Existence and nonexistence of positive solutions for the fractional coupled system involving generalized p Laplacian

R E S E A R C H

Open Access

Existence and nonexistence of positive

solutions for the fractional coupled system

involving generalized

p

-Laplacian

Ying Wang

_{and Jiqiang Jiang}

*_{Correspondence:}

[email protected]

3_{School of Mathematical Sciences,}

Qufu Normal University, Qufu, Shandong 273165, People’s Republic of China

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

Abstract

In this article, we study a class of fractional coupled systems with Riemann-Stieltjes integral boundary conditions and generalizedp-Laplacian which involves two different parameters. Based on the Guo-Krasnosel’skii fixed point theorem, some new results on the existence and nonexistence of positive solutions for the fractional system are received, the impact of the two different parameters on the existence and nonexistence of positive solutions is also investigated. An example is then given to illuminate the application of the main results.

MSC: 26A33; 34B18

Keywords: positive solutions; fractional coupled system; Riemann-Stieltjes integral conditions; generalizedp-Laplacian operator

1 Introduction

In this paper, our main research is the existence and nonexistence of positive solutions for the following fractional coupled system with generalizedp-Laplacian involving Riemann-Stieltjes integral conditions.

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Dβ

+(φ(Dα_+u(t))) +λf(t,u(t),v(t)) = ,

Dβ

+(φ(D

α

+v(t))) +λf(t,u(t),v(t)) = ,  <t< ,

u() =u() =· · ·u(n–)= , φ(Dα

+u()) = (φ(Dα_+u()))= ,

v() =v() =· · ·v(m–)_{= , φ(D}α

+v()) = (φ(D

α

+v()))= ,

u() =μ



g(s)v(s)dA(s), v() =μ



g(s)u(s)dA(s),

()

whereλi>  (i= , ) is a parameter,  <βi≤,n–  <α≤n,m–  <α≤m,n,m≥,

Dαi

+,D

βi

+are the standard Riemann-Liouville derivatives.μi>  is a constant,gi: (, )→ [, +∞) is continuous withgi∈L(, ),Aiis right continuous on [, ), left continuous at

t= , and nondecreasing on [, ],Ai() = ,



x(s)dAi(s) denotes the Riemann-Stieltjes integrals ofxwith respect toAi,φis a generalizedp-Laplacian operator and satisfies the following condition (H).

The positive solution (u,v) of system () means that (u,v)∈C[, ]×C[, ], (u,v) satis-fies system () andu(t) > ,v(t) >  for allt∈(, ].

(H) φ:R→Ris an odd, increasing homeomorphism, and there exist two increasing homeomorphismsψ,ψ: (, +∞)→(, +∞)such that

ψ(x)φ(y)≤φ(xy)≤ψ(x)φ(y), x,y> .

Moreover,φ,φ–∈C(R), whereφ–denotes the inverse ofφandR= (–∞, +∞).

Lemma .([]) Assume that(H)holds.Then

ψ_–(x)y≤φ–xφ(y)≤ψ_–(x)y, x,y> .

Forφsatisfying (H), we call it a generalizedp-Laplacian operator, it contains two im-portant special cases:φ(u) =uandφ(u) =|u|p–_{u(p> ) (see []). Many researchers have} studied the existence of positive solutions for two above cases due to their great applica-tion background (see [–]). Combined with the fracapplica-tional calculus, the applicaapplica-tion of the above two kinds of special circumstances becomes more extensive and practical. For the sake of considering the turbulent flow in a porous medium, the governing equation

∂ ∂x

∂um

∂x

∂_∂u_xmp–=g t,u,∂u

∂t

, m≤,

≤p≤ ()

was presented by Leibenson (see []). Ifp= ,m> , it is used as a nonlinear model for the dispersion of animals and insects (see []).

