Jiang Boundary Value Problems (2015) 2015:39 DOI 10.1186/s13661-015-0299-3

RESEARCH

Open Access

The existence of solutions for impulsive p-Laplacian boundary value problems at resonance on the half-line Weihua Jiang* *

Correspondence: [email protected] College of Sciences, Hebei University of Science and Technology, Shijiazhuang, Hebei 050018, P.R. China

Abstract By using the continuous theorem of Ge and Ren and constructing suitable Banach spaces and operators, we investigate the existence of solutions for an impulsive p-Laplacian boundary value problem with integral boundary condition at resonance on the half-line. An example is given to illustrate our main results. MSC: 34B40 Keywords: impulsive; p-Laplacian operator; boundary value problem; integral boundary condition; resonance

1 Introduction Boundary value problems on the half-line arise in various applications such as in the study of the unsteady flow of a gas through semi-infinite porous medium, in analyzing the heat transfer in radial flow between circular disks, in the study of plasma physics, in an analysis of the mass transfer on a rotating disk in a non-Newtonian fluid, etc. [] Many dynamical systems have an impulsive dynamical behavior due to abrupt changes at certain instants during the evolution process. The mathematical description of these phenomena leads to impulsive differential equations. For some general and recent works on the theory of impulsive differential equations we refer the reader to [–]. Impulsive differential equations occur in biology, medicine, mechanics, engineering, chaos theory, etc. [–]. Impulsive boundary value problems have been studied by many papers; see [–]. For example, in [], the authors studied the existence of solutions for the problem ⎧ (p(t)u (t)) = f (t, u(t)), t ∈ (, ∞)\{t , t , . . . , tn }, ⎪ ⎪ ⎪ ⎨u (t ) = I (u(t )), k = , , . . . , n, k k k ⎪αu() – β limt→+ p(t)u (t) = , ⎪ ⎪ ⎩ γ limt→∞ u(t) + δ limt→∞ p(t)u (t) = . In [], the impulsive boundary value problem on the half-line ⎧  (p(t)x (t)) = f (t, xt ), t ∈ (, ∞)\{t , t , . . . , tn }, ⎪ p(t) ⎪ ⎪ ⎨  x (tk ) = Ik (xtk ), k = , , . . . , m, ⎪ λx() – β limt→+ p(t)x (t) = a, ⎪ ⎪ ⎩ γ x(∞) + δ limt→∞ p(t)x (t) = b was studied. © 2015 Jiang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

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A boundary value problem is said to be a resonance one if the corresponding homogeneous boundary value problem has a non-trivial solution. The boundary value problems at resonance have been studied by many papers; see [–]. In [], the author gave the existence of solutions for the p-Laplacian boundary value problem at resonance on the half-line  (ϕp (u )) (t) = ψ(t)f (t, u(t), u (t)), t ∈ [, +∞),  +∞ u() =  h(t)u(t) dt, u (+∞) = , where ϕp (s) = |s|p– s, p > . As far as we know, the impulsive p-Laplacian boundary value problems at resonance on the half-line have not been investigated. In this paper, we will discuss the existence of solutions for the problem ⎧    ⎪ ⎨(ϕp (u (t))) + f (t, u(t), u (t)) = , t ∈ [, ∞)\{t , t , . . . , tk },   ϕp (u (ti )) = Ii (u(ti ), u (ti )), i = , , . . . , k, ⎪  +∞ ⎩ u() = , ϕp (u (+∞)) =  h(t)ϕp (u (t)) dt,

(.)

where  < t < t < · · · < tk < +∞, ϕp (u (ti )) = ϕp (u (ti + )) – ϕp (u (ti – )). In this paper, we will always suppose that the following conditions hold.  +∞ (H ) h(t) ≥ , t ∈ [, +∞),  h(t) dt = , f : [, +∞) × R → R, and Ii : R → R, i = , , . . . , k are continuous. (H ) For any constant r > , there exist a function hr ∈ L[, +∞) and a constant Mr > , such that |f (t, ( + t)u, v)| ≤ hr (t), t ∈ [, +∞), |u| < r, |v| < r, |Ii (u, v)| ≤ Mr , i = , , . . . , k, |u| ≤ r( + tk ), |v| ≤ r.

