Skip to main content

Positive solutions for impulsive fractional differential equations with generalizedperiodic boundary value conditions

Abstract

By constructing Green’s function, we give the natural formulae of solutions forthe following nonlinear impulsive fractional differential equation with generalizedperiodic boundary value conditions:

{ D t q c u ( t ) = f ( t , u ( t ) ) , t J = J { t 1 , , t m } , J = [ 0 , 1 ] , Δ u ( t k ) = I ( u ( t k ) ) , Δ u ( t k ) = J k ( u ( t k ) ) , k = 1 , , m , a u ( 0 ) b u ( 1 ) = 0 , a u ( 0 ) b u ( 1 ) = 0 ,

where 1<q<2 is a real number, D t q c is the standard Caputo differentiation. We present theproperties of Green’s function. Some sufficient conditions for the existence ofsingle or multiple positive solutions of the above nonlinear fractional differentialequation are established. Our analysis relies on a nonlinear alternative of theSchauder and Guo-Krasnosel’skii fixed point theorem on cones. As applications,some interesting examples are provided to illustrate the main results.

MSC: 34B10, 34B15, 34B37.

1 Introduction

In recent years, the fractional order differential equation has aroused great attentiondue to both the further development of fractional order calculus theory and theimportant applications for the theory of fractional order calculus in the fields ofscience and engineering such as physics, chemistry, aerodynamics, electrodynamics ofcomplex medium, polymer rheology, Bode’s analysis of feedback amplifiers,capacitor theory, electrical circuits, electron-analytical chemistry, biology, controltheory, fitting of experimental data, and so forth. Many papers and books on fractionalcalculus differential equation have appeared recently. One can see [117] and the references therein.

In order to describe the dynamics of populations subject to abrupt changes as well asother phenomena such as harvesting, diseases, and so on, some authors have used animpulsive differential system to describe these kinds of phenomena since the lastcentury. For the basic theory on impulsive differential equations, the reader can referto the monographs of Bainov and Simeonov [18], Lakshmikantham et al.[19] and Benchohra et al.[20].

In this article, we consider the following nonlinear impulsive fractional differentialequation with generalized periodic boundary value conditions (for short BVPs (1.1)):

{ D t q c u ( t ) = f ( t , u ( t ) ) , t J = J { t 1 , , t m } , J = [ 0 , 1 ] , Δ u ( t k ) = I k ( u ( t k ) ) , Δ u ( t k ) = J k ( u ( t k ) ) , k = 1 , , m , a u ( 0 ) b u ( 1 ) = 0 , a u ( 0 ) b u ( 1 ) = 0 ,
(1.1)

where a, b are real constants with a>b>0. D 0 + q c is the Caputo fractional derivative of order1<q<2. f:J× R + R + is jointly continuous. I k , J k C( R + , R + ), R + =[0,+). The impulsive point set { t k } k = 1 m satisfies 0= t 0 < t 1 << t m < t m + 1 =1. u( t k + )= lim h 0 + u( t k +h) and u( t k )= lim h 0 u( t k +h) represent the right and left limits ofu(t) at the impulsive point t= t k . Let us set J 0 =[0, t 1 ], J k =( t k , t k + 1 ], 1km. The goal of this paper is to study the existence ofsingle or multiple positive solutions for the impulsive BVPs (1.1) by a nonlinearalternative of the Schauder and Guo-Krasnosel’skii fixed point theorem oncones.

The rest of the paper is organized as follows. In Section 2, we present some usefuldefinitions, lemmas and the properties of Green’s function. In Section 3, wegive some sufficient conditions for the existence of a single positive solution for BVPs(1.1). In Section 4, some sufficient criteria for the existence of multiplepositive solutions for BVPs (1.1) are obtained. Finally, some examples are provided toillustrate our main results in Section 5.

2 Preliminaries

For the convenience of the reader, we present here the necessary definitions fromfractional calculus theory. These definitions and properties can be found in theliterature.

Definition 2.1 (see [21, 22])

The Riemann-Liouville fractional integral of order α>0 of a function f:(0,+)R is given by

I 0 + α f(t)= 1 Γ ( α ) 0 t ( t s ) α 1 f(s)ds,

provided that the right-hand side is pointwise defined on (0,+).

Definition 2.2 (see [21, 22])

The Caputo fractional derivative of order α>0 of a continuous function f:(0,+)R is given by

D 0 + α c f(t)= 1 Γ ( n α ) 0 t f ( n ) ( s ) ( t s ) α n + 1 ds,

where n1<αn, provided that the right-hand side is pointwise defined on(0,+).

Lemma 2.1 (see [21])

Assume thatuC(0,1)L(0,1)with a Caputo fractional derivative oforderq>0that belongs tou C n [0,1], then

I 0 + q D 0 + q u(t)=u(t)+ c 0 + c 1 t++ c n 1 t n 1

for some c i R, i=0,1,2,,n1 (n=[q]) and[q]denotes the integer part of the real number q.

Lemma 2.2 (see [23])

Let E be a Banach space. AssumethatT:EEis a completely continuous operator and thesetV={uEu=μTu,0<μ<1}is bounded. Then T has a fixedpoint in E.

Lemma 2.3 (Schauder fixed point theorem, see [24])

If U is a close bounded convex subset of a Banachspace E andT:UUis completely continuous,then T has at least one fixed point in U.

Lemma 2.4 (see [25])

Let E be a Banach space, PEbe a cone, and Ω 1 , Ω 2 be two bounded open balls of E centeredat the origin with0 Ω 1 and Ω ¯ 1 Ω 2 . Suppose thatA:P( Ω ¯ 2 Ω 1 )Pis a completely continuous operator such thateither

  1. (i)

    Auu, uP Ω 1 andAuu, uP Ω 2 , or

  2. (ii)

    Auu, uP Ω 1 andAuu, uP Ω 2

hold. Then A has at least one fixed pointinP( Ω ¯ 2 Ω 1 ).

Now we present Green’s function for a system associated with BVPs (1.1).

Lemma 2.5 GivenhC(J, R + )and1<q<2, the unique solution of

{ D t q c u ( t ) = h ( t ) , t J , Δ u ( t k ) = I k ( u ( t k ) ) , Δ u ( t k ) = J k ( u ( t k ) ) , k = 1 , , m , a u ( 0 ) b u ( 1 ) = 0 , a u ( 0 ) b u ( 1 ) = 0 , a > b > 0 ,
(2.1)

is formulated by

u(t)= 0 1 G 1 (t,s)h(s)ds+ i = 1 m G 2 (t, t i ) J i ( u ( t i ) ) + i = 1 m G 3 (t, t i ) I i ( u ( t i ) ) ,tJ,

where

G 1 (t,s)= { ( t s ) q 1 Γ ( q ) + b ( 1 s ) q 1 ( a b ) Γ ( q ) + b ( q 1 ) t ( 1 s ) q 2 ( a b ) Γ ( q ) + b 2 ( q 1 ) ( 1 s ) q 2 ( a b ) 2 Γ ( q ) , 0 s t 1 , b ( 1 s ) q 1 ( a b ) Γ ( q ) + b ( q 1 ) t ( 1 s ) q 2 ( a b ) Γ ( q ) + b 2 ( q 1 ) ( 1 s ) q 2 ( a b ) 2 Γ ( q ) , 0 t s 1 ,
(2.2)
G 2 (t, t i )= { a b ( a b ) 2 + a ( t t i ) a b , 0 t i < t 1 , i = 1 , 2 , , m , a b ( a b ) 2 + b ( t t i ) a b , 0 t t i 1 , i = 1 , 2 , , m ,
(2.3)
G 3 (t, t i )= { a a b , 0 t i < t 1 , i = 1 , 2 , , m , b a b , 0 t t i 1 , i = 1 , 2 , , m .
(2.4)