In [], Lu et al. studied the existence of positive solution for the fractional boundary value problem with ap-Laplacian operator:

⎧ ⎨ ⎩

Dβ_+(φp(Dα_+u(t))) =f(t,u(t)),  <t< ,

u() =u() =u() = , D_α+u() =Dα_+u() = ,

()

where  <α≤,  <β≤,Dα

+,Dβ_+are the standard Riemann-Liouville fractional

deriva-tives,φp(s) =|s|p–s,p> ,φp–=φq,_p+_q= ,f : [, ]×[, +∞)→[, +∞) is a continuous function. By the properties of Green’s function and the Guo-Krasnosel’skii fixed point the-orem, some results on the existence of positive solutions are obtained.

In [], Wang et al. investigated the same equation as () for  <α≤,  <β≤, with boundary value conditionu() = ,Dα

+u() = ,u() =au(ξ), where ≤a≤,  <ξ< ,

f : [, ]×[, +∞)→[, +∞) is a continuous function. Through the application of the Guo-Krasnosel’skii fixed point theorem and the Leggett-Williams theorem, sufficient con-ditions for the existence of positive solutions are received.

boundary value conditions. Henderson and Luca in [] considered the following system:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Dα

+u(t) +λf(t,u(t),v(t)) = ,

Dα

+v(t) +λf(t,u(t),v(t)) = ,  <t< ,

u() =u() =· · ·u(n–)_{= , u() =}p

i=aiu(ξi),

v() =v() =· · ·v(m–)= , v() =q_i=biv(ηi),

()

whereλi>  (i= , ) is a parameter,n–  <α≤n,m–  <α≤m,n,m≥,D αi

+,D

βi

+

are the standard Riemann-Liouville derivatives. ai> ,bi>  are constants,fi: [, ]× [, +∞)×[, +∞)→[, +∞) is a continuous function. By the Guo-Krasnosel’skii fixed point theorem, the authors in [] got the existence of positive solutions on system (). System () with uncoupled and coupled multi-point boundary value conditions

⎧ ⎨ ⎩

u() =u() =· · ·u(n–)_{= , u() =μ} 



u(s)dA(s),

v() =v() =· · ·v(m–)_{= , v() =μ} 



v(s)dA(s),

⎧ ⎨ ⎩

u() =u() =· · ·u(n–)_{= , u() =μ} 



v(s)dA(s),

v() =v() =· · ·v(m–)_{= , v() =μ} 



u(s)dA(s)

has been studied in many papers, whereμi>  is a constant, forμi=  as an exceptional case ( see [–] and the references therein). However, these articles only study the exis-tence of positive solutions for the system, and do not relate to the nonexisexis-tence of positive solutions.

Up to now, coupled boundary value conditions for a fractional differential system with generalized p-Laplacian like system () have seldom been considered whenλ,λare dif-ferent. Motivated by the results mentioned above, in this paper, we obtain several new existence and nonexistence results for positive solutions in terms of different values of the parameterλiby using the properties of Green’s function and the Guo-Krasnosel’skii fixed point theorem on cone. Especially, paying attention to the nonlinear operatorDβ_+(φ(Dα_+))

with the discussion in (), we can convert it to the linear operatorDβ_+Dα_+, ifφ(u) =u, and

the additive index law

Dβ_+Dα_+u(t) =Dα+β_+ u(t)

holds under some reasonable constraints on the functionu(see []). Therefore, our ar-ticle promotes, includes and improves the previous results in a certain degree.

2 Preliminaries and lemmas

For convenience of the reader, we present some necessary definitions about fractional cal-culus theory.

Definition .([, ]) Letα>  andube piecewise continuous on (, +∞) and

inte-grable on any finite subinterval of [, +∞). Then, fort> , we call

Iα +u(t) =



(α)

t



(t–s)α–u(s)ds,

Definition . ([, ]) The Riemann-Liouville fractional derivative of order α > ,

n– ≤α<n,n∈N, is defined as

Dα_+u(t) =



(n–α)

d dt

n t



(t–s)n–α–u(s)ds,

whereNdenotes the natural number set, the functionu(t) isntimes continuously differ-entiable on [, +∞).

