Transformation technique, fixed point theorem and positive solutions for second order impulsive differential equations with deviating arguments

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R E S E A R C H

Open Access

Transformation technique, ﬁxed point

theorem and positive solutions for

second-order impulsive differential equations

with deviating arguments

Xuemei Zhang

1*

_{and Meiqiang Feng}

2

*_{Correspondence: [email protected]} 1_{Department of Mathematics and} Physics, North China Electric Power University, Beijing, 102206, Republic of China

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

Abstract

This paper investigates the boundary value problems of second-order impulsive diﬀerential equations with deviating arguments

⎧ ⎨ ⎩

x(t) +

ω

(t)f(t,x(

α

(t))) = 0, t∈J,t=tk, x(t+

k) –x(tk) =ckx(tk), k= 1, 2,. . .,n, ax(0) –bx(0) =ax(1) –bx(1) =₀1h(s)x(t)dt,

where{ck}is a real sequence withck> –1,k= 1, 2,. . .,n,

ω

may be singular att= 0 and/ort= 1. Several new and more general results are obtained for the existence of positive solutions for the above problem by using transformation technique and Krasnosel’skii’s fixed point theorem. We discuss our problems under two cases when the deviating arguments are delayed and advanced. The approach to deal with the impulsive term is different from earlier approaches. It is the first paper where the transformation technique and a fixed point theorem for cones are applied to second-order differential equations with impulsive effects and deviating arguments. An example is included to verify the theoretical results.

Keywords: advanced and delayed arguments; impulsive diﬀerential equations; transformation technique; ﬁxed point theorem; positive solutions

1 Introduction

Impulsive diﬀerential equations, which provide a natural description of observed evolu-tion processes, are regarded as important mathematical tools for better understanding of several real world problems in applied sciences, such as population dynamics, ecology, biological systems, biotechnology, industrial robotic, pharmacokinetics, optimal control,

etc.Therefore, the study of this class of impulsive differential equations has gained promi-nence and it is a rapidly growing field. For the general theory of impulsive differential equations, we refer the reader to [–], whereas the applications of impulsive differential equations can be found in [–]. Nieto and O’Regan [] pointed out that in a second-order differential equationu=f(t,u,u) one usually considers impulses in the position

uand the velocityu. However, in the motion of spacecraft one has to consider instanta-neous impulses depending on the position that result in jump discontinuities in velocity,

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but with no change in position []. The impulses only on velocity occur also in impulsive mechanics [].

Some classical tools such as bifurcation theory [, ], fixed point theorems in cones [–], the method of lower and upper solutions [, ], the theory of critical point theory and variational methods [, , –] and the technique via appropriate trans-formation [–] have been widely used to study impulsive differential equations. But it is quite difficult to apply these approaches to an impulsive differential equation with de-viating arguments; therefore, there was no result in this area for a long time. Only in the recent eight years, there appeared a few articles which dealt with some impulsive differen-tial equations with deviating arguments by using fixed point theorems in cones [–]. Motivated by [–], in this article we shall use a different approach to discuss the ex-istence of positive solutions for a class of impulsive differential equations with deviating arguments.

Consider the second-order nonlinear impulsive diﬀerential equation of the type

⎧ ⎪ ⎨ ⎪ ⎩

x(t) +ω(t)f(t,x(α(t))) = , t∈J,t=tk,

x(t+

k) –x(tk) =ckx(tk), k= , , . . . ,n,

ax() –bx() =ax() –bx() =_h(s)x(t)dt,

(.)

whereJ= [, ],f ∈C(J×R+_,_R+_),_R+_{= [, +}_∞_),_t

k(k= , , . . . ,n, herenis a ﬁxed posi-tive integer) are ﬁxed points with  <t<t<· · ·<tk<· · ·<tn< ,a,b> ,{ck}is a real sequence withck> –,k= , , . . . ,n,x(t+k) (k= , , . . . ,n) represent the right-hand limit of

x(t) attk,h∈C[, ] is nonnegative.