2 Preliminaries For convenience, we introduce some notations and a theorem. For more details see []. Definition . [] Let X and Y be two Banach spaces with norms · X , · Y , respectively. A continuous operator M : X ∩ dom M → Y is said to be quasi-linear if (i) Im M := M(X ∩ dom M) is a closed subset of Y , (ii) Ker M := {x ∈ X ∩ dom M : Mx = } is linearly homeomorphic to Rn , n < ∞, where dom M denote the domain of the operator M. Let X = Ker M and X be the complement space of X in X, then X = X ⊕ X . On the other hand, suppose Y is a subspace of Y and that Y is the complement of Y in Y , i.e. Y = Y ⊕ Y . Let P : X → X and Q : Y → Y be two projectors and ⊂ X an open and bounded set with the origin θ ∈ . Definition . [] Suppose that Nλ : → Y , λ ∈ [, ] is a continuous operator. Denote N by N . Let  λ = {x ∈ : Mx = Nλ x}. Nλ is said to be M-compact in if there exist a vector subspace Y of Y satisfying dim Y = dim X and an operator R : × [, ] → X being continuous and compact such that for λ ∈ [, ], (a) (I – Q)Nλ ( ) ⊂ Im M ⊂ (I – Q)Y , (b) QNλ x = θ , λ ∈ (, ) ⇔ QNx = θ ,

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(c) R(·, ) is the zero operator and R(·, λ)| λ = (I – P)| λ , (d) M[P + R(·, λ)] = (I – Q)Nλ . Theorem . [] Let X and Y be two Banach spaces with the norms · X , · Y , respectively, and ⊂ X an open and bounded nonempty set. Suppose that M : X ∩ dom M → Y is a quasi-linear operator and Nλ : → Y , λ ∈ [, ] is M-compact. In addition, if the following conditions hold: (C ) Mx = Nλ x, ∀x ∈ ∂ ∩ dom M, λ ∈ (, ), (C ) deg{JQN, ∩ Ker M, } = , then the abstract equation Mx = Nx has at least one solution in dom M ∩ , where N = N , J : Im Q → Ker M is a homeomorphism with J(θ ) = θ .

3 Main results In the following, we will always suppose that q satisfies /p + /q = .  +∞ Let R+ = [, +∞), J  = R+ \{t , t , . . . , tk }, Y = L(R+ ) with norm y  =  |y(t)| dt, 

 PC  R+ = u : u ∈ C  J  , u (ti – ), u (ti + ) exist and  u (ti – ) = u (ti ), i = , , . . . , k ,

  |u(t)| < +∞, lim u (t) exists X = u : u() = , u ∈ C R+ ∩ PC  R+ , sup t→+∞ t∈R+  + t u with norm u = max{ +t ∞ , u ∞ }, where u ∞ = supt∈R+ |u(t)|. k Let Z = Y × R , with norm (y, c , c , . . . , ck ) = max{ y  , |c |, |c |, . . . , |ck |}. Then (X, · ) and (Z, · ) are Banach spaces. Define the operators M : X ∩ dom M → Z, Nλ : X → Z as follows:

⎡

(ϕp (u )) (t) ⎢ ϕ (u (t )) ⎢ p  Mu = ⎢ ⎣ ··· ϕp (u (tk ))

⎤ ⎥ ⎥ ⎥, ⎦

⎡ –λf (t, u(t), u (t)) ⎢ λI (u(t ), u (t )) ⎢    Nλ u = ⎢ ⎣ ··· λIk (u(tk ), u (tk ))

⎤ ⎥ ⎥ ⎥, ⎦

 +∞ where dom M = {u ∈ X : (ϕp (u )) ∈ Y , ϕp (u (+∞)) =  h(t)ϕp (u (t)) dt}. It is clear that u ∈ dom M is a solution of the problem (.) if it satisfies Mu = Nu, where  a N = N . For convenience, let (a, b)T := b , denote J = [, t ], Ji = (ti , ti+ ], i = , , . . . , k – , Jk = (tk , +∞). Lemma . M is a quasi-linear operator. Proof It is easy to get Ker M = {at | a ∈ R} := X . For u ∈ X ∩ dom M, if Mu = (y, c , c , . . . , ck )T , then     ϕp u (t) = y(t),

 ϕp u (ti ) = ci ,

i = , , . . . , k.