Lemma .([, ]) Letα> ,if the fractional derivatives Dα–_+ u(t)and Dα_+u(t)are

con-tinuous on[, +∞),then

Iα

+Dα_+u(t) =u(t) +c_tα–+c_tα–+· · ·+c_ntα–n,

where c,c, . . . ,cn∈(–∞, +∞),n is the smallest integer greater than or equal toα.

Similarly to the proof in [], it enables us to obtain the following Lemmas ., . and Remark ..

Lemma . Assume that the following condition(H)holds.

(H)

k=





g(t)tα–_{dA(t) > , k}

=





g(t)tα–_{dA(t) > ,}

 –μμkk> .

Let hi∈C(, )∩L(, ) (i= , ),then the system with the coupled boundary conditions

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Dα

+u(t) +h(t) = , D α

+v(t) +h(t) = ,  <t< ,

u() =u() =· · ·u(n–)_{= , u() =μ} 



g(s)v(s)dA(s),

v() =v() =· · ·v(n–)_{= , v() =μ} 



g(s)u(s)dA(s)

()

has a unique integral representation ⎧

⎨ ⎩

u(t) =_K(t,s)h(s)ds+



H(t,s)h(s)ds,

v(t) =_K(t,s)h(s)ds+_H(t,s)h(s)ds, ()

where

K(t,s) = μμkt α–

 –μμkk





g(t)G(t,s)dA(t) +G(t,s),

H(t,s) =

μtα–  –μμkk





g(t)G(t,s)dA(t),

K(t,s) = μμkt α–

 –μμkk





g(t)G(t,s)dA(t) +G(t,s),

H(t,s) = μt α–

 –μμkk





g(t)G(t,s)dA(t),

and

Gi(t,s) = 

(αi)

⎧ ⎨ ⎩

[t( –s)]αi–_{– (t–s)}αi–_{, }≤_s≤_t≤_,

[t( –s)]αi–_{, }≤_t≤_s≤_,

i= , .

Lemma . For t,s∈[, ],the functions Ki(t,s)and Hi(t,s) (i= , )defined as()satisfy

K(t,s),H(t,s)≤ρs( –s)α–_{, K(t,s),H(t,s)}≤_{ρs( –s)}α–_,

K(t,s),H(t,s)≤ρtα–_{, K(t,s),H(t,s)}≤_ρtα–_,

K(t,s)≥tα–_{s( –s)}α–_{, H(t,s)}≥_tα–_{s( –s)}α–_,

K(t,s)≥tα–s( –s)α–, H(t,s)≥tα–s( –s)α–,

where

ρ=max



(α– )

μμk  –μμkk





g(t)dA(t) + 

,

μ

(α– )( –μμkk)





g(t)dA(t),



(α– )

μμk  –μμkk





g(t)dA(t) + 

,

μ

(α– )( –μμkk)





g(t)dA(t)

,

=min

μμk

(α)( –μμkk)





g(t)( –t)tα–_dA(t),

μ

(α)( –μμkk)





g(t)( –t)tα–_dA(t),

μμk

(α)( –μμkk)





g(t)( –t)tα–_dA(t),

μ

(α)( –μμkk)





g(t)( –t)tα–dA(t)

.

Remark . From Lemma ., fort,t,s∈[, ], we have

Ki(t,s)≥ωtαi–Ki(t,s), Hi(t,s)≥ωtαi–Hi(t,s), i= , ,

K(t,s)≥ωtα–H(t,s), H(t,s)≥ωtα–K(t,s),

K(t,s)≥ωtα–_{H(t,s), H(t,s)}≥_ωtα–_K(t,s),

whereω=_ρ,,ρare defined as in Lemma .,  <ω< .

