Hu et al. Advances in Difference Equations 2013, 2013:312 http://www.advancesindifferenceequations.com/content/2013/1/312

RESEARCH

Open Access

Existence of solutions for a coupled system of fractional p-Laplacian equations at resonance Zhigang Hu* , Wenbin Liu and Jiaying Liu * Correspondence: [email protected] Department of Mathematics, China University of Mining and Technology, Xuzhou, 221008, P.R. China

Abstract In this paper, by using the extension of Mawhin’s continuation theorem due to Ge, we study the existence of solutions for a coupled system of fractional p-Laplacian equations at resonance. A new result on the existence of solutions for a fractional boundary value problem is obtained. MSC: 34B15 Keywords: fractional p-Laplacian equation; coupled system; boundary value problem; degree theory; resonance

1 Introduction In the recent years, fractional differential equations have played an important role in many fields such as physics, electrical circuits, biology, control theory, etc. (see [–]). Recently, many scholars have paid more attention to boundary value problems for fractional differential equations (see [–]). In [], by means of a fixed point theorem on a cone, Agarwal et al. considered a twopoint boundary value problem at nonresonance given by ⎧ ⎨Dα+ x(t) + f (t, x(t), Dμ+ x(t)) = , 



⎩x() = x() = , where  < α < , μ >  are real numbers, α – μ ≥  and Dα+ is the Riemann-Liouville fractional derivative. By using the coincidence degree theory, Bai (see []) considered m-point fractional boundary value problems at resonance in the form ⎧ ⎨Dα+ u(t) = f (t, u(t), Dα–  < t < ,  + u(t)) + e(t),  m– –α ⎩I + u(t)|t= = , u() = i= βi u(ηi ),  where  < α ≤  is a real number, βi ∈ R, ηi ∈ (, ) are given constants such that m– m– = , and Dα+ , Iα+ are the Riemann-Liouville differentiation and integration. i= βi ηi Moreover, the existence of solutions to a coupled system of fractional differential equations have been studied by many authors (see [–]). In [], relying on Schauder’s fixed point theorem, Ahmad et al. considered a threepoint boundary value problem for a coupled system of nonlinear fractional differential ©2013 Hu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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equations at nonresonance given by ⎧ p α ⎪ ⎪ ⎨D+ u(t) = f (t, v(t), D+ v(t)),

q β D v(t) = g(t, u(t), D+ u(t)), ⎪ +

⎪ ⎩ u() = ,

u() = γ u(η),

 < t < ,  < t < , v() = ,

v() = γ v(η),

where  < α, β < , p, q, γ > ,  < η < , α – q ≥ , β – p ≥ , γ ηα– < , γ ηβ– < , D is the standard Riemann-Liouville differentiation and f , g : [, ] × R × R → R are given continuous functions. In [], by using the coincidence degree theory due to Mawhin, Jiang discussed the existence of solutions to a coupled system of fractional differential equations at resonance ⎧ ⎨Dα+ u(t) = f (t, v(t), Dδ + v(t)),

  ⎩Dβ+ v(t) = g(t, u(t), Dγ + u(t)),  

γ

u() = ,

D+ u() =

v() = ,

Dδ+ v() =

n

γ i= ai D+ u(ξi ),

n

δ i= ai D+ v(ηi ),

where t ∈ [, ],  < α, β ≤ ,  < γ ≤ α – ,  < δ ≤ β – ,  < ξ < ξ < · · · < ξn < ,  < η < η < · · · < ηm < . The turbulent flow in a porous medium is a fundamental mechanics problem. For studying this type of problems, Leibenson (see []) introduced the p-Laplacian equation as follows:      φp x(t) = f t, x(t), x(t) , where φp (s) = |s|p– s, p > . Obviously, φp is invertible and its inverse operator is φq , where q >  is a constant such that p + q = . In the past few decades, many important results relative to a p-Laplacian equation with certain boundary value conditions have been obtained. We refer the reader to [–] and the references cited therein. We noticed that φp is a quasi-linear operator. So, Mawhin’s continuation theorem is not suitable for a p-Laplacian operator. In [], Ge and Ren extended Mawhin’s continuation theorem, which is used to deal with more general abstract operator equations. Motivated by all the works above, in this paper, we consider the following boundary value problem (BVP for short) for a coupled system of fractional p-Laplacian equations given by ⎧ β α δ ⎪ ⎪ ⎨D+ φp (D+ u(t)) = f (t, v(t), D+ v(t)), t ∈ (, ), γ D+ φp (Dδ+ v(t)) = g(t, u(t), Dα+ u(t)), t ∈ (, ), ⎪ ⎪ ⎩ α D+ u() = Dα+ u() = Dδ+ v() = Dδ+ v() = , β

