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R E S E A R C H

Open Access

Existence on positive solutions for boundary

value problems of nonlinear fractional

differential equations with

p

-Laplacian

Hongling Lu, Zhenlai Han

*

, Shurong Sun and Jian Liu

*Correspondence:

hanzhenlai@163.com

School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P.R. China

Abstract

In this paper, we study the existence of positive solutions for the nonlinear fractional boundary value problem with ap-Laplacian operator

0+

p

(

0+u(t)

))

=f

(

t,u(t)

)

, 0 <t< 1,

u(0) =u(0) =u(1) = 0, 0+u(0) =Dα0+u(1) = 0,

where 2 <

α

≤3, 1 <

β

≤2,0+,0+are the standard Riemann-Liouville fractional derivatives,

φ

p(s) =|s|p–2s,p> 1,

φ

p–1=

φ

q, 1/p+ 1/q= 1, and

f(t,u)∈C([0, 1]×[0, +∞), [0, +∞)). By the properties of Green’s function, the Guo-Krasnosel’skii fixed-point theorem, the Leggett-Williams fixed-point theorem, and the upper and lower solutions method, some new results on the existence of positive solutions are obtained. As applications, examples are presented to illustrate the main results.

MSC: 34A08; 34B18; 35J05

Keywords: fractional boundary value problem; positive solution; upper and lower solutions; fixed-point theorems;p-Laplacian operator

1 Introduction

Recently, fractional differential equations have been of great interest. The motivation for

those works stems from both the intensive development of the theory of fractional cal-culus itself and the applications such as economics, engineering and other fields [–]. Many people pay attention to the existence and multiplicity of solutions or positive solu-tions for boundary value problems of nonlinear fractional differential equasolu-tions by means of some fixed-point theorems [–] (such as the Schauder fixed-point theorem, the Guo-Krasnosel’skii fixed-point theorem, the Leggett-Williams fixed-point theorem) and the

up-per and lower solutions method [–].

To the best of our knowledge, there are few papers devoted to the study of fractional differential equations with ap-Laplacian operator [–, –]. Its theories and appli-cations seem to be just being initiated.

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Wanget al.[] considered the followingp-Laplacian fractional differential equations boundary value problems:

+φp

+u(t)

=ft,u(t),  <t< ,

u() = , u() =au(ξ),

+u() = , +u() =bDα+u(η),

where  <α,γ ≤, ≤a,b≤,  <ξ,η< , and

+is the standard Riemann-Liouville fractional derivative.φp(s) =|s|p–s,p> ,φp–=φq, /p+ /q= . They obtained the

exis-tence of at least one positive solution by means of the upper and lower solutions method. Wanget al.[] investigated the existence and multiplicity of concave positive solu-tions of a boundary value problem of a fractional differential equation with ap-Laplacian operator as follows:

+φp

+u(t)+ft,u(t),+u(t)= ,  <t< ,

u() =u() = , u() = , +u() = , where  <α< ,  <γ < ,  <ρ≤,

+is the Caputo derivative. By using a fixed-point theorem, some results for multiplicity of concave positive solutions are obtained.

Chenet al.[] considered the boundary value problem for a fractional differential equa-tion with ap-Laplacian operator at resonance

+φp

+x(t)=ft,x(t),+x(t), t∈[, ],

+x() =+x() = ,

where  <α,β≤,  <α+β≤, and

+is the Caputo fractional derivative. By using the coincidence degree theory, a new result on the existence of solutions is obtained.

Guoqing Chai [] investigated the existence and multiplicity of positive solutions for a class of boundary value problems of fractional differential equations with ap-Laplacian operator

+φp

+u(t)+ft,u(t)= ,  <t< ,

u() = , u() +σDγ+u() = , +u() = ,

where  <α≤,  <β≤,  <γ ≤, ≤αγ– ,σis a positive constant number,+, +,+are the standard Riemann-Liouville derivatives. By means of the fixed-point the-orem on cones, some existence and multiplicity results of positive solutions are obtained. Motivated by all the works above, in this paper, we deal with the followingp-Laplacian fractional differential equation boundary value problem:

+φp

+u(t)

=ft,u(t),  <t< , (.)

u() =u() =u() = , +u() =+u() = , (.) where  <α≤,  <β≤,

+,D

β

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the upper and lower solutions method, some new results on the existence of positive so-lutions are obtained for the fractional differential equation boundary value problem (.) and (.).

