Han and Yang Boundary Value Problems (2017) 2017:78 DOI 10.1186/s13661-017-0808-7

RESEARCH

Open Access

Existence and multiplicity of positive solutions for a system of fractional differential equation with parameters Xiaoling Han* and Xiaojuan Yang *

Correspondence: [email protected] Department of Mathematics, Northwest Normal University, Lanzhou, Gansu 730070, People’s Republic of China

Abstract In this paper, we study the existence and multiplicity of positive solutions for a class of systems of fractional differential equation with parameters. By applying the Krasnosel’skii fixed point theorem for a cone map, we conclude to the existence of at least one and two solutions for our considered system. MSC: 34B15; 34B18 Keywords: system of fractional differential equation; Green’s function; cone; Krasnosel’skii fixed point theorem

1 Introduction Fractional differential equations can describe some phenomena in various fields of engineering and scientific, disciplines such as control theory, chemistry, physics, biology, economics, mechanics and electromagnetic. Especially in recent years, a large number of papers dealt with the existence of positive solutions of boundary value problems for nonlinear differential equations of fractional order; for details, see [–]. In addition, the existence of positive solutions to fractional differential equations and their systems, especially coupled systems, were well studied by many authors; for details, see [–]. In [], Su studied the existence of solutions for a coupled system of fractional differential equations ⎧ γ α ⎪ ⎪D+ u(t) = f (t, v(t), D+ v(t)), ⎨

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

⎪ ⎩

 < t < ,  < t < ,

u() = u() = v() = v(),

where  < α, β < , γ , η > , α – η ≥ , β – γ ≥ , f , g : [, ] × R → R are given functions and D+ is the standard Riemann-Liouville fractional derivative. In [], Dunninger and Wang considered the existence and multiplicity of positive radial solutions for elliptic systems of the form ⎧ ⎪ ⎪ ⎨u + λk (|x|)f (u, v) = , v + μk (|x|)g(u, v) = , ⎪ ⎪ ⎩ u|∂ = v|∂ = , © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided 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.

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where (u, v) ∈ C  () × C  (), with  = {x ∈ RN : R < |x| < R , R , R > } an annulus with boundary ∂. Motivated by [] and [], in this paper, we consider the system of fractional differential equations with parameters ⎧ p ⎪ ⎪ ⎪D x(t) + λ w(t, x(t), y(t)) = , t ∈ J = [, ],  < p ≤ , ⎪ ⎪ ⎨Dq y(t) + λ h(t, x(t), y(t)) = , t ∈ J,  < q ≤ ,  q ⎪  ⎪ D x() = Dp x() = Dγ x() = , x() = α x(η), ⎪ ⎪ ⎪ ⎩ q p γ D y() = D y() = D y() = , y() = α y(ξ ),

()

where Dp is the standard Riemann-Liouville derivative. Moreover, in the rest of this paper we always suppose that the following assumptions hold. (A) (i) w, h : [, ] × [, +∞) × [, +∞) → [, +∞) are continuous; (ii) λ and λ are positive parameters; (iii) qi ∈ (, ), pi ∈ (, ), γi ∈ (, ), η, ξ ∈ (, ) (i = , ),  < α ηp– < ,  < α ξ p– < . (A) w(t, x, y), h(t, x, y) >  for x, y > , t ∈ J. By applying the Krasnosel’skii fixed point theorem for a cone map, we obtained the existence of at least one and two positive solutions for the system ().

2 Preliminaries For the sake of convenience, we introduce following notations: w = lim max

(x,y)→ t∈[,]

w∞ = lim

w(t, x, y) , x+y

min

(x,y)→∞ t∈[,]

h = lim max

(x,y)→ t∈[,]

h∞ = lim

w(t, x, y) , x+y

h(t, x, y) , x+y

min

(x,y)→∞ t∈[,]

h(t, x, y) . x+y

Theorem A ([]) Let X be a Banach space, and let K ⊂ X be a cone. Assume  ,  are two open bounded subsets of X with  ∈  ,  ⊂  , and let T : K ∩ ( \  ) → K be a completely continuous operator such that (i) Tx ≤ x , x ∈ K ∩ ∂ , and Tx ≥ x , x ∈ K ∩ ∂ ; or (ii) Tx ≥ x , x ∈ K ∩ ∂ , and Tx ≤ x , x ∈ K ∩ ∂ . Then T has a fixed point in K ∩ ( \  ). t  Definition  ([]) We call Dp w(x) = (m–p) ( dtd )m  (t–s)w(t) p–m+ dt, p > , m = [p] +  is the Riemann-Liouville fractional derivative of order p. [p] denotes the integer part of number p. Definition  ([]) We call I p w(x) = Liouville fractional integral of order p.