Lemma . Let <βi≤,n–  <α≤n,m–  <α≤m,hi∈C(, )∩L(, ) (i= , ),

the following system of fractional differential equations

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Dβ

+(φ(Dα_+u(t))) +λh(t) = , Dβ+(φ(Dα_+v(t))) +λh(t) = ,  <t< ,

u() =u() =· · ·u(n–)_{= , φ(D}α

+u()) = (φ(D

α

+u()))= ,

v() =v() =· · ·v(m–)= , φ(Dα

+v()) = (φ(Dα_+v()))= ,

u() =μ



v(s)dA(s), v() =μ



u(s)dA(s)

has a unique integral representation

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

u(t) =_K(t,s)φ–(λ



G(s,τ)h(τ)dτ)ds +_H(t,s)φ–(λ



G(s,τ)h(τ)dτ)ds,

v(t) =_K(t,s)φ–_(λ 



G(s,τ)h(τ)dτ)ds +_H(t,s)φ–(λ



G(s,τ)h(τ)dτ)ds,

where

Gi(s,τ) = 

(βi)

⎧ ⎨ ⎩

s[s( –τ)]βi–_{– (s–τ)}βi–_{, }≤_τ≤_s≤_,

s[s( –τ)]βi–_{, }≤_s≤_τ≤_,

i= , . ()

Lemma .([]) The function Gi(s,τ)defined as()is continuous on[, ]×[, ],and

for s,τ∈[, ],Gi(s,τ)satisfies

() Gi(s,τ)≥; () Gi(s,τ)≤Gi(τ,τ); () Gi(s,τ)≥sβi–Gi(,τ).

In the rest of the paper, we always suppose that the following assumption holds:

(H) fi: [, ]×[, +∞)×[, +∞)→[, +∞)is continuous.

LetX=C[, ]×C[, ], thenXis a Banach space with the norm

(u,v)=max u , v , u =max

t∈[,]

u(t), v =max

t∈[,]

v(t).

Denote

K=(u,v)∈X:u(t)≥ωtα–_(u,v),v(t)≥_ωtα–_(u,v),t∈_{[, ],}

whereωis defined as Remark .. It is easy to see thatKis a positive cone inX. Under the above conditions (H)(H)(H), for any (u,v)∈K, we can define an integral operator

T:K→Xby

T(u,v)(t) =

Lemma . Assume that(H)(H)(H)hold.Then T:K→K is a completely continuous

operator.

Proof By the routine discussion, we know thatT:K→Xis well defined, so we only prove

T(K)⊆K. For any (u,v)∈K, ≤t,t≤, by Remark ., we have

On the other hand,

Then we have

T(u,v)(t)≥ωtα–T(u,v), T(u,v)(t)≥ωtα–T(u,v),

i.e.,

T(u,v)(t)≥ωtα–T(u,v),T(u,v).

In the same way as () and (), we can prove that

T(u,v)(t)≥ωtα–T(u,v),T(u,v). Therefore, we haveT(K)⊆K.

According to the Ascoli-Arzela theorem, we can easily get thatT:K→Kis completely

continuous. The proof is completed.

In order to obtain the existence of the positive solutions of system (), we will use the following cone compression and expansion fixed point theorem.

Lemma .([]) Let P be a positive cone in a Banach space E,andare bounded

open sets in E,θ∈,⊂,A:P∩\→P is a completely continuous operator.If

the following conditions are satisfied:

Ax ≤ x , ∀x∈P∩∂, Ax ≥ x , ∀x∈P∩∂,

or

Ax ≥ x , ∀x∈P∩∂, Ax ≤ x , ∀x∈P∩∂,

then A has at least one fixed point in P∩(\).

3 Main results

Denote

f=lim inf

x→+ _t∈[a,inf_{b]⊂(,)}

y∈[,+∞)

f(t,x,y)

φ(x) , f 

 =lim sup x→+

sup

t∈[,] y∈[,+∞)

f(t,x,y)

φ(x) ,

f=lim inf

y→+ _t∈[a,inf_{b]⊂(,)}

x∈[,+∞)

f(t,x,y)

φ(y) , f 

 =lim sup y→+

sup

t∈[,] x∈[,+∞)

f(t,x,y)

φ(y) ,

f∞=lim inf

x→+∞t∈[a,infb]⊂(,) y∈[,+∞)

f(t,x,y)

φ(x) , f ∞

 =lim sup x→+∞

sup

t∈[,] y∈[,+∞)

f(t,x,y)

φ(x) ,

f∞=lim inf

y→+∞t∈[a,infb]⊂(,) x∈[,+∞)

f(t,x,y)