Proof Let u be a general solution of (2.1) on each interval( t k , t k + 1 ] (k=0,1,,m). Applying Lemma 2.1, Eq. (2.1) is translated intothe following equivalent integral equation (2.5):

u(t)= 1 Γ ( q ) 0 t ( t s ) q 1 h(s)ds c k d k t,t( t k , t k + 1 ],
(2.5)

where t 0 =0, t m + 1 =1. Then we have

u (t)= 1 Γ ( q 1 ) 0 t ( t s ) q 2 h(s)ds d k ,t( t k , t k + 1 ].

In the light of the generalized periodic boundary value conditions of Eq. (2.1), we get

b 0 1 ( 1 s ) q 1 Γ ( q ) h(s)ds+a c 0 b c m b d m =0,
(2.6)
b 0 1 ( 1 s ) q 2 Γ ( q 1 ) h(s)ds+a d 0 b d m =0.
(2.7)

Next, using the right impulsive condition of Eq. (2.1), we derive

c k 1 c k = I k ( u ( t k ) ) J k ( u ( t k ) ) t k ,
(2.8)
d k 1 d k = J k ( u ( t k ) ) .
(2.9)

By (2.7) and (2.9), we have

d 0 = b a b 0 1 ( 1 s ) q 2 Γ ( q 1 ) h(s)ds b a b i = 1 m J i ( u ( t i ) ) ,
(2.10)
d m = b a b 0 1 ( 1 s ) q 2 Γ ( q 1 ) h(s)ds a a b i = 1 m J i ( u ( t i ) ) .
(2.11)

By (2.9) we have

d k = d 0 i = 1 k J i ( u ( t i ) ) = b a b 0 1 ( 1 s ) q 2 Γ ( q 1 ) h ( s ) d s b a b i = 1 m J i ( u ( t i ) ) i = 1 k J i ( u ( t i ) ) .
(2.12)

From (2.6), (2.8) and (2.11), we have

c 0 = b a b 0 1 ( 1 s ) q 1 Γ ( q ) h ( s ) d s b 2 ( a b ) 2 0 1 ( 1 s ) q 2 Γ ( q 1 ) h ( s ) d s a b ( a b ) 2 i = 1 m J i ( u ( t i ) ) b a b i = 1 m ( I i ( u ( t i ) ) J i ( u ( t i ) ) t i ) .
(2.13)

According to (2.8), we obtain

c k = c 0 i = 1 k ( I i ( u ( t i ) ) J i ( u ( t i ) ) t i ) = b a b 0 1 ( 1 s ) q 1 Γ ( q ) h ( s ) d s b 2 ( a b ) 2 0 1 ( 1 s ) q 2 Γ ( q 1 ) h ( s ) d s a b ( a b ) 2 i = 1 m J i ( u ( t i ) ) b a b i = 1 m I i ( u ( t i ) ) i = 1 k I i ( u ( t i ) ) + b a b i = 1 m J i ( u ( t i ) ) t i + i = 1 k J i ( u ( t i ) ) t i .
(2.14)

Hence, for k=1,2,,m, (2.12) and (2.14) imply

c k + d k t = b a b 0 1 ( 1 s ) q 1 Γ ( q ) h ( s ) d s ( a b b 2 ) t + b 2 ( a b ) 2 0 1 ( 1 s ) q 2 Γ ( q 1 ) h ( s ) d s 1 ( a b ) 2 i = 1 m J i ( u ( t i ) ) [ a b + b ( a b ) ( t t i ) ] b a b i = 1 m I i ( u ( t i ) ) i = 1 k I i ( u ( t i ) ) i = 1 k J i ( u ( t i ) ) ( t t i ) .
(2.15)

Now substituting (2.10) and (2.13) into (2.5), for t J 0 =[0, t 1 ], we obtain

u ( t ) = 0 t ( t s ) q 1 Γ ( q ) h ( s ) d s + b a b 0 1 ( 1 s ) q 1 Γ ( q ) h ( s ) d s + ( a b b 2 ) t + b 2 ( a b ) 2 0 1 ( 1 s ) q 2 Γ ( q 1 ) h ( s ) d s + 1 ( a b ) 2 i = 1 m J i ( u ( t i ) ) [ a b + b ( a b ) ( t t i ) ] + b a b i = 1 m I i ( u ( t i ) ) = 0 t ( t s ) q 1 Γ ( q ) h ( s ) d s + b a b ( 0 t + t 1 ) ( 1 s ) q 1 Γ ( q ) h ( s ) d s + ( a b b 2 ) t + b 2 ( a b ) 2 ( 0 t + t 1 ) ( 1 s ) q 2 Γ ( q 1 ) h ( s ) d s + 1 ( a b ) 2 i = 1 m J i ( u ( t i ) ) [ a b + b ( a b ) ( t t i ) ] + b a b i = 1 m I i ( u ( t i ) ) = 0 t [ ( t s ) q 1 Γ ( q ) + b ( 1 s ) q 1 ( a b ) Γ ( q ) + [ ( a b b 2 ) t + b 2 ] ( 1 s ) q 2 ( a b ) 2 Γ ( q 1 ) ] h ( s ) d s + t 1 [ b ( 1 s ) q 1 ( a b ) Γ ( q ) + [ ( a b b 2 ) t + b 2 ] ( 1 s ) q 2 ( a b ) 2 Γ ( q 1 ) ] h ( s ) d s + 1 ( a b ) 2 i = 1 m J i ( u ( t i ) ) [ a b + b ( a b ) ( t t i ) ] + b a b i = 1 m I i ( u ( t i ) ) = 0 1 G 1 ( t , s ) h ( s ) d s + i = 1 m G 2 ( t , t i ) J i ( u ( t i ) ) + i = 1 m G 3 ( t , t i ) I i ( u ( t i ) ) ,

where G 1 (t,s), G 2 (t, t i ) and G 3 (t, t i ) are defined by (2.2)-(2.4).

