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R E S E A R C H

Open Access

Existence of solution for integral

boundary value problems of fractional

differential equations

Xueying Yue

1

and Kemei Zhang

1*

*Correspondence:

zhkm90@126.com

1School of Mathematical Sciences,

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

Abstract

In this paper, we discuss the existence of positive solutions of fractional differential equations on the infinite interval (0, +∞). The positive solution of fractional differential equations is gained by using the properties of the Green’s function, Leray–Schauder’s fixed point theorems, and Guo–Krasnosel’skii’s fixed point theorem. As an application, two examples are given to prove our conclusions.

Keywords: Positive solutions; Fractional differential equations; Leray–Schauder fixed point theorem; Guo–Krasnosel’skii’s fixed point theorem

1 Introduction

In this article, we investigate the positive solutions for fractional differential equations with infinite-point boundary value conditions:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

0+u(t) +q(t)f(t,u(t)) = 0, t∈(0, +∞),

ui(0) = 0, i= 0, 1, 2, . . . ,n– 3,

–1

0+ u(+∞) = 0, 0+–2u(0) =

i=1βiu(ξi),

(1.1)

where 1 <n– 1 <αn, n= [α] + 1,

0+ is the Riemann–Liouville fractional

deriva-tive, 0 <ξ1<ξ2<· · ·<ξi<· · ·< +∞,βi≥0,i= 0, 1, 2, . . . ,n– 3,f : [0, +∞)×[0, +∞)→

[0, +∞).

Recently, many kinds of fractional differential equations have been generally studied, see [1–15]. As everyone knows, fractional differential equations are used in many fields, for instance, control theory, mechanics, polymer rheology and engineering, and so on, for the details, see [16–21]. From the references, we can obtain that many authors have used the cone expansion and cone compression fixed point theorem to prove their conclusions, see [5,22–26].

(2)

In [4], the authors studied the following equations:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

0+u(t) +f(t,u(t),0+u(t)) = 0, t∈(0, +∞), u(0) = 0,

0+–1u(∞) = τ

0 g1(s)u(s)ds+a,

–2 0+ u(0) =

τ

0 g2(s)u(s)ds+b,

where 2 <α≤3,R+= [0, +),f :R+×(R+)2R+,f(t,u,v)0, disturbance parameters

a,bR+,g

1,g2∈L1[0,τ) are nonnegative.

Li et al. in [7] investigated the following fractional:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

0+u(t) +f(t,u(t)) = 0, t∈(0, 1),

u(0) = 0,

0+u(1) =αDβ0+u(ξ),

where 1 <α≤2 and

0+ is the Riemann–Liouville fractional derivative. The existence

results of positive solutions are gained by using fixed point theorems.

Liang and Zhang [23] considered the following fractional boundary value problem:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

0+u(t) +a(t)f(t,u(t)) = 0, t∈(0, +∞),

u(0) =u(0) = 0,

–1

0+ u(+∞) = m–2

i=1 βiu(ξi),

where 2 <α< 3,

0+ is the Riemann–Liouville fractional derivative, 0 <ξ1<ξ2<· · ·<

ξm–2< +∞,βi≥0,i= 1, 2, . . . ,m– 2, satisfies 0 <

m–2

i=1 βiξiα–1<(α).

By using Banach contraction mapping principle, the writers gave some new results for existence and uniqueness of positive solutions for the above problem in [12]. As far as we can see, there are still very few papers (such as [22]) discussing the fractional coupled sys-tems. To prove our conclusions, we put forward some necessary basic concepts in Sect.2. In Sect.3, we obtain the positive solutions of BVP (1.1) by using the relevant fixed point theorem. In Sect.4, two examples are presented to verify our results.

2 Preliminaries and correlative lemmas

To prove our conclusion, we first introduce some basic concepts, see [5,9].

