Anti periodic fractional boundary value problems for nonlinear differential equations of fractional order

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R E S E A R C H

Open Access

Anti-periodic fractional boundary value

problems for nonlinear differential equations

of fractional order

Fang Wang

1,2*

_{and Zhenhai Liu}

3

*_{Correspondence:}

[email protected]

1_{School of Mathematical Science}

and Computing Technology, Central South University, Changsha, Hunan 410075, P.R. China

2_{School of Mathematics and}

Computing Science, Changsha University of Science and Technology, Changsha, Hunan 410076, P.R. China

Full list of author information is available at the end of the article

Abstract

By using Schauder’s ﬁxed point theorem and the contraction mapping principle, we discuss the existence of solutions for nonlinear fractional diﬀerential equations with fractional anti-periodic boundary conditions. Some examples are given to illustrate the main results.

Keywords: fractional diﬀerential equations; boundary value problem; anti-periodic; ﬁxed point theorem

1 Introduction

Fractional calculus has been recognized as an effective modeling methodology by re-searchers. Fractional differential equations are generalizations of classical differential equations to an arbitrary order. They have broad application in engineering and sciences such as physics, mechanics, chemistry, economics and biology,etc.[–]. For some recent development on the topic, see [–] and the references therein.

In [], Ahmadet al.considered the following anti-periodic fractional boundary value problems:

⎧ ⎨ ⎩

c_Dq_x₍_t_{) =}_f_t_,_x₍_t₎_, _t_∈_[,_T_],_T_{> ,  <}_q_≤_,

x() = –x(T), cDpx() = –cDpx(T),  <p< ,

()

wherec_Dq_{denotes the Caputo fractional derivative of order}_q_{, and}_f _{is a given continuous}

function. The results are based on some standard ﬁxed point principles.

In recent years, there has been a great deal of research into the questions of existence and uniqueness of solutions to anti-periodic boundary value problems for differential equa-tions. First, second and higher-order differential equations with anti-periodic boundary value conditions have been considered in papers [–]. The existence of solutions for anti-periodic boundary value problems for fractional differential equations was studied in [–].

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In this paper, we investigate the existence and uniqueness of solutions for an anti-periodic fractional boundary value problem given by

⎧ ⎨ ⎩

c_Dα_x₍_t_{) =}_f_t_,_x₍_t_),c_Dq_x₍_t₎_, _t_∈_[,_T_],

x() = –x(T), c_Dp_x_{() = –}c_Dp_x₍_T_), ()

wherecDα_{denotes the Caputo fractional derivative of order}_α_,_T _{is a positive constant,}  <α≤,  <p,q< ,α–q≥ andf is a given continuous function.

2 Preliminaries

Theorem .([]) Let E be a closed, convex and nonempty subset of a Banach space X, let F :E→E be a continuous mapping such that FE is a relatively compact subset of X. Then F has at least one ﬁxed point in E.

Theorem .([]) Let p and q be two positive numbers such that_p+_q= . If|f(x)|p_and

|g(x)|qare Riemann integrable on[a,b], then

b

a

f(x)g(x)dx≤

b

a

_f₍_x₎p

dx

 p b

a

_g₍_x₎q

dx

 q .

Lemma .([]) For any y∈C[,T], a unique solution of the linear fractional boundary value problem

⎧ ⎨ ⎩

c_Dα_x₍_t_{) =}_y₍_t_), _t_∈_[,_T_],_T_{> ,  <}_α_≤_,

x() = –x(T), c_Dp_x_{() = –}c_Dp_x₍_T_), ()

is

x(t) =

T



G(t,s)y(s)ds, ()

where G(t,s)is the Green’s function given by

G(t,s) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

(t–s)α–_–

(T–s)

α–

(α) +

( –p)(T– t)(T–s)α–p–

(α–p)T–p , s≤t,

–(T–s) α–

(α) +

( –p)(T– t)(T–s)α–p–

(α–p)T–p , t≤s.

()

Remark . Forp→– _{the solution of the classical anti-periodic problem (}c_Dα_x₍_t_{) =}

f(t,x(t),c_Dq_x₍_t_)),_x_{() = –}_x₍_T_),_x_{() = –}_x₍_T_{), }_≤_t_≤_T_{,  <}_α_≤_{,  <}_q_{< ,}_α_–_q_≥_)

is given in [].

3 Main results

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Theorem . Let f :J×R×R→R be a continuous function. Assume that (H) There exist a constant l∈(,α– )and a real-valued function m(t)∈L



l_([,_T_{], (,}

∞))such that

f(t,x,y)≤m(t) +d|x|ρ+d|y|ρ,

where d,d≥,≤ρ,ρ< . Then the problem () has at least a solution on[,T].