φ(y) , f ∞

 =lim sup y→+∞ tsup∈[,]

x∈[,+∞)

f(t,x,y)

φ(y) ,

L=max

ρϕ_–





G(τ,τ)dτ

, ρϕ–_





G(τ,τ)dτ

L=min

3.1 Existence of system (1)

Theorem . Assume that(H)(H)(H)hold and fi∞ϕ(L– ) >fiϕ(L– ),then system()

has at least one positive solution for

λi∈

where we impose 

Therefore, we obtain

T(u,v)=maxT(u,v),T(u,v)≥r=(u,v) for any (u,v)∈∂Kr. ()

It follows from the above discussion, (), (), Lemmas . and . that, for anyλi∈ (ϕ(L– )

fi∞ , ϕ(L– )

f_i ),Thas a fixed point (u,v)∈Kr\Kr, so system () has at least one positive

solution (u,v); moreover, (u,v) satisfiesr≤ (u,v) ≤r. The proof is completed.

Remark . From the proof of Theorem ., if we choose

L=ωθ





ϕ_–sβ–_sβ–_ϕ–

 



G(,τ)dτ

ds, θ= min

t∈[a,b]

tα–_,tα–_{, ()}

then forλ∈(ϕ(L

– )

f∞ , ϕ(L– )

f_ ),λ∈(, ϕ(L– )

f_ ), the conclusion of Theorem . is valid. Or we choose

L=ωθ





ϕ_–sβ–_sβ–_ϕ–

 



G(,τ)dτ

ds, θ= min

t∈[a,b]

tα–_,tα–_{, ()}

then, forλ∈(, ϕ(L–)

f_ ),λ∈( ϕ(L– )

f∞ ,

ϕ(L–)

f_ ), the conclusion of Theorem . is valid.

Theorem . Assume that(H)(H)(H)hold and fiϕ(L– ) >fi∞ϕ(L– ),then system()

has at least one positive solution for

λi∈

ϕ(L– )

fi ,ϕ(L

–  )

f_i∞

,

where we impose _f

i= if fi= +∞and



f_i∞ = +∞if fi∞= ,i= , .

The proof of Theorem . is similar to that of Theorem ., and so we omit it.

Remark . Similar to Remark ., if we chooseLas (), then forλ∈(

ϕ(L– )

f ,

ϕ(L–)

f∞ ),

λ∈(, ϕ(L– )

f_∞ ), the conclusion of Theorem . is valid. Or we chooseLas (), then forλ∈(,

ϕ(L– )

f_∞ ),λ∈( ϕ(L– )

f ,

ϕ(L–)

f_∞ ), the conclusion of Theorem . is valid.

Theorem . Assume that(H)(H)(H)hold and there exist R>r> such that

λi min t∈[a,b]⊂(,)

ωθr≤x,y≤r

fi(t,x,y)≥φ

r L

, λi max t∈[,] ≤x,y≤R

fi(t,x,y)≤φ

R L

, i= , . ()

Then system () has at least one positive solution (u,v); moreover, (u,v) satisfies r ≤

(u,v) ≤R.

Proof SetKr={(u,v)∈K: (u,v) <r}. For any (u,v)∈∂Kr, by the definition of · , we have

ωθr≤ωtα–_r=ωtα–_(u,v)≤_u(t)≤_r,

≤φ–

λ max

τ∈[,] ≤u(τ),v(τ)≤R

f

τ,u(τ),v(τ)L 

+φ_–

λ max

τ∈[,] ≤u(τ),v(τ)≤R

fi

τ,u(τ),v(τ)L 

≤R=(u,v). ()

Similarly to (), for any (u,v)∈∂KR, we also have

T(u,v)<R=(u,v).

Consequently, we have

T(u,v)=maxT(u,v),T(u,v)<R=(u,v), (u,v)∈∂KR. ()

It follows from the above discussion, (), (), Lemmas . and . thatT has a fixed point (u,v)∈KR\Kr, so system () has at least one positive solution (u,v); moreover, (u,v) satisfiesr≤ (u,v) ≤R. The proof is completed.