(α)

t

 (t

– s)p– w(s) ds, t > , p >  is Riemann-

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Lemma  ([]) Let p > , then, for ∀Ci ∈ R, i = , , , . . . , m, m = [p] + , we have I p Dp x(t) = x(t) + C t p– + C t p– + · · · + Cm t p–m . Lemma  Suppose that ϕ ∈ C(J) and (A) holds, then the unique solution of the linear boundary value problem ⎧ ⎨Dp x(t) + ϕ(t) = ,

t ∈ J,  < p ≤ ,

⎩Dq x() = Dp x() = Dγ x() = ,

() x() = α x(η),

is provided by 



G (t, s)ϕ(s) ds,

x(t) = 

where G (t, s) is the Green’s function defined by ⎧ t p– ⎪ ⎪ [( – s)p– – α (η – s)p– ] – ⎪ (–α ηp– ) (p) ⎪ ⎪ ⎪ t p– ⎨ [( – s)p– – α (η – s)p– ], p– G (t, s) = (–α ηp– ) (p) p– ⎪ ⎪ (–α ηt p– ) (p) ( – s)p– – (t–s) , ⎪

(p)  ⎪ ⎪ ⎪ t p– p– ⎩ ( – s) , (–α ηp– ) (p)

(t–s)p– ,

(p)



 ≤ s ≤ t ≤ η ≤ ,  ≤ t ≤ s ≤ η ≤ ,  ≤ η ≤ s ≤ t ≤ ,

()

 ≤ η ≤ t ≤ s ≤ .

Proof Let x(t) = –I p ϕ(t) + C t p– + C t p– + C t p– + C t p– , then Dq x(t) = –I p–q ϕ(t) + C + C

(p – )t p–q –

(p – )t p–q – + C ,

(p – q – )

(p – q – )

Dp x(t) = –I p–p ϕ(t) + C + C

(p)t p–p –

(p – )t p–p – + C

(p – p )

(p – p – )

(p – )t p–p –

(p – )t p–p – + C ,

(p – p – )

(p – p – )

Dγ x(t) = –I p–γ ϕ(t) + C + C

(p)t p–q –

(p – )t p–q – + C

(p – q )

(p – q – )

(p)t p–γ –

(p – )t p–γ – + C

(p – γ )

(p – γ – )

(p – )t p–γ –

(p – )t p–γ – + C ,

(p – γ – )

(p – γ – )

Dq x() =  implies that C = . In fact, if t = , we see that t p–q – = , t p–q – = , t p–q – = , t p–q – = t+q –p is not well defined. Similarly, Dp x() =  implies that C =  and Dγ x() =  implies that C = . Thus, x(t) = –I p ϕ(t) + C t p– . Now, by using boundary condition x() = α x(η), we get C = –α ηp– (I p ϕ() – α I p ϕ(t)). Hence, we get the solution  as follows: t p–  p I ϕ() – α I p ϕ(t)  – α ηp–

    η t p– p– p– = ( – s) ϕ(s) ds – α (η – s) ϕ(s) ds  ( – α ηp– ) (p)  

x(t) = –I p ϕ(t) +

Han and Yang Boundary Value Problems (2017) 2017:78

–  =



(p)



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t

(t – s)p– ϕ(s) ds 



G (t, s)ϕ(s) ds. 