Proof Let the condition (H) be valid. According to Lemma ., the problem () is

equiv-alent to the following integral equation:

x(t) =

t



(t–s)α– (α) f

s,x(s),cDqx(s)ds– 

T



(T–s)α– (α) f

s,x(s),cDqx(s)ds

+( –p)(T– t) T–p

T



(T–s)α–p– (α–p) f

s,x(s),cDqx(s)ds.

Deﬁne

(Fx)(t) =

t



(t–s)α– (α) f

T



(T–s)α– (α) f

s,x(s),cDqx(s)ds

+( –p)(T– t) T–p

T



(T–s)α–p– (α–p) f

s,x(s),cDqx(s)ds,

Br=

x(t)∈X,x ≤r,t∈J, where

r≥max(Ad)



–ρ_{, (}_Ad_₎  –ρ_{, }_K_,

K=MT α–l (α)

 –l α–l

–l

+( –p)MT α–l (α–p)

 –l α–p–l

–l

+ M(α–l)T α–q–l

(α– )(α–q–l+ )

×

 –l α–l– 

–l

+M( –p)T α–q–l

(α–p)( –q)

 –l α–p–l

–l ,

A= T

α–q

(α–q+ )+

( –p)Tα–q

( –q)(α–p+ )+ Tα (α+ )+

( –p)Tα (α–p+ ),

andM= (_T(m(s))l _ds₎l_{. Observe that}_B_r_{is a closed, bounded and convex subset of} Ba-nach spaceX. Now, we prove thatF:Br→Br. For anyx∈Br, by Theorem . (Hölder inequality), we have

(Fx)(t)=

t



(t–s)α– (α) f

T



(T–s)α– (α) f

s,x(s),cDqx(s)ds

+( –p)(T– t) T–p

T



(T–s)α–p– (α–p) f

s,x(s),cDqx(s)ds

≤

t



(t–s)α– (α) f

s,x(s),cDqx(s)ds+ 

T



(T–s)α– (α) f

s,x(s),cDqx(s)ds

+( –p)T p



T



(T–s)α–p– (α–p) f

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+ ( –p)

( –q)(α–p)T–p

t



(t–s)–q

× T



(T–τ)α–p––l_dτ

–l T



m(τ)l _dτ

l

ds

+dr ρ₊_d

rρ ( –q)(α)

t



(t–s)–qsα–ds+(dr ρ₊_d

rρ)( –p)Tα–q ( –q)(α–p+ )

× M

(α– )( –q)

 –l α–l– 

–l t



(t–s)–qsα–l–ds

+M( –p)T α–l– (α–p)( –q)

 –l α–p–l

–l t



(t–s)–qds+(dr ρ₊_d

rρ)Tα–q (α–q+ )

+(dr ρ₊_d

rρ)( –p)Tα–q ( –q)(α–p+ )

≤ M(α–l)Tα–q–l (α– )(α–q–l+ )

 –l α–l– 

–l

(α–p)( –q)

 –l α–p–l

–l

+ T α–q

(α–q+ ) +

( –p)Tα–q

( –q)(α–p+ )

drρ+drρ

.

Thus,

(Fx)(t)=max t∈J

(Fx)(t)+max t∈J

c_Dq₍_Fx₎₍_t₎

≤MTα–l

(α)

 –l α–l

–l

+( –p)MT α–l (α–p)

 –l α–p–l

–l

+ M(α–l)T α–q–l

(α– )(α–q–l+ )

×

 –l α–l– 

–l

(α–p)( –q)

 –l α–p–l

–l

+ T α–q

(α–q+ )

+ ( –p)T α–q

( –q)(α–p+ ) + Tα (α+ )+

( –p)Tα (α–p+ )

drρ+drρ

=K+drρ+drρ

A≤ r

+

r

+

r

=r.

Notice that (Fx)(t),Dq(Fx)(t) are continuous onJ; therefore,F:Br→Br. In view of the continuity off, it is easy to know that the operatorFis continuous. Now, we show thatF

is a completely continuous operator. For eachx∈Br, we ﬁxN=maxt∈J|f(t,x(t),cDqx(t))|, for anyε> , setting

δ=min

(α)(α–p+ )ε

NTα–₍₍_α_–_p_{+ ) +}_{( –}_p₎₍_α₎₎,

 

(α)(α–p+ )ε

NTα–₍₍_α_–_p_{+ ) +}_{( –}_p₎₍_α₎₎

 –q

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For eachx∈Br, we will prove that ift,t∈Jand  <t–t<δ, then

₍_Fx₎₍_t__{) – (}_Fx₎₍_t_₎_<ε.