The rest of this paper is organized as follows. In Section , we introduce some defini-tions and lemmas to prove our main results. In Section , we investigate the existence of a single positive solution for boundary value problems (.) and (.) by the upper and lower solutions method. In Section , we establish the existence of single and multiple positive solutions for boundary value problems (.) and (.) by fixed-point theorems. As appli-cations, examples are presented to illustrate our main results in Section  and Section , respectively.

2 Preliminaries and lemmas

For the convenience of the reader, we give some background material from fractional cal-culus theory to facilitate the analysis of problem (.) and (.). These materials can be found in the recent literature, see [, , , , –].

Definition .[] The fractional integral of orderα>  of a functiony: (, +∞)→Ris given by

I+α y(t) = 

(α)

t

(ts)α–y(s)ds

provided the right-hand side is pointwise defined on (, +∞).

Definition .[] The fractional derivative of orderα>  of a continuous functiony: (, +∞)→Ris given by

+y(t) = 

(nα)

d dt

n t

y(s) (ts)αn+ds,

wherenis the smallest integer greater than or equal toα, provided that the right-hand side is pointwise defined on (, +∞).

Lemma .[] Letα> .If we assume u+uL(, ),then the fractional differential equation

+u(t) = 

has

u(t) =ctα–+ctα–+· · ·+cntαn, ci∈R,i= , , . . . ,n as a unique solution,

where n is the smallest integer greater than or equal toα.

Lemma .[] Assume that Dα+uL(, )with a fractional derivative of order α> . Then

I+α +u(t) =u(t) +ctα–+ctα–+· · ·+cntαn

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Lemma . [] Let yC[, ] and  <α ≤. Then fractional differential equation boundary value problem

+u(t) +y(t) = ,  <t< , (.)

u() =u() =u() =  (.)

has a unique solution

u(t) =

G(t,s)y(s)ds,

where

G(t,s) =

⎧ ⎨ ⎩

–(–s)α––(ts)α–

(α) , st,

–(–s)α–

(α) , ts.

(.)

Lemma . Let yC[, ]and <α≤,  <β≤.Then the fractional differential equa-tion boundary value problem

+φp

+u(t)

=y(t),  <t< , (.)

u() =u() =u() = , +u() =+u() =  (.)

has a unique solution

u(t) =

G(t,s)φq

H(s,τ)y(τ)

ds,

where

H(t,s) =

⎧ ⎨ ⎩

[t(–s)]β––(ts)β–

(β) , st, [t(–s)]β–

(β) , ts,

(.)

G(t,s)is defined as(.).

Proof From Lemma . and  <β≤, we have

I+β +φp

+u(t)=φp

+u(t)+ctβ–+ctβ– for somec,c∈R.

In view of (.), we obtain

I+β +φp

+u(t)

=I+β y(t).

Therefore,

φp

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that is,

By the boundary conditions

+u() =+u() = , we have

Therefore, the solution u(t) of fractional differential equation boundary value problem (.) and (.) satisfies boundary value problem (.) and (.) is equivalent to the following problem:

+u(t) +φq

Lemma . implies that fractional differential equation boundary value problem (.) and (.) has a unique solution

u(t) =

The proof is complete.

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In the following, we consider the existence ofδ(s) andδ(s). Firstly, for givens∈(, ),

Similar to Lemma . in [], we choose

δ(s) =

The proof is complete.

Lemma . Let <α≤.If y(t)∈C[, ]and y(t)≥,then fractional differential equa-tion boundary value problem(.)and(.)has a unique solution u(t)≥,t∈[, ].

Proof From Lemma ., the fractional differential equation boundary value problem (.) and (.) has a unique solution

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Definition .[] A functionη(t) is called an upper solution of fractional differential equation boundary value problem (.) and (.) ifη(t)∈Eandη(t) satisfies

+φp

+η(t)≥ft,η(t),  <t< ,  <α≤,  <β≤,

η()≥, η()≥, η()≥,

+η()≤, +η()≤.

Definition .[] A function ξ(t) is called a lower solution of fractional differential equation boundary value problem (.) and (.) ifξ(t)∈Eandξ(t) satisfies

+φp

+ξ(t)≤ft,ξ(t),  <t< ,  <α≤,  <β≤,

ξ()≤, ξ()≤, ξ()≤, +ξ()≥, +ξ()≥.

Definition .[] The mapθis said to be a nonnegative continuous concave functional

on a conePof a real Banach spaceEprovided thatθ:P→[, +∞) is continuous and

θtx+ ( –t)y(x) + ( –t)θ(y)

for allx,yPand ≤t≤.