Then G (t, s) can easily be obtained. Notation  Similarly, we can get 

   ξ t q– q– q– ( – s) ϕ(s) ds – α (ξ – s) ϕ(s) ds y(t) = ( – α ξ q– ) (q)    t  – (t – s)q– ϕ(s) ds

(q)    G (t, s)ϕ(s) ds = 

and ⎧ t q– ⎪ ⎪ [( – s)q– – α (ξ – s)q– ] – ⎪ ⎪ (–α ξ q– ) (q) ⎪ ⎪ t q– ⎨ [( – s)q– – α (ξ – s)q– ], (–α ξ q– ) (q) G (t, s) = q– q– t ⎪ ⎪ ( – s)q– – (t–s) , ⎪

(q) (–α ξ q– ) (q) ⎪ ⎪ q– ⎪ t q– ⎩ ( – s) , (–α ξ q– ) (q)

(t–s)q– ,

(q)



 ≤ s ≤ t ≤ ξ ≤ ,  ≤ t ≤ s ≤ ξ ≤ ,  ≤ ξ ≤ s ≤ t ≤ ,  ≤ ξ ≤ t ≤ s ≤ .

In view of Lemma , the system () is equivalent to the Fredholm integral system of ⎧ ⎨x(t) = λ   G (t, s)w(s, x(s), y(s)) ds,    ⎩y(t) = λ   G (t, s)h(s, x(s), y(s)) ds, 

where Gi (t, s) is the Green’s function defined by Lemma . Define X = {x(t) | x ∈ C(J)}, endowed with the norm x = maxt∈J |x(t)|, further the norm for the product space X ×X, we define as x+y = x + y . Obviously, (X, · ) is a Banach space. We define the cone K ⊂ X × X by  

 K = (x, y) ∈ X × X : x, y ≥ , min x(t) + y(t) ≥ θ (x, y) , t∈J

  θ = min θ = δ p– , θ = δ q– . Define an operator T : X × X → X × X as         T(x, y)(t) = λ G (t, s)w s, x(s), y(s) ds, λ G (t, s)h s, x(s), y(s) ds 

 = T (x, y), T (x, y) .



The solutions of the system () and the fixed points of operator T coincide with each other. Lemma  The Green’s function Gi (t, s) (i = , ) are continuous on J × J and satisfy the following properties:

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() Gi (t, s) ∈ C(J × J) and Gi (t, s) ≥ , ∀t, s ∈ J; () maxt∈J Gi (t, s) = Gi (, s); mint∈[δ,–δ] Gi (t,s) () ≥ θi (s), δ ∈ (, ). Gi (,s) Proof () If  ≤ s ≤ t ≤ η ≤ , then

G (t, s) = ≥

 (t – s)p–

t p– ( – s)p– – α (η – s)p– – p– ( – α η ) (p)

(p) t p– [( – s)p– – α ηp– ( – ηs )p– ] – ( – α ηp– )t p– ( – ts )p– ( – α ηp– ) (p)

≥

t p– [( – s)p– – α ηp– ( – s)p– ] – ( – α ηp– )t p– ( – ts )p– ( – α ηp– ) (p)

=

t p– [( – s)p– – ( – ts )p– ] .

(p)

Since s ≤ t, G (t, s) ≥ . If  ≤ t ≤ s ≤ η ≤ , then

G (t, s) = ≥

 t p– ( – s)p– – α (η – s)p– p– ( – α η ) (p) t p– [( – s)p– – α ηp– ( – ηs )p– ] ( – α ηp– ) (p)

≥

t p– ( – α ηp– )( – s)p– ( – α ηp– ) (p)

=

t p– ( – s)p– .

(p)

Hence G (t, s) ≥ . If  ≤ η ≤ s ≤ t ≤ , then

G (t, s) =

(t – s)p– t p– ( – s)p– – ( – α ηp– ) (p)

(p)

=

t p– ( – s)p– – ( – α ηp– )(t – s)p– ( – α ηp– ) (p)

≥

t p– ( – s)p– α ηp– ( – α ηp– ) (p)

≥ . If  ≤ η ≤ t ≤ s ≤ , then

G (t, s) =

t p– ( – s)p– ≥ . ( – α ηp– ) (p)

Similarly, we can obtain G (t, s) ≥ . Thus, Gi (t, s) ≥  for every t, s ∈ J.

