Existence of solutions to fourth-order differential equations with deviating arguments

R E S E A R C H

Open Access

Existence of solutions to fourth-order

differential equations with deviating

arguments

Mostepha Naceri

1

_{, Ravi P Agarwal}

2,3*

_{, Erbil Çetin}

2,4

_{and El Haffaf Amir}

5

*_{Correspondence:} [email protected]

2_{Department of Mathematics, Texas} A&M University-Kingsville, 700 University Blvd., Kingsville, TX 78363-8202, USA

3_{Department of Mathematics,} Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

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

Abstract

In this paper, we consider fourth-order differential equations on a half-line with deviating arguments of the formu(4)_{(t) +q(t)f(t, [u(t)], [u(t)], [u(t)],u(t)) = 0,}

0 <t< +∞, with the boundary conditionsu(0) =A,u(0) =B,u(t) –au(t) =

θ

(t), –τ≤t≤0;u(+∞) =C. We present sufficient conditions for the existence of a solution between a pair of lower and upper solutions by using Schäuder’s fixed point theorem. Also, we establish the existence of three solutions between two pairs of lower and upper solutions by using topological degree theory. An important feature of our existence criteria is that the obtained solutions may be unbounded. We illustrate the importance of our results through two simple examples.

MSC: 34B15; 34B40

Keywords: fourth-order; boundary value problem; half-line; upper solution; lower solution

1 Introduction

In recent years considerable attention has been focused on the existence of solutions to boundary value problems involving differential equations with deviating arguments (DEDA) [–]. While most of these works deal with problems on finite intervals and the literature is satisfactory, study of infinite interval problems has been just initiated in [, –]. This study compare to boundary value problems for second and higher order or-dinary differential equations over infinite intervals (and their wide variety of applications to real world problems) [–] is far from complete, and needs attention. To fill some of this gap, in this paper we shall provide existence criteria for fourth-order differential equations with deviating arguments of the form

u()(t) +q(t)ft,u(t),u(t),u(t),u(t)= ,  <t< +∞, (.)

where

u(t)=u(t),ut–τ,(t)

, . . . ,ut–τ,n(t)

,

u(t)=u(t),ut–τ,(t)

, . . . ,ut–τ,n(t)

,

u(t)=u(t),ut–τ,(t)

, . . . ,ut–τ,n(t)

,

andq: (, +∞)→(, +∞),f : [, +∞)×Rn+_×Rn+_×Rn+_{×R→Rare continuous, and}

τj,i: [, +∞)→(, +∞) are continuous for allj= , , ,i= , , . . . ,n. In what follows we

shall always assume thatlimt→+∞(t–τj,i(t)) = +∞,j= , , ,i= , , . . . ,n. We define the

positive real numberτas

τ= – min

≤j≤,≤i≤nmint≥

t–τj,i(t)

and seek the solutions of (.) which satisfy the boundary conditions

u() =A, u() =B, u(t) –au(t) =θ(t), –τ≤t≤;

u(+∞) =C, (.)

whereθ∈C[–τ, ],A,B∈R,a,C≥, andu(+∞) =limt→+∞u(t). For this, following as in several above works, and inspired by the contributions in [–], we shall employ the method of upper and lower solutions.

The plan of this paper is as follows: In Section , we state some definitions and lemmas which are needed to prove the main results. In Section , we show that in the presence of a pair of upper and lower solutions the problem (.)-(.) has at least one solution. Also in this section, we establish that in the presence of two pairs of upper and lower solutions the problem (.)-(.) has at least three solutions. Finally, in Section , we illustrate two examples which show the importance of our results.

2 Preliminaries

In this section we introduce some necessary definitions, lemmas, and preliminary results that will be used in main results which give the existence of solutions of the problem (.)-(.). First, we construct the Green’s function for the linear boundary value problem

⎧ ⎪ ⎨ ⎪ ⎩

u()_{(t) +e(t) = ,  <t< +∞;}

u() =A, u() =B, u(t) –au(t) =θ(t), –τ≤t≤; u(+∞) =C.

(.)

Lemma . Let e∈L_{[, +∞).Then the solution u∈C}_{[–τ, +∞)∩C}_{(, +∞)of the}

prob-lem(.)can be expressed as

u(t) =

φ(t), –τ≤t≤;

A+Bt+ (aC+θ())t_ +C_!t +_∞G(t,s)e(s)ds, ≤t< +∞, (.) where

G(t,s) =

a t+

st  –

s_t  +

s

!, ≤s≤t< +∞; a

t+ t

!, ≤t≤s< +∞,

(.)

and

φ(t) =A+Bt+

θ() +aC+a

∞



e(s)ds–at–a+aeat

+



t

Proof Sincee∈L_{[, +∞), we can integrate (.) fromtto +∞, and useu(+∞) =C, to}

get

u(t) =C+

∞

t

e(s)ds, t≥.

Integrating the above equation on [,t], and applying Fubini’s theorem and usingu() – au() =θ(), we obtain

u(t) =aC+a

_∞



e(s)ds+θ() +Ct+ t



se(s)ds+

_∞

t

te(s)ds. (.)

Again integrating (.) twice on [,t], and applying Fubini’s theorem and usingu() =A andu() =B, we find

u(t) =A+Bt+aC+θ()t



 +C t !+

t



a t

₊st

 – st

 + s !

e(s)ds

+

∞

t

a t

₊t

!

e(s)ds,

fort∈[, +∞). Now we consider the following third order linear differential equation:

u(t) –au(t) =θ(t), t∈[–τ, ].

If the above equation is rearranged, we have

u(t) –  au

_{(t) = –}

aθ(t), t∈[–τ, ], and solving this linear equation on [t, ], we find

u(t) =u()eat ₊ 

a



t

et–as_{θ(s)ds. (.)}

Next, integrating (.) twice on [t, ], and applying Fubini’s theorem and using the follow-ing boundary conditions:

u() =A, u() =B and u() =θ() +au() =θ() +aC+a

∞



e(s)ds,

we obtain

u(t) =A+Bt+

θ() +aC+a

_∞



e(s)ds–at–a+aeat

+



t

s–a–t+aet–as_θ(s)ds

Remark . G(t,s) defined in (.) is the Green’s function of the BVP

–u()_{(t) = ,  <t< +∞;}

u() =u() = , u() =au(), u(+∞) = .