Substituting (2.15) into (2.5), for t J k =( t k , t k + 1 ], k=1,2,,m, we have

u ( t ) = 0 t ( t s ) q 1 Γ ( q ) h ( s ) d s + b a b 0 1 ( 1 s ) q 1 Γ ( q ) h ( s ) d s + ( a b b 2 ) t + b 2 ( a b ) 2 0 1 ( 1 s ) q 2 Γ ( q 1 ) h ( s ) d s + 1 ( a b ) 2 i = 1 m J i ( u ( t i ) ) [ a b + b ( a b ) ( t t i ) ] + b a b i = 1 m I i ( u ( t i ) ) + i = 1 k I i ( u ( t i ) ) + i = 1 k J i ( u ( t i ) ) ( t t i ) = 0 t ( t s ) q 1 Γ ( q ) h ( s ) d s + b a b ( 0 t + t 1 ) ( 1 s ) q 1 Γ ( q ) h ( s ) d s + ( a b b 2 ) t + b 2 ( a b ) 2 ( 0 t + t 1 ) ( 1 s ) q 2 Γ ( q 1 ) h ( s ) d s + 1 ( a b ) 2 ( i = 1 k + i = k + 1 m ) J i ( u ( t i ) ) [ a b + b ( a b ) ( t t i ) ] + b a b ( i = 1 k + i = k + 1 m ) I i ( u ( t i ) ) + i = 1 k I i ( u ( t i ) ) + i = 1 k J i ( u ( t i ) ) ( t t i ) = 0 t [ ( t s ) q 1 Γ ( q ) + b ( 1 s ) q 1 ( a b ) Γ ( q ) + [ ( a b b 2 ) t + b 2 ] ( 1 s ) q 2 ( a b ) 2 Γ ( q 1 ) ] h ( s ) d s + t 1 [ b ( 1 s ) q 1 ( a b ) Γ ( q ) + [ ( a b b 2 ) t + b 2 ] ( 1 s ) q 2 ( a b ) 2 Γ ( q 1 ) ] h ( s ) d s + i = 1 k [ a b ( a b ) 2 + b ( a b ) + ( a b ) 2 ( a b ) 2 ( t t i ) ] J i ( u ( t i ) ) + i = k + 1 m a b + b ( a b ) ( t t i ) ( a b ) 2 J i ( u ( t i ) ) + i = 1 k [ b a b + 1 ] I i ( u ( t i ) ) + i = k + 1 m b a b I i ( u ( t i ) ) = 0 t [ ( t s ) q 1 Γ ( q ) + b ( 1 s ) q 1 ( a b ) Γ ( q ) + [ ( a b b 2 ) t + b 2 ] ( 1 s ) q 2 ( a b ) 2 Γ ( q 1 ) ] h ( s ) d s + t 1 [ b ( 1 s ) q 1 ( a b ) Γ ( q ) + [ ( a b b 2 ) t + b 2 ] ( 1 s ) q 2 ( a b ) 2 Γ ( q 1 ) ] h ( s ) d s + i = 1 k a b + a ( a b ) ( t t i ) ( a b ) 2 J i ( u ( t i ) ) + i = k + 1 m a b + b ( a b ) ( t t i ) ( a b ) 2 J i ( u ( t i ) ) + i = 1 k a a b I i ( u ( t i ) ) + i = k + 1 m b a b I i ( u ( t i ) ) = 0 1 G 1 ( t , s ) h ( s ) d s + i = 1 m G 2 ( t , t i ) J i ( u ( t i ) ) + i = 1 m G 3 ( t , t i ) I i ( u ( t i ) ) ,

where G 1 (t,s), G 2 (t, t i ) and G 3 (t, t i ) are defined by (2.2)-(2.4). The proof iscomplete. □

Lemma 2.6 Let0<b<a<+, then Green’sfunctions G 1 (t,s), G 2 (t, t i )and G 3 (t, t i )defined by (2.2), (2.3) and (2.4) arecontinuous and satisfy the following:

  1. (i)

    G 1 (t,s)C(J×J, R + ), G 2 (t, t i ), G 3 (t, t i )C(J×J, R + ), and G 1 (t,s), G 2 (t, t i ), G 3 (t, t i )>0for allt, t i ,s(0,1), whereJ=[0,1].

  2. (ii)

    The functions G 1 (t,s), G 2 (t, t i )and G 3 (t, t i )have the following properties:

    b a M(s) G 1 (t,s)M(s),tJ,s(0,1),
    (2.16)
b 2 ( a b ) 2 G 2 (t, t i ) a 2 ( a b ) 2 ,t, t i J,
(2.17)
b a b G 3 (t, t i ) a a b ,t, t i J,
(2.18)

where

M(s)= a [ ( 1 s ) a ( 2 s q ) b ] ( 1 s ) q 2 ( a b ) 2 Γ ( q ) >0,s[0,1).
(2.19)

Proof From the expressions of G 1 (t,s), G 2 (t, t i ) and G 3 (t, t i ), it is obvious that G 1 (t,s), G 2 (t, t i ), G 3 (t, t i )C(J×J, R + ) and G 1 (t,s), G 2 (t, t i ), G 3 (t, t i )>0 for all t, t i ,s(0,1). Next, we will prove (ii). From the definition of G 1 (t,s), we can know that, for given s(0,1), G 1 (t,s) is increasing with respect to t fortJ. We let

g 1 ( t , s ) = ( a b ) 2 ( t s ) q 1 + [ ( a b ) ( 1 s ) + [ ( a b ) t + b ] ( q 1 ) ] b ( 1 s ) q 2 ( a b ) 2 Γ ( q ) , t [ s , 1 ] , g 2 ( t , s ) = [ ( a b ) ( 1 s ) + [ ( a b ) t + b ] ( q 1 ) ] b ( 1 s ) q 2 ( a b ) 2 Γ ( q ) , t [ 0 , s ] .

Hence, we derive

min t [ 0 , 1 ] G 1 ( t , s ) = min { min t [ s , 1 ] g 1 ( t , s ) , min t [ 0 , s ] g 2 ( t , s ) } = min { g 1 ( s , s ) , g 2 ( 0 , s ) } = g 2 ( 0 , s ) = [ ( a b ) ( 1 s ) + b ( q 1 ) ] b ( 1 s ) q 2 ( a b ) 2 Γ ( q ) = b [ ( 1 s ) a ( 2 s q ) b ] ( 1 s ) q 2 ( a b ) 2 Γ ( q ) m ( s ) , s [ 0 , 1 ) , max t [ 0 , 1 ] G 1 ( t , s ) = max { max t [ s , 1 ] g 1 ( t , s ) , max t [ 0 , s ] g 2 ( t , s ) } = max { g 1 ( 1 , s ) , g 2 ( s , s ) } = g 1 ( 1 , s ) = [ ( a b ) ( 1 s ) + b ( q 1 ) ] a ( 1 s ) q 2 ( a b ) 2 Γ ( q ) = a [ ( 1 s ) a ( 2 s q ) b ] ( 1 s ) q 2 ( a b ) 2 Γ ( q ) M ( s ) , s [ 0 , 1 ) .

Thus, we have

b a M(s)=m(s) G 1 (t,s)M(s).

It is obvious that

b 2 ( a b ) 2 = G 2 (0,1) G 2 (t, t i ) G 2 (1,0)= a 2 ( a b ) 2 , b a b G 3 (t, t i ) a a b .

The proof is completed. □

3 Existence of single positive solutions

In this section, we discuss the existence of positive solutions for BVP (1.1).