In fact,

(Fx)(t) – (Fx)(t)=

_t(t–s)α– (α) f

s,x(s),cDqx(s)ds

–

t



(t–s)α– (α) f

s,x(s),cDqx(s)ds

+(t–t)( –p)

(α–p)T–p

T



(T–s)α–p–_f_s_,_x₍_s_),c_Dq_x₍_s₎_ds

≤

t



(t–s)α–– (t–s)α–

(α) f

s,x(s),cDqx(s)ds

+

t t

(t–s)α– (α) f

s,x(s),cDqx(s)ds

+(t–t)( –p)

(α–p)T–p

T



(T–s)α–p–fs,x(s),cDqx(s)ds

≤N

t



(t–s)α–– (t–s)α–

(α) ds+N

t t

(t–s)α– (α) ds

+N(t–t)( –p)

(α–p)T–p

T



(T–s)α–p–_ds

= N

(α+ )

t_α–t_α+N( –p)T α– (α–p+ ) (t–t).

By mean value theorem, we have

(Fx)(t) – (Fx)(t)≤ N (α+ )

t_α–t_α+N( –p)T α– (α–p+ ) (t–t)

≤ N

(α+ )αT α–₍_t

–t) +

N( –p)Tα– (α–p+ ) (t–t)

≤

N (α)T

α–₊N( –p)Tα– (α–p+ )

δ<ε

and

c_Dq₍_Fx₎₍_t

) –cDq(Fx)(t)

=

t



(t–s)–q ( –q)(Fx)

₍_s₎_ds_– t 

(t–s)–q ( –q)(Fx)

₍_s₎_ds

=

t



(t–s)–q ( –q)

s



(s–τ)α– (α– )f

τ,x(τ),cDqx(τ)dτ

–( –p)

T–p

T



(T–τ)α–p– (α–p) f

τ,x(τ),cDqx(τ)dτ

ds

+

t t

(t–s)–q ( –q)

s



(s–τ)α– (α– )f

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–( –p)

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≤

NTα– (α) +

N( –p)Tα– (α–p+ )

δ–q+ ( –q)δ–q(t–t)

<

NTα– (α) +

N( –p)Tα– (α–p+ )

( –q)δ–q<

 

–q_ε <

ε

.

Case . For ≤t<δ,t< δ, we have

c_Dq₍_Fx₎₍_t

) –cDq(Fx)(t)≤

NTα– (α) +

N( –p)Tα– (α–p+ )

(t–t)–q+

t–_ q–t_–q

≤

NTα– (α) +

N( –p)Tα– (α–p+ )

t–_ q

<

NTα– (α) +

N( –p)Tα– (α–p+ )

(δ)–q<ε .

Hence,

(Fx)(t) – (Fx)(t)<ε.

Therefore,Fis equicontinuous and uniformly bounded. The Arzela-Ascoli theorem im-plies thatFis compact onBr, so the operatorFis completely continuous. Thus the con-clusion of Theorem . implies that the anti-periodic boundary value problem () has at least one solution on [,T]. This completes the proof.

Corollary . Let f :J×R×R→R be a continuous function. Assume that (H) There exist a constant l∈(,α– )and a real-valued function m(t)∈L



l_([,_T_{], (,}

∞))such that

f(t,x,y)≤m(t) +d|x|+d|y|,

and(d+d)A< , where d,d≥, A is deﬁned in the proof of Theorem .. Then the problem () has at least a solution on[,T].

The proof of Corollary . is similar to Theorem ..

Theorem . Assume that

(H) There exist a constant r∈(,α– )and a real-valued functionμ(t)∈L



r_([,_T_{], (,}

∞))such that

f(t,x,y) –f(t,u,v)≤μ(t)|x–u|+|y–v|,

for any t∈[,T], x,y,u,v∈R, and if

μ*_Tα–r (α)

 –r α–r

–r

+( –p)μ

*_Tα–r (α–p)

 –r α–p–r

–r

+ (α–r)μ

*_Tα–q–r

(α– )(α–q–r+ )

×

 –r α–r– 

–r

+( –p)μ

*_Tα–q–r

( –q)(α–p)

 –r α–r–p

–r

< , ()

whereμ*_{= (}T (μ(s))



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(10)

≤ t



(t–s)–q

( –q)

s



(s–τ)α– (α– )f

τ,x(τ),cDqx(τ)–fτ,y(τ),cDqy(τ)dτ

ds

+( –p)

