Guo Boundary Value Problems (2016) 2016:70 DOI 10.1186/s13661-016-0577-8

RESEARCH

Open Access

Existence of two positive solutions for a class of third-order impulsive singular integro-differential equations on the half-line in Banach spaces Dajun Guo* *

Correspondence: [email protected] Department of Mathematics, Shandong University, Jinan, Shandong 250100, People’s Republic of China

Abstract In this paper, we discuss the existence of two positive solutions for an infinite boundary value problem of third-order impulsive singular integro-differential equations on the half-line in Banach spaces by means of the fixed point theorem of cone expansion and compression with norm type. MSC: 45J05; 47H10 Keywords: impulsive singular integro-differential equation in a Banach space; infinite boundary value problem; fixed point theorem of cone expansion and compression with norm type

1 Introduction The theory of impulsive differential equations has been emerging as an important area of investigation in recent years (see [–]). In papers [] and [], we have discussed two infinite boundary value problems for nth-order impulsive nonlinear singular integrodifferential equations of mixed type on the half-line in Banach spaces. By constructing a bounded closed convex set, apart from the singularities, and using the Schauder fixed point theorem, we obtain the existence of positive solutions for the infinite boundary value problems. But such equations are sublinear, and there are no results on existence of two positive solutions. In a recent paper [], we discussed the existence of two positive solutions for a class of second order superlinear singular equations by means of different method, that is, by using the fixed point theorem of cone expansion and compression with norm type, which was established by the author in [] (see also [–]). Now, in this paper, we extend the results of [] to third-order equations in Banach spaces. The difficulty of this extension appears in two sides: we must introduce a new cone such that we can still use the fixed point theorem of cone expansion and compression with norm type, and, on the other hand, we need to introduce a suitable condition to guarantee the compactness of the corresponding operator. In addition, the construction of an example to show the application of our theorem to an infinite system of scalar equations is also difficult. © 2016 Guo. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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Let E be a real Banach space, and P be a cone in E that defines a partial ordering in E by x ≤ y if and only if y – x ∈ P. P is said to be normal if there exists a positive constant N such that θ ≤ x ≤ y implies x ≤