Let E={u(t):u(t)C(J)} denote a real Banach space with the norm defined by u= max t J |u(t)|. Let

P C ( J ) = { u E u : J R + , u C ( J ) , u ( t k )  and  u ( t k + ) exist with  u ( t k ) = u ( t k ) , 1 k m } , K = { u P C ( J ) : u ( t ) b 2 a 2 u , t J } ,
(3.1)
K r = { u K : u < r } , K r = { u K : u = r } .
(3.2)

Obviously, PC(J)E is a Banach space with the norm u= max t J |u(t)|. KPC(J) is a positive cone.

In the following, we need the assumptions and some notations as follows:

(B1) 0<b<a<1, 0< σ 1 , σ 2 <+, where σ 1 = 0 1 M(s)ds, σ 2 = b 3 a 3 0 1 M(s)ds.

(B2) fC(J× R + , R + ) and f(t,0)=0 for all tJ.

(B3) I k (u( t k )), J k (u( t k ))C( R + , R + ), k=1,2,,m.

Let

N ¯ = max { a 2 ( a b ) 2 i = 1 m J i ( u ( t i ) ) , a a b i = 1 m I i ( u ( t i ) ) } , f δ = lim sup u δ max t J f ( t , u ) u , f δ = lim inf u δ min t J f ( t , u ) u ,

where δ denotes 0 or +∞. In addition, we introduce the followingweight functions:

Φ ( r ) = max { f ( t , u ( t ) ) : ( t , u ) [ 0 , 1 ] × [ b 2 a 2 r , r ] } , ϕ ( r ) = min { f ( t , u ( t ) ) : ( t , u ) [ 0 , 1 ] × [ b 2 a 2 r , r ] } .

From Lemma 2.4, we can obtain the following lemma.

Lemma 3.1 Suppose thatf(t,u)is continuous,thenuPC(J)is a solution of BVPs (1.1) if and onlyifuPC(J)is a solution of the integral equation

u(t)= 0 1 G 1 (t,s)f ( s , u ( s ) ) ds+ i = 1 m G 2 (t, t i ) J i ( u ( t i ) ) + i = 1 m G 3 (t, t i ) I i ( u ( t i ) ) ,tJ.

Define T:PC(J)PC(J) to be the operator defined as

(Tu)(t)= 0 1 G 1 (t,s)f ( s , u ( s ) ) ds+ i = 1 m G 2 (t, t i ) J i ( u ( t i ) ) + i = 1 m G 3 (t, t i ) I i ( u ( t i ) ) .
(3.3)

Then, by Lemma 3.1, the existence of solutions for BVPs (1.1) is translated intothe existence of the fixed point for u=Tu, where T is given by (3.3). Thus, the fixed pointof the operator T coincides with the solution of problem (1.1).

Lemma 3.2 Assume that (B1)-(B3) hold,thenT:PC(J)PC(J)andT:KKdefined by (3.3) are completelycontinuous.

Proof Firstly, we shall show that T:PC(J)PC(J) is completely continuous through three steps.

Step 1. Let uPC(J), in view of the nonnegativity and continuity of functions G 1 (t,s), G 2 (t, t i ), G 3 (t, t i ), f(t,u(t)), I k , J k and a>b>0, we conclude that T:PC(J)PC(J) is continuous.

Step 2. We will prove that T maps bounded sets into bounded sets. Indeed, it isenough to show that for any r>0 there exists a positive constant L such that, foreach u Ω r ={uPC(J):ur}, TuL when |f(t,u)| l 1 , | J k | l 2 , | I k | l 3 , where l i (i=1,2,3) are some fixed positive constants. In fact, for eacht J k , u Ω r , k=0,1,2,,m, by Lemma 2.5, we have

| ( T u ) ( t ) | 0 1 | G 1 ( t , s ) f ( s , u ( s ) ) | d s + i = 1 m | G 2 ( t , t i ) J i ( u ( t i ) ) | + i = 1 m | G 3 ( t , t i ) I i ( u ( t i ) ) | σ 1 l 1 + a 2 m l 2 ( a b ) 2 + a m l 3 a b L ,

which imply that TuL.

Step 3. T is equicontinuous. In fact, since G 1 (t,s), G 2 (t, t i ), G 3 (t, t i ) are continuous on J×J, they are uniformly continuous on J×J. Thus, for fixed sJ and for any ε>0, there exists a constant δ>0 such that for any t 1 , t 2 J k with | t 1 t 2 |<δ, 0km, we have

| G 1 ( t 1 , s ) G 2 ( t 2 , s ) | < ε 3 l 1 , | G 2 ( t 1 , t i ) G 2 ( t 2 , t i ) | < ε 3 m l 2 , | G 3 ( t 1 , t i ) G 3 ( t 2 , t i ) | < ε 3 m l 3 .

Then

| T u ( t 2 ) T u ( t 1 ) | = | 0 1 ( G 1 ( t 2 , s ) G 1 ( t 1 , s ) ) f ( s , u ( s ) ) d s + i = 1 m ( G 2 ( t 2 , t i ) G 2 ( t 1 , t i ) ) J i ( u ( t i ) ) + i = 1 m ( G 3 ( t 2 , t i ) G 3 ( t 1 , t i ) ) I i ( u ( t i ) ) | l 1 0 1 | G 1 ( t 2 , s ) G 2 ( t 1 , s ) | d s + m l 2 | G 2 ( t 2 , t i ) G 2 ( t 1 , t i ) | + m l 3 | G 3 ( t 2 , t i ) G 3 ( t 1 , t i ) | < ε 3 + ε 3 + ε 3 = ε ,

which means that T( Ω r ) is equicontinuous on all the subintervalst J k , k=0,1,,m. Thus, by means of the Arzela-Ascoli theorem, we have thatT:PC(J)PC(J) is completely continuous.

Next, we will show that T:KK is completely continuous. Indeed, for eacht J k , every uC( J k , R + ), k=0,1,2,,m, Lemma 2.5 implies that

(Tu)(t) b a 0 1 M(s)f ( s , u ( s ) ) ds+ b 2 ( a b ) 2 i = 1 m J i ( u ( t i ) ) + b a b i = 1 m I i ( u ( t i ) ) .

On the other hand,

Tu= max t J k (Tu)(t) 0 1 M(s)f ( s , u ( s ) ) ds+ a 2 ( a b ) 2 i = 1 m J i ( u ( t i ) ) + a a b i = 1 m I i ( u ( t i ) ) .

Thus,

b 2 a 2 T u = b 2 a 2 max t J k ( T u ) ( t ) b 2 a 2 0 1 M ( s ) f ( s , u ( s ) ) d s + b 2 ( a b ) 2 i = 1 m J i ( u ( t i ) ) + b 2 a ( a b ) i = 1 m I i ( u ( t i ) ) ( T u ) ( t ) .

So (Tu)(t) b 2 a 2 Tu for every uC(J, R + ), which implies T(K)K. Similar to the above arguments, we can easily concludethat T:KK is a completely continuous operator. The proof iscomplete. □

Theorem 3.1 Assume that (B1)-(B3) hold,and suppose that the following assumptions hold:

(A1) There exists a constant L 1 >0such that|f(t,u)f(t,v)| L 1 |uv|for eachtJand allu,v R + .