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For t ∈ J , we get 

 ϕp u (t) =

t

y(s) ds + a. 

For t ∈ J , considering ϕp (u (t )) = c , we get 

 ϕp u (t) =

t

y(s) ds + a + c . 

For t ∈ Ji , i = , , . . . , k, considering ϕp (u (ti )) = ci , we get 

 ϕp u (t) =

t

y(s) ds + a +





 +∞

By ϕp (u (+∞)) = isfies 



+∞



tk

y(s) ds +

h(t)

 +∞

h(t)ϕp (u (t)) dt and





+∞

ci .

ti


t





h(t) dt = , we find that (y, c , c , . . . , ck )T sat-

ci h(t) dt = .

(.)

ti ≥t

On the other hand, if (y, c , c , . . . , ck )T satisfies (.), take  u(t) =



t

s

ϕq 

  y(r) dr + ci ds.



ti
By a simple calculation, we get u ∈ X ∩ dom M and Mu = (y, c , c , . . . , ck )T . Thus

 Im M = (y, c , c , . . . , ck )T | y ∈ Y , c , c , . . . , ck satisfies (.) . Obviously, Im M ⊂ Z is closed. So, M is quasi-linear. The proof is completed.



Take projectors P : X → X and Q : Z → Z as follows: (Pu)(t) = u (+∞)t,   +∞ 

T

Q(y, c , c , . . . , ck ) =

h(t)

 +∞ t

t  T y(s) ds dt +  k ti ≥t ci h(t) dt –t  +∞ e , , . . . ,  , h(t)e–t dt 

where Z = {(ce–t , , . . . , )T | c ∈ R}. Obviously, QZ = Z and dim Z = dim X . Define an operator R : X × [, ] → X as 



t

R(u, λ)(t) =

+∞

ϕq 

s

  λ f r, u(r), u (r)

t   f (s, u(s), u (s)) ds dt –  k ti ≥t Ii (u(ti ), u (ti ))h(t) dt –r  +∞ dr e – h(t)e–t dt      + ϕp u (+∞) – λ Ij u(tj ), u (tj ) ds – u (+∞)t, t ∈ Ji , i = , , . . . , k,  +∞ 

h(t)

 +∞ t

tj ≥s

where X ⊕ X = X.

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By [, ], we get the following lemma. : u ∈ V } and {u (t) : u ∈ Lemma . Assume that V ⊂ X is bounded. V is compact if { u(t) +t V } are both equicontinuous on Ji , i = , , . . . , k – , and JT = (tk , T], for any given T > tk , respectively, and equiconvergent at infinity. Lemma . R : × [, ] → X is continuous and compact, where ⊂ X is an open bounded set. Proof By (H ), (H ), the continuity of ϕq and Lebesgue’s dominated convergence theorem, we find that R is continuous and {R(u, λ) | u ∈ , λ ∈ [, ]} is bounded. We will prove that R( × [, ]) is compact. Since ⊂ X is bounded, there exists a constant r >  such that u ≤ r, u ∈ . It follows from (H ) that there exist a function hr ∈ L(R+ ) and a constant Mr >  such that |f (t, u(t), u (t))| ≤ hr (t), |Ii (u(ti ), u (ti ))| ≤ Mr , i = , , . . . , k, t ∈ R+ , u ∈ . For any given T > tk , x , x ∈ Ji , i = , , . . . , k – , T, x < x , we have    R(u, λ)(x ) R(u, λ)(x )    –  +x  + x     x  +∞      ϕq λ f x, u(x), u (x) ≤  + x  s  +∞ t   +∞  h(t) t f (s, u(s), u (s)) ds dt –  k ti ≥t Ii (u(ti ), u (ti ))h(t) dt –x   +∞ dx e – h(t)e–t dt      + ϕp u (+∞) – λ Ij u(tj ), u (tj ) ds tj ≥s