Definition 2.1([9]) The Riemann–Liouville fractional integral of orderα> 0 of a function

u:R+Ris given by

I0α+u(t) =

1

(α)

t

0

(ts)α–1u(s)ds.

Definition 2.2([9]) The Riemann–Liouville fractional derivative of orderα> 0 of a func-tionu:R+Ris given by

0+u(t) =DnI0n+–αu(t) =

1

(nα)

d dt

n t

0

(ts)nα–1u(s)ds,

(3)

Remark2.1 ([1]) In this way, we need the following conclusions: 1.

0+–1=(β(βα))α–1,α,β≥0;

2.

0+j= 0,j= 1, 2, . . . , [α] + 1.

Lemma 2.1([9,12]) Forα> 0,the equality Dα

0+u(t) = 0is well-founded if and only if

u(t) =

n

i=1

citαi,

where ciR,i= 1, 2, . . . ,n,n= [α] + 1.

Lemma 2.2([9,12]) If uC(R+)and Dα

0+uL1(R+),then

I0α+0+u(t) =u(t) +c1–1+c2–2+· · ·+cntαn,

where ciR,i= 1, 2, . . . ,n,n= [α] + 1.

Lemma 2.3 Let hL1[0, +),the following boundary value problem:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

0+u(t) +h(t) = 0, t∈(0, +∞),

ui(0) = 0, i= 0, 1, 2, . . . ,n– 3,

–1

0+ u(+∞) = 0, –2

0+ u(0) = ∞

i=1βiu(ξi),

(2.1)

where1 <n– 1 <αn,n= [α] + 1, 0 <ξ1<ξ2<· · ·<ξi<· · ·< +∞,βi≥0,i= 0, 1, 2, . . . ,n

3,satisfies0 <∞i=1βiξiα–2<(α),has a unique solution

u(t) =

+∞

0

G(t,s)h(s)ds, (2.2)

where

G(t,s) =G1(t,s) +G2(t,s), (2.3)

G1(t,s) = 1

(α)

⎧ ⎨ ⎩

–1– (ts)α–1, 0st< +,

–1, 0ts< +, (2.4)

G2(t,s) =

i=1βitα–2

(α– 1) –∞i=1βiξiα–2

G1(ξi,s). (2.5)

Proof From Lemma2.2and (2.1), we have

u(t) = – 1

(α)

t

0

(ts)α–1h(s)ds+c1–1+c2–2+· · ·+cntαn.

Fromui(0) = 0,i= 0, 1, 2, . . . ,n– 3, we know thatc

3=c4=· · ·=cn= 0,

0+–1u(t) = – t

0

(4)

and

–2 0+ u(t) = –

t

0

(ts)h(s)ds+c1(α)t+c2(α– 1).

By the boundary conditions, we can get

⎧ ⎨ ⎩

0+∞h(s)ds+c1(α) = 0,

c2(α– 1) =

i=1βi(–(1α)

ξi

0 (ξis)

α–1h(s)ds+c

1ξiα–1+c2ξiα–2),

then

c1= 1

(α)

+∞

0

h(s)ds,

and

c2=

i=1βi[(1α)

+∞

0 ξ

α–1

i h(s)ds1(α)

ξi

0 (ξis)

α–1h(s)ds]

(α– 1) –∞i=1βiξiα–2

.

So

u(t) = – 1

(α)

t

0

(ts)α–1h(s)ds+ 1

(α)

+∞

0

–1h(s)ds

+

i=1βitα–2[1(α)

+∞

0 ξ

α–1

i h(s)ds1(α)

ξi

0 (ξis)α–1h(s)ds]

(α– 1) –∞i=1βiξiα–2

=

+∞

0

G1(t,s)h(s)ds+

i=1βitα–2

(α– 1) –∞i=1βiξiα–2

+∞

0

G1(ξi,s)h(s)ds

=

+∞

0

G1(t,s)h(s)ds+

+∞

0

G2(t,s)h(s)ds

=

+∞

0

G(t,s)h(s)ds,

whereG(t,s),G1(t,s), andG2(t,s) are defined by (2.3)–(2.5).