(A2) There exists a constant L 2 >0such that| J k (u) J k (v)| L 2 |uv|for allu,v R + , k=1,2,,m.

(A3) There exists a constant L 3 >0such that| I k (u) I k (v)| L 3 |uv|for allu,v R + , k=1,2,,m.

Ifρ= σ 1 L 1 + m a 2 L 2 ( a b ) 2 + m a L 3 a b <1, then problem (1.1) has a unique solutionin K ρ .

Proof Let the operator T: K ρ K ρ be defined by (3.3). For all u,v K ρ , from Lemma 2.5, we obtain

| ( T u ) ( t ) ( T v ) ( t ) | 0 1 G 1 ( t , s ) | f ( s , u ( s ) ) f ( s , v ( s ) ) | d s + i = 1 m G 2 ( t , t i ) | J i ( u ( t i ) ) J i ( v ( t i ) ) | + i = 1 m G 3 ( t , t i ) | I i ( u ( t i ) ) I i ( v ( t i ) ) | σ 1 L 1 u v + m a 2 L 2 ( a b ) 2 u v + m a L 3 a b u v = ρ u v ,

where ρ= σ 1 L 1 + m a 2 L 2 ( a b ) 2 + m a L 3 a b <1. Consequently, T is a contraction mapping.Moreover, from Lemma 3.2, T is completely continuous. Therefore, by theBanach contraction map principle, the operator T has a unique fixed point in K ρ which is the unique positive solution of system (1.1).This completes the proof. □

Theorem 3.2 Assume that (B1)-(B3) hold,and suppose that the following assumptions hold:

(A4) There exists a constant N 1 >0such that|f(t,u)| N 1 for eachtJand allu R + .

(A5) There exists a constant N 2 >0such that| J k (u)| N 2 for allu R + , k=1,2,,m.

(A6) There exists a constant N 3 >0such that| I k (u)| N 3 for allu R + , k=1,2,,m.

Then BVPs (1.1) have at least one positive solutioninPC(J).

Proof Let T:PC(J)PC(J) be cone preserving completely continuous that is definedby (3.3). According to Lemma 2.2, now it remains to show that the set

Ω= { u P C ( J ) u = λ T u  for some  0 < λ < 1 }
(3.4)

is bounded.

Let uΩ, then u=λTu for some 0<λ<1. Thus, by Lemma 2.5, for each t J k , k=0,1,,m, we have

| u ( t ) | = | λ T u | 0 1 | G 1 ( t , s ) f ( s , u ( s ) ) | d s + i = 1 m | G 2 ( t , t i ) J i ( u ( t i ) ) | + i = 1 m | G 3 ( t , t i ) I i ( u ( t i ) ) | σ 1 N 1 + a 2 m N 2 ( a b ) 2 + a m N 3 a b .

Thus, for every tJ, we have u(t) σ 1 N 1 + a 2 m N 2 ( a b ) 2 + a m N 3 a b , which indicates that the set Ω is bounded. Accordingto Lemma 2.2, T has a fixed point uPC(J). Therefore, BVPs (1.1) have at least one positivesolution. The proof is complete. □

In the following, we present an existence result when the nonlinearity and the impulsefunctions have sublinear growth.

Theorem 3.3 Assume that (B1)-(B3) hold andsuppose that the following assumptions hold:

(A7) There exist a 1 PC(J), b 1 >0andα[0,1)such that|f(t,u)| a 1 (t)+ b 1 | u | α for eachtJand allu R + .

(A8) There exist constants a 2 , b 2 >0andα[0,1)such that| J k (u)| a 2 + b 2 | u | α for allu R + , k=1,2,,m.

(A9) There exist constants a 3 , b 3 >0andα[0,1)such that| I k (u)| a 3 + b 3 | u | α for allu R + , k=1,2,,m.

(A10) b <1, a + b 1, where a =p σ 1 + a 2 m a 2 ( a b ) 2 + a m a 3 a b , b = b 1 σ 1 + a 2 m b 2 ( a b ) 2 + a m b 3 a b .

Then BVPs (1.1) have at least one positive solutioninPC(J).

Proof Let T:PC(J)PC(J) and Ω be defined by (3.3) and (3.4), respectively.Denote p= max t J | a 1 (t)|. If uΩ, then for tJ we have

| u ( t ) | = | λ T u | 0 1 | G 1 ( t , s ) ( a 1 ( s ) + b 1 | u ( s ) | α ) | d s + i = 1 m | G 2 ( t , t i ) ( a 2 + b 2 | u | α ) | + i = 1 m | G 3 ( t , t i ) ( a 3 + b 3 | u | α ) | p 0 1 M ( s ) d s + b 1 u α 0 1 M ( s ) d s + a 2 m ( a 2 + b 2 u α ) ( a b ) 2 + a m ( a 3 + b 3 u α ) a b = a + b u α ,

which imply that u a + b u α . When 0<u1, then u a + b . When u>1, then u a 1 b . Taking C=max{ a + b , a 1 b ,}, we have uC for any solution of (3.4). This shows that the set Ωis bounded. According to Lemma 2.2, T has at least one fixed point inPC(J). Therefore, BVPs (1.1) have at least one positive solutionin PC(J). The proof is complete. □

Theorem 3.4 Assume that (B1)-(B3) hold.And suppose that one of the following conditions is satisfied:

(H1) f < 1 σ 1 (particularly, f =0).

(H2) There exists a constantM>0such thatf(t,u) M σ 1 fortJ, u[M,+).

(H3) There exists a constantN>0such thatΦ(N) N 3 σ 1 fortJ, u[ b 2 a 2 N,N].

Then BVPs (1.1) have at least one positive solution.

Proof Case 1. Considering f < 1 σ 1 , there exists r ¯ 1 >0 such that f(t,u)( f + ε 1 )u for all u( r ¯ 1 ,+), tJ, where ε 1 satisfies σ 1 ( f + ε 1 )1.

Choose r 1 >max{ r ¯ 1 ,2 N ¯ ( 1 σ 1 ( f + ε 1 ) ) 1 }, let u Ω 1 K r 1 . We can easily know that Ω 1 is a close bounded convex subset of a Banach spacePC(J). Then, for tJ, u Ω 1 , in view of the nonnegativity and continuity of functions G 1 (t,s), G 2 (t, t i ), G 3 (t, t i ), f(t,u(t)), I k , J k and a>b>0, we conclude that TuP, Tu0, tJ. By Lemma 2.5, we can obtain the followinginequality:

b 2 a 2 T u = b 2 a 2 max t J ( T u ) ( t ) b 2 a 2 [ 0 1 M ( s ) f ( s , u ( s ) ) d s + a 2 ( a b ) 2 i = 1 m J i ( u ( t i ) ) + a a b i = 1 m I i ( u ( t i ) ) ] b 2 a 2 [ 0 1 M ( s ) f ( s , u ( s ) ) d s + a 2 ( a b ) 2 i = 1 m J i ( u ( t i ) ) + a a b i = 1 m I i ( u ( t i ) ) ] b a 0 1 M ( s ) f ( s , u ( s ) ) d s + b 2 ( a b ) 2 i = 1 m J i ( u ( t i ) ) + b a b i = 1 m I i ( u ( t i ) ) ( T u ) ( t ) , t J .