+∞    ϕq λ f x, u(x), u (x)  + x  s  +∞ t   +∞  h(t) t f (s, u(s), u (s)) ds dt –  k ti ≥t Ii (u(ti ), u (ti ))h(t) dt –x   e dx – +∞ h(t)e–t dt          x    x       + ϕp u (+∞) – λ Ij u(tj ), u (tj ) ds +  –  u (+∞)  + x  + x   t ≥s



x



–

j

 +∞     +∞      hr (t) dt + ki= |Ii (u(ti ), u (ti ))| –x   Tϕq  – (x) + h dx e ≤  r +∞  + x  + x  h(t)e–t dt    + ϕp (r) + kMr  +∞

  hr (t) dt + ki= |Ii (u(ti ), u (ti ))| –x  +∞ hr (x) + e dx h(t)e–t dt       x x   r + ϕp (r) + kMr +  –  + x  + x           hr  + kMr  + ϕ ≤  – – x ) ϕ + (r) + kM h T + (x   q r  p r +∞  + x  + x  h(t)e–t dt     x x  – r. +   + x  + x  x – x ϕq +  + x



+∞ 



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 t Since t, +t , and +t are equicontinuous on Ji , i = , , . . . , k – , T, we find that { R(u,λ)(t) ,u ∈ +t , λ ∈ [, ]} are equicontinuous on Ji , i = , , . . . , k – , T. We have

  R(u, λ) (x ) – R(u, λ) (x )   +∞    = ϕq λ f s, u(s), u (s) x

 +∞

t   f (s, u(s), u (s)) ds dt –  k ti ≥t Ii (u(ti ), u (ti ))h(t) dt –s  +∞ – ds e h(t)e–t dt      + ϕp u (+∞) – λ Ij u(tj ), u (tj ) 

h(t)

 – ϕq

 +∞ t

tj ≥x

+∞   λ f s, u(s), u (s)

x

t   f (s, u(s), u (s)) ds dt –  k ti ≥t Ii (u(ti ), u (ti ))h(t) dt –s  +∞ ds e – h(t)e–t dt       + ϕp u (+∞) – λ Ij u(tj ), u (tj ) .  +∞ 

h(t)

 +∞ t

tj ≥x

For u ∈ , λ ∈ [, ], define 

+∞

F(u, λ)(t) = t

  λ f s, u(s), u (s)

t   f (s, u(s), u (s)) ds dt –  k ti ≥t Ii (u(ti ), u (ti ))h(t) dt –s  +∞ – ds e h(t)e–t dt     + ϕp u (+∞) – λ Ij u(tj ), u (tj ) .  +∞ 

h(t)

 +∞ t

tj ≥t

Obviously,   r  + kMr F(u, λ)(t) ≤ hr  +  h + ϕp (r) + kMr := K, +∞ h(t)e–t dt    F(u, λ)(x ) – F(u, λ)(x )   x    =  λ f s, u(s), u (s) x

 +∞ 

–  ≤

x x

h(t)

 +∞ t

u ∈ , λ ∈ [, ], t ∈ R+ ,

t     f (s, u(s), u (s)) ds dt –  k ti ≥t Ii (u(ti ), u (ti ))h(t) dt –s  +∞ e ds –t h(t)e dt 

hr  + kMr  –x e – e–x , hr (t) dt +  +∞ –t dt h(t)e 

u ∈ , λ ∈ [, ].