Lemma . The Green’s function G(t,s)has the following properties: () G(t,s)is twice continuously differentiable on[, +∞)×[, +∞)and

∂G(t,s)

∂t

t=s+–

∂G(t,s)

∂t

t=s–= –;

() ∂iG_∂t(it,s)≥,∀(t,s)∈[, +∞)×[, +∞),fori= , , , ;

() sup_{t∈[,+∞)}G(t,s)

+t ≤(a  √

+

 ),supt∈[,+∞)(+t ∂G(t,s)

∂t )≤( a+

 ),

sup_{t∈[,+∞)}(_+_t∂_∂G_t(t,s))≤(a+ ),supt∈[,+∞)∂ _G(t,s)

∂t ≤.

Proof () and () are obvious. Here we shall prove the first inequality of (). We note that for all integerskandl

sup

t∈[,+∞)

tk

 +tl = ⎧ ⎪ ⎨ ⎪ ⎩

l–k l (

k l–k)

k l_{, k<l;}

, k=l; +∞, k>l.

Fors≤t, we have

sup

t∈[,+∞)

G(t,s)  +t = sup

t∈[,+∞) a

t+ st

 – st

 + s



 +t

≤ sup

t∈[,+∞) at



 +t + t



 +t

≤a t∈[,+sup∞)

t

 +t +

 t∈sup[,+∞)

t

 +t ≤

a√ +  

and fors≥t

sup

t∈[,+∞)

G(t,s)  +t = sup

t∈[,+∞) a

t ₊t



 +t

≤ sup

t∈[,+∞) at



 +t+ t



 +t

≤a t∈[,+sup∞)

t

 +t +

 t∈sup[,+∞)

t

 +t ≤

a√  +   .

The other parts can be proved similarly.

We consider the spaceXdefined by

X=

u∈C[–τ, +∞) : sup

t∈[,+∞)

|u(t)|

 +t < +∞, sup t∈[,+∞)

|u(t)|  +t < +∞,

sup

t∈[,+∞)

|u_(t)|

 +t < +∞,t→lim+∞u

_{(t) exists}

with the norm

u=maxu,u,u,u∞,u∞,u∞,u∞

where

u= max t∈[–τ,]

u(t), u_∞= sup

t∈[,+∞)

|u(t)|  +t,

u= max t∈[–τ,]

u(t), u_∞= sup

t∈[,+∞)

|u_(t)|  +t,

u= max t∈[–τ,]

u(t), u_∞= sup

t∈[,+∞)

|u_(t)|

 +t , u



∞= sup

t∈[–τ,+∞) u(t).

It is clear that (X, · ) is a Banach space. Next we define the mapping T :X → C_{[–τ, +∞)∩C}_{(, +∞) by}

Tu(t) =

ψ(t), –τ≤t≤;

l(t) +_∞G(t,s)q(s)f(s, [u(s)], [u(s)], [u(s)],u(s))ds, ≤t< +∞, (.)

where

ψ(t) =A+Bt+

θ() +aC+a

∞



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

×–at–a+aeat₊



t

s–a–t+aet–as_θ(s)ds

and

l(t) =A+Bt+aC+θ()t



 +C t

!. (.)

Lemma . The mapping T:X→C_{[–τ, +∞)∩C}_{(, +∞)in(.)has the following}

prop-erties:

() Tu() =A,(Tu)() =B,(Tu)(t) –a(Tu)(t) =θ(t)fort∈[–τ, ], () Tu(t)is three-times continuously differentiable ont∈[–τ, +∞),

() (Tu)()_{(t) = –q(t)f(t, [u(t)], [u(t)], [u(t)],u(t)),t∈(, +∞),}

() fixed points ofTare solutions of BVP(.)-(.).

When applying the Schäuder fixed point theorem to prove the existence result, it is nec-essary to show that the operatorT(defined later) is completely continuous. For this, we

need the following modified version of the Arzela-Ascoli lemma (see [, ]).

Lemma . M⊂X is relatively compact if the following conditions hold: () all functions belonging toMare uniformly bounded,

() all functions belonging toMare equi-continuous on any compact sub-interval of [–τ, +∞),

() all functions fromMare equi-convergent at infinity,that is,for any> ,there exists aT=T() > such that,for allt≥Tand anyu∈M,

_{ +}u(t)_t–t→lim+∞ u(t)  +t

<, u _(t)

 +t –t→lim+∞ u(t)  +t <,

u_{ +}(t)_t – lim

t→+∞

u(t)  +t

< and u(t) – lim

t→+∞u

Definition . A functionα∈X∩C_{(, +∞) is called a lower solution of (.)-(.)}

pro-vided

α()(t) +q(t)ft,α(t),α(t),α(t),α(t)≥,  <t< +∞; (.)

α()≤A, α() =B, α(t) –aα(t)≤θ(t), –τ≤t≤;

α(+∞)≤C.

(.)

Similarly, a functionβ∈X∩C(, +∞) is called an upper solution of (.)-(.) provided

β()(t) +q(t)ft,β(t),β(t),β(t),β(t)≤,  <t< +∞; (.)

β()≥A, β() =B, β(t) –aβ(t)≥θ(t), –τ≤t≤;

β(+∞)≥C.

(.)

Also, we sayα(β) is a strict lower solution (strict upper solution) for problem (.)-(.) if all the above inequalities are strict.

Remark . If

α(t)≤β(t), for everyt∈[–τ, +∞), (.) then on integrating (.) and using the boundary restrictions in Definition ., we find thatα(t)≤β(t),α(t)≤β(t) for allt∈[, +∞) andβ(t)≤α(t),α(t)≤β(t) for allt∈ [–τ, ).

Definition . Letα,βbe lower and upper solutions for the problem (.)-(.) satisfying

α(t)≤β(t), for allt∈[–τ, +∞).