γ

(.)

where Dα+ , D+ , D+ , Dδ+ are the standard Caputo fractional derivatives,  < α, δ, β, γ ≤ ,  < α + β < ,  < δ + γ <  and f , g : [, ] × R → R is continuous. The rest of this paper is organized as follows. Section  contains some necessary notations, definitions and lemmas. In Section , we establish a theorem on the existence of solutions for BVP (.) under nonlinear growth restriction of f and g, based on the extension of Mawhin’s continuation theorem due to Ge (see []). Finally, in Section , an example is given to illustrate the main result.

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2 Preliminaries In this section, we introduce some notations, definitions and preliminary facts which are used throughout this paper. Definition . Let X and Y be two Banach spaces with norms ·X and ·Y , respectively. A continuous operator M : X ∩ dom M → Y is said to be quasi-linear if (i) Im M := M(X ∩ dom M) is a closed subset of Y , (ii) Ker M := {X ∩ dom M : Mu = } is linearly homeomorphic to Rn , n < ∞. Definition . Let X be a real Banach space and X ⊂ X. The operator P : X → X is said to  be a projector provided P = P, P(λ x + λ x ) = λ P(x ) + λ P(x ) for x , x ∈ X, λ , λ ∈ R. The operator Q : X → X is said to be a semi-projector provided Q = Q. Definition . ([]) Let X = Ker M and

X be the complement space of X in X, then X = X ⊕

X . On the other hand, suppose that Y is a subspace of Y and

Y is the complement space of Y in Y so that Y = Y ⊕

Y . Let P : X → X be a projector and Q : Y → Y be a semiprojector, and let ⊂ X be an open and bounded set with origin θ ∈ , where θ is the origin of a linear space. Suppose that Nλ : → Y , λ ∈ [, ] is a continuous operator. Denote N by N . Let λ = {u ∈ : Mu = Nλ u}. Nλ is said to be M-compact in if there is Y ⊂ Y with dim Y = dim X and an operator R : × [, ] → X continuous and compact such that for λ ∈ [, ], (I – Q)Nλ ( ) ⊂ Im M ⊂ (I – Q)Y ,

(.)

λ ∈ (, )

(.)

QNλ x = θ ,

⇔

QNx = θ ,

R(·, ) is the zero operator and R(·, λ)| λ = (I – P)| λ ,  M P + R(·, λ) = (I – Q)Nλ .

(.) (.)

Lemma . ([], Ge-Mawhin’s continuation theorem) Let X and Y be two Banach spaces with norms  · X and  · Y , respectively. ⊂ X is an open and bounded nonempty set. Suppose that M : X ∩ dom M → Y is a quasi-linear operator and Nλ : → Y ,

λ ∈ [, ]

is M-compact in . In addition, if (C ) Mx = Nλ x, ∀(x, λ) ∈ (dom M ∩ ∂ ) × (, ), (C ) QNx = , for x ∈ dom M ∩ ∂ , (C ) deg(JQN, Ker M ∩ , ) = ,

Hu et al. Advances in Difference Equations 2013, 2013:312 http://www.advancesindifferenceequations.com/content/2013/1/312

where N = N and J : Y → X is a homeomorphism with J(θ ) = θ , then the equation Mu = Nu has at least one solution in . Definition . The Riemann-Liouville fractional integral operator of order α >  of a function x is given by Iα+ x(t) =