Throughout this paper we assume thatα(t)≡tonJ= [, ]. In addition,ω,f,ck,αand

hsatisfy

(H) ω∈C((, ), [, +∞))with < 

ω(s)ds<∞andωdoes not vanish on any

subinter-val of(, );

(H) f ∈C([, ]×[, +∞), [, +∞)),α∈C(J,J);

(H) {ck}is a real sequence withck> –,k= , , . . . ,n,c(t) :=

<tk<t( +ck); (H) h∈C[, ]is nonnegative withν∈[,a), where

ν=





h(t)c(t)dt. (.)

Remark . Throughout this paper, we always assume that a productc(t) :=_<_tk_<_t( +ck) equals unity if the number of factors is equal to zero, and let

cM=max

t∈J c(t), cm=mint∈J c(t), c

–₍_t_{) =} <tk<t

( +ck)–.

Remark . Combining (H) and the deﬁnition ofc(t), we know thatc(t) is a step function

and bounded onJ, and

c(t) > , ∀t∈J, c(t) = , ∀t∈[,t].

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point theories in cones, the authors proved the existence of positive solutions for problem (.).

At the same time, a class of boundary value problems with delay has been investigated; for example, see [–]. It is not diﬃcult to see that the corresponding functionsf ap-pearing on the right-hand side depend onx(t–τ),τ> , where initial functionsxare given on the initial set, for example, [–τ, ]. Jankowski [, ] pointed out that in such cases

α(t) =t–τ, there are some problems with a constant delayτ. If we consider the differential problem on intervals [,k], wherek≤τ, then it means that we have no delays; we have such a situation in paper []. Ifk>τ, then it is easy to solve the differential equation on [,τ], since we have the solution on the initial set [–τ, ]. Continuing this process, we can find a solution on the whole interval [,k] by using the method of steps. In the present paper, for example, the deviating argument α can have a form α(t) =ρt=t– ( –ρ)t

with a fixed numberρ∈(, ), so the delay ( –ρ)tis a function oft. In this case, the initial set reduces to one pointt= , and we cannot apply the step method. To the au-thors’ knowledge, it is the first paper when positive solutions have been investigated for a class of second-order impulsive differential equations with deviating arguments both of advanced and delayed type.

Remark . There are almost no papers, except [–], studying second-order impul-sive diﬀerential equations with deviating arguments using ﬁxed point theory. However, in [–], Jankowski only consideredω∈C([, ],∞), notωis singular att=  and/or

t= , and dealt with the nonlinear term that is in the form off(y(t)), notf(t,c(t)y(t)); see (.).

Remark . Comparing with [–], we transform problem (.) into a differential sys-tem without impulse,i.e., the technique to deal with impulses is completely different from that of [–]. According to the authors’ knowledge, it is probably the first paper where this technique is applied to second-order impulsive boundary value problems with devi-ating arguments.

Remark . The technique to deal withf(t,c(t)y(t)) is completely diﬀerent from that of Zhanget al.[], Zhanget al.[] and Sunet al.[].

Being directly inspired by [–], the authors will prove several new and more general results for the existence of positive solutions for problem (.) by using fixed point theories. The organization of this paper is as follows. In Section , we present some definitions and lemmas which are needed throughout this paper. In particular, we transform problem (.) into a differential system without impulse. In Section , we use a fixed point theorem to obtain the existence of positive solutions for problem (.) with advanced argumentα. Finally, in Section , we formulate sufficient conditions under which delayed problem (.) has positive solutions. In particular, our results in these sections are new whenα(t)≡ton

t∈J.

2 Preliminaries

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Deﬁnition . (see []) LetEbe a real Banach space overR. A nonempty closed set

P⊂Eis said to be a cone provided that

(i) au+bv∈Pfor allu,v∈Pand alla≥,b≥, and (ii) u, –u∈Pimpliesu= .