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() If  ≤ s ≤ t ≤ η ≤ , then

G (, s) = ≥ ≥

( – s)p– – α ηp– ( – ηs )p– – ( – α ηp– )( – s)p– ( – α ηp– ) (p) ( – s)p– – α ηp– ( – ηs )p– – ( – α ηp– )( – ηs )p– ( – α ηp– ) (p) ( – s)p– – ( – ηs )p– ( – α ηp– ) (p)

and G (t, s) = ≤ ≤

t p– [( – s)p– – α ηp– ( – ηs )p– ] – ( – α ηp– )( – ts )p– ( – α ηp– ) (p) ( – α ηp– )[( – ηs )p– – ( – ts )p– ] ( – α ηp– ) (p) ( – α ηp– )[( – s)p– – ( – ts )p– ] . ( – α ηp– ) (p)

Therefore, G (t, s) ≤ G (, s). If  ≤ t ≤ s ≤ η ≤ , then G (, s) = ≥

( – s)p– – α (η – s)p– ( – α ηp– ) (p) t p– [( – s)p– – α (η – s)p– ] ( – α ηp– ) (p)

= G (t, s). If  ≤ η ≤ s ≤ t ≤ , then G (, s) = =

( – s)p– – ( – α ηp– )( – s)p– ( – α ηp– ) (p) α ηp– ( – s)p– ( – α ηp– ) (p)

and G (t, s) =

t p– ( – s)p– – t p– ( – ts )p– ( – α ηp– ) ( – α ηp– ) (p)

≤

( – s)p– – ( – ts )p– ( – α ηp– ) ( – α ηp– ) (p)

≤

( – ts )p– α ηp– ( – α ηp– ) (p)

≤

( – s)p– α ηp– ( – α ηp– ) (p)

= G (, s).

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If  ≤ η ≤ t ≤ s ≤ , then G (, s) =

t p– ( – s)p– ( – s)p– ≥ = G (t, s). ( – α ηp– ) (p) ( – α ηp– ) (p)

Similarly, we can obtain G (t, s) ≤ G (, s). () If  ≤ s ≤ t ≤ η ≤ , then G (, s) ≤

( – s)p– ( – α ηp– ) (p)

G (t, s) ≥

t p– [( – s)p– – α (η – s)p– ] – t p– ( – α ηp– )( – ts )p– ( – α ηp– ) (p)

≥

t p– [( – s)p– – α ηp– ( – ts )p– – ( – α ηp– )( – ts )p– ] ( – α ηp– ) (p)

=

t p– [( – s)p– – ( – ts )p– ] . ( – α ηp– ) (p)

and

Let σ be a positive number such that mint∈[δ,–δ] G (t, s) ≥ σ G (, s). Then we can obtain σ ≤

  t p– ( – s)p– – (t – s)p– t – s p– p– = t – ≤ t p– . ( – s)p– –s

If  ≤ t ≤ s ≤ η ≤ , then G (, s) ≤

( – s)p– ( – α ηp– ) (p)

G (t, s) ≥

t p– ( – s)p– .

(p)

and

Let σ be a positive number such that mint∈[δ,–δ] G (t, s) ≥ σ G (, s). Then we can obtain σ ≤ t p– ( – α ηp– ). If  ≤ η ≤ s ≤ t ≤ , then G (, s) ≤

( – s)p– ( – α ηp– ) (p)

G (t, s) ≥

t p– ( – s)p– α ηp– . ( – α ηp– ) (p)

and

Let σ be a positive number such that mint∈[δ,–δ] G (t, s) ≥ σ G (, s). Then we can obtain σ ≤ t p– α ηp– .

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If  ≤ η ≤ t ≤ s ≤ , then G (, s) ≤

( – s)p– ( – α ηp– ) (p)

G (t, s) =

t p– ( – s)p– . ( – α ηp– ) (p)

and

Let σ be a positive number such that mint∈[δ,–δ] G (t, s) ≥ σ G (, s). Then we can obtain σ ≤ t p– . mint∈[δ,–δ] G (t,s) Define θ = min{σ , σ , σ , σ }. Then ≥ θ (s), δ ∈ (, ). Similarly, we can G (,s) prove

mint∈[δ,–δ] G (t,s) G (,s)

≥ θ (s), δ ∈ (, ).