 (α)

t

(t – s)α– x(s) ds, 

provided that the right-hand side integral is pointwise defined on (, +∞). Definition . The Caputo fractional derivative of order α >  of a continuous function x is given by Dα+ x(t) = In–α +

 dn x(t) = n dt (n – α)

t

(t – s)n–α– x(n) (s) ds, 

where n is the smallest integer greater than or equal to α, provided that the right-hand side integral is pointwise defined on (, +∞). Lemma . [] Assume that Dα+ x ∈ C[, ], α > . Then Iα+ Dα+ x(t) = x(t) + c + c t + c t  + · · · + cn– t n– , where ci = – x to α.

(i) ()

i!

, i = , , , . . . , n – , here n is the smallest integer greater than or equal

Lemma . [] Assume that α >  and x ∈ C[, ]. Then Dα+ Iα+ x(t) = x(t). In this paper, we denote Y = C[, ] with the norm yY = y∞ , X = {x|x, Dα+ x ∈ Y } with the norm xX = max{x∞ , Dα+ x∞ } and X = {x|x, Dδ+ x ∈ Y } with the norm xX = max{x∞ , Dδ+ x∞ }, where x∞ = maxt∈[,] |x(t)|. Then we denote X = X × X with the norm (u, v)X = max{uX , vX } and Y = Y × Y with the norm (x, y)Y = max{xY , yY }. Obviously, both X and Y are Banach spaces. Define the operator M : dom M ⊂ X → Y by   β M u = D+ φp Dα+ u , where     β dom M = u ∈ X|D+ φp Dα+ u ∈ Y , Dα+ u() = Dα+ u() =  . Define the operator M : dom M ⊂ X → Y by   γ M v = D+ φp Dδ+ v ,

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where     γ dom M = v ∈ X |D+ φp Dδ+ v ∈ Y , Dδ+ v() = Dδ+ v() =  . Define the operator M : dom M ⊂ X → Y by M(u, v) = (M u, M v),

(.)

where   dom M = (u, v) ∈ X|u ∈ dom M , v ∈ dom M . Define the operator N : X → Y by   N(u, v) = N  v, N  u , where N  : X → Y   N  v(t) = f t, v(t), Dδ+ v(t) and N  : X → Y   N  u(t) = g t, u(t), Dα+ u(t) . Then BVP (.) is equivalent to the operator equation M(u, v) = N(u, v),

(u, v) ∈ dom M.

3 Main result In this section, a theorem on the existence of solutions for BVP (.) will be given. Theorem . Let f , g : [, ] × R → R be continuous. Assume that (H ) there exist nonnegative functions pi , qi , ri ∈ C[, ] (i = , ) with    p– Q p– Q  + R + R   < (β + )(γ + ) ((δ + ))p– ((α + ))p– such that for all (u, v) ∈ R , t ∈ [, ],   f (t, u, v) ≤ p (t) + q (t)|u|p– + r (t)|v|p– and   g(t, u, v) ≤ p (t) + q (t)|u|p– + r (t)|v|p– , where Pi = pi ∞ , Qi = qi ∞ , Ri = ri ∞ (i = , );

(.)

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(H ) there exists a constant B >  such that for all t ∈ [, ], |u| > B, v ∈ R either uf (t, u, v) > ,

ug(t, u, v) > 

uf (t, u, v) < ,

ug(t, u, v) < .

or

Then BVP (.) has at least one solution. In order to prove Theorem ., we need to prove some lemmas below. Lemma . Let M be defined by (.), then   Ker M = (Ker M , Ker M ) = (u, v) ∈ X|(u, v) = (a, b), a, b ∈ R ,

(.)