Every coneP⊂Einduces an ordering inEgiven byx≤yif and only ify–x∈P. Deﬁnition . The mapβis said to be a nonnegative continuous concave functional on a conePof a real Banach spaceEprovided thatβ:P→[,∞) is continuous and

βtx+ ( –t)y≥tβ(x) + ( –t)β(y) for allx,y∈Pand ≤t≤.

Deﬁnition . A functionx(t) is said to be a solution of problem (.) onJif: (i) x(t)is absolutely continuous on each interval(,t]and(tk,tk+],k= , , . . . ,n;

(ii) for anyk= , , . . . ,n,x(t_k+),x(t–

k)exist andx(tk–) =x(tk); (iii) x(t)satisﬁes (.).

We shall reduce problem (.) to a system without impulse. To this goal, ﬁrstly by means of the transformation

x(t) =c(t)y(t), (.)

we convert problem (.) into

⎧ ⎪ ⎨ ⎪ ⎩

–y(t) =ω(t)c–(t)f(t,c(α(t))y(α(t))), t∈J,

ay() –by() =_h(s)c(s)y(s)ds,

ac()y() –bc()y() =_h(s)c(s)y(s)ds.

(.)

The following lemmas will be used in the proof of our main results.

Lemma . Assume that(H)-(H)hold.Then

(i) Ify(t)is a solution of problem(.)onJ,thenx(t) =c(t)y(t)is a solution of problem

(.)onJ;

(ii) Ifx(t)is a solution of problem(.)onJ,theny(t) =c–(t)x(t)is a solution of problem

(.)onJ.

Proof The proof is similar to that of Lemma . in []. Lemma . If(H)-(H)hold,then problem(.)has a solution y,and y can be expressed

in the form

y(t) =





H(t,s)c–(s)ft,cα(s)yα(s)ds, (.)

where

H(t,s) =G(t,s) + 

a–ν 



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G(t,s) = 

a(a+ b)

(b+at)(a( –s) +b), ≤t≤s≤,

(b+as)(a( –t) +b), ≤s≤t≤. (.)

Proof The proof is similar to that of Lemma . in []. Lemma . Letξ∈(, ),G and H be given as in Lemma..Then we have the following results:

G(t,s)≤G(s,s), H(t,s)≤H(s,s) = a

a–νG(s,s), ∀t,s∈J, (.)

G(t,s)≥δG(s,s), H(t,s)≥δH(s,s) = δa

a–νG(s,s), ∀t∈[ξ, ],s∈J, (.)

where

δ= b

b+a.

Proof Relation (.) is simple to prove. Note that

G(t,s)

G(s,s)=

b+at b+as ≥

b+aξ

b+a fors≤t, G(t,s)

G(s,s)=

b+a( –t)

b+a( –s)≥

b

b+a fort≤s,

fort∈[ξ, ],s∈J.

Similarly, we can prove thatH(t,s)≥δH(s,s) fort∈[ξ, ],s∈J. Hence, it follows from

G(t,s)≥δG(s,s) that

H(t,s)≥δH(s,s) = δa

a–νG(s,s), ∀t∈[ξ, ],s∈J.

This gives the proof of Lemma ..

Remark . Noticing thata,b> , it follows from (.) and (.) that

 <β≤G(t,s)≤β, ∀t,s∈J, (.)

and

 <β∗≤H(t,s)≤β, ∀t,s∈J, (.)

where

β=

b

a(a+ b), β=

a+ b

a , β

∗₌ b

a(a+ b)(a–ν), β=

a+ b

(a–ν). (.) LetE=C[, ]. ThenEis a real Banach space with the norm · deﬁned by

y=max

t∈J

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Deﬁne a coneKinEby

K=

y∈E:y(t)≥,min

t∈[ξ,]y(t)≥δy

. (.)

Also, deﬁne forra positive numberrby

r=

y∈E:y<r.

Note that∂r={y∈E:y=r}. DeﬁneT:K→Kby

(Ty)(t) =





H(t,s)ω(s)c–(s)fs,cα(s)yα(s)ds. (.)