It follows from the absolute continuity of integral and the equicontinuity of e–t that {F(u, λ)(t), u ∈ , λ ∈ [, ]} are equicontinuous on Ji , i = , , . . . , k – , T. By the uniform continuity of ϕq (t) in [–K, K], we find that {R(u, λ) (t), u ∈ , λ ∈ [, ]} are equicontinuous on Ji , i = , , . . . , k – , T.

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For any u ∈ , λ ∈ [, ], since    

  λ f s, u(s), u (s)

+∞ t

 +∞ – 



+∞

≤ t

h(t)

 +∞ t

t     f (s, u(s), u (s)) ds dt –  k ti ≥t Ii (u(ti ), u (ti ))h(t) dt –s  +∞ ds e –t h(t)e dt 

hr  + kMr –s e ds →  hr (s) +  +∞ h(t)e–t dt 

(t → ∞)

and ϕq (u) is uniform continuous on [–K – ϕp (r), K + ϕp (r)], for any ε > , there exists a constant T > tk such that    ϕq 

s

+∞



 λ f r, u(r), u (r)

 +∞

t   f (s, u(s), u (s)) ds dt –  k ti ≥t Ii (u(ti ), u (ti ))h(t) dt –r  +∞ – dr e h(t)e–t dt     ε    + ϕp u (+∞) – u (+∞) ≤ , s > T , u ∈ , λ ∈ [, ].  

h(t)

 +∞ t

Obviously, there exists a constant T > T such that, for any t > T,   ε ϕq (K) + r T < . +t  Thus, for any x , x > T, we have    R(u, λ)(x ) R(u, λ)(x )    –  +x  + x     x  +∞     ϕq λ f r, u(r), u (r) =   + x  s  +∞ t   +∞  h(t) t f (s, u(s), u (s)) ds dt –  k ti ≥t Ii (u(ti ), u (ti ))h(t) dt –r   +∞ – e dr h(t)e–t dt 

       + ϕp u (+∞) – λ Ij u(tj ), u (tj ) ds – u (+∞)x 

tj ≥s



x +∞    ϕq λ f r, u(r), u (r) –  + x  s  +∞ t   +∞  h(t) t f (s, u(s), u (s)) ds dt –  k ti ≥t Ii (u(ti ), u (ti ))h(t) dt –r   +∞ dr e – h(t)e–t dt  

     + ϕp u (+∞) – λ Ij u(tj ), u (tj ) ds – u (+∞)x  tj ≥s

  T  +∞     ≤  ϕq λ f r, u(r), u (r)  + x  s  +∞ t   +∞  h(t) t f (s, u(s), u (s)) ds dt –  k ti ≥t Ii (u(ti ), u (ti ))h(t) dt –r   +∞ – e dr h(t)e–t dt 

Jiang Boundary Value Problems (2015) 2015:39

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        + ϕp u (+∞) – λ Ij u(tj ), u (tj ) ds – u (+∞)T       + +x 

tj ≥s



x

+∞

ϕq T

s

t   f (s, u(s), u (s)) ds dt –  k ti ≥t Ii (u(ti ), u (ti ))h(t) dt –r  +∞ dr e – h(t)e–t dt  

   + ϕp u (+∞) ds – u (+∞)(x – T )    T  +∞     +  ϕq λ f r, u(r), u (r)  + x  s  +∞ t   +∞  h(t) t f (s, u(s), u (s)) ds dt –  k ti ≥t Ii (u(ti ), u (ti ))h(t) dt –r   +∞ e dr – h(t)e–t dt 

         + ϕp u (+∞) – λ Ij u(tj ), u (tj ) ds – u (+∞)T   +∞ 

h(t)

 +∞

  λ f r, u(r), u (r)

t

     + +x 

tj ≥s



x

+∞

ϕq

T

s

t   f (s, u(s), u (s)) ds dt –  k ti ≥t Ii (u(ti ), u (ti ))h(t) dt –r  +∞ – dr e h(t)e–t dt  

   + ϕp u (+∞) ds – u (+∞)(x – T )   +∞ 

h(t)