Im M = (Im M , Im M )  

    ( – s)β– x(s) ds = , ( – s)γ – y(s) ds =  , = (x, y) ∈ Y 

(.)





and M is a quasi-linear operator. β

Proof By Lemma ., M u = D+ φp (Dα+ u) =  has the solution u(t) = u() + Iα+ φq (c ) = u() +

φq (c ) α t , (α + )

  c = φp Dα+ u() ,

which satisfies Dα+ u(t) = φq (c ). Combining with the boundary value condition Dα+ u() = , we have Ker M = {u ∈ X |u = a, a ∈ R}. For x ∈ Im M , there exists u ∈ dom M such that x = M u ∈ Y . By Lemma ., we have β  Dα+ u(t) = φq I+ x(t) + c  

t  β– (t – s) x(s) ds + c . = φq (β)  From the condition Dα+ u() = , one has c = . By the condition Dα+ u() = , we obtain that



( – s)β– x(s) ds = . 

(.)

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 On the other hand, suppose that x ∈ Y and satisfies  ( – s)β– x(s) ds = . Let u(t) = β Iα+ φq (I+ x(t)), then u ∈ dom M . By Lemma ., we have Dα+ u(t) = x(t). So that x ∈ Im M . Then we have      β– Im M = x ∈ Y  ( – s) x(s) ds =  . 

Then we have dim Ker M =  and M (dom M ∩ X ) ⊂ Y closed. Therefore, M is a quasilinear operator. Similarly, we can get Ker M = {v ∈ X |v = b, b ∈ R},      ( – s)γ – y(s) ds =  , Im M = y ∈ Y  

and M is a quasi-linear operator. Then the proof is complete.



Lemma . Let ⊂ X be an open and bounded set, then Nλ is M-compact in . X and the semi-projector Q : Y → Y Proof Define the continuous projector P : X →   P(u, v) = (P u, P v) = u(), v() ,   

 β– γ – ( – s) x(s) ds, γ ( – s) y(s) ds , Q(x, y) = (Q x, Q y) = β 



where X = Ker M and Y = Im Q. Obviously, Im P = Ker M and P (u, v) = P(u, v). It follows from (u, v) = ((u, v) – P(u, v)) + P(u, v) that X = Ker P + Ker M. By a simple calculation, we can get that Ker M ∩ Ker P = {(, )}. Then we get X = Ker M ⊕ Ker P = X ⊕

X. For (x, y) ∈ Y , we have   Q (x, y) = Q(Q x, Q y) = Q x, Q y . By the definition of Q , we can get

Q x = Q x · β



( – s)β– ds = Q x. 

Similar proof can show that Q y = Q y. Thus, we have Q (x, y) = Q(x, y). Let (x, y) = ((x, y) – Q(x, y)) + Q(x, y), where (x, y) – Q(x, y) ∈ Ker Q = Im M, Q(x, y) ∈ Im Q. It follows from Ker Q = Im M and Q (x, y) = Q(x, y) that Im Q ∩ Im M = {(, )}. Then we have Y = Im Q ⊕ Im M = Y ⊕

Y.

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Thus dim X = dim Ker M = dim Im Q = dim Y. Let ⊂ X be an open and bounded set with (θ , θ ) ∈ . For each (u, v) ∈ , we can get Q[(I – Q)Nλ (u, v)] = . Thus, (I – Q)Nλ (u, v) ∈ Im M = Ker Q. Take any (x, y) ∈ Im M in the type (x, y) = ((x, y) – Q(x, y)) + Q(x, y). Since Q(x, y) = , we can get (I – Q)(x, y) ∈ Y . So (.) holds. It is easy to verify (.). Furthermore, define R = (R , R ) : × [, ] →

X by 

s    β–  (t – s) φq (s – τ ) (I – Q )Nλ v(τ ) dτ ds, (β)    

t

s     δ– γ –  (t – s) φq (s – τ ) (I – Q )Nλ u(τ ) dτ ds. R (v, λ)(t) = (δ)  (γ )   R (u, λ)(t) = (α)



t

α–

By the continuity of f and g, it is easy to get that R(u, v, λ) is continuous on × [, ]. Moreover, for all (u, v) ∈ , there exists a constant T >  such that γ β max{|I+ (I – Q )Nλ v(τ )|, |I+ (I – Q )Nλ u(τ )|} ≤ T, so we can easily obtain that R( , λ) is uniformly bounded. By the Arzela-Ascoli theorem, we just need to prove that R :