Lemma 2.4 The function G(t,s)defined by(2.3)satisfies

(1) G1(t,s)≥0is continuous for allt,sRR+;

(2) 0≤ G1(t,s)

1+–1 ≤(1α) for allt,sR +×R+;

(3) 0≤1+G(t,s–1) ≤(Lα) for allt,sRR+,L= 1 +

i=1βiξiα–2 (α–1)–∞i=1βiξiα–2

.

Proof (1) From (2.4), we know G1(t,s) is continuous on [0, +∞)×[0, +∞), obviously

G1(t,s)≥0 forst. For 0≤st< +∞, we have

–1– (ts)α–1=–1 1 – ts

t

α–1

≥0.

By (2.4), we knowG1(t,s)≥0,t,s∈[0, +∞). (2) From (2.4), it is easy to show that 0≤G1(t,s)

(5)

(3) From (2), we have

0≤ G1(t,s) 1 +–1 ≤

1

(α).

Furthermore, we have

0≤G1(ξi,s) 1 +–1 ≤

1

(α).

Therefore

0≤ G(t,s) 1 +–1 =

G1(t,s) 1 +–1+

G2(t,s) 1 +–1 ≤

1

(α)+ 1

(α)

i=1βiξiα–2

(α– 1) –∞i=1βiξiα–2

= L

(α)

and

L= 1 +

i=1βiξiα–2

(α– 1) –∞i=1βiξiα–2

.

Lemma 2.5([23], Leray–Schauder fixed point theorem) Let B be a bounded,nonempty,

convex,and closed subset of Banach space E,and let F:BE be a completely continuous operator with F(B)⊂B.Then F has a fixed point in B.

Lemma 2.6([9]) Let k> 1and denoteλ(k) =min{ 1

4k2(1+kα–1),kα–2(1+1kα–1)}for a fixed number kR+.Then functions G

1,G2,and G defined by(2.3)(2.5)satisfy:

min

t∈[1k,k]

G1(t,s) 1 +–1 ≥

1

4k2(1 +kα–1)· sup

t∈[0,+∞)

G1(t,s)

1 +–1, (2.6)

min

t∈[1k,k]

G2(t,s) 1 +–1 ≥

1

–2(1 +kα–1)· sup

t∈[0,+∞)

G2(t,s)

1 +–1, (2.7)

min

t∈[1k,k]

G(t,s)

1 +–1 ≥λ(k)· sup

t∈[0,+∞)

G(t,s)

1 +–1, (2.8)

where t,s∈[0, +∞).

Proof According to Lemma 3.3 in [17], (2.6) is established. From the definition ofG2, we have

min

t∈[1k,k]

G2(t,s) 1 +–1 = min

t∈[1k,k]

–2 1 +–1·

i=1βiG1(ξi,s)

(α– 1) –∞i=1βiξiα–2

1

–2

1 +–1 ·

i=1βiG1(ξi,s)

(α– 1) –∞i=1βiξiα–2

= 1

–2

1 +–1 · sup

t∈[0,+∞)

–2 1 +–1 ·

i=1βiG1(ξi,s)

(α– 1) –∞i=1βiξiα–2

≥ 1

–2(1 +kα–1)· sup

t∈[0,+∞)

G2(t,s) 1 +–1.

(6)

Lemma 2.7([25]) Let V ={u∈E:u<l,l> 0},V0={1+ut(αt–1) :uV}.If V0is

equicontin-uous in any finite subinterval of[0, +∞)and equiconvergent at infinity,then V is relatively compact on E.

Lemma 2.8([26], Guo–Krasnosel’skii’s fixed point theorem) Let E be a Banach space,

PE be a cone,1,2be two bounded open sets of E centered at the origin with1⊂2.