Remark . From the proof of Theorem ., if we choose

L=θ





ϕ_–sβ–_{s( –s)}α–_ϕ–

 



G(,τ)dτ

ds,

θ= min

t∈[a,b]

tα–_,tα–_{, ()}

then for

λ min t∈[a,b]⊂(,)

ωr≤x,y≤r

f(t,x,y)≥φ r L

, λi max t∈[,] ≤x,y≤R

fi(t,x,y)≤φ

R L

, i= , ,

the conclusion of Theorem . is valid. Or we choose

L=θ





ϕ_–sβ–_{s( –s)}α–_ϕ–

 



G(,τ)dτ

ds,

θ= min

t∈[a,b]

tα–_,tα–_{. ()}

Then, for

λ min t∈[a,b]⊂(,)

ωr≤x,y≤r

f(t,x,y)≥φ r L

, λi max t∈[,] ≤x,y≤R

fi(t,x,y)≤φ

R L

, i= , ,

the conclusion of Theorem . is valid.

Theorem . Assume that(H)(H)(H)hold and fi=fi∞= +∞,then there existsλ∗i > 

Proof Chooser> , define

χi(r) =sup r>

φ(r)

ϕ(L)maxt∈[,]

≤x,y≤rfi(t,x,y)

, i= , .

In view of the continuity offiandfi=fi∞= +∞, we knowχi(r) : (, +∞)→(, +∞) is continuous and

lim

t→+χi(r) =_t→lim_+∞χi(r) = .

So, there existsr∗∈(, +∞) such thatχi(r∗) =supr>χi(r) =λ∗i. Therefore, forλi∈(,λ∗i), we can findr,r ( <r<r∗<r< +∞) satisfyingχi(r) =λ,χi(r) =λ. Thus, by (H), we have

λ max t∈[,] ≤x,y≤r

fi(t,x,y)≤

φ(r)

ϕ(L)≤φ

r

L

, ()

λ max t∈[,] ≤x,y≤r

fi(t,x,y)≤

φ(r)

ϕ(L)≤φ

r

L

. ()

From the conditionfi=fi∞= +∞, there existR,R( <R<r<r∗<r<R< +∞) satisfying

f(t,x,y)

φ(x) ≥



λϕ(ωθ)ϕ(L)

, (x,y)∈[,R]∪[R, +∞),t∈[a,b]⊂(, ),

f(t,x,y)

φ(y) ≥



λϕ(ωθ)ϕ(L)

, (x,y)∈[,R]∪[R, +∞),t∈[a,b]⊂(, ).

Hence, by (H), we get

λi min t∈[a,b]⊂(,) ωθR≤x,y≤R

fi(t,x,y)≥φ

R

L

, ()

λi min t∈[a,b]⊂(,) ωθR≤x,y≤R

fi(t,x,y)≥φ

R

L

. ()

By () and (), () and (), combining with Lemmas ., . and Theorem ., sys-tem () has at least two positive solutions forλi∈(,λ∗i),i= , . The proof is completed.

Remark . From the proof of Theorem ., assume that (H)(H)(H) hold, iffi= +∞

orfi∞= +∞, then there existsλ∗i >  such that system () has at least one positive solution forλi∈(,λ∗i),i= , .

3.2 Nonexistence of system (1)

Theorem . Assume that(H)(H)(H)hold and f_i∞< +∞,f

i < +∞,then there exists

Proof From the definitions off_i∞,f_i, which are finite, there exist positive constantsM

Similarly to () (), we also have

v <(u,v). ()

Hence, by () (), we get

_(u,v)=max u , v <(u,v), ()

which is a contradiction. Therefore, system () has no positive solution. The proof is

Theorem . Assume that(H)(H)(H)hold and fi∞ > ,fi> ,fi(t,x,y) > for t∈ [a,b]⊂(, ),x≥,y> or t∈[a,b]⊂(, ),x> ,y≥,then there existsλi∗> such

that forλi∈(λi∗, +∞) (i= , ),system()has no positive solution.