Thus TuK.

Next, we prove Tu r 1 . Indeed, for tJ, u K r 1 , we get

T u = max t J ( T u ) ( t ) 0 1 M ( s ) f ( s , u ( s ) ) d s + a 2 ( a b ) 2 i = 1 m J i ( u ( t i ) ) + a a b i = 1 m I i ( u ( t i ) ) 0 1 M ( s ) ( f + ε 1 ) u ( s ) d s + a 2 ( a b ) 2 i = 1 m J i ( u ( t i ) ) + a a b i = 1 m I i ( u ( t i ) ) σ 1 ( f + ε 1 ) u + 2 N ¯ < σ 1 ( f + ε 1 ) r 1 + r 1 σ 1 ( f + ε 1 ) r 1 = r 1 .

Therefore, T( Ω 1 ) Ω 1 . From Lemma 3.2, we have that T: Ω 1 Ω 1 is completely continuous. Thus BVPs (1.1) have at least apositive solution by Lemma 2.3.

Case 2. Condition (H2) holds. Let u Ω 2 K d , where d>0 satisfies d1+M+ σ 1 max t J , u [ 0 , M ] f(t,u)+2 N ¯ . By the ways of Case 1, we can also getTuK. Now we prove Tud. In fact,

T u = max t J ( T u ) ( t ) 0 1 M ( s ) f ( s , u ( s ) ) d s + a 2 ( a b ) 2 i = 1 m J i ( u ( t i ) ) + a a b i = 1 m I i ( u ( t i ) ) s J , u ( s ) > M M ( s ) f ( s , u ( s ) ) d s + s J , 0 u ( s ) M M ( s ) f ( s , u ( s ) ) d s + 2 N ¯ 0 1 M ( s ) M σ 1 d s + 0 1 M ( s ) d s max t J , u [ 0 , M ] f ( t , u ) + 2 N ¯ = M + σ 1 max t J , u [ 0 , M ] f ( t , u ) + 2 N ¯ < d .

Therefore, T( Ω 2 ) Ω 2 . From Lemma 3.2 we have that T: Ω 2 Ω 2 is completely continuous. Thus BVPs (1.1) have at least apositive solution by Lemma 2.3.

Case 3. Condition (H3) holds. Let u Ω 3 K N , where N>0 satisfies N3 N ¯ , we get b 2 a 2 uu(t)u. By the ways of Case 1, we can also getTuK. Now we prove TuN. By assumption (H3), we have

f(t,u)Φ(N) N 3 σ 1 ,tJ,u [ b 2 a 2 N , N ] .

In view of Lemma 2.6, we have

T u = max t J ( T u ) ( t ) 0 1 M ( s ) f ( s , u ( s ) ) d s + a 2 ( a b ) 2 i = 1 m J i ( u ( t i ) ) + a a b i = 1 m I i ( u ( t i ) ) 0 1 M ( s ) N 3 σ 1 d s + 2 N ¯ N 3 + 2 N 3 = N .

Therefore, T( Ω 3 ) Ω 3 . From Lemma 3.2 we have T: Ω 3 Ω 3 is completely continuous. Thus BVPs (1.1) have at least apositive solution by Lemma 2.3. We complete the proof ofTheorem 3.4. □

4 Existence of multiple positive solutions

In this section, we discuss the multiplicity of positive solutions for BVPs (1.1) by theGuo-Krasnoselskii fixed point theorem.

Theorem 4.1 Assume that (B1)-(B3) hold,and suppose that the following two conditions are satisfied:

(H4) f 0 > 1 σ 2 and f > 1 σ 2 (particularly, f 0 = f =).

(H5) There exists a constantc3 N ¯ such thatΦ(c)< c 3 σ 1 fortJ, u[ b 2 a 2 c,c].

Then for BVPs (1.1) there exist at least two positivesolutions u 1 , u 2 , which satisfy

0< u 1 <c< u 2 .
(4.1)

Proof Choose r, R with 0<r<c<R. Considering f 0 > 1 σ 2 , there exists r>0 such that f(t,u)( f 0 ε 2 )u for tJ, u[0,r], where ε 2 >0 satisfies ( f 0 ε 2 ) σ 2 1. Then, for u K r , tJ, we have

T u = max t J ( T u ) ( t ) b a 0 1 M ( s ) f ( s , u ( s ) ) d s + b 2 ( a b ) 2 i = 1 m J i ( u ( t i ) ) + b a b i = 1 m I i ( u ( t i ) ) b a 0 1 M ( s ) f ( s , u ( s ) ) d s b a 0 1 M ( s ) ( f 0 ε 2 ) u ( s ) d s b a 0 1 M ( s ) ( f 0 ε 2 ) b 2 a 2 u d s = ( f 0 ε 2 ) σ 2 u u .

Therefore,

Tuu,u K r .
(4.2)

Considering f > 1 σ 2 , there exists R>0 such that f(t,u)( f ε 3 )u for tJ, u[R,), where ε 3 >0 satisfies ( f ε 3 ) σ 2 1. Then, for u K R , tJ, we have

T u = max t J ( T u ) ( t ) b a 0 1 M ( s ) f ( s , u ( s ) ) d s + b 2 ( a b ) 2 i = 1 m J i ( u ( t i ) ) + b a b i = 1 m I i ( u ( t i ) ) b a 0 1 M ( s ) f ( s , u ( s ) ) d s b a 0 1 M ( s ) ( f ε 3 ) u ( s ) d s b a 0 1 M ( s ) ( f ε 3 ) b 2 a 2 u d s = ( f ε 3 ) σ 2 u u .

So

Tuu,u K R .
(4.3)

On the other hand, by assumption (H5), we have

f(t,u)Φ(c)< c 3 σ 1 ,for tJ,u [ b 2 a 2 c , c ] .

For u K c , where c>0 satisfies c3 N ¯ . In view of Lemma 2.6, we have

T u = max t J ( T u ) ( t ) 0 1 M ( s ) f ( s , u ( s ) ) d s + a 2 ( a b ) 2 i = 1 m J i ( u ( t i ) ) + a a b i = 1 m I i ( u ( t i ) ) < 0 1 M ( s ) c 3 σ 1 d s + 2 N ¯ c 3 + 2 c 3 = c = u .

Therefore,

Tu<u,u K c .
(4.4)

Thus, applying Lemma 2.4 to (4.2)-(4.4) yields that T has the fixed point u 1 K( K ¯ c K r ) and the fixed point u 2 K( K ¯ R K c ). Thus it follows that problem (1.1) has at least twopositive solutions u 1 and u 2 . Noticing (4.4), we have u 1 c and u 2 c. Therefore (4.1) holds. The proof iscomplete. □

Theorem 4.2 Assume that (B1)-(B3) hold.Further suppose that there exist three positivenumbers ξ i (i=1,2,3) with3 N ¯ ξ 1 < ξ 2 < ξ 3 such that one of the following conditions issatisfied:

(H6) ϕ( ξ 1 ) ξ 1 σ 2 , Φ( ξ 2 ) ξ 2 3 σ 1 , ϕ( ξ 3 ) ξ 3 σ 2 .