Suppose that A:P∩(2\1)→P is a completely continuous operator such that either

(i) Ax ≥ x,∀x∈P1andAx ≤ x,∀x∈P2,or

(ii) Ax ≤ x,∀x∈P1andAx ≥ x,∀x∈P2

holds.Then A has at least one fixed point in P∩(2\1).

In this article, we use the following spaceEto study (1.1), which is denoted by

E=

uC[0, +∞) :sup

tR+ |u(t)| 1 +–1 < +∞

,

with the normu=suptR+ |u(t)|

1+–1. Obviously, (E, · ) is a Banach space.

LetP={u∈E:u(t)≥0,tR+},PEis a cone ofE.

DenotePr={u∈P: 0≤ u<r},∂Pr={x∈P:x=r}andPr={x∈P:x ≤r}where

r> 0.

3 Main results

Define an operatorT:PEas follows:

Tu(t) =

+∞

0

G(t,s)q(s)fs,u(s)ds.

For anyuPE,t,s∈[0, +∞), we have

G(t,s)q(s)fs,u(s)≤G(t,s)q(s)ϕl(s)≤

L

(α)q(s)ϕl(s).

So, for any t0∈[0, +∞), we have|Tu(t) –Tu(t0)| →0. Thus,Tu(t) =

+∞

0 G(t,s)q(s)f(s,

u(s))dsis convergent, thenT:PEis well defined. Therefore, the solution of BVP (1.1) is equivalent to the fixed point of operatorT.

If the following conditions are established, then f : [0, +∞)×[0, +∞)→[0, +∞) is known as anL1-Carathéodoryfunction:

(1) for eachu∈[0, +∞),tf(t,u)is measurable ont∈[0, +∞); (2) for a.e.t∈[0, +∞),uf(t,u)is continuous onu∈[0, +∞);

(3) for eachr> 0, there existsϕrL1[0, +∞)withϕr(t)≥0ont∈[0, +∞)such that

ft,1 +–1uϕr(t), for allur, and a.e.t∈[0, +∞).

Throughout this article, we hypothesize that the condition holds: (H1) qL1(R+)is nonnegative and

0 q(s)ϕr(s)ds<∞for anyr> 0. Denote

f0=lim sup

u→0+ sup

tR+

f(t, (1 +–1)u)

u , ρ=

(α)

(7)

Theorem 3.1 Assume that(H1)holds,q(t)f(t, 0)≡0in any subinterval(0, +∞),if f0<ρ,

then BVP(1.1)has at least one positive solution.

Proof Step 1: We show thatT:PP.

Obviously, Tu(t) is continuous with respect to tR+. For any uPE, there ex-istsl> 0 such thatul,f is anL1-Carathéodoryfunction. Sincesup

tR+ u(t)

1+–1 < +∞,

then

Tu(t) 1 +–1 <

L (α)

+∞

0

q(s)ϕl(s)ds< +∞.

SoTuE. BecauseG,f,qare nonnegative, in that wayTu(t)≥0, which signifiesTuP

for anyuP.

Step 2: we prove thatTis a continuous operator.

Letun,uP,n= 1, 2, . . . , such thatunu →0 asn→ ∞, that is, 1+untα(t–1) →

u(t) 1+–1.

Then there existsr> 0 such thatunr,ur. Sincef is anL1-Carathéodoryfunction,

then

fs,un(s)

fs,u(s)→0, n→ ∞,

and

fs,un(s)

fs,u(s)=f s,1 +–1 un(s)

1 +–1

f s,1 +–1 u(s) 1 +–1

≤2ϕr(s).

By using the Lebesgue dominated convergence theorem, we get

+∞

0

q(s)fs,un(s)

fs,u(s)ds→0, n→ ∞.

So, we have

|Tun(t) –Tu(t)|

1 +–1 =

0+∞1 +G(tt,αs–1) q(s)

fs,un(s)

fs,u(s)ds

≤ +∞

0

G(t,s) 1 +–1q(s)f

s,un(s)

fs,u(s)ds

L

(α)

+∞

0

q(s)fs,un(s)

fs,u(s)ds→0, n→ ∞.