Lemma . Assume that(H)-(H)hold.Then T(K)⊂K and T:K→K is completely

continuous.

Proof Fory∈K, it follows from (.) and (.) that

(Ty)(t) =





H(t,s)ω(s)c–(s)fs,cα(s)yα(s)ds

≤





H(s,s)ω(s)c–(s)fs,cα(s)yα(s)ds, t∈J. (.)

It follows from (.), (.) and (.) that

min

t∈[ξ,](Ty)(t) =tmin∈[ξ,] 



≥δ 



H(s,s)ω(s)c–(s)fs,cα(s)yα(s)ds

≥δTy.

Thus,T(K)⊂K.

Next, by arguments similar to those of Theorem  in [], one can prove thatT:K→K

is completely continuous. So it is omitted, and the lemma is proved.

Remark . From (.), we know thaty∈Eis a solution of problem (.) if and only ify

is a ﬁxed point of the operatorT.

From Lemma . and Remark ., we can obtain the following results.

Lemma . Assume that(H)-(H)hold.Then

(i) Ifx(t)is a solution of problem(.)onJ,theny(t) =c–₍_t₎_x₍_t₎_{is a ﬁxed point of}_T_;

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In the rest of this section, we state a well-known ﬁxed point theorem which we need later.

Lemma .(see []) Let P be a cone in a real Banach space E.Assume that,are

bounded open sets in E with∈,¯⊂.If

A:P∩(¯\)→P

is completely continuous such that either

(i) Ax ≤ x,∀x∈P∩∂andAx ≥ x,∀x∈P∩∂,or

(ii) Ax ≥ x,∀x∈P∩∂andAx ≤ x,∀x∈P∩∂,

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

3 Existence of positive solutions for problem (1.1) under

α

(t)≥tonJ For convenience, we introduce the following notations:

f=lim sup

y→ max

t∈J

f(t,y)

y , f=lim infy→ mint∈J

f(t,y)

y , f∞=lim sup

y→∞

max

t∈J

f(t,y)

y , f∞=lim infy→∞ mint∈J

f(t,y)

y .

We also deﬁne as []i= number of zeros in the set{f,f∞}andi∞= number of

inﬁni-ties in the set{f,f∞}. Sun and Li [] pointed out thati,i∞= ,  or , and there are six

possible cases: (i)i=  andi∞= ; (ii)i=  andi∞= ; (iii)i=  andi∞= ; (iv)i= 

andi∞= ; (v)i=  andi∞= ; and (vi)i=  andi∞= . By using Krasnosel’skii’s ﬁxed

point theorem in a cone, some results are obtained for the existence of at least one or two positive solutions of problem (.) forα(t)≥tonJunder the above six possible cases. 3.1 For the case

α

(t)≥tonJunderi0= 1 andi∞= 1

In this subsection, we discuss the existence of a single positive solution for problem (.) forα(t)≥tonJunderi=  andi∞= .

For convenience, we introduce the following notations:

γ =





ω(s)ds, γ= 

ξ

ω(s)ds.

Theorem . Assume that(H)-(H)hold.If i= and i∞= ,then problem(.)has at

least one positive solution.

Proof First, we consider the casef_{=  and}_f

∞=∞. Sincef= , then there existsr> 

such that

f(t,y)≤ cm

βγcM

y (∀t∈J, ≤y≤r).

Since ≤t≤α(t)≤ onJ, it follows from ≤y(t)≤ronJthat

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which implies

Ty ≥ y, ∀y∈K∩∂R. (.)

Thus by (i) of Lemma ., it follows thatThas a ﬁxed pointyinK∩(¯R\r) with

r≤ y ≤R.

Lemma . implies that problem (.) has at least one positive solutionxwith

cmr≤ x ≤cMR.

This gives the proof of Theorem ..