 +∞

  λ f r, u(r), u (r)

t

  x – T ε x – T ε   ϕq (K) + r T + ϕq (K) + r T + + < ε,  + x  + x   + x  + x    R(u, λ) (x ) – R(u, λ) (x )   +∞    λ f s, u(s), u (s) ≤ ϕq ≤

x

t   f (s, u(s), u (s)) ds dt –  k ti ≥t Ii (u(ti ), u (ti ))h(t) dt –s  +∞ – ds e h(t)e–t dt     +∞       λ f s, u(s), u (s) + ϕp u (+∞) – u (+∞) + ϕq  +∞ 

h(t)

 +∞ t



h(t)

 +∞

x

t   f (s, u(s), u (s)) ds dt –  k ti ≥t Ii (u(ti ), u (ti ))h(t) dt –s  +∞ – e ds h(t)e–t dt     ε ε    + ϕp u (+∞) – u (+∞) < + < ε.    +∞

t

By Lemma ., we find that {R(u, λ) | u ∈ , λ ∈ [, ]} is compact. The proof is completed.  Lemma . Assume that ⊂ X is an open bounded set. Then Nλ is M-compact in . Proof By (H ), we get Nλ : → Y , λ ∈ [, ] is continuous. It is clear that Im P = Ker M, QNλ x = θ , λ ∈ (, ) ⇔ QNx = θ , i.e. Definition .(b) holds.

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For u ∈ , it follows from Q(I – Q)Nλ u = θ that (I – Q)Nλ u satisfies (.). So, (I – Q)Nλ u ∈ Im M, i.e. (I –Q)Nλ ( ) ⊂ Im M. Furthermore, by Im M = Ker Q and z = Qz +(I –Q)z we find that z ∈ Im M implies z = (I – Q)z ∈ (I – Q)Z, i.e. Im M ⊂ (I – Q)Z. Thus, (I – Q)Nλ ( ) ⊂ Im M ⊂ (I – Q)Z, i.e. Definition .(a) holds. Obviously, R(·, ) = . For u ∈  λ = {u ∈ ∩ dom M : Mu = Nλ u}, we get QNλ u = θ and  ϕp u (t) =



+∞

    λf s, u(s), u (s) ds + ϕp u (+∞) – λ Ii u(ti ), u (ti ) .

t

ti ≥t

So, we have 

t

R(u, λ) =

  ϕq ϕp u (s) ds – u (+∞)t = (I – P)u,



i.e. Definition .(c) holds. For u ∈ , λ ∈ [, ], t ∈ Ji , i = , , , . . . , k, we have     ϕp Pu + R(u, λ) (t)  = –λf t, u(t), u (t)  +∞ t   +∞ h(t) t –λf (s, u(s), u (s)) ds dt + λ  k ti ≥t Ii (u(ti ), u (ti ))h(t) dt –t   +∞ e – h(t)e–t dt  and   ϕp Pu + R(u, λ) (t)  +∞   = λ f r, u(r), u (r) t

t   f (s, u(s), u (s)) ds dt –  k ti ≥t Ii (u(ti ), u (ti ))h(t) dt –r  e dr – +∞ h(t)e–t dt     + ϕp u (+∞) – λ Ij u(tj ), u (tj ) .  +∞ 

h(t)

 +∞ t

tj ≥t

By a simple calculation, we can get   M Pu + R(u, λ) = (I – Q)Nλ u. So, Definition .(d) holds. These, together with Lemma ., mean that Nλ is M-compact  in . The proof is completed. Theorem . Assume that (H ), (H ), and the following conditions hold: (H ) There exist nonnegative functions a(t), b(t), c(t), and nonnegative constants di , gi , ei ,  i = , , . . . , k with ( + t)p– a(t), b(t), c(t) ∈ Y , and a(t)( + t)p–  + b  + ki= [di ( + ti )p– + gi ] <  such that       f (t, x, y) ≤ a(t)ϕp (x) + b(t)ϕp (y) + c(t), a.e. t ∈ [, +∞), x, y ∈ R,       Ii (x, y) ≤ di ϕp (x) + gi ϕp (y) + ei , i = , , . . . , k, x, y ∈ R.