Hence,TunTu →0 asn→ ∞, thusTis a continuous operator.

Step 3:Tis relatively compact.

LetBPbe a nonempty bounded closed subset, that is, there existsk> 0 such that

u ≤kfor alluB, so there existsϕkL1(R+) such that

fs,u(s)=f s,1 +–1 u(s) 1 +–1

(8)

(1) For anyuB,

1 +Tut(αt–1)

=

+∞

0

G(t,s) 1 +–1q(s)f

s,u(s)ds

L

(α)

+∞

0

q(s)ϕk(s)ds

< +∞.

So,T(B) is uniformly bounded. For notational convenience, denote

V1=

z:z= x(t)

1 +–1,xT(B)

.

(2) For anyA> 0, letI= [0,A] be a compact interval andt1,t2∈I,t2>t1.

Tu(t1)

1 +–1 1

Tu(t2) 1 +–1

2 ≤ +∞ 0 G1(t1,s)

1 +–1 1

G1(t2,s) 1 +–1

2

q(s)fs,u(s)ds

+

i=1βi

(α– 1) –∞i=1βiξiα–2

1–2

1 +–1 1

t

α–2 2 1 +–1

2 × +∞ 0

G1(ξi,s)q(s)f

s,u(s)ds

≤ +∞

0

G1(t1,s)

1 +t1α–1 –

G1(t2,s) 1 +1–1

q(s)ϕk(s)ds

+

+∞

0

G1(t2,s)

1 +–1 1

G1(t2,s) 1 +–1

2

q(s)ϕk(s)ds

+

i=1βi

(α)((α– 1) –∞i=1βiξiα–2)

1–2

1 +–1 1

t

α–2 2 1 +–1

2 × +∞ 0

q(s)ϕk(s)ds.

On the other hand, we have

+∞

0

G1(t1,s)

1 +–1 1

G1(t2,s) 1 +–1

1

q(s)ϕk(s)ds

t1

0

G1(t1,s)

1 +–1 1

G1(t2,s) 1 +–1

1

q(s)ϕk(s)ds

+

t2

t1 G1(t1,s)

1 +–1 1

G1(t2,s) 1 +–1

1

q(s)ϕk(s)ds

+

+∞

t2 G1(t1,s)

1 +–1 1

G1(t2,s) 1 +–1

1

q(s)ϕk(s)ds

ϕk

t1

0

(t1α–1–2–1) + ((t2–s)α–1– (t1–s)α–1) 1 +1–1

q(s)ds

+ϕk

t2

t1

(–1

1 –t2α–1) + (t2–s)α–1 1 +–1

1

(9)

+ϕk

+∞

t2 –1

1 –2–1 1 +–1

1

q(s)ds

→0 uniformly ast1→t2.

Similarly, we have

+∞

0

G1(t2,s)

1 +–1 1

G1(t2,s) 1 +–1

2

q(s)ϕk(s)ds→0 uniformly ast1→t2.

So, we have

Tu(t1)

1 +–1 1

Tu(t2) 1 +–1

2

→0 uniformly ast1→t2.

Therefore,V1is locally equicontinuous onR+. (3) For anyuB, we have

+∞

0

q(s)fs,u(s)dsϕk

+∞

0

q(s)ds< +∞

and

lim

t→∞

1 1 +–1

+∞

0

G1(t,s)q(s)f

s,u(s)ds

= lim

t→∞

–1 1 +–1

+∞

t

q(s)fs,u(s)ds

+ lim

t→∞

1 1 +–1

t

0

–1– (ts)α–1q(s)fs,u(s)ds

ϕk lim t→∞

+∞

t

q(s)ds

+ϕk lim t→∞

t

0

(–1– (ts)α–1) 1 +–1 q(s)ds

= 0

sincelimt→∞(–1–(ts)α–1)

1+–1 = 0 andlimt→∞

+∞

t q(s)ds= 0.