× [, ] →

X is equicontinuous. Furthermore, for  ≤ t < t ≤ , (u, v, λ) ∈ × [, ] = (  ,  ) × [, ], we have   R(u, v, λ)(t ) – R(u, v, λ)(t )    β γ =  Iα+ φq I+ (I – Q )Nλ v(t ) , Iδ+ φq I+ (I – Q )Nλ u(t )    β γ – Iα+ φq I+ (I – Q )Nλ v(t ) , Iδ+ φq I+ (I – Q )Nλ u(t )     β β =  Iα+ φq I+ (I – Q )Nλ v(t ) – Iα+ φq I+ (I – Q )Nλ v(t ) ,   γ γ Iδ+ φq I+ (I – Q )Nλ u(t ) – Iδ+ φq I+ (I – Q )Nλ u(t ) . β

By |I+ (I – Q )Nλ v| ≤ T, we have α β    I + φq I + (I – Q )N  v(t ) – I α+ φq I β+ (I – Q )N  v(t )   λ  λ     β   t ≤ (t – s)α– φq I+ (I – Q )Nλ v(s) ds  (α)  

t   β α–  (t – s) φq I+ (I – Q )Nλ v(s) ds – 

≤

φq (T) (α)



t

(t – s)α– – (t – s)α– ds +



t

 (t – s)α– ds

t

φq (T)  α α  = t –t . (α + )   Since t α is uniformly continuous on [, ], so R (  , λ) is equicontinuous. Similarly, we β can get I+ ((I – Q )Nλ v(τ )) ⊂ C[, ] is equicontinuous. Considering that φq (s) is uniformly β continuous on [–T, T], we have Dα+ R (  , λ) = I+ ((I – Q )Nλ ( )) is also equicontinuous. So, we can obtain that R (  , λ) →

X is compact.

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Similarly, we can get that R (  , λ) →

X is compact. So, we can obtain that R : × [, ] →

X is compact. β For each (u, v) ∈ λ = {(u, v) ∈ : M(u, v) = Nλ (u, v)}, we have (D+ φp (Dα+ u(t)), γ D+ φp (Dδ+ v(t))) = Nλ (u(t), v(t)) ∈ Im M. Thus, R (u, λ)(t) = =

 (α)  (α)



t

(t – s)α– φq





t

(t – s)α– φq 

 (β)  (β)



s

   (s – τ )β– (I – Q )Nλ v(τ ) dτ ds

s

   β (s – τ )β– D+ φp Dα+ u(τ ) dτ ds,





which together with Dα+ u() =  yields that  R (u, λ)(t) = u(t) – u() = (I – P )u (t). It is easy to verify that R (u, )(t) is the zero operator. Similarly, we can get R (v, λ)(t) = [(I – P )v](t) and R (v, )(t) is the zero operator. So (.) holds. On the other hand,  M P u + R (u, λ) (t)    

t

s     α– β–  (t – s) φq (s – τ ) (I – Q )Nλ v(τ ) dτ ds + u() = M (α)  (β)    = (I – Q )Nλ v (t). Similarly, we have M [P v + R (v, λ)](t) = [((I – Q )Nλ )u](t). So, (.) holds. Then we have  that Nλ is M-compact in . The proof is complete. Lemma . Suppose that (H ), (H ) hold, then the set  

 = (u, v) ∈ dom M \ Ker M | M(u, v) = λN(u, v), λ ∈ (, ) is bounded. Proof Take (u, v) ∈  , then N(u, v) ∈ Im M. By (.), we have



  ( – s)β– f s, v(s), Dδ+ v(s) ds = ,



  ( – s)γ – g s, u(s), Dα+ u(s) ds = .





Then, by the integral mean value theorem, there exist constants ξ , η ∈ (, ) such that f (ξ , v(ξ ), Dδ+ v(ξ )) =  and g(η, u(η), Dα+ u(η)) = . So, from (H ), we get |v(ξ )| ≤ B and |u(η)| ≤ B. By Lemma ., v(t) = v() + Iδ+ Dδ+ v(t)

t  (t – s)δ– Dδ+ v(s) ds. = v() + (δ) 

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Take t = ξ , we have v(ξ ) = v() +

 (δ)

ξ 

(ξ – s)δ– Dδ+ v(s) ds.