(H7) Φ( ξ 1 ) ξ 1 3 σ 1 , ϕ( ξ 2 )> ξ 2 σ 2 , Φ( ξ 3 ) ξ 3 3 σ 1 .

Then BVPs (1.1) have at least two positivesolutions u 1 , u 2 with

ξ 1 u 1 < ξ 2 < u 2 ξ 3 .
(4.5)

Proof Because the proofs are similar, we prove only case (H6).Considering ϕ( ξ 1 ) ξ 1 σ 2 , we have f(t,u)ϕ( ξ 1 ) ξ 1 σ 2 for tJ, u[ b 2 a 2 ξ 1 , ξ 1 ]. Then, for u K ξ 1 , tJ, we have

| T u = max t J ( T u ) ( t ) b a 0 1 M ( s ) f ( s , u ( s ) ) d s + b 2 ( a b ) 2 i = 1 m J i ( u ( t i ) ) + b a b i = 1 m I i ( u ( t i ) ) b a 0 1 M ( s ) f ( s , u ( s ) ) d s b a 0 1 M ( s ) ξ 1 σ 2 d s b 3 a 3 0 1 M ( s ) d s ξ 1 σ 2 = ξ 1 = u .

Therefore,

Tuu,u K ξ 1 .
(4.6)

Considering Φ( ξ 2 ) ξ 2 3 σ 1 , we have f(t,u)Φ( ξ 2 ) ξ 2 3 σ 1 for tJ, u[ b 2 a 2 ξ 2 , ξ 2 ]. Then, for u K ξ 2 , tJ, we derive

T u = max t J ( T u ) ( t ) 0 1 M ( s ) f ( s , u ( s ) ) d s + a 2 ( a b ) 2 i = 1 m J i ( u ( t i ) ) + a a b i = 1 m I i ( u ( t i ) ) 0 1 M ( s ) ξ 2 3 σ 1 d s + 2 N ¯ ξ 2 3 + 2 ξ 1 3 < ξ 2 3 + 2 ξ 2 3 = ξ 2 = u .

So,

Tu<u,u K ξ 2 .
(4.7)

Considering ϕ( ξ 3 ) ξ 3 σ 2 , we have f(t,u)ϕ( ξ 3 ) ξ 3 σ 2 for tJ, u[ b 2 a 2 ξ 3 , ξ 3 ]. Then, for u K ξ 3 , tJ, we have

T u = max t J ( T u ) ( t ) b a 0 1 M ( s ) f ( s , u ( s ) ) d s + b 2 ( a b ) 2 i = 1 m J i ( u ( t i ) ) + b a b i = 1 m I i ( u ( t i ) ) b a 0 1 M ( s ) f ( s , u ( s ) ) d s b a 0 1 M ( s ) ξ 3 σ 2 d s b 3 a 3 0 1 M ( s ) d s ξ 3 σ 2 = ξ 3 = u .

Therefore,

Tuu,u K ξ 3 .
(4.8)

Thus, applying Lemma 2.4 to (4.6)-(4.8) yields that T has the fixed point u 1 K( K ¯ ξ 2 K ξ 1 ) and the fixed point u 2 K( K ¯ ξ 3 K ξ 2 ). Thus it follows that BVPs (1.1) have at least twopositive solutions u 1 and u 2 . Noticing (4.7), we have u 1 ξ 2 and u 2 ξ 2 . Therefore (4.5) holds. The proof iscomplete. □

Similar to the above proof, we can obtain the general theorem.

Theorem 4.3 Assume that (B1)-(B3) hold.Suppose that there existn+1positive numbers ξ i (i=1,2,,n+1) with3 N ¯ ξ 1 < ξ 2 << ξ n + 1 such that one of the following conditions issatisfied:

(H8) ϕ( ξ 2 j 1 )> ξ 2 j 1 σ 2 , Φ( ξ 2 j )< ξ 2 j 3 σ 1 , j=1,2,,[ n + 2 2 ];

(H9) Φ( ξ 2 j 1 )< ξ 2 j 1 3 σ 1 , ϕ( ξ 2 j )> ξ 2 j σ 2 , j=1,2,,[ n + 2 2 ].

Then BVPs (1.1) have at least n positivesolutions u i (i=1,2,,n) with

ξ i < u i < ξ i + 1 .
(4.9)

5 Illustrative examples

Example 5.1 Consider the BVPs of impulsive nonlinear fractional orderdifferential equations:

{ D t q c u ( t ) = f ( t , u ( t ) ) , t J , t 1 2 , Δ u ( 1 2 ) = I ( u ( 1 2 ) ) , Δ u ( 1 2 ) = J ( u ( 1 2 ) ) , a u ( 0 ) b u ( 1 ) = 0 , a u ( 0 ) b u ( 1 ) = 0 .
(5.1)

If we let q= 3 2 , a=2, b=1, f(t,u)= Γ ( 3 2 ) cos t ( t + 2 5 ) 2 u ( t ) 1 + u ( t ) , (t,u)[0,1]×[0,), I(u)= u 5 + u , J(u)= u 10 + u , u[0,).

For u,v[0,), t[0,1],

| f ( t , u ) f ( t , v ) | | Γ ( 3 2 ) cos t ( t + 2 5 ) 2 | | u v ( 1 + u ) ( 1 + v ) | Γ ( 3 2 ) 20 | u v | , | I ( u ) I ( v ) | 5 ( 5 + u ) ( 5 + v ) | u v | 1 5 | u v | , | J ( u ) J ( v ) | 10 ( 10 + u ) ( 10 + v ) | u v | 1 10 | u v | .

Clearly, L 1 = Γ ( 3 2 ) 20 , L 2 = 1 10 , L 3 = 1 5 . Therefore,

ρ= σ 1 L 1 + m a 2 L 2 ( a b ) 2 + m a L 3 a b = 10 3 Γ ( 3 2 ) Γ ( 3 2 ) 20 + 2 5 + 2 5 = 29 30 <1.

Thus, all the assumptions of Theorem 3.1 are satisfied. Hence, BVPs (5.1) have aunique solution on [0,1].

In addition, in this case, let N 1 = Γ ( 3 2 ) 20 , N 2 = N 3 =1. It is clear that |f(t,u)| N 1 , | J k (u)| N 2 , | I k (u)| N 3 . Thus, BVPs (5.1) have at least one solution on[0,1] by Theorem 3.2.

Example 5.2 Consider the BVPs of impulsive nonlinear fractional orderdifferential equations:

{ D t q c u ( t ) ) = f ( t , u ( t ) , t J , t 1 2 , Δ u ( 1 2 ) = I ( u ( 1 2 ) ) , Δ u ( 1 2 ) = J ( u ( 1 2 ) ) , a u ( 0 ) b u ( 1 ) = 0 , a u ( 0 ) b u ( 1 ) = 0 .
(5.2)

Let a=2, b=1, q= 3 2 , f(t,u)=| u ( t ) ln u ( t ) 5 ( 1 + t 2 ) |, I(u)=J(u)= 1 16 ( 1 + u ) . It is easy to see that (H4) holds. By a simplecomputation, we have

f 0 = lim inf u 0 min t [ 0 , 1 ] | u ln u 5 ( 1 + t 2 ) u | = lim inf u 0 | ln u | 10 = + , f = lim inf u min t [ 0 , 1 ] | u ln u 5 ( 1 + t 2 ) u | = lim inf u | ln u | 10 = + .