Lemma  If (A) holds, then T(K) ⊂ K and T : K → K is a completely continuous operator. Proof The continuity of T is obvious. To prove T(K) ⊂ K , let us choose (x, y) ∈ K . Since Gi (t, s) ≤ Gi (, s) for  ≤ s ≤ , and Gi (t, s) ≥ θ Gi (, s) for δ ≤ t ≤  – δ, we have  min T (x, y)(t) ≥ λ δ p–

t∈[δ,–δ]



 G (, s)w s, x(s), y(s) ds



 G (, s)w s, x(s), y(s) ds



 ≥ δ p– λ



≥δ

  T (x, y).

p– 

Similarly,   min T (x, y)(t) ≥ δ q– T (x, y).

t∈[δ,–δ]

Thus,  min T (x, y)(t) + T (x, y)(t) ≥ min T (x, y)(t) + min T (x, y)(t)

t∈[δ,–δ]

t∈[δ,–δ]

t∈[δ,–δ]

  ≥ θ  T (x, y), T (x, y) .

Since Gi (t, s) ≥ , ∀t, s ∈ J and (A) holds, we conclude that T(K) ⊂ K . It is not difficult to show that T is uniformly bounded. Combining this with the Arzelà-Ascoli Theorem, we see that T : K → K is a completely continuous operator. 

3 Main results and proofs Theorem  Assume that (A) holds, then, for all λi > , i = , , the system () has at least one positive solution in the following cases: (a) w = h = , and either w∞ = ∞ or h∞ = ∞ (superlinear). (b) w∞ = h∞ = , and either w = ∞ or h = ∞ (sublinear).

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Proof (a) Since w = h = , we may choose H >  such that w(t, x, y) ≤ ε(x + y) and h(t, x, y) ≤ ε(x + y) for  < x + y ≤ H , t ∈ J, where the constant ε >  satisfies  ελ



 G (, s) ds ≤ ,



ελ



G (, s) ds ≤ .



Set  = {(x, y) : (x, y) ∈ X × X, (x, y) < H }. If (x, y) ∈ K ∩ ∂ , (x, y) = H , we have 



T (x, y)(t) ≤ λ

 G (t, s)w s, x(s), y(s) ds







≤ ελ

 G (t, s) x(s) + y(s) ds





≤ ελ x + y





G (, s) ds 

≤

(x, y)

. 

Similarly, T (x, y)(t) ≤

(x,y)

. 

Hence,

          T(x, y) = T (x, y), T (x, y) = T (x, y) + T (x, y) ≤ (x, y)  >  such that for (x, y) ∈ K ∩ ∂ . If we further assume that w∞ = ∞, then there exists H   w(t, x, y) ≥ β(x + y) for (x + y) ≥ H , t ∈ J, where β >  is chosen so that λ β  G (, s) ds ≥ .  } and set  = {(x, y) : (x, y) ∈ X × X, (x, y) < H }. If (x, y) ∈ K ∩ Let H = max{H , δ –p H  and for ∀t[δ,  – δ], ∂ , we have mint∈[δ,–δ] (x(t) + y(t)) ≥ δ p– (x, y) ≥ H  min T (x, y)(t) ≥ min λ

t∈[δ,–δ]

t∈[δ,–δ]

 ≥ λ β



 G (t, s)w s, x(s), y(s) ds



–δ

 G (t, s) x(s) + y(s) ds

δ

≥ λ βδ

  (x, y)

p– 





G (, s) ds 

  ≥ (x, y).