Lemma .[] Let E be a Banach space,PE be a cone,and,be two bounded

open balls of E centered at the origin with¯⊂.Suppose thatA:P∩(¯\)→P is a completely continuous operator such that either

(i) Axx,xPandAxx,xPor (ii) Axx,xPandAxx,xPholds.ThenAhas a fixed point in P∩(¯\).

Leta,b,c>  be constants,Pc={uP:u<c},P(θ,b,d) ={uP:bθ(u),ud}.

Lemma .[] Let P be a cone in a real Banach space E,Pc={xP| xc},θ be a

nonnegative continuous concave functional on P such thatθ(x)≤ xfor all x∈ ¯Pc,and

P(θ,b,d) ={xP|bθ(x),xd}.SupposeB:P¯c→ ¯Pcis completely continuous and

there exist constants <a<b<dc such that

(C) {xP(θ,b,d)|θ(x) >b} =∅andθ(Bx) >bforxP(θ,b,d); (C) Bx<aforxa;

(C) θ(Bx) >bforxP(θ,b,c)withBx>d. ThenBhas at least three fixed points x,x,and xwith

x<a, b<θ(x), a<x with θ(x) <b.

LetE=C[, ] be endowed withu=max≤t≤|u(t)|. Define the conePEby

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Let the nonnegative continuous concave functionalθ on the conePbe defined by

Hence,T() is uniformly bounded.

On the other hand, sinceG(t,s) is continuous on [, ]×[, ], it is uniformly continuous

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3 Existence of a single positive solution

In this section, for the sake of simplicity, we assume that

(H) f(t,u)is nonincreasing tou;

Theorem . Assume that(H)and(H)hold.Then the fractional differential equation boundary value problem(.)and(.)has at least one positive solutionγ(t).

Proof From Lemma ., we obtainT(P)⊆P. By direct computations, we have of fractional differential equation boundary value problem (.) and (.), respectively.

From (H) and (H), we have

that is, η(t) and ξ(t) are upper and lower solutions of fractional differential equation boundary value problem (.) and (.), respectively.

Next, we show that the fractional differential equation boundary value problem

+φp

+u(t)=gt,u(t),  <t< ,  <α≤,  <β≤, (.)

u() =u() =u() = ,

+u() =+u() =  (.)

has a positive solution, where

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Thus, we consider the operatorB:PEdefined as follows:

Bu(t) :=

G(t,s)φq

H(s,τ),u(τ)

ds,

whereG(t,s) andH(t,s) are defined as (.) and (.), respectively. It is clear thatBu(t)≥ for alluPand a fixed point of the operatorBis a solution of the fractional differential equation boundary value problem (.) and (.).

Similar to Lemma ., we know thatBis a compact operator. By the Schauder fixed-point theorem, the operatorBhas a fixed point, that is, the fractional differential equation boundary value problem (.) and (.) has a positive solution.

Finally, we will prove that fractional differential equation boundary value problem (.) and (.) has at least one positive solution.

Suppose thatγ(t) is a solution of (.) and (.). Now, to complete the proof, it suffices to show thatξ(t)≤γ(t)≤η(t),t∈[, ].

Letγ(t) be a solution of (.) and (.). We have

γ() =γ() =γ() = , +γ() =+γ() = . (.) From (H), we have

ft,η(t)≤gt,γ(t)≤ft,ξ(t), t∈[, ]. (.)

By (H) and (.), we obtain

ft,q(t)≤gt,γ(t)≤ft,p(t), t∈[, ]. (.)

Byp(t)∈Pand (.), we can get

+φp

+η(t)=+φp

+(Tp)(t)=ft,p(t), t∈[, ]. (.)

Combining (.), (.)-(.), we have

+φp

+η(t)–+φp

+γ(t)=ft,p(t)–gt,γ(t)≥, t∈[, ], (.)

(ηγ)() = (ηγ)() = (ηγ)() = , +(ηγ)() =+(ηγ)() = .

(.)

Letx(t) =φp(+η(t)) –φp(+γ(t)). By (.), we obtainx() =x() = . By Lemma ., we knowx(t)≤,t∈[, ], which implies that

φp

+η(t)≤φp

+γ(t), t∈[, ].

Sinceφpis monotone increasing, we obtain+η(t)≤+γ(t), that is,+(ηγ)(t)≤. By Lemma ., (.) and (.), we have (ηγ)(t)≥. Therefore,η(t)≥γ(t),t∈[, ].