Remark . Fori=  andi∞= , there is another casef∞=  andf=∞. However, at

the moment, we give no information on the existence of a positive solution for problem (.) if we changef=  andf∞=∞intof∞=  andf=∞in Theorem ..

3.2 For the case

α

(t)≥tonJunderi0= 0 andi∞= 0

In this subsection, we discuss the existence for the positive solutions of problem (.) under

i=  andi∞= . For convenience, we introduce the following notations:

fρ=max

max

t∈J

f(t,y)

ρ :y∈[,ρ]

and

l= cm

βγ, L=

cM

cmβ∗δγ

.

Now, we shall state and prove the following main result.

Theorem . Suppose that(H)-(H)hold andα(t)≥t on J.In addition,let the following

two conditions hold:

(H) There existl> andρ> such thatfρ≤l;

(H) There existη> andρ> such thatf(t,y)≥ηfort∈J,y≥ρ;furthermore,ρ=ρ.

Then problem(.)has at least one positive solution.

Proof Without loss of generality, we may assume thatρ<ρ. Consideringf ρ

 ≤l, we have

f(t,y)≤lρfor ≤y≤ρ,t∈J.

Since ≤t≤α(t)≤ onJ, it follows from ≤y(t)≤ρonJthat

≤yα(t)≤ρ.

Letρ=min{ρ,_cM ρ}. Then, fory∈K∩∂ρ, we have ≤y(t)≤ρ≤ρfort∈J, and

then

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Consequently, for anyt∈Jandy∈K∩∂ρ, (.) and (.) imply

(Ty)(t) =





≤βc–_mlρ 

 ω(s)ds

=βc–_mlργ

=ρ,

which implies

Ty ≤ y, ∀y∈K∩∂ρ. (.)

On the other hand, from (H), whenρis ﬁxed, there existsη>  such that

f(t,y)≥η≥max

ρ, ρ δcm

× cM

β∗γ

fort∈Jandy≥ρ. Since ≤t≤α(t)≤ onJ, it follows fromy(t)≥ρonJthat

yα(t)≥ρ.

Letρ¯=max{ρ,δρcm }. Then, fory∈K∩∂ρ¯, we have

c(t)y(t)≥cmy(t)≥cmδy ≥ρ, t∈[ξ, ].

Hence, fory∈K∩∂ρ¯, it follows from (.) and (.) that

(Ty)(t) =





≥β∗ 



ω(s)c–(s)fs,cα(s)yα(s)ds

≥β∗c–_M



ξ

ω(s)ηds

≥β∗c–_Mη 

ξ ω(s)ds

=β∗c–_Mηγ

≥β∗c–_Mγρ¯

cM

β∗γ

=ρ¯,

which implies

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Thus by (i) of Lemma ., it follows thatThas a ﬁxed pointyinK∩(¯ρ\ρ) with

ρ≤ y ≤ ¯ρ.

Thus, it follows from Lemma . that problem (.) has at least one positive solutionx

with

cmρ≤ x ≤cMρ¯.

This ﬁnishes the proof of Theorem ..

We remark that condition (H) in Theorem . can be replaced by the following

condi-tion:

(H) f≤l,

which is a special case of (H).

Corollary . Suppose that(H)-(H), (H), (H)hold andα(t)≥t on J.Then problem

(.)has at least one positive solution.

Proof We show that (H)implies (H). Suppose that (H)holds. Then there exists a

pos-itive numberρ=ρsuch that

f(t,y)

y ≤l, t∈J,  <y≤ρ.

Hence, we obtain

f(t,y)≤ly≤lρ, t∈J,  <y≤ρ.

Therefore, (H) holds. Hence, by Theorem ., problem (.) has at least one positive

solution.

Theorem . Suppose that(H)-(H)hold andα(t)≥t on J.In addition,let the following

condition hold: (H) f∞≥L.

Proof The proof is similar to those of (.) and (.), respectively. Corollary . Suppose that(H)-(H), (H), (H)hold andα(t)≥t on J.Then problem

α

(t)≥tonJunderi0= 1 andi∞= 0 ori0= 0 andi∞= 1

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Theorem . Suppose that(H)-(H)hold,α(t)≥t on J and f∈[,l)and f∞∈(L,∞).