A continuous functionf is said to satisfy Nagumo’s condition with respect to the pair of functionsα,βif there exist positive functionsϕandh∈C[, +∞) such that

f(t,x, . . . ,xn,y, . . . ,yn,z, . . . ,zn,w)≤ϕ(t)h

|w|

for all t∈[, +∞), and (x, . . . ,xn)∈[[α(t)], [β(t)]],yi(t–τ,i(t))∈[α(t–τ,i(t)),β(t–

τ,i(t))] ift–τ,i(t) > ,yi(t–τ,i(t))∈[β(t–τ,i(t)),α(t–τ,i(t))] ift–τ,i(t)≤, ≤i≤n,

τ,= , (z, . . . ,zn)∈[[α(t)], [β(t)]],w∈R, and

∞



q(s)ϕ(s)ds< +∞,

∞



s

h(s)ds= +∞.

3 Main results

In this section we state and prove our existence results. We begin with the following lemma.

(H) BVP(.)-(.)has a pair of lower and upper solutionsα,βsatisfying

α(t)≤β(t), fort∈[–τ, +∞),

andf satisfies Nagumo’s condition with respect to the pair of functionsα,β. (H) There exists a constantγ > such that

sup

≤t<+∞( +t)

γ

q(t)ϕ(t) < +∞,

whereϕis the function in Nagumo’s condition off.

Then there exists a constant R>  (depending onα,β,h,and C)such that every solution u of(.)-(.)with

α(t)≤u(t)≤β(t), α(t)≤u(t)≤β(t),

α(t)≤u(t)≤β(t) for all t∈[, +∞)

(.)

and

α(t)≤u(t)≤β(t), β(t)≤u(t)≤α(t),

α(t)≤u(t)≤β(t) for all t∈[–τ, )

(.)

satisfiesu

∞<R.

Proof We can chooseR>ηsuch that

η≥max

sup

t∈[,+∞)

β(t), sup

t∈[,+∞)

α(t),β _–θ



a ,

α–θ

a ,C

and

R

η

s

h(s)ds>M

sup

t∈[,+∞)

β(t)

( +t)γ –_t∈[,+inf_∞)

α(t) ( +t)γ +

γN

γ– 

,

whereCis the nonhomogeneous boundary value, and

M= sup

t∈[,+∞)

( +t)γq(t)ϕ(t), N=maxβ_∞,α_∞.

Letube a solution of the differential equation (.) satisfying (.) and (.). If|u(t)|<R, for allt∈[, +∞), there is nothing to prove. If this is not true, there exists at∈[, +∞)

such that|u(t)| ≥R. Sincelimt→+∞u(t) =C<R, there exists aT>  such that u(t)<R for allt≥T.

Lett =inf{t≤T :|u(s)|<Rfor alls∈[t, +∞)}. Then|u(t)|=Rand|u(t)|<Rfor

to consider u(t) =Randu(t)≥Rfort∈[t,t], oru(t) = –Randu(t)≤–Rfor

t∈[t,t]. We assume thatu(t) =Randu(t)≥Rfort∈[t,t], then we have

R

η

s h(s)ds≤

R

C

s h(s)ds

= –

∞

t

u(s)u()_(s)

h(u(s)) ds

= –

∞

t

–q(s)f(s, [u(s)], [u(s)], [u(s)],u(s))u(s)

h(u(s)) ds

≤ ∞

t

q(s)ϕ(s)u(s)ds

≤M

∞

t

u(s) ( +s)γ ds

=M

∞

t

u(s) ( +s)γ

ds+

∞

t u(s)

 +s ·

γ

( +s)γ ds

≤M

sup

t∈[,+∞)

β(t)

( +t)γ –_t∈[,+inf_∞)

α(t) ( +t)γ +

γN

γ– 

<

R

η

s h(s)ds,

which is a contradiction. In the caseu(t) = –Randu(t)≤–Rfort∈[t,t], we obtain

a similar contradiction. Thus,|u(t)|<Rfor allt∈[, +∞). From the boundary condition (.) we also have

–R< –η≤α

_{(t) –θ(t)}

a ≤u

_{(t) =} u(t) –θ(t)

a ≤

β(t) –θ(t) a ≤η<R for allt∈[–τ, ]. Therefore,|u(t)|<Rfort∈[–τ, ). To sum up, we haveu

∞<R.

Theorem . Suppose conditions(H)and(H)hold.Suppose further that

(H) For any fixedt∈[, +∞),yi,zi,w∈R,i= , . . . ,n,when

αt–τ,i(t)

≤xi≤β

t–τ,i(t)

, i= , , . . . ,n, ft,x,x, . . . ,α

t–τ,i(t)

, . . . ,xn,y, . . . ,yn,z, . . . ,zn,w

≤f(t,x,x, . . . ,xi, . . . ,xn,y, . . . ,yn,z, . . . ,zn,w)

≤ft,x,x, . . . ,β

t–τ,i(t)

, . . . ,xn,y, . . . ,yn,z, . . . ,zn,w

.

(H) For any fixedt∈[, +∞),xi,zi,w∈R,i= , . . . ,nwhen

αt–τ,i(t)

≤yi≤β

t–τ,i(t)

, t–τ,i(t) > ,

or when

βt–τ,i(t)

≤yi≤α

t–τ,i(t)

ft,x, . . . ,xn,y, . . . ,α

t–τ,i(t)

, . . . ,yn,z, . . . ,zn,w

≤f(t,x, . . . ,xn,y, . . . ,yi, . . . ,yn,z, . . . ,zn,w)

≤ft,x, . . . ,xn,y, . . . ,β

t–τ,i(t)

, . . . ,yn,z, . . . ,zn,w

.

(H) For any fixedt∈[, +∞),xi,yi,w∈R,i= , . . . ,nwhen

αt–τ,i(t)

≤zi≤β

t–τ,i(t)

, i= , , . . . ,n, ft,x, . . . ,xn,y, . . . ,yn,z, . . . ,α

t–τ,i(t)

, . . . ,zn,w

≤f(t,x, . . . ,xn,y, . . . ,yn,z, . . . ,zi, . . . ,zn,w)

≤ft,x, . . . ,xn,y, . . . ,yn,z, . . . ,β

t–τ,i(t)

, . . . ,zn,w

,

whereτ,=τ,=τ,= .

(H) _∞



max{s, }q(s)ds< +∞,

∞



max{s, }q(s)ϕ(s)ds< +∞.

Then BVP(.)-(.)has at least one solution u∈X∩C_{(, +∞)satisfying(.)-(.)and}

u

∞<R.