Thus

lim

t→∞ 1 +Tut(αt)–1

= lim

t→∞

1 1 +–1

+∞

0

G(t,s)q(s)fs,u(s)ds

= lim

t→∞

1 1 +–1

+∞

0

G1(t,s)q(s)f

s,u(s)ds

+ lim

t→∞

i=1βitα–2

(1 +–1)((α– 1) –

i=1βiξiα–2)

+∞

0

ξiα–1 (α)q(s)f

s,u(s)ds

– lim

t→∞

i=1βitα–2

(1 +–1)((α– 1) –

i=1βiξiα–2)

ξi

0

(ξis)α–1

(α) q(s)f

(10)

i=1βiξiα–1

(α)((α– 1) –∞i=1βiξiα–2)

+∞

0

q(s)ϕk(s)ds

i=1βi

(α)((α– 1) –∞i=1βiξiα–2)

ξi

0

(ξis)α–1q(s)ϕk(s)ds

< +∞,

that is, for anyε> 0, there existsN> 0 such that, for allt1,t2>NandTuP, we have

Tu(t2)

1 +–1 2

Tu(t1) 1 +–1

1

<ε.

Hence,V1isequiconvergentat infinity. SoTis a completely continuous operator. Step 4:T(B)⊂B.

Becausef0<ρ, there existsr1> 0 such thatf(t,(1+t

α–1)u)

u <ρu<ρr1,tR

+, andu(0,r 1]. SetB={u∈P:u ≤r1}, then, for anyuB,

fs,u(s)=f s,1 +–1 u(s)

1 +–1

ρ u(s)

1 +–1 ≤ρr1,

and

Tu(t) 1 +–1 =

+∞

0

G(t,s) 1 +–1q(s)f

s,u(s)ds

Lρr1

(α)

+∞

0

q(s)ds

=r1.

So T(B)⊂B. According to Lemma2.5, we get that BVP (1.1) has at least one positive

solution.

Theorem 3.2 f is an L1-Carathéodoryfunction.Let F(t,u) =f(t, (1 +tα–1)u),r

2>r1> 0,

k> 1, (H1)hold.Suppose that F satisfies one of the following conditions:

(A1) F(t,1+t–1)≥rd1 for all(t,u)∈[k1,k]×[0,r1]andF(t,1+t–1)≤rD2 for all

(t,u)∈[0, +∞]×[0,r2];

(A2) F(t,1+t–1)≤

r1

d for all(t,u)∈[0, +∞]×[0,r1]andF(t, u

1+–1)≥

r2

D for all

(t,u)∈[1k,k]×[0,r2].

Then BVP(1.1)has at least one positive solution uwith r1≤ u∗ ≤r2.

Proof Firstly, we assume that (A1) holds.

According to Theorem 3.1,T :Pr2\Pr1 →P is a completely continuous operator. By

Lemma2.8, the proof is as follows. Step 1. Letu∂Pr1,t∈[0, +∞), then

u(t)

1+–1 ≤r1. We get the following conclusion by using (A1):

min

t∈[1k,k]

|(Tu)(t)| 1 +–1 ≥

+∞

0

min

t∈[1k,k]

G(t,s) 1 +–1q(s)f

s,u(s)ds

λ(k)

+∞

0

sup

t∈[0,+∞)

G(t,s) 1 +–1q(s)f

(11)

λ(k)

+∞

0

G(t,s) 1 +–1q(s)f

s,u(s)ds

λ(k)

–2(1 +kα–1)(α)

k

1 k

q(s)F s, u(s) 1 +–1

ds

r1

d ·

λ(k)

–2(1 +kα–1)(α)

k

1 k

q(s)ds

=r1,

which means

Tu ≥ u, u∂Pr1.