Therefore, T(x, y) = T (x, y), T (x, y) = T (x, y) + T (x, y) ≥ (x, y) for (x, y) ∈ K ∩ ∂ . An analogous estimate holds for h∞ = ∞. Now by Theorem A, T has a fixed point (x, y) ∈ K ∩ ∂( \  ) such that H ≤ (x, y) ≤ H and the system () has a positive solution. (x + y) for  < x + y ≤ H , t ∈ J, (b) If w = ∞, we choose H >  so that w(t, x, y) ≥ β  p–    where β satisfies λ β δ  G (, s) ds ≥ . Let  = {(x, y) : (x, y) ∈ X × X, (x, y) < H }, if (x, y) ∈ K ∩ ∂ , (x, y) = H , and for ∀t ∈ [δ,  – δ],  min T (x, y)(t) ≥ min λ

t∈[δ,–δ]

t∈[δ,–δ]

  ≥ λ β δ

–δ



 G (t, s)w s, x(s), y(s) ds



 G (t, s) x(s) + y(s) ds

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  δ p– (x, y) ≥ λ β





G (, s) ds 

  ≥ (x, y).

Therefore, T(x, y) = T (x, y), T (x, y) = T (x, y) + T (x, y) ≥ (x, y) for (x, y) ∈ K ∩ ∂ An analogous estimate holds for h = ∞. Set w∗ (t) = max≤x+y≤t w(t, x, y) and h∗ (t) = max≤x+y≤t h(t, x, y). Then w∗ and h∗ are nondecreasing in their respective arguments. Moreover, from w∞ = h∞ = , we see that ∗ ∗ limt→∞ w t(t) = , limt→∞ h t(t) = . Therefore, there is an H > H such that w∗ (t) ≤ εt, h∗ (t) ≤ εt for t ≥ H , where the constant ε >  satisfies  ελ



 G (, s) ds ≤ ,



ελ



G (, s) ds ≤ .



Set  = {(x, y) : (x, y) ∈ X × X, (x, y) < H }. If (x, y) ∈ K ∩ ∂ , (x, y) = H , we have 



T (x, y)(t) ≤ λ 

 G (t, s)w s, x(s), y(s) ds

 

≤ λ 

G (t, s)w∗ (H ) ds 



≤ ελ H

G (, s) ds 

≤

(x, y)

. 

Similarly, T (x, y)(t) ≤

(x,y)

. 

Hence,

          T(x, y) = T (x, y), T (x, y) = T (x, y) + T (x, y) ≤ (x, y) for (x, y) ∈ K ∩ ∂ . Applying Theorem A, we conclude to the existence of a positive solution (x, y) ∈ K ∩ ( \  ) for the system ().  Theorem  Assume that (A) and (A) hold. (a) If w = h = w∞ = h∞ = , then there is a positive constant σ such that () has at least two positive solutions for all λ , λ ≥ σ . (b) If w = ∞ or h = ∞, and either w∞ =  or h∞ = , then there is a positive constant σ such that the system () has at least two positive solutions for all λ , λ ≥ σ . Proof (a) For (x, y) ∈ K and (x, y) = l, let   m(l) = min λ

–δ

 G (, s)w s, x(s), y(s) ds, λ

δ



–δ

  G (, s)h s, x(s), y(s) ds .

δ

By assumption m(l) >  for l > . Choose two numbers  < H < H , and let  σ = max

 H H , , m(H ) m(H )

    (i = , ). i = (x, y) : (x, y) ∈ X × X, and (x, y) < Hi

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Then, for λ , λ ≥ σ , (x, y) ∈ K ∩ ∂i (i = , ), and (x, y) = Hi , we have   min T (x, y)(t) ≥ λ β

t∈[δ,–δ]

–δ δ

 Hi G (t, s)w s, x(s), y(s) ds ≥ λ m(Hi ) ≥ 

(i = , ).

Similarly, mint∈[δ,–δ] T (x, y)(t) ≥ Hi (i = , ). This implies that T(x, y) ≥ Hi = (x, y)

for (x, y) ∈ K ∩ ∂i (i = , ). Since w = h = w∞ = h∞ = , it follows from the proof of Theorem (a) and (b), respectively, we can choose H < H and H > H such that

T(x, y) ≤ (x, y) for (x, y) ∈ K ∩ ∂i (i = , ), where i = {(x, y) : (x, y) ∈ X × X, (x, y) < Hi } (i = , ). Applying Theorem A to  ,  and  ,  we get a positive solution (x , y ) such that H ≤ (x , y ) ≤ H and another positive solution (x , y ) such that H ≤ (x , y ) ≤ H . Since H < H , these two solutions are distinct. (b) For (x, y) ∈ K and (x, y) = L, let         M(L) = max λ G (, s)w s, x(s), y(s) ds, λ G (, s)h s, x(s), y(s) ds . 