In a similar way, we can prove thatξ(t)≤γ(t),t∈[, ]. Consequently,γ(t) is a positive solution of fractional differential equation boundary value problem (.) and (.). This

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Example . We consider the following fractional differential equation boundary value

uis nonincreasing relative tou. This shows that (H) holds.

Letm(t) =t/. From Lemma ., we have

Choosing a positive numberd≤ {,d}and combining the monotonicity ofT, we have

Tdm(t)≥Tm(t)≥dm(t)≥dm(t), Tdm(t)≥Tdm(t)≥dm(t).

That is, the condition (H) holds. By Theorem ., the fractional differential equation boundary value problem (.) and (.) has at least one positive solution.

4 Existence of single and multiple positive solutions

In this section, for convenience, we denote

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(A) f(t,u)≥φp(Na)for(t,u)∈[, ]×[,a];

(A) f(t,u)≤φp(Ma)for(t,u)∈[, ]×[,a].

Then the fractional differential equation boundary value problem(.)and(.)has at least one positive solution u such that a≤ ua.

Proof From Lemmas ., ., and ., we get thatT :PPis completely continuous and fractional differential equation boundary value problem (.) and (.) has a solution u=u(t) if and only ifusolves the operator equationu=Tu(t). In order to apply Lemma ., we divide our proof into two steps.

Step . Let:={uP| u<a}. Foru, we have ≤u(t)≤afor allt∈[, ]. It

Then, by (ii) of Lemma ., we complete the proof.

Example . We consider the following fractional differential equation boundary value

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Letp= . By a simple computation, we obtainM= .,N≈.. Choosinga=

With the use of Theorem ., the fractional differential equation boundary value problem (.) and (.) has at least one solutionusuch that .≤ u ≤..

Theorem . Let f(t,u)be continuous on[, ]×[, +∞).Assume that there exist

con-stants <a<b<c such that the following assumptions hold: (B) f(t,u) <φp(Ma)for(t,u)∈[, ]×[,a];

(B) f(t,u)≥φp(Nb)for(t,u)∈[/, /]×[b,c];

(B) f(t,u)≤φp(Mc)for(t,u)∈[, ]×[,c].

Then the fractional differential equation boundary value problem(.)and(.)has at least three positive solutions u,u,and uwith

max fractional differential equation boundary value problem (.) and (.) has a solutionu= u(t) if and only ifusatisfies the operator equationu=Tu(t).

Therefore, condition (C) of Lemma . is satisfied.

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Nb

/

/

δ(s)G(,s)φq

/

/

δ(τ)H(τ,τ)

ds

=b,

i.e., θ(Tu) >b for all uP(θ,b,c). Choosing d=c, this shows that condition (C) of Lemma . is also satisfied.

In the same way, ifuP(θ,b,c) andTu>c=d, we also obtainθ(Tu) >b. Then condi-tion (C) of Lemma . is also satisfied.

By Lemma ., the fractional differential equation boundary value problem (.) and (.) has at least three positive solutionsu,u, andu, satisfying

max

≤t≤u(t)<a, b</min≤t≤/

u(t),

a<max

≤t≤u(t), /min≤t≤/u(t)<b.

The proof is complete.

Example . We consider the following fractional differential equation boundary value

problem:

D/+φp

D/+u(t)=f(t,u),  <t< , (.)

u() =u() =u() = , D/+u() =D/+u() = , (.) where

f(t,u) =

⎧ ⎨ ⎩

t

+ u foru≤,

t

+u+  foru> .

Letp= . We obtainM= .,N≈.. Choosinga= .,b= ,c= , therefore

f(t,u) = t

+ u

. <φ

(Ma) = . for (t,u)∈[, ]×[, .],

f(t,u) = t

+u+ ≥. >φ(Nb)

≈. for (t,u)∈[/, /]×[, ],

f(t,u) = t

+u+ ≤. <φ(Mc) =  for (t,u)∈[, ]×[, ].

With the use of Theorem ., the fractional differential equation boundary value problem (.) and (.) has at least three positive solutionsu,u, anduwith

max ≤t≤

u(t)< .,  < min /≤t≤/

u(t)<max ≤t≤

u(t)≤,

. < max ≤t≤

u(t)≤, min /≤t≤/

u(t)< .

Competing interests

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Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (11071143, 60904024, 61174217), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2010AL002, ZR2011AL007), also supported by Natural Science Foundation of Educational Department of Shandong Province (J11LA01).

Received: 31 October 2012 Accepted: 21 January 2013 Published: 11 February 2013 References

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