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(H ) There exists a constant e >  such that if inft∈R+ |u (t)| > e , then one of the following inequalities holds: 





+∞

() u (t)

+∞

      f s, u(s), u (s) ds – Ii u(ti ), u (ti ) dt > ;

+∞

      f s, u(s), u (s) ds – Ii u(ti ), u (ti ) dt < ,

h(t) 



() u (t)

t



+∞

ti ≥t

h(t) 

t

ti ≥t

where t ∈ [, +∞). Then boundary value problem (.) has at least one solution. In order to prove Theorem ., we show two lemmas. Lemma . Suppose that (H )-(H ) hold. Then the set

  = u ∈ dom M | Mu = Nλ u, λ ∈ (, ) is bounded in X. Proof For u ∈  , we have QNλ u = , i.e. 



+∞



+∞









t



+∞

=

tk

f s, u(s), u (s) ds dt –

h(t) 



+∞

h(t) 

  Ii u(ti ), u (ti ) h(t) dt ti ≥t

    f s, u(s), u (s) ds – Ii u(ti ), u (ti ) dt = .

t

ti ≥t

By (H ), there exists a constant t ∈ R+ such that |u (t )| ≤ e . Assume t ∈ Jm , m = , , . . . , k. It follows from Mu = Nλ u that ⎧ t m     ⎪ ⎪ t λf (s, u(s), u (s)) ds + ϕp (u (t )) – λ j=i+ Ij (u(tj ), u (tj )), ⎪ ⎪ ⎪ ⎪ t ∈ Ji , i = , , . . . , m – ,   ⎨  t ϕp u (t) = t λf (s, u(s), u (s)) ds + ϕp (u (t )), t ∈ Jm , ⎪  t i ⎪    ⎪ ⎪ ⎪ t λf (s, u(s), u (s)) ds + ϕp (u (t )) + λ j=m+ Ij (u(tj ), u (tj )), ⎪ ⎩ t ∈ Ji , i = m + , m + , . . . , k. Since u(t) =

t 

(.)

u (s) ds,

 |u(t)|  ≤ u ∞ , +t

t ∈ [, +∞).

By (.), (H ), and (.), we obtain     ϕp u (t)  ≤



+∞ 

       a(t)ϕp u(t)  + b(t)ϕp u (t)  + c(t) dt + ϕp (e )



+ 

k         di ϕp u(ti )  + gi ϕp u (ti )  + ei i=

   k  u      di ( + ti )p– ϕp  ≤ a(t)( + t)p–  +  + t  ∞ i=

(.)

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 + b  +

k 

k    gi ϕp u ∞ + c  + ϕp (e ) + ei

i=



i= k 

  ≤ a(t)( + t)p–  + b  +



di ( + ti )p– + gi



  ϕp u ∞

i=

+ c  + ϕp (e ) +

k 

ei .

i=

Thus   u  ≤ ϕq ∞



  c  + ϕp (e ) + ki= ei .   – ( a(t)( + t)p–  + b  + ki= [di ( + ti )p– + gi ]) 

This, together with (.), means that  is bounded in X. Lemma . Assume that (H ), (H ), and (H ) hold. Then  = {u ∈ Ker M | QNu = } is bounded in X, where N = N . Proof For u ∈  , we have u = at, a ∈ R, and Q(Nu) = , i.e. 



+∞



+∞









t



+∞

=

tk

f (s, as, a) ds dt –

h(t)

+∞

f (s, as, a) ds –

h(t) 

t

Ii (ati , a)h(t) dt

ti ≥t



 Ii (ati , a) dt = .

ti ≥t

By (H ), we get u = |a| = |u (t)| ≤ e . So,  is bounded. The proof is completed.