T–p

t



(t–s)–q

( –q)

×

T



(T–τ)α–p– (α–p) f

τ,x(τ),cDqx(τ)–fτ,y(τ),cDqy(τ)dτ

ds

≤ x–y

(α– )( –q)

t



(t–s)–q

s



(s–τ)α–μ(τ)dτ

ds

+ x–y( –p)

T–p_{( –}_q₎₍_α_–_p₎

t



(t–s)–q

T



(s–τ)α–p–μ(τ)dτ

ds

≤ x–yμ* (α– )( –q)

 –r α–r– 

–r t



(t–s)–qsa–r–ds

+x–yμ

*_{( –}_p₎_Tα–r–

( –q)(α–p)

 –r α–r–p

–r t



(t–s)–qds

≤x–yμ*Tα–q–r(α–r) (α– )(α–q–r+ )

 –r α–r– 

–r

+x–yμ

*_{( –}_p₎_Tα–q–r

( –q)(α–p)

 –r α–r–p

–r

≤ μ*Tα–q–r(α–r) (α– )(α–q–r+ )

 –r α–r– 

–r

+μ

*_{( –}_p₎_Tα–q–r

( –q)(α–p)

 –r α–r–p

–r

x–y.

Hence, we obtain

Fx–Fy ≤ μ *_Tα–r (α)

 –r α–r

–r

+( –p)μ

*_Tα–r (α–p)

 –r α–p–r

–r

+ (α–r)μ

*_Tα–q–r

(α– )(α–q–r+ )

×

 –r α–r– 

–r

+( –p)μ

*_Tα–q–r

( –q)(α–p)

 –r α–r–p

–r

x–y.

From the assumption (), it follows thatFis a contraction mapping. Therefore, the Banach ﬁxed point theorem yields thatFhas a unique ﬁxed point which is the unique solution of

the problem ().

4 Examples

Example . Letα=

,p=q=



,T= . Consider the following anti-periodic fractional

boundary value problem:

⎧ ⎨ ⎩

c_D__x₍_t_{) =}_f_t_,_x₍_t_),c_D__x₍_t₎_, _t_∈_{[, ],}

x() = –x(), cD_x_{() = –}c_D_x_().

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We have

ft,x(t),cD_x₍_t₎₌_m₍_t_{) +}

t– 



x(t)ρ+cD_x₍_t₎ρ_,

m(t)∈L_{([, ], (,}_∞_{)), }_≤_ρ

,ρ≤.

Since

ft,x(t),cD_x₍_t₎≤_m₍_t₎₊

t– 



x(t)ρ+

t– 



D_x₍_t₎ρ

≤m(t)+  x(t)

ρ + 



c_D__x₍_t₎ρ ,

therefore, by Theorem ., the problem () has at least a solution on [, ].

Example . Consider the following anti-periodic fractional boundary value problem:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

c_D__x₍_t_{) =}  (t+ )

|x+c_D__x_|  +|x+c_D_x|

+ t

,

x() = –x(), c_D__x_{() = –}c_D__x_().

()

We have

ft,x,cD_x_–_f_t_,_y_,c_D_y≤  

|x–y|+cD_x_–c_D_y_.

Obviously,μ(t)≡ 

∈L

_{([, ], (,}_∞_)),_r₌ 

 andμ

*_{= (}T

(μ(s))



rds)r= (

(  )

_ds₎_ ₌



. Note that( 

)≈.,( 

)≈.,( 

)≈., we have

μ*Tα–r (α)

 –r α–r

–r

+( –p)μ

*_Tα–r (α–p)

 –r α–p–r

–r

+ (α–r)μ

*_Tα–q–r

(α– )(α–q–r+ )

×

 –r α–r– 

–r

+( –p)μ

*_Tα–q–r

( –q)(α–p)

 –r α–r–p

–r

= (

 )

  (_)+

(_)  +

(_)

(_)(_)+  

≈. + . + . + . = . < .

Therefore, () has a unique solution on [, ] by Theorem ..

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The author Zhenhai Liu contributed to each part of this study equally and read and approved the ﬁnal version of the manuscript.

Author details

1_{School of Mathematical Science and Computing Technology, Central South University, Changsha, Hunan 410075,}

P.R. China.2_{School of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha,}

Hunan 410076, P.R. China.3_{School of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning,}

(12)

Acknowledgement

The authors are highly grateful for the referee’s careful reading and comments on this note.

Received: 8 April 2012 Accepted: 30 June 2012 Published: 20 July 2012 References

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doi:10.1186/1687-1847-2012-116