Then we have     v() ≤ v(ξ ) +

 (δ)

ξ 

  (ξ – s)δ– Dδ+ v(s) ds

   D δ + v  ·  ξ δ  ∞ δ (δ)  δ   D + v . ≤B+ (δ + )  ∞

  ≤ v(ξ ) +

So, we get     v(t) ≤ v() +

 (δ)

t 

  (t – s)δ– Dδ+ v(s) ds

   D δ + v  ·  ∞ (δ)    D δ + v  , ≤B+  ∞ (δ + )

  ≤ v() +

 δ t δ ∀t ∈ [, ].

That is, v∞ ≤ B +

   Dδ + v . (δ + )  ∞

(.)

Similarly, we can get u∞ ≤ B +

 α   D + u . ∞ (α + ) 

By M(u, v) = λN(u, v) and Dα+ u() = Dδ+ v() = , we get   β φp Dα+ u(t) = λI+ N  v(t)

t   λ (t – s)β– f s, v(s), Dδ+ v(s) ds. = (β)  So, from (H ), we have   α  φp D + u(t)  ≤  ≤

 (β)  (β)



t

   (t – s)β– f s, v(s), Dδ+ v(s)  ds

t

 p–  (t – s)β– p (s) + q (s)v(s)





 p–  + r (s)Dδ+ v(s) ds

 δ p–   β     p ∞ + q ∞ vp– t ∞ + r ∞ D+ v ∞ · (β) β  δ p–      P + Q vp– ≤ ∞ + R D+ v ∞ , (β + )

≤

(.)

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which together with |φp (Dα+ u(t))| = |Dα+ u(t)|p– and (.) yields that  α p– D + u ≤ 

∞

     p–  δ p–    Dδ + v   P + Q B + . + R v D  + ∞ (β + ) (δ + )  ∞

(.)

Similarly, we can get  δ p– D + v ≤ 

∞

     α  p–  α p–   D + u D + u P + Q B + + R .   ∞ ∞ (γ + ) (α + ) 

(.)

Then from (.), (.) and (.), we can see that there exists a constant M >  such that  α   δ  D + u , D + v ≤ M .   ∞ ∞

(.)

Thus, from (.) and (.), we get  u∞ , v∞ ≤ max B +

 M M ,B + := M . (α + ) (δ + )

(.)

Combining (.) and (.), we have   (u, v) ≤ max{M , M } := M. X So,  is bounded. The proof is complete.



Lemma . Suppose that (H ) holds, then the set  

 = (u, v)|(u, v) ∈ Ker M, N(u, v) ∈ Im M is bounded. Proof For (u, v) ∈  , we have (u, v) = (a, b). Then, from N(u, v) ∈ Im M, we get



( – s)β– f (s, b, ) ds = , 



( – s)γ – g(s, a, ) ds = , 

which together with (H ) implies |a|, |b| ≤ B. Thus, we have   (u, v) ≤ B. X Hence,  is bounded. The proof is complete. Lemma . Suppose that the first part of (H ) holds, then the set  

 = (u, v) ∈ Ker M|λJ – (u, v) + ( – λ)QN(u, v) = (, ), λ ∈ [, ]



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is bounded, where J – : Ker M → Im Q is a homeomorphism defined by a, b ∈ R.

J – (a, b) = (b, a),

Proof For (u, v) ∈  , we have (u, v) = (a, b) and



λb + ( – λ)β

( – s)β– f (s, b, ) ds = ,

(.)

( – s)γ – g(s, a, ) ds = .

(.)





λa + ( – λ)γ 

If λ = , then a = b = . For λ ∈ [, ), we can obtain |a|, |b| ≤ B. Otherwise, if |a| or |b| > B, in view of the first part of (H ), one has



λb + ( – λ)β

( – s)β– bf (s, b, ) ds > , 

or



λa + ( – λ)γ

( – s)γ – ag(s, a, ) ds > , 

which contradicts (.) or (.). Therefore,  is bounded. The proof is complete.