Proof The proof is similar to that of Theorem .. Theorem . Suppose that(H)-(H)hold,α(t)≥t on J and f∈(L,∞)and f∞∈[,l).

Proof Considerf∈(L,∞), then there existsρ>  such thatf(t,y) >Lyfor ≤y≤ρ,

t∈J.

Since ≤t≤α(t)≤ onJ, it follows from ≤y(t)≤ρonJthat

≤yα(t)≤ρ.

Letρ=min{ρ,_cM ρ}. Then, fory∈K∩∂ρ, we have ≤y(t)≤ρ≤ρfort∈J, and

then

cα(t)yα(t)≤cMy=cMρ≤ρ, t∈J.

Consequently, fory∈K∩∂ρ, it follows from (.) and (.) that

(Ty)(t) =





≥β∗ 



ω(s)c–(s)fs,cα(s)yα(s)ds

≥β∗c–_M



ξ

ω(s)Lcα(s)yα(s)ds

≥β∗c–_MLcmδy



ξ ω(s)ds

=β∗c–_MLcmδyγ

≥ y,

which implies

Ty ≥ y, ∀y∈K∩∂ρ. (.)

Next, turn tof∞∈[,l). In fact, we can show thatf∞∈[,l) implies (H).

Letτ∈(f∞,l). Then there existsr>τ such thatmaxt∈Jf(t,y)≤τyfory∈[r,∞). Let

β=maxmax

t∈J f(t,y) : ≤y≤r

and ρ_∗>max

β

l–τ,ρ,cMρ

.

Then we have

max

≤t≤f(t,y)≤τy+β≤τρ ∗

 +β<lρ∗, ∀y∈

,ρ∗_.

This implies thatfρ∗

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Similarly to the proof of (.), we have

Ty ≤ y, ∀y∈K∩∂ρ∗, (.)

whereρ∗=min{ρ_∗,_cM ρ_∗}.

Thus by (ii) of Lemma ., it follows thatThas a ﬁxed pointyinK∩(¯ρ∗\ρ) with

ρ≤ y ≤ρ∗.

This ﬁnishes the proof of Theorem ..

From Theorems . and ., we have the following result.

Corollary . Suppose that f_{= }_{and condition}_(H

)in Theorem.hold.Then problem

Theorem . Suppose that(H)-(H),α(t)≥t on J,f∈(,l)and f∞=∞.Then problem

Proof The proof is similar to that of Theorem .. Theorem . Suppose(H)-(H),α(t)≥t on J,f=∞and f∞∈(,l).Then problem(.)

has at least one positive solution.

Proof The proof is similar to that of Theorem .. From Theorems . and ., the following corollaries are easily obtained.

Corollary . Suppose that f₌_∞_{and condition}_(H

)in Theorem.hold.Then problem

Corollary . Suppose that f∞=∞and condition(H)in Theorem.hold.Then

prob-lem(.)has at least one positive solution.

α

(t)≥tonJunderi0= 0 andi∞= 2 ori0= 2 andi∞= 0

In this subsection we study the existence of multiple positive solutions for problem (.) for the caseα(t)≥tonJunderi=  andi∞=  ori=  andi∞= .

Combining the proofs of Theorems . and ., the following theorem is easily proved.

Theorem . Suppose that(H)-(H),α(t)≥t on J,i= and i∞= and condition(H)

of Theorem.hold.Then problem(.)has at least two positive solutions.

Corollary . Suppose that(H)-(H),α(t)≥t on J,i= and i∞= and condition(H)

of Corollary.hold.Then problem(.)has at least two positive solutions.

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4 Positive solutions of problem (1.1) for the case of

α

(t)≤tonJ

Now we deal with problem (.) for the case ofα(t)≤tonJ. Similarly as in Lemmas . and ., we can prove the following results.

