

62500−3136

625 ·213749 24890 , u∈

56 25,3

, 1

2 _t

1070313 31250 |v|

10⁶, u∈3,∞,

f₂t, u, v

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩ 1

5 t

u² 10³ v²

10¹⁰, u∈

0,56

25

, 1

5 _t

14740614

2489000 u² v²

10¹⁰ 3136

625000−3136

625 ·14740614 2489000, u∈

56 25,3

, 1

5 _t

2359 100 v²

10¹⁰, u∈3,∞.

5.11

Here, we haveα₁ 3/2,α₂7/4,μ₁1/2,μ₂1/4. By choosingθ1/4 and the definition ofσandρij,i1,2,j1,2,3,4, one gets

σmax

1

Γ11/2, 1 Γ11/4

1

Γ11/2 1 1/2√

π ≈1.12,

ρ11 3/2−1^3/2−1

3/2^3/2Γ3/21 8 9√

3π ≈0.28,

ρ₁₂ 1/41−1/4^3/2−1 Γ3/21 1

2√

3π ≈0.16,

ρ_{13 3/4}

1/4

γ₁sG1s, sds 2

√π _3/4

1/4

1 2

√1−sds 2

√π

₁₋^√_3/6

1/4

3

4−sds≈0.12,

ρ₁₄ 3−1/4

3/2Γ3/2−1/41 88

15Γ1/4 ≈1.61,

ρ21≈0.18, ρ22≈0.12, ρ23≈0.06, ρ24≈1.53.

5.12

Takingl2,b3,andL1000, we have

f₁t, u, v≤min σL

ρ11

, L ρ14

≈621.11, fort, u, v∈0,1×0,1120×−1000,1000,

f1t, u, v> b

ρ₁₃ ≈16.67, fort, u, v∈ 1

4,3 4

×3,1120×−1000,1000,

f₁t, u, v<min σl

ρ₁₁, l ρ₁₄

≈1.24, fort, u, v∈0,1×

0,56 25

×−3,3,

f₁t, u, v> b

ρ12 ≈18.75, fort, u, v∈0,1×3,12×−1000,1000,

5.13

that is,f1 satisfies the conditionsH1–H4ofTheorem 5.1. Similarly, we can show thatf2

satisfiesH1–H4. Thus, byTheorem 5.1, the BVP5.10has at least three positive solutions x x1, x₂,y y1, y₂,andz z1, z₂such that

0≤xit≤2.24, 0≤yit≤1120, 2.24≤zit≤1120, t∈0,1, i1,2, D₀^1/4x1

∞≤2, D^1/2₀ x2

∞≤2, D₀^1/4y1

∞≤1000, D₀^1/2y2

∞≤1000,

−2≤D^1/4₀ z1t≤1000, −2≤D₀^1/2z2t≤1000, t∈0,1, y_it>3, z_it≤3, t∈

1 4,3

4

, i1,2.

5.14

Remark 5.3. The particular case N 2 has been studied by 12 for the existence of one solution, our paper generalizes12 for the obtaining of one and three positive solutions.

ForN1, we develop13–15by the nonlinear termsfiinvolved in theμi-order Riemann- Liouville derivative explicitly.

Acknowledgments

This work is supported by National Natural Science Foundation of ChinaNNSF 10671012 and the Specialized Research Fund for the Doctoral Program of Higher EducationSRFDP of China20050007011.

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