Remark . If the second part of (H ) holds, then the set  

 = (u, v) ∈ Ker M|–λJ – (u, v) + ( – λ)QN(u, v) = (, ), λ ∈ [, ] is bounded. Proof of Theorem . Set = {(u, v) ∈ X|(u, v)X < max{M, B} + }. It follows from Lemmas . and . that M is a quasi-linear operator and Nλ is M-compact on . By Lemmas . and ., we get that the following two conditions are satisfied: (C ) Mx = Nλ x, ∀(x, λ) ∈ (dom M ∩ ∂ ) × (, ), , for x ∈ dom M ∩ ∂ . (C ) QNx = Take   H (u, v), λ = ±λ(u, v) + ( – λ)JQN(u, v). According to Lemma . (or Remark .), we know that H((u, v), λ) =  for (u, v) ∈ Ker M ∩ ∂ . Therefore     deg JQN|Ker M , ∩ Ker M, (, ) = deg H(·, ), ∩ Ker M, (, )   = deg H(·, ), ∩ Ker M, (, )   = deg ±I, ∩ Ker M, (, ) = .

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So, condition (C ) of Lemma . is satisfied. By Lemma ., we can get that M(u, v) = N(u, v) has at least one solution in dom M ∩ . Therefore BVP (.) has at least one solution. The proof is complete. 

4 Example Example . Consider the following BVP: ⎧   ⎪    ⎪ ⎪ ⎨D+ φ (D+ u(t)) = –  +   +   +

  +

 



   v (t) + te–|D+ v(t)| ,  

  u (t) + sin (D+ u(t)), 

D φ (D v(t)) = + ⎪ ⎪    ⎪ ⎩ D u() = D+ u() = D+ v() = D+ v() = .

t ∈ (, ), (.)

t ∈ (, ),

Corresponding to BVP (.), we have that p = , α =  , δ =  , β =  , γ =

 

and

   + u + te–|v| ,      + u + sin v. g(t, u, v) =  

f (t, u, v) = –

Choose p (t) = p (t) = , q (t) =  , q (t) =  , r (t) = r (t) = , B = . Then we have P = P = , Q =  , Q =  , R (t) = R (t) = . By a simple calculation, we get         < . (  + )(  + ) ((  + )) ((  + )) Then (H ) and the first part of (H ) hold. By Theorem ., we obtain that BVP (.) has at least one solution.

Competing interests The authors declare that they have no competing interests. Authors’ contributions The authors contributed equally in this article. All authors read and approved the final manuscript. Acknowledgements The authors are grateful to those who gave useful suggestions about the original manuscript. This research was supported by the Fundamental Research Funds for the Central Universities (2013QNA33). Received: 24 May 2013 Accepted: 2 October 2013 Published: 08 Nov 2013 References 1. Metzler, R, Klafter, J: Boundary value problems for fractional diffusion equations. Physica A 278, 107-125 (2000) 2. Scher, H, Montroll, E: Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 12, 2455-2477 (1975) 3. Mainardi, F: Fractional diffusive waves in viscoelastic solids. In: Wegner, JL, Norwood, FR (eds.) Nonlinear Waves in Solids, pp. 93-97. ASME/AMR, Fairfield (1995) 4. Diethelm, K, Freed, AD: On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity. In: Keil, F, Mackens, W, Voss, H, Werther, J (eds.) Scientific Computing in Chemical Engineering II Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, pp. 217-224. Springer, Heidelberg (1999) 5. Gaul, L, Klein, P, Kempfle, S: Damping description involving fractional operators. Mech. Syst. Signal Process. 5, 81-88 (1991) 6. Glockle, WG, Nonnenmacher, TF: A fractional calculus approach of self-similar protein dynamics. Biophys. J. 68, 46-53 (1995) 7. Mainardi, F: Fractional calculus: some basic problems in continuum and statistical mechanics. In: Carpinteri, A, Mainardi, F (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 291-348. Springer, Wien (1997)

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