Lemma . Letξ∈(, ),G and H be given as in Lemma..Then we have the following results:

G(t,s)≥δG(s,s), H(t,s)≥δH(s,s) = δa

a–νG(s,s), ∀t∈[,ξ],s∈J, (.)

where

δ= b

b+a. Proof Note that

G(t,s)

G(s,s)=

b+at b+as ≥

b

b+a fors≤t, G(t,s)

G(s,s)=

b+a( –t)

b+a( –s)≥

b+a( –ξ)

b+a fort≤s,

fort∈[,ξ],s∈J.

Similarly, we can prove thatH(t,s)≥δH(s,s) fort∈[,ξ],s∈J. Hence, it follows from

G(t,s)≥δG(s,s) that

H(t,s)≥δH(s,s) = δa

a–νG(s,s), ∀t∈[,ξ],s∈J.

This gives the proof of Lemma ..

LetEbe as deﬁned in Section . We deﬁne a coneK∗inEby

K∗=

u∈E:u≥, min

t∈[,ξ]u(t)≥δu

.

It is easy to see thatK∗is a closed convex cone ofE. DeﬁneT∗:K∗→Eby

T∗y(t) =





H(t,s)ω(s)c–(s)fs,cα(s)yα(s)ds. (.)

It is clear thaty(t) is a positive solution of problem (.) if and only ofyis a ﬁxed point ofT∗.

By analogous methods, we have the following results. Here we only give the proof of Lemma ..

Lemma . Assume that(H)-(H)hold.Then T∗(K∗)⊂K∗and T∗:K∗→K∗ is

com-pletely continuous.

Lemma . Assume that(H)-(H)hold.Then

(i) Ifx(t)is a solution of problem(.)onJ,theny(t) =c–₍_t₎_x₍_t₎_{is a ﬁxed point of}_T∗_;

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Similar to the proof of that in Section , we have the following results.

α

(t)≤tonJunderi0= 1 andi∞= 1

For convenience, we introduce the following notation:

γ_∗=

ξ



ω(s)ds.

Theorem . Assume that(H)-(H)hold.If i= and i∞= ,then problem(.)has at

least one positive solution.

Proof First, we consider the casef_{=  and}_f

∞=∞. Sincef= , then there existsr> 

such that

f(t,y)≤ cm

βγcM

y (∀t∈J, ≤y≤r).

Since ≤α(t)≤t≤ onJ, it follows from ≤y(t)≤ronJthat

≤yα(t)≤r fort∈J.

Letr=min{r,_cM r}. Then, fory∈K∗∩∂r, we have ≤y(t)≤r≤rfort∈J, and then

cα(t)yα(t)≤cMy=cMr≤r, t∈J.

Consequently, for anyt∈Jandy∈K∗∩∂r, (.) and (.) imply

T∗y(t) =





≤βc–

m





ω(s) cm

γ βcM

cα(s)yα(s)ds

≤βc–_m cm

γ βcM

cM





ω(s)yα(s)ds

≤ 

γ 



ω(s)dsy

=y,

which implies

T∗y≤ y, ∀y∈K∗∩∂r. (.)

Next we considerf∞=∞, there existsrˆsatisfying  <r<rˆsuch that

f(t,y)≥ cM

cmδαγ

y, ∀t∈[,ξ],y≥ ˆr.

Sinceα(t)≤t≤ξonJ, it follows fromy(t)≥ ˆron [,ξ] that

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LetR>max{ˆr,_δ_cmrˆ }. Then, fory∈K∗∩∂R, we have

cα(t)yα(t)≥cmy

α(t)≥cmδy ≥ ˆr, t∈[,ξ].

Hence, fory∈K∗∩∂R, it follows from (.) and (.) that

T∗y(t) =





≥αc–_M

ξ



ω(s) cM

cmδαγ

cα(s)yα(s)ds

≥αc–_M cM cmδαγ

cm

ξ



ω(s)yα(s)ds

≥ 

δγ_∗ ξ



ω(s)dsδy

=y,

which implies

T∗y≥ y, ∀y∈K∗∩∂R. (.)