Proof From the definitions offi∞,fi, which are finite, there exist positive constantsmi,

m_i andR,R(R<R) such that

f(t,x,y)≥mφ(x), ≤x,y≤R,t∈[a,b]⊂(, ),

f(t,x,y)≥mφ(x), x,y≥R,t∈[a,b]⊂(, ).

Setm_ =min{m_,m_,mint∈[a,b]⊂(,),R≤x,y≤R

f(t,x,y)

φ(x) }, we have

f(t,x,y)≥mφ(x), x,y≥,t∈[a,b]⊂(, ).

Similarly, setm_=min{m_,m_,mint∈[a,b]⊂(,),R≤x,y≤R

f(t,x,y)

φ(y) }, we have

f(t,x,y)≥mφ(y), x,y≥,t∈[a,b]⊂(, ).

Assume that (u,v) is a positive solution of system (), we will show that this leads to a contradiction. Defineλi∗= (mi)

–

ϕ(L–

 ), sinceλi∈(λi∗, +∞), by Lemmas ., . and ., we conclude that

u =T(u,v)

≥ min

t∈[a,b]





tα–_{s( –s)}α–_φ– _λ







sβ–_{G(,τ)m}

φ

u(τ)dτ

ds

+





tα–_{s( –s)}α–_φ– _λ 





sβ–_G

(,τ)mφ

v(τ)dτ

ds

≥ωθ(u,v)





ϕ–_ sβ–_sβ–_ϕ–

 



G(,τ)dτ

dsψ_–λm

+ωθ(u,v)





ϕ_–sβ–_sβ–_ϕ–

 



G(,τ)dτ

dsψ_–λm

≥(u,v)ϕ_–λm

L

 +(u,v)ϕ – 

λm

L

 >(u,v). ()

Similarly to (), we also have

v >(u,v). ()

Hence, by () (), we get

_(u,v)=max u , v >(u,v), ()

which is a contradiction. Therefore, system () has no positive solution. The proof is

com-pleted.

Theorem . Assume that(H)(H)(H)hold and f∞> ,f> ,f(t,x,y) > for t∈

Remark . From the proof of Theorems .-., if we choose

f=lim inf

then all the conclusion of Theorems .-. are valid.

4 Example

Consider the fractional differential system

⎧

So, condition (H) holds. Next, in order to demonstrate the application of our main results obtained in Section , we choose two different sets of functionsfi (i= , ) such thatfi satisfies the conditions of Theorems . and ..

Case . Letf(t,x,y) = _+x_t +xsiny,f(t,x,y) = y_et +ysinx, choose [_,_]⊂[, ], we know

fi∞= +∞,fi= . Then, by Theorem ., system () has at least one positive solution for

Case . Letf(t,x,y) =(x_(++x_t)()(+_x+)siny),f(t,x,y) =(y_e+ty₍)(+_y+)sinx), therefore, we havef_∞= ,

f

 = ,f∞= ,f= , and forx,y≤, we getx≤f(t,x,y)≤x,y≤f(t,x,y)≤y. By calculation, we obtain L= .

 

τ(–τ)–  ()

dτ ≈.. Then, by Theorem ., system () has no positive solution forλ∈(, .),λ∈(, .).

Acknowledgements

The authors were supported financially by the National Natural Science Foundation of China (11626125, 11701252), Natural Science Foundation of Shandong Province of China (ZR2016AP04), China Postdoctoral Science Foundation funded project (2017M612231), a Project of Shandong Province Higher Educational Science and Technology Program (J16LI03), the Science Research Foundation for Doctoral Authorities of Linyi University (LYDX2016BS080), and the Applied Mathematics Enhancement Program of Linyi University.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Author details

1_{School of Mathematics and Statistics, Linyi University, Linyi, Shandong 276000, People’s Republic of China.}2_Key

Laboratory of Complex Systems and Intelligent Computing, Universities of Shandong (Linyi University), Linyi, Shandong 276000, People’s Republic of China. 3_{School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165,}

People’s Republic of China.

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Received: 5 July 2017 Accepted: 2 October 2017 References

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