Step 2. Letu∂Pr1,t∈[0, +∞), then

u(t)

1+–1≤r2. Using (A1) again, we get

|(Tu)(t)| 1 +–1 =

+∞

0

G(t,s) 1 +–1q(s)f

s,u(s)ds

L

(α)

+∞

0

q(s)F s, u(s) 1 +–1

ds

r2

D· L (α)

+∞

0

q(s)ds

=r2,

which means

Tu ≤ u, u∂Pr2.

Therefore, from Lemma2.8(i), we get a positive solutionu∗withr1≤ u∗ ≤r2of BVP (1.1).

Similarly, condition (A2) has the same conclusion when it is established.

4 Example

Example4.1 Consider the problem

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ D

9 2

0+u(t) +ett

7 2

u2= 0, t∈(0, +∞),

ui(0) = 0, i= 0, 1, 2, . . . ,n– 3,

D 7 2

0+u(+∞) = 0,

D 5 2

0+u(0) =

i=1i22u(1 –i+11 ).

(4.1)

Letα =92, q(t) =et, f(t,u) =et72 u2,β

i=i22, xii= 1 – i+11 . It is easy to calculate that

i=1βiξiα–2≈1.643,(α) =(92)≈11.6319,(α–1) =( 7

2)≈3.3234, whereϕr(t) =r 2(1+

t72)2et 7 2

(12)

Then

+∞

0

q(s)ϕr(s)ds< +∞,

L= 1 +

i=1βiξiα–2

(α– 1) –∞i=1βiξiα–2

≈1.9777,

ρ= (α)

L0+∞q(s)ds≈5.8815,

f0=lim sup

u→0+ sup

tR+

f(t, (1 +t72)u)

u =lim supu→0+ sup

tR+ et

7 2

(1 +t72)2u2 u = 0 <ρ.

According to Theorem3.1, we get that (4.1) has a positive solution.

Example4.2 Consider the following problem:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ D

9 2

0+u(t) + (t0.5+e2)–1e

πt(1.302×107+u(t) +sinu(t)) = 0,

ui(0) = 0, i= 0, 1, 2, . . . ,n– 3,

D 7 2

0+u(+∞) = 0,

D 5 2

0+u(0) =

i=1i22u(1 –i+11 ),

(4.2)

where t∈ (0, +∞), α = 9 2, (

9

2)≈ 11.6319, ( 7

2)≈ 3.3234, βi = 2

i2, ξi = 1 – i+11 (i=

0, 1, 2, . . . ,n– 3), and∞i=1βiξiα–2≈1.643.

Letq(t) =t–0.5eπtandf(t,u) = 1

1+t1.5(1.302×107+u+sinu). Obviously, +∞

0 q(s)ds= 1. So condition (H1) holds. Setk= 2. By calculation, we haveL≈1.9777,λ≈4.9783×10–3,

d≈6.2653×10–6, andD0.2008. Chooser

1=12andr2= 8×107. Then we get

f(t,u) = 1 1 +t1.5

1.302×107+u+sinu

> 1.2305×107>r1

d ≈7.9804×10

4

for all (t,u)∈[12, 2]×[0,12] and

f(t,u) = 1 1 +t1.5

1.302×107+u+sinu

< 1.0209×108<r2

D≈3.9841×10

8

for all (t,u)∈[0, +∞]×[0, 8×107].

So, according to Theorem3.2, problem (4.2) has at least one solution.

Acknowledgements

The authors are grateful to the referees for carefully reading the paper and for their comments and suggestions.

Funding

The paper is supported by the National Science Foundation of Shandong (No. ZR2016FM10) and by the National Natural Science Foundation of China (No. 11571197).