Proof LetRbe a positive number as in Lemma . and define the auxiliary functions,

F,F,F,F: [, +∞)×Rn+×Rn+×Rn+×R→R

as follows:

F(t,x, . . . ,xn,y, . . . ,yn,z, . . . ,zn,w)

= ⎧ ⎪ ⎨ ⎪ ⎩

f(t,β,x, . . . ,xn,y, . . . ,yn,z, . . . ,zn,w), x>β(t),

f(t,x,x, . . . ,xn,y, . . . ,yn,z, . . . ,zn,w), α(t)≤x≤β(t),

f(t,α,x, . . . ,xn,y, . . . ,yn,z, . . . ,zn,w), x<α(t),

F(t,x, . . . ,xn,y, . . . ,yn,z, . . . ,zn,w)

= ⎧ ⎪ ⎨ ⎪ ⎩

F(t,x, . . . ,xn,β,y, . . . ,yn,z, . . . ,zn,w), y>β(t),

F(t,x, . . . ,xn,y,y, . . . ,yn,z, . . . ,zn,w), α(t)≤y≤β(t),

F(t,x, . . . ,xn,α,y, . . . ,yn,z, . . . ,zn,w), y<α(t),

F(t,x, . . . ,xn,y, . . . ,yn,z, . . . ,zn,w)

= ⎧ ⎪ ⎨ ⎪ ⎩

F(t,x, . . . ,xn,y, . . . ,yn,β,z, . . . ,zn,w) – z–β

+|z–β|, z>β _(t),

F(t,x, . . . ,xn,y, . . . ,yn,z,z, . . . ,zn,w), α(t)≤z≤β(t),

F(t,x, . . . ,xn,y, . . . ,yn,α,z, . . . ,zn,w) + z–α

+|z–α|, z<α

_(t),

F(t,x,x, . . . ,xn,y,y, . . . ,yn,z, . . . ,zn,w)

= ⎧ ⎪ ⎨ ⎪ ⎩

F(t,x, . . . ,xn,y, . . . ,yn,z, . . . ,zn,R), w>R,

F(t,x, . . . ,xn,y, . . . ,yn,z, . . . ,zn,w), –R≤w≤R,

where, fori= , , . . . ,n,

xi= ⎧ ⎪ ⎨ ⎪ ⎩

β, xi>β(t–τ,i(t));

xi, α(t–τ,i(t))≤xi≤β(t–τ,i(t));

α, xi<α(t–τ,i(t)),

ift–τ,i(t) > ,

yi= ⎧ ⎪ ⎨ ⎪ ⎩

β, yi>β(t–τ,i(t));

yi, α(t–τ,i(t))≤yi≤β(t–τ,i(t));

α, yi<α(t–τ,i(t)),

ift–τ,i(t)≤,

yi= ⎧ ⎪ ⎨ ⎪ ⎩

α, yi>α(t–τ,i(t));

yi, β(t–τ,i(t))≤yi≤α(t–τ,i(t));

β, yi<β(t–τ,i(t)),

and

zi= ⎧ ⎪ ⎨ ⎪ ⎩

β, zi>β(t–τ,i(t));

zi, α(t–τ,i(t))≤zi≤β(t–τ,i(t));

α, zi<α(t–τ,i(t)).

We consider the modified boundary value problem

u()(t) +q(t)F

t,u(t),u(t),u(t),u(t)= ,  <t< +∞, (.)

with the boundary conditions (.). We will show that the problem (.)-(.) has at least one solutionuinX. Now foru∈X, we define two operatorsT,Tby

Tu(t) =

_∞



G(t,s)q(s)F

s,u(s),u(s),u(s),u(s)ds

and

Tu(t) =

ψ(t), –τ≤t≤;

l(t) +Tu(t), ≤t< +∞,

(.)

wherel(t) is as in (.) and

ψ(t) =A+Bt+

θ() +aC+a

∞



q(s)F

s,u(s),u(s),u(s),u(s)ds

×–at–a+aeat₊



t

s–a–t+aet–as_θ(s)ds.

We want to show that the operatorTis completely continuous. We split the proof in the

()T:X→Xis well defined. Obviously, for anyu∈Xby direct calculation, we have

(Tu)(t) –a(Tu)(t) =θ(t) fort∈[–τ, ] and

(Tu)() =B, (Tu)() =A

and fort∈(, +∞),

(Tu)(t) =l(t) +

∞



∂G(t,s)

∂t q(s)F

s,u(s),u(s),u(s),u(s)ds,

(Tu)(t) =l(t) +

∞



∂_G(t,s)

∂t q(s)F

s,u(s),u(s),u(s),u(s)ds,

(Tu)(t) =C+

∞

t

q(s)F

s,u(s),u(s),u(s),u(s)ds,

which show thatTu(t)∈C[–τ, +∞). Further, we have

∞



q(s)F

s,u(s),u(s),u(s),u(s)ds

≤

∞



max{s, }q(s)Hϕ(s) + ds< +∞, (.)

whereH=max≤s≤sup_{t∈[,+∞)}|u(t)|h(s). Now from (.) it follows that

∞



sq(s)Hϕ(s) + ds≤

∞



max{s, }q(s)Hϕ(s) + ds< +∞,

which implies

lim

t→+∞tq(t)

Hϕ(t) + = . (.)

Next since

∞

t

q(s)Hϕ(s) + ds≤

∞

t

sq(s)Hϕ(s) + ds< +∞, t≥,

we also have

lim

t→+∞

∞

t

q(s)Hϕ(s) + ds= . (.)

By Lebesgue’s dominated convergent theorem, L’Hopital’s rule, and (.), (.), we obtain

lim

t→+∞

(Tu)(t)

 +t

≤ lim

t→+∞

∞



|G(t,s)|  +t q(s)F

s,u(s),u(s),u(s),u(s)ds

≤ lim

t→+∞

∞



|G(t,s)|  +t q(s)

= lim

t→+∞

t



(a_t₊st  –

s_t  +

s )

 +t q(s)

Hϕ(s) + ds

+

∞

t

(a_t₊t )

 +t q(s)

Hϕ(s) + ds

= lim

t→+∞

t



(at+st–s_) t q(s)

Hϕ(s) + ds+(

a t+

t )

t q(t)

Hϕ(t) + 

+ lim

t→+∞

∞

t

(at+t_) t q(s)

Hϕ(s) + ds–(

a t+

t )

t q(t)

Hϕ(t) + 

= lim

t→+∞ t



(at+st–s_) t q(s)

Hϕ(s) + ds+ lim

t→+∞

(a_t+t_) t tq(t)

Hϕ(t) + 

+ lim

t→+∞

∞

t

(at+t_) t q(s)

Hϕ(s) + ds– lim

t→+∞

(a_t+t_) t tq(t)

Hϕ(t) + 

= lim

t→+∞ t



(a+s) t q(s)

Hϕ(s) + ds+(at+

t )

t q(t)

Hϕ(t) + 

+ lim

t→+∞

∞

t

a+t t q(s)

Hϕ(s) + ds–(at+

t )

t q(t)

Hϕ(t) + 

= lim

t→+∞ t



(a+s) t q(s)

Hϕ(s) + ds+ lim

t→+∞ (a+t_)

t tq(t)

Hϕ(t) + 

+ lim

t→+∞

∞

t

a+t t q(s)

Hϕ(s) + ds– lim

t→+∞ (a+_t)

t tq(t)

Hϕ(t) + 

= lim

t→+∞

∞

t

q(s)Hϕ(s) + ds– lim

t→+∞

(a+t)  q(t)

Hϕ(t) + 

= lim

t→+∞

∞

t

q(s)Hϕ(s) + ds– lim

t→+∞ (a+t)

t tq(t)

Hϕ(t) + 

= t→lim+∞

∞

t

q(s)Hϕ(s) + ds= ,

that is,limt→+∞(T_+u_t)(t)= , and

lim

t→+∞

(Tu)(t)

 +t =t→lim+∞ l(t)

 +t +t→lim+∞

(Tu)(t)

 +t

= lim

t→+∞

A+Bt+ (aC+θ())t_+Ct_!

 +t =

C ,

which implies thatsup_{t∈[,+∞)}|(Tu)(t)|

+t < +∞. Similarly, we have

lim

t→+∞

(Tu)(t)

 +t

≤ lim

t→+∞

  +t

_∞



∂G(t,s)

∂t q(s)F

s,u(s),u(s),u(s),u(s)ds

≤ t→lim+∞

∞

t

that is, limt→+∞(T_+u_t)(t) =  and limt→+∞(T_+u_t)(t) =limt→+∞_+l(_tt) +limt→+∞(T_+u_t)(t) = C_, which implies thatsup_{t∈[,+∞)}|(Tu)(t)|

+t < +∞,

lim

t→+∞

(Tu)(t)

 +t

≤ lim

t→+∞   +t

∞



∂G(t,s)

∂t q(s)F

s,u(s),u(s),u(s),u(s)ds

≤ lim

t→+∞

∞

t

q(s)Hϕ(s) + ds= ,

that is, limt→+∞(T+u)t(t) =  andlimt→+∞ (Tu)(t)

+t =limt→+∞ l(t)

+t +limt→+∞ (Tu)(t)

+t =C,

which implies thatsup_{t∈[,+∞)}|(Tu)(t)|

+t < +∞, and by (.)

lim

t→+∞

∞

t

q(s)F

s,u(s),u(s),u(s),u(s)ds

≤ lim

t→+∞

∞

t

q(s)Hϕ(s) + ds= ,

then

lim

t→+∞(Tu)

_{(t) = lim}

t→+∞

C+

∞

t

q(s)F

s,u(s),u(s),u(s),u(s)ds

=C< +∞.

ThereforeTu∈X.

()T:X→Xis continuous. For any convergent sequenceum→uinX, we have

um(t)→u(t), um(t)→u(t), um(t)→u(t),

u_m(t)→u(t), m→+∞,t∈[–τ, +∞).

Thus the continuity ofFimplies that F

s,um(s)

,u_m(s),u_m(s),u_m(s)–F

s,u(s),u(s),u(s),u(s)→,

m→+∞.

Sinceu_m(t)→u(t),σ=sup{s:s=supt∈[,+∞)|um(t)|,m∈N}< +∞.

LetH=max≤s≤max{sup_{t∈[,+∞)}|u(t)|,σ}h(s). Then we have

_∞



q(s)F

s,um(s)

,u_m(s),u_m(s),u_m(s)

–F

s,u(s),u(s),u(s),u(s)ds

≤

∞



q(s)Hφ(s) + 

ds< +∞.

Thus, we find

Tum–Tu

= max

t∈[–τ,]

= max

t∈[–τ,]

_∞–a–at+aeat_{q(s)Fs,um(s),u}

m(s)

,u_m(s),u_m(s)

–F

s,u(s),u(s),u(s),u(s)ds

≤–a+aτ+ae–aτ

_∞



q(s)F

s,um(s)

,u_m(s),u_m(s),u_m(s)

–F

s,u(s),u(s),u(s),u(s)ds,

that is,

Tum–Tu→, (.)

asm→+∞.

Tum–Tu_∞ = sup t∈[,+∞)

(Tum)(t)

 +t –

(Tu)(t)

 +t

= sup

t∈[,+∞)

∞



G(t,s)  +tq(s)

F

s,um(s)

,u_m(s),u_m(s),u_m(s)

–F

s,u(s),u(s),u(s),u(s)ds

≤

∞



a√ +  

q(s)F

s,um(s)

,u_m(s),u_m(s),u_m(s)

–F

s,u(s),u(s),u(s),u(s)ds,

that is,

Tum–Tu∞→, (.)

asm→+∞.

Tum–Tu = max t∈[–τ,]

(Tum)(t) – (Tu)(t)

= max

t∈[–τ,]

_∞–a+aeat_{q(s)Fs,um(s),u}

m(s)

,u_m(s),u_m(s)

–F

s,u(s),u(s),u(s),u(s)ds

≤–a+ae–aτ

∞



q(s)F

s,um(s)

,u_m(s),u_m(s),u_m(s)

–F

s,u(s),u(s),u(s),u(s)ds,

that is,

Tum–Tu→, (.)

Tum–Tu∞

= sup

t∈[,+∞)

(Tum)(t)

 +t –

(Tu)(t)

 +t

= sup

t∈[,+∞) _{ +}_t

∞



∂G(t,s)

∂t q(s)

F

s,um(s)

,u_m(s),u_m(s),u_m(s)

–F

s,u(s),u(s),u(s),u(s)ds

≤

∞



a+  

q(s)F

s,um(s)

,u_m(s),u_m(s),u_m(s)

–F

s,u(s),u(s),u(s),u(s)ds,

that is,

Tum–Tu∞→, (.)

asm→+∞.

Tum–Tu = max t∈[–τ,]

(Tum)(t) – (Tu)(t)

= max

t∈[–τ,]

∞



aeat_q(s)F



s,um(s)

,u_m(s),u_m(s),u_m(s)

–F

s,u(s),u(s),u(s),u(s)ds

≤a

∞



q(s)F

s,um(s)

,u_m(s),u_m(s),u_m(s)

–F

s,u(s),u(s),u(s),u(s)ds,

that is,

Tum–Tu→, (.)

asm→+∞.

Tum–Tu∞

= sup

t∈[,+∞)

(Tum)(t)

 +t –

(Tu)(t)

 +t

= sup

t∈[,+∞)

_{ +}_t ∞ 

∂G(t,s)

∂t q(s)

F

s,um(s)

,u_m(s),u_m(s),u_m(s)

–F

s,u(s),u(s),u(s),u(s)ds

≤

∞



(a+ )q(s)F

s,um(s)

,u_m(s),u_m(s),u_m(s)

–F

that is,

Tum–Tu∞→, (.)

asm→+∞.

To showTum–Tu_∞→, asm→+∞, we need the following:

sup

t∈[,+∞)

(Tum)(t) – (Tu)(t)

= sup

t∈[,+∞)

(Tum)(t) – (Tu)(t)

= sup

t∈[,+∞)

_t∞q(s)F

s,um(s)

,u_m(s),u_m(s),u_m(s)

–F

s,u(s),u(s),u(s),u(s)ds

≤

∞



q(s)F

s,um(s)

,u_m(s),u_m(s),u_m(s)

–F

s,u(s),u(s),u(s),u(s)ds

and

sup

t∈[–τ,)

(Tum)(t) – (Tu)(t)

= sup

t∈[–τ,)

 a(Tum)

_{(t) – (T}

u)(t)

≤

∞



q(s)F

s,um(s)

,u_m(s),u_m(s),u_m(s)

–F

s,u(s),u(s),u(s),u(s)ds.

Hence, it follows that

Tum–Tu∞

= sup

t∈[–τ,+∞)

(Tum)(t) – (Tu)(t)

≤ sup

t∈[,+∞)

(Tum)(t) –Tu)(t)+ sup t∈[–τ,)

(Tum)(t) – (Tu)(t)

≤

∞



q(s)F

s,um(s)

,u_m(s),u_m(s),u_m(s)

–F

s,u(s),u(s),u(s),u(s)ds,

that is,

Tum–Tu∞→, (.)

Combining (.)-(.), we find(Tum) – (Tu) →, asm→+∞; soT:X→Xis

continuous.

() T :X →X is compact. The operator T is compact if T maps bounded

sub-sets of X into relatively compact sets. Let K be any bounded subset of X, then r =

sup_≤s≤{sup_{t∈[,+∞)}|u(t)|,u∈K}h(s) < +∞. For anyu∈K, we have the following:

Tu≤ |A|+τ|B|+aC+aθ()+τθ

–a+τ+ae–aτ

+–a+aτ+ae–aτ

∞



q(s)rϕ(s) + 

ds,

Tu∞≤ |A|+|B|+aC+θ()+C+

a√  +  

∞



q(s)rϕ(s) + 

ds, (Tu)_≤ |B|+aC+aθ()+τθ

 –e–aτ

+a–ae–aτ

∞



q(s)rϕ(s) + 

ds, (Tu)



∞≤ |B|+aC+θ()+C+

a+  

∞



q(s)rϕ(s) + 

ds, (Tu)_≤aC+θ()+θ+a

∞



q(s)rϕ(s) + 

ds, (Tu)_∞≤aC+θ()+C+ (a+ )

_∞



q(s)rϕ(s) + 

ds, (Tu)



∞≤ 

aaC+θ()+ 

aθ+C+ 

∞



q(s)rϕ(s) + 

ds,

which implies that

Tu ≤ |A|+ξ|B|+υaC+θ()+C+γθ+χ

∞



q(s)rϕ(s) + 

ds,

where

χ=max–a+aτ+ae–aτ_,

a√ +  

,a–ae–aτ_{, (a+ ), }

,

υ=max–a+aτ+ae–aτ_,a–ae–aτ_,

a, 

, ξ=max{τ, },

γ =max–aτ+τ+aτe–aτ_,τ–τe–aτ_{, ,}

a

.

Therefore,TKis uniformly bounded. We also know thatψ(t) andψ(t) are continuous

on [–τ, ]. Thus in view of [–τ, ] compact,ψ(t) andψ(t) are also uniformly continuous.

Thus it follows that fort,t∈[–τ, ],

further since

(Tu)(t) =ψ(t)

=

θ() +aC+a

∞



q(s)F

s,u(s),u(s),u(s),u(s)ds

eta

+ a



t

et–as_θ(s)ds

is continuous on [–τ, ], we find

(Tu)(t) – (Tu)(t)=ψ(t) –ψ(t)→ ast→t.

Next, fort,t∈[,ε] withε>  a constant, we have

(Tu)(t)

 +t 

–(Tu)(t)  +t



=l(t)  +t_ –

l(t)

 +t_ +

_∞



G(t,s)

 +t_ – G(t,s)

 +t_

q(s)

×F

s,u(s),u(s),u(s),u(s)ds

≤ l(t)

 +t_ – l(t)

 +t_ +

_∞

 G(t,s)

 +t_ – G(t,s)

 +t_

q(s)rϕ(s) + 

ds

→ ast→t,

(Tu)(t)

 +t_ –

(Tu)(t)

 +t_

=l _(t

)

 +t 

– l _(t

)

 +t 

+

∞



∂G(t,s) ∂t

 +t 

–

∂G(t,s) ∂t

 +t 

q(s)

×F

s,u(s),u(s),u(s),u(s)ds

≤l(t)

 +t_ – l(t)

 +t_ +

∞

 ∂G∂(tt,s)

 +t_ –

∂G(t,s) ∂t

 +t_

q(s)rϕ(s) + 

ds

→ ast→t,

(Tu)(t)

 +t

–(Tu) _(t

)

 +t

=l _(t

)

 +t

–l _(t

)

 +t

+

∞



∂G(t,s) ∂t  +t



–

∂G(t,s)

∂ t



 +t 

q(s)

×F

s,u(s),u(s),u(s),u(s)ds

≤l(t)

 +t

–l _(t

)

 +t + ∞  ∂

_G(t ,s) ∂t  +t

–

∂G(t,s) ∂t  +t

q(s)rϕ(s) + 

ds

and

(Tu)(t) – (Tu)(t)

=

∞

t q(s)F

s,u(s),u(s),u(s),u(s)ds

–

∞

t q(s)F

s,u(s),u(s),u(s),u(s)ds

≤ t

t

q(s)rϕ(s) + 

ds

→ ast→t.

Thus,TKis equi-continuous. Finally, we will show thatTKis equi-convergent at infinity.

In fact, whent>  we have

(Tu)(t)

 +t –t→lim+∞

(Tu)(t)

 +t

=(Tu)(t)  +t –

C 

→, ast→+∞,

(Tu)(t)

 +t –_t→lim_+∞

(Tu)(t)

 +t

=(Tu) _(t)

 +t –

C 

→, ast→+∞,

(Tu)(t)

 +t –t→lim+∞

(Tu)(t)

 +t

=(Tu) _(t)

 +t –C

→, ast→+∞,

and

(Tu)(t) – lim t→+∞(Tu)

_(t)=(T

u)(t) –C≤

∞

t

q(s)rϕ(s) + 

ds→

ast→+∞.

Hence all conditions of Lemma . are fulfilled, soTKis relatively compact. Therefore,

T:X→Xis completely continuous.

()T:X→Xhas at least one fixed point. Let={u∈X,u ≤N}where

N=|A|+ξ|B|+υaC+θ()+C+γθ+χ

∞



q(s)Hϕ(s) + ds. (.)

For anyu∈, it is easy to see thatTu ≤, and thusT⊂. The Schäuder fixed point

theorem now guarantees that the operatorThas at least one fixed point in, which is a

solution of BVP (.)-(.). Now we shall show that this solutionusatisfies the inequalities (.) and (.) which in view of the definitions ofF,F, F, andFwill imply thatuis

in fact a solution of (.)-(.). For this, we only prove thatu(t)≤β(t),t∈[–τ, +∞).

A similar argument can be used to proveα(t)≤u(t),t∈[–τ, +∞). If not true, we set ω(t) =u(t) –β(t), then there existst∗∈[–τ, +∞) such thatω(t∗) =sup_{–τ≤t<+∞}ω(t) > . Obviously, ift∗= –τthenω(t∗)≤, and ift∗∈(–τ, ] thenω(t∗) = . However, from the boundary condition, we haveω(t∗) = 

aω(t∗) > , which gives a contraction. Ift∗∈(, +∞),

then we have

By the definition of auxiliary functions andR>sup_{t∈[,+∞)}|β(t)|, we have

u()t∗= –qt∗F

t∗,ut∗,ut∗,ut∗,ut∗ = –qt∗F

t∗,ut∗,ut∗,ut∗,βt∗ = –qt∗F

t∗,ut∗,ut∗,βt∗,ut∗–τ,

t∗, . . . ,βt∗

– u

_(t∗_{) –β(t}∗₎  +|u(t∗) –β(t∗)|

.

Now ifu(t∗–τ,(t∗)) >β(t∗–τ,(t∗)) from the definition ofzit follows that

u()t∗= –qt∗F

t∗,ut∗,ut∗,βt∗,βt∗–τ,

t∗,ut∗–τ,

t∗, . . . ,

βt∗+qt∗ u

_(t∗_{) –β(t}∗₎  +|u(t∗) –β(t∗)|

and ifu(t∗–τ,(t∗))≤β(t∗–τ,(t∗)) from the condition (H), we have

u()t∗≥–qt∗F

t∗,ut∗,ut∗,βt∗,βt∗–τ,

t∗,ut∗–τ,

t∗, . . . ,

βt∗+qt∗ u

_(t∗_{) –β(t}∗₎  +|u(t∗) –β(t∗)|.

Similarly, we consider the casesu(t∗–τ,i(t∗)) >β(t∗–τ,i(t∗)) oru(t∗–τ,i(t∗))≤β(t∗–

τ,i(t∗)),i= , , . . . ,n, and obtain the inequality

u()t∗≥–qt∗F

t∗,ut∗,ut∗,βt∗,βt∗+qt∗ u

_(t∗_{) –β(t}∗₎  +|u(t∗) –β(t∗)|. Next, ifu(t∗) >β(t∗) from the definition ofFit follows that

u()t∗≥–qt∗F

t∗,ut∗,βt∗,ut∗–τ,

t∗, . . . ,βt∗,βt∗ +qt∗ u

_(t∗_{) –β(t}∗₎  +|u(t∗) –β(t∗)|

and ifu(t∗)≤β(t∗) from the definition ofFand the condition (H) we have

u()t∗≥–qt∗F

t∗,ut∗,βt∗,ut∗–τ,

t∗, . . . ,βt∗,βt∗ +qt∗ u

_(t∗_{) –β(t}∗₎  +|u(t∗) –β(t∗)|.

Ift∗–τ,(t∗) > , while discussing the casesu(t∗–τ,(t∗)) >β(t∗–τ,(t∗)) we use the

definition ofy, and when discussing the casesu(t∗–τ,(t∗))≤β(t∗–τ,(t∗)) we use

(H), and obtain

u()t∗≥–qt∗F

t∗,ut∗,βt∗,βt∗–τ,

t∗, . . . ,βt∗,βt∗ +qt∗ u

Similarly, ift∗–τ,(t∗)≤, while discussing the casesu(t∗–τ,(t∗))≥β(t∗–τ,(t∗))

we use (H), while when discussing the casesu(t∗–τ,(t∗)) <β(t∗–τ,(t∗)) we use the

definition ofy, to again find

u()t∗≥–qt∗F

t∗,ut∗,βt∗,βt∗–τ,

t∗, . . . ,βt∗,βt∗ +qt∗ u

_(t∗_{) –β(t}∗₎  +|u(t∗) –β(t∗)|.