Thus by (i) of Lemma ., it follows thatT∗has a ﬁxed pointyinK∗∩(¯R\r) with

r≤y(t)≤R.

This ﬁnishes the proof of Theorem ..

Remark . Fori=  andi∞= , there is another casef∞=  andf=∞. However, at

the moment, we give no information on the existence of a positive solution for problem (.) if we changef_{=  and}_f

∞=∞intof∞=  andf=∞in Theorem ..

α

(t)≤tonJunderi0= 0 andi∞= 0

In this subsection, we discuss the existence for the positive solutions of problem (.) under

i=  andi∞= . For convenience, we introduce the following notation:

L∗= cM

cmαδγ∗

.

Now, we shall state and prove the following main result.

Theorem . Suppose that(H)-(H)hold andα(t)≤t on J.In addition,let the following

condition hold:

(H∗_) There exist η∗ > and ρ>  such thatf(t,y)≥η∗ for t∈J, y≥ρ;furthermore, ρ=ρ.

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Corollary . Suppose that(H)-(H), (H), (H∗)hold andα(t)≤t on J.Then problem

Theorem . Suppose that(H)-(H)hold andα(t)≤t on J.In addition,let the following

condition hold: (H∗_) f∞≥L∗.

Corollary . Suppose that(H)-(H), (H), (H∗)hold andα(t)≥t on J.Then problem

α

(t)≤tonJunderi0= 1 andi∞= 0 ori0= 0 andi∞= 1

Theorem . Suppose that(H)-(H)hold,α(t)≤t on J and f∈[,l)and f∞∈(L∗,∞).

Theorem . Suppose that(H)-(H)hold,α(t)≤t on J and f∈(L∗,∞)and f∞∈[,l).

α

(t)≤tonJunderi0= 0 andi∞= 2 ori0= 2 andi∞= 0

Combining the proofs of Theorems . and ., the following theorem is easily proved.

Theorem . Suppose that(H)-(H),α(t)≤t on J,i= and i∞= and condition(H)

of Theorem.hold.Then problem(.)has at least two positive solutions.

Corollary . Suppose that(H)-(H),α(t)≤t on J,i= and i∞= and condition(H)

of Corollary.hold.Then problem(.)has at least two positive solutions.

5 An example

To illustrate how our main results can be used in practice, we present an example.

Example . Consider the following boundary value problem:

⎧ ⎪ ⎨ ⎪ ⎩

x(t) +ω(t)f(t,x(α(t))) = , t∈J,t=_,

x(_+) –x(_) = _x(_), k= ,

x() –x() = x() –x() =_x(t)dt,

(.)

whereα∈C(J,J),α(t)≥tonJand

ω(t) = √

t, f(t,x) =

n √

 +tn_xn_,

heren≥ is a positive integral number.

This means that problem (.) involves the advanced argumentα. For example, we can takeα(t) =√

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Problem (.) can be regarded as a problem of the form (.), wherea= ,b= ,h(t)≡,

Hence, by Theorem ., the conclusion follows, and the proof is complete.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

XZ completed the main study and carried out the results of this article. MF checked the proofs and veriﬁed the calculation. All the authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics and Physics, North China Electric Power University, Beijing, 102206, Republic of China.} 2_{School of Applied Science, Beijing Information Science & Technology University, Beijing, 100192, Republic of China.}

Acknowledgements

This work is sponsored by the project NSFC (11301178, 11171032), the Fundamental Research Funds for the Central Universities (2014ZZD10, 2014MS58) and the improving project of graduate education of Beijing Information Science and Technology University (YJT201416). The authors are grateful to anonymous referees for their constructive comments and suggestions, which has greatly improved this paper.

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Cite this article as:Zhang and Feng:Transformation technique, ﬁxed point theorem and positive solutions for