Availability of data and materials

(13)

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 27 June 2018 Accepted: 24 September 2018

References

1. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematic Studies. Elsevier, Amsterdam (2006)

2. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

3. Zhou, Y.: Theory of Fractional Differential Equations. World Scientific, Singapore (2014)

4. Li, X.C., Liu, X.P., Jia, M., Li, X., Zhang, S.: Existence of positive solutions for integral boundary value problems of fractional differential equations on infinite interval. Math. Methods Appl. Sci.40, 1892–1904 (2010)

5. Zhang, K.M.: Nontrivial solutions of fourth-order singular boundary value problems with sign-changing nonlinear terms. Topol. Methods Nonlinear Anal.40, 53–70 (2012)

6. Wang, G.: Explicit iteration and unbounded solutions for fractional integral boundary value problem on an infinite interval. Appl. Math. Lett.47, 1–7 (2015)

7. Li, C.F., Luo, X.N., Zhou, Y.: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Comput. Math. Appl.59, 1363–1375 (2010)

8. Guan, Y.L., Zhao, Z.Q., Lin, X.L.: On the existence of positive solutions and negative solutions of singular fractional differential equations via global bifurcation techniques. Bound. Value Probl.2016, 141 (2016)

9. Li, X., Liu, X., Jia, M., Zhang, L.: The positive solutions of infinite-point boundary value problem of fractional differential equations on the infinite interval. Adv. Differ. Equ.2017, 126 (2017)

10. Zhang, X.: Positive solutions for a class of singular fractional differential equation with infinite-point boundary value conditions. Appl. Math. Lett.39, 22–27 (2015)

11. Liu, Y.: Existence and unboundedness of positive solution for singular boundary value problem on half-line. Appl. Math. Comput.144, 543–556 (2003)

12. Zhang, K.M.: On sign-changing solution for some fractional differential equations. Bound. Value Probl.2017, 59 (2017) 13. Denton, Z., Ramirez, J.D.: Existence of minimal and maximal solutions to RL fractional integro-differential initial value

problems. Opusc. Math.37, 705–724 (2017)

14. Kumar, S., Kumar, D., Singh, J.: Fractional modelling arising in unidirectional propagation of long waves in dispersive media. Adv. Nonlinear Anal.5, 383–394 (2016)

15. Lyons, J.W., Neugebauer, J.T.: Positive solutions of a singular fractional boundary value problem with a fractional boundary condition. Opusc. Math.37, 421–434 (2017)

16. Gao, H., Han, X.: Existence of positive solutions for fractional differential equation with nonlocal boundary condition. Int. J. Differ. Equ.2011, 10 (2011)

17. Liang, S., Zhang, J.: Existence of three positive solutions of m-point boundary value problems for some nonlinear fractional differential equations on an infinite interval. Comput. Math. Appl.61, 3343–3354 (2011)

18. Smart, D.R.: Fixed Point Theorems. Cambridge University Press, Cambridge (1974) 19. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

20. Liu, L.S., Li, H.D., Liu, C., Wu, Y.H.: Existence and uniqueness of positive solutions for singular fractional differential systems with coupled integral boundary value problems. J. Nonlinear Sci. Appl.10, 243–262 (2017)

21. Lin, X.L., Zhao, Z.Q.: Sign-changing solution for a third-order boundary value problem in ordered Banach space with lattice structure. Bound. Value Probl.2014, 132 (2014)

22. Wang, J., Xiang, H., Liu, Z.: Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations. Int. J. Differ. Equ.2010, Article ID 186928 (2010)

23. Liang, S.H., Zhang, J.H.: Existence of multiple positive solutions for m-point fractional boundary value problems on an interval. Math. Comput. Model.54, 1334–1346 (2011)

24. Gao, Y., Chen, P.: Existence of solutions for a class of nonlinear higher-order fractional differential equation with fractional nonlocal boundary condition. Adv. Differ. Equ.2016, 314 (2016)

25. Liu, Y.: Existence and unboundedness of positive solutions for singular boundary value problems on half-line. Appl. Math. Comput.144, 543–556 (2003)

References

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