Take c=1, it is clear that 3 N ¯ < 3 4 <c. For 1 4 u1, f(t,u)= u ln u 5 ( 1 + t 2 ) , we can obtain that f(t,u) arrives at maximum at u= 1 e [ 1 4 ,1], t=0. Thus, we have

Φ(1)= max t [ 0 , 1 ] , u [ 1 4 , 1 ] f(t,u)=f ( 0 , 1 e ) = 1 5 e 0.0736< 1 3 σ 1 = π 20 0.0886.

Thus it follows that BVPs (5.2) have at least two positive solutions u 1 , u 2 with 0< u 1 <1< u 2 by Theorem 4.1.

References

  1. Lakshmikantham V, Leela S: Nagumo-type uniqueness result for fractional differential equations. Nonlinear Anal. 2009, 71: 2886-2889. 10.1016/j.na.2009.01.169

    MathSciNet  Article  MATH  Google Scholar 

  2. Chen F, Zhou Y: Attractivity of fractional functional differential equations. Comput. Math. Appl. 2011, 62: 1359-1369. 10.1016/j.camwa.2011.03.062

    MathSciNet  Article  MATH  Google Scholar 

  3. Chang Y, Nieto J: Some new existence results for fractional differential inclusions with boundaryconditions. Math. Comput. Model. 2009, 49: 605-609. 10.1016/j.mcm.2008.03.014

    MathSciNet  Article  MATH  Google Scholar 

  4. Kilbas AA, Trujillo JJ: Differential equations of fractional order: methods, results and problems-I. Appl. Anal. 2001, 78: 153-192. 10.1080/00036810108840931

    MathSciNet  Article  MATH  Google Scholar 

  5. Kilbas AA, Trujillo JJ: Differential equations of fractional order: methods, results and problems-II. Appl. Anal. 2002, 81: 435-493. 10.1080/0003681021000022032

    MathSciNet  Article  MATH  Google Scholar 

  6. Bai Z: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 2010, 72: 916-924. 10.1016/j.na.2009.07.033

    MathSciNet  Article  MATH  Google Scholar 

  7. Zhang X, Liu L, Wu Y: Multiple positive solutions of a singular fractional differential equation withnegatively perturbed term. Math. Comput. Model. 2012, 55(3):1263-1274.

    MathSciNet  Article  MATH  Google Scholar 

  8. Zhao Y, Sun S, Han Z, et al.: The existence of multiple positive solutions for boundary value problems ofnonlinear fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 2011, 16(4):2086-2097. 10.1016/j.cnsns.2010.08.017

    MathSciNet  Article  MATH  Google Scholar 

  9. Bai Z, Qiu T: Existence of positive solution for singular fractional differential equation. Appl. Math. Comput. 2009, 215(7):2761-2767. 10.1016/j.amc.2009.09.017

    MathSciNet  Article  MATH  Google Scholar 

  10. Zhao Y, Sun S, Han Z, et al.: Positive solutions for boundary value problems of nonlinear fractionaldifferential equations. Appl. Math. Comput. 2011, 217(16):6950-6958. 10.1016/j.amc.2011.01.103

    MathSciNet  Article  MATH  Google Scholar 

  11. Goodrich C: Existence of a positive solution to a class of fractional differentialequations. Appl. Math. Lett. 2010, 23: 1050-1055. 10.1016/j.aml.2010.04.035

    MathSciNet  Article  MATH  Google Scholar 

  12. Salem H: On the existence of continuous solutions for a singular system of nonlinearfractional differential equations. Appl. Math. Comput. 2008, 198: 445-452. 10.1016/j.amc.2007.08.063

    MathSciNet  Article  MATH  Google Scholar 

  13. Bai Z, Lv H: Positive solutions for boundary value problem of nonlinear fractional differentialequation. J. Math. Anal. Appl. 2005, 311: 495-505. 10.1016/j.jmaa.2005.02.052

    MathSciNet  Article  Google Scholar 

  14. Zhao Y, Chen H, Zhang Q: Existence and multiplicity of positive solutions for nonhomogeneous boundary valueproblems with fractional q -derivatives. Bound. Value Probl. 2013., 2013: Article ID 103

    Google Scholar 

  15. Li X, Chen F, Li X: Generalized anti-periodic boundary value problems of impulsive fractionaldifferential equations. Commun. Nonlinear Sci. Numer. Simul. 2013, 18: 28-41. 10.1016/j.cnsns.2012.06.014

    MathSciNet  Article  Google Scholar 

  16. Tian Y, Bai Z: Impulsive boundary value problem for differential equations with fractionalorder. Differ. Equ. Dyn. Syst. 2013, 21(3):253-260. 10.1007/s12591-012-0150-6

    MathSciNet  Article  MATH  Google Scholar 

  17. Zhao KH, Gong P: Existence of positive solutions for a class of higher-order Caputo fractionaldifferential equation. Qual. Theory Dyn. Syst. 2014. 10.1007/s12346-014-0121-0

    Google Scholar 

  18. Bainov DD, Simeonov PS: Impulsive Differential Equations: Periodic Solutions and Applications. Longman, New York; 1993.

    MATH  Google Scholar 

  19. Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.

    Book  MATH  Google Scholar 

  20. Benchohra M, Henderson J, Ntouyas SK 2. In Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York; 2006.

    Chapter  Google Scholar 

  21. Kilbas A, Srivastava H, Trujillo J North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.

    Google Scholar 

  22. Podlubny I: Fractional Differential Equations. Academic Press, New York; 1993.

    MATH  Google Scholar 

  23. Sun JX: Nonlinear Functional Analysis and Its Application. Science Press, Beijing; 2008. (in Chinese)

    Google Scholar 

  24. Hale JK: Theory of Functional Differential Equations. Springer, New York; 1977.

    Book  MATH  Google Scholar 

  25. Guo D, Lakshmikantham V, Liu X Mathematics and Its Applications 373. In Nonlinear Integral Equations in Abstract Spaces. Kluwer Academic, Dordrecht; 1996.

    Chapter  Google Scholar 

Download references

Acknowledgements

The authors thank the referees for their valuable comments and suggestions for theimprovement of the manuscript. This work is supported by the National NaturalSciences Foundation of Peoples Republic of China under Grant No. 11161025 and YunnanProvince Natural Scientific Research Fund Project under Grant No. 2011FZ058.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Kaihong Zhao.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authorsread and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhao, K., Gong, P. Positive solutions for impulsive fractional differential equations with generalizedperiodic boundary value conditions. Adv Differ Equ 2014, 255 (2014). https://doi.org/10.1186/1687-1847-2014-255

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-1847-2014-255

Keywords

  • impulsive fractional differential equation
  • positive solutions
  • boundary value problems
  • fixed point theorem