H Then M(L) >  for L > . Choose two numbers  < H < H , let σ = min{ M(H , H }  ) M(H ) and set i = {(x, y) : (x, y) ∈ X × X, (x, y) < Hi } (i = , ). Then, for λ , λ ≤ σ and (x, y) ∈ K ∩ ∂i (i = , ), (x, y) = Hi , we have T (x, y)(t) ≤ λ M(Hi ) ≤ Hi (i = , ), and, T (x, y)(t) ≤ Hi (i = , ), which implies T(x, y) ≤ Hi = (x, y) for (x, y) ∈ K ∩ ∂i (i = , ). Since either w = ∞ or h = ∞, and either w∞ = ∞ or h∞ = ∞, it follows from the proof of Theorem (a) and (b), we can choose H < H and H > H such that

T(x, y) ≥ (x, y) for (x, y) ∈ K ∩ ∂i (i = , ), where i = {(x, y) : (x, y) ∈ X × X, (x, y) < Hi } (i = , ). Once again, we conclude to the existence of two distinct positive solutions. 

Theorem  Assume (A) and (A) hold. (a) If w = h =  or w∞ = h∞ = , then there is a positive constant σ such that () has at least two positive solutions for all λ , λ ≥ σ . (b) If w = ∞ or h = ∞, or if w∞ = ∞ or h∞ = ∞, then there is a positive constant σ such that the system () has at least two positive solutions for all λ , λ ≤ σ . Example  Consider the system of fractional differential equation provided by ⎧  ⎪ D  x(t) + [x(t) + y(t)] = , t ∈ [, ], ⎪ ⎪ ⎪ ⎪ ⎨D  y(t) + [x(t) + y(t)] = , t ∈ [, ],    ⎪ ⎪ D  x() = D  x() = D  x() = , x() =  x(  ), ⎪ ⎪ ⎪   ⎩  D x() = D  x() = D  x() = , y() =  y(  ),

()

where for ∀x, y > , w(t, x(t), y(t)) = [x(t) + y(t)] > , h(t, x(t), y(t)) = [x(t) + y(t)] > , q =  , q =  ∈ (, ); p =  , p =  ∈ (, ); γ =  , γ =  ∈ (, ); λ = , λ = , α =  , α =  , η =  , ξ =  , and α ηp– = p , α ξ p– = p ∈ (, ). By direct calculation we obtain w = h =  and w∞ = h∞ = ∞. Then, by Theorem (a), the system () has at least one positive solution.

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Example  Consider the system of fractional differential equation provided by ⎧   ⎪ D  x(t) +  x(t) + y(t) = , t ∈ [, ], ⎪ ⎪ ⎪ ⎪ ⎨D  y(t) +   x(t) + y(t) = , t ∈ [, ],   ⎪D  x() = D  x() = D  x() = , ⎪ x() = x(  ), ⎪ ⎪ ⎪    ⎩ y() = y(  ), D  x() = D  x() = D  x() = ,

()

  where ∀x, y > , w(t, x(t), y(t)) =  x(t) + y(t) > , h(t, x(t), y(t)) =  x(t) + y(t) > , q = q =  ∈ (, ); p = p =  ∈ (, ); γ = γ =  ∈ (, ); λ = λ = , α = α = , η = ξ =  , and   ∈ (, ). By direct calculation we see that w = h = ∞ and w∞ = α ηp– = α ξ p– = p– h∞ = . Then, by Theorem (b), the system () has at least one positive solution.

4 Conclusions In this research, by using the Krasnosel’skii fixed point theorem for a cone map, we studied the existence and multiplicity of positive solutions for a class of systems of fractional differential equations with parameters, and we obtained the existence of at least one and two solutions for our considered system. Acknowledgements The authors would like to thank very much the referees for their expert and constructive comments, which have further enriched our article. This work is supported by the National Natural Science Foundation of China (Nos. 11561063, 11401479, 71561024). Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

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