Following exactly as above, using the definition ofyiand (H), we consider the casesi=

, . . . ,nto finally obtain

u()t∗≥–qt∗F

t∗,ut∗,βt∗,βt∗,βt∗+qt∗ u

_(t∗_{) –β(t}∗₎  +|u(t∗) –β(t∗)|. Next, ifu(t∗) >β(t∗) from the definition ofFit follows that

u()t∗≥–qt∗ft∗,βt∗,ut∗–τ,

t∗, . . . ,ut∗,βt∗,βt∗ +qt∗ u

_(t∗_{) –β(t}∗₎  +|u(t∗) –β(t∗)|

and ifu(t∗)≤β(t∗) from the definition ofFand the condition (H) we have

u()t∗≥–qt∗ft∗,βt∗,ut∗–τ,

t∗, . . . ,ut∗,βt∗,βt∗ +qt∗ u

_(t∗_{) –β(t}∗₎  +|u(t∗) –β(t∗)|.

Similarly, we use the definition ofxiand (H) while discussing the casesu(t∗–τ,i(t∗)) >

β(t∗–τ,i(t∗)) oru(t∗–τ,i(t∗))≤β(t∗–τ,i(t∗)),i= , . . . ,n, to get

u()t∗≥–qt∗ft∗,βt∗,βt∗,βt∗,βt∗+qt∗ u

_(t∗_{) –β(t}∗₎  +|u(t∗) –β(t∗)|, which implies that

ωt∗≥qt∗ u

_(t∗_{) –β(t}∗₎  +|u(t∗) –β(t∗)|> , which is a contractions.

Ift∗= +∞thenω(+∞) =sup_{t∈[–τ,+∞)}ω(t) > . From the boundary conditions, we also haveω(+∞) =u(+∞) –β(+∞)≤. But this implies thatω(+∞)≤ andω(+∞) = . However, now following as above, we findω(+∞) > , which is a contradiction. Thus,

u(t)≤β(t),t∈[–τ, +∞).

Consequently, we have

α(t)≤u(t)≤β(t), t∈[–τ, +∞),

which on integration and using boundary conditions gives

and now a further integration leads to

α(t)≤u(t)≤β(t), t∈[–τ, +∞).

Further, since all conditions of the Lemma . are satisfied,u

∞<R. Consequently, we have

u()(t) = –q(t)F

t,u(t),u(t),u(t),u(t) = –q(t)ft,u(t),u(t),u(t),u(t)

and hence,uis a solution of (.)-(.).

Theorem . Assume that there exist two pairs of upper and lower solutionsβk,αk,k= , 

of BVP(.)-(.),whereα,βare strict and

α(_i)(t)β_(i)(t), i= , , ,

α_(i)(t)≤α_(i)(t)≤β_(i)(t), α_(i)(t)≤β_(i)(t)≤β_(i)(t), t∈[–τ, +∞),i= , ,

β_(t)≤α_(t)≤α_(t), β_(t)≤β_(t)≤α_(t), t∈[–τ, ),

α_(t)≤α_(t)≤β_(t), α_(t)≤β_(t)≤β_(t), t∈[, +∞),

(.)

and f satisfies Nagumo’s condition with respect toα,β.Suppose further that conditions

(H)-(H)hold withαandβreplaced byαandβ,respectively.Then the problem

(.)-(.)has at least three solutions u,u,and usuch that

α_k(t)≤u_k(t)≤β_k(t), αk(t)≤uk(t)≤βk(t), t∈[–τ, +∞),k= , ,

β_k(t)≤u_k(t)≤α_k(t), t∈[–τ, ), α_k(t)≤u_k(t)≤β_k(t), t∈[, +∞),k= , , u(_i)(t)β_(i)(t), u_(i)(t)α(_i)(t), t∈[–τ, +∞),i= , , .

Proof First we define the truncated functionsF,F,F,Fthe same asF,F,F,Fin

The-orem . withα,βreplaced byαandβ, respectively. Consider the modified differential

equation

u()(t) +q(t)F

t,u(t),u(t),u(t),u(t)= , ≤t< +∞. (.)

To show that (.)-(.) has at least three solutions, we define operatorsT,Tas

Tu(t) =

∞



G(t,s)q(s)F

s,u(s),u(s),u(s),u(s)ds

and

Tu(t) =

ψ(t), –τ ≤t≤;

wherel(t) is as in (.) and

ψ(t) =A+Bt

+

θ() +aC+a

∞



q(s)F

s,u(s),u(s),u(s),u(s)ds

×–at–a+aeta₊



t

s–a–t+aet–as_θ(s)ds.

As in Theorem .,T is completely continuous. By using the degree theory, we will

show thatThas at least three fixed points which are solutions of (.)-(.). We note

thatRin Lemma . instead ofα,β now depends onα,β. Set={u∈X,u<N}

whereN is as in (.) then for anyu∈, it follows thatTu<N, thusT⊂,

and so we havedeg(I–T,, ) = . Set

α=

u∈:u(t) >α(t),t∈[–τ, +∞)

,

β_=u∈

:u(t) <β(t),t∈[–τ, +∞)

.

Sinceα_β_,α_(t)≤α_(t)≤β_(t),α_(t)≤β_(t)≤β_(t), we findα =∅ = β_,

α∩

β₌∅,_\_α

∪β=∅. Now sinceα,βare strict lower and upper solutions there is no solution in∂α∪∂

β_{. The additivity of degree implies that}

deg(I–T,, ) =deg

I–T,\α∪β, 

+deg(I–T,α, ) +deg

I–T,β, 

.

We will show thatdeg(I–T,α, ) =deg(I–T,

β_{, ) = . For this, we define new}

op-eratorsT:→andT:→as

Tu(t) =

∞



q(s)F

s,u(s),u(s),u(s),u(s)ds

and

Tu(t) =

ψ(t), –τ ≤t≤;

l(t) +Tu(t), ≤t< +∞,

wherel(t) is as in (.) and

ψ(t) =A+Bt

+

θ() +aC+a

∞



q(s)F

s,u(s),u(s),u(s),u(s)ds

×–at–a+aeta₊



t

s–a–t+aet–as_θ(s)ds.

Here the functionsF,F,F,F are same asF,F,F,Fexcept thatα is replaced by