Proof of Theorem . Let = {u ∈ X | u < r}, where r > e is large enough such that ⊃  ∪  . By Lemmas . and ., we have Mu = Nλ u, u ∈ dom M ∩ ∂ , and QNu = , u ∈ Ker M ∩ ∂ . Let H(u, δ) = ρδu + ( – δ)JQNu, δ ∈ [, ], u ∈ Ker M ∩ , where J : Im Q → Ker M is a !–, if (H )() holds, homeomorphism with J(ae–t , , . . . , )T = at, ρ = , if (H )() holds.  For u ∈ Ker M ∩ ∂ , we have u = at = . Thus  +∞ H(u, δ) = ρδat – ( – δ)



h(t)

 +∞ t

t  f (s, as, a) ds dt –  k ti ≥t Ii (ati , a)h(t) dt  +∞ t. h(t)e–t dt 

If δ = , H(u, ) = ρat = . If δ = , by QNu = , we get H(u, ) = JQN(at) = . For  < δ < , we now prove that H(u, δ) = . Otherwise, if H(u, δ) = , then 



+∞





+∞

f (s, as, a) ds dt –

h(t) t



tk

 ti ≥t

Ii (ati , a)h(t) dt =

ρδa –δ



+∞

h(t)e–t dt. 

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Thus 



+∞



+∞

f (s, as, a) ds –

h(t)

a

t

  ρδa +∞ Ii (ati , a) dt = h(t)e–t dt.  – δ  t ≥t

 i

Since |u (t)| = |a| = u = r > e , this is a contradiction with (H ) and the definition of ρ. So, H(u, δ) = , u ∈ Ker M ∩ ∂ , δ ∈ [, ]. By the homotopy of degree, we get " # " # deg JQN, ∩ Ker M,  = deg H(·, ), ∩ Ker M,  " # = deg H(·, ), ∩ Ker M,  " # = deg ρI, ∩ Ker M,  = . By Theorem ., we can find that Mu = Nu has at least one solution in . The proof is completed. 

4 Example Let us consider the following impulsive p-Laplacian boundary value problems at resonance on the half-line ⎧    ⎪ ⎨(ϕp (u (t))) + f (t, u(t), u (t)) = , t ∈ [, ∞)\{t , t , . . . , tk }, ϕp (u (ti )) = ci , i = , , . . . , k, ⎪  +∞ ⎩ u() = , ϕp (u (+∞)) =  e–t ϕp (u (t)) dt, where  < t < t < · · · < tk < +∞, p =  , f (t, x, y) =

e–t  (+t) 

√ 

(.)

√ sin x + e–t  y + e–t .

Corresponding to the problem (.), we have h(t) = e–t , Ii (u, v) = ci , i = , , . . . , k. Take   –t hr (t) = (( + t)–  + r  + )e–t , a(t) = e  , b(t) = c(t) = e–t , di = gi = , ei = ci , i = , , . . . , k, (+t)   e = e(+tk ) ( +  ki= |ci |) , Mr = max≤i≤k {|ci |}. By a simple calculation, we find that (H )-(H ) and (H )() hold. By Theorem ., we find that the problem (.) has at least one solution. Competing interests The author declares that she has no competing interests. Acknowledgements This work is supported by the Natural Science Foundation of China (11171088) and the Natural Science Foundation of Hebei Province (A2013208108). The author is grateful to anonymous referees for their constructive comments and suggestions, which led to improvement of the original manuscript. Received: 1 October 2014 Accepted: 28 January 2015 References 1. Agarwal, RP, O’Regan, D: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht (2001) 2. Benchohra, M, Henderson, J, Ntouyas, S: Impulsive Equations and Inclusions. Contemporary Mathematics and Its Applications, vol. 2. Hindawi Publishing Corporation, New York (2006) 3. Lakshmikantham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differential Equations. Series in Modern Applied Mathematics, vol. 6. World Scientific, Hackensack (1994) 4. Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations. World Scientific, Singapore (1995) 5. Zhang, X, Shuai, Z, Wang, K: Optimal impulsive harvesting policy for single population. Nonlinear Anal., Real World Appl. 4, 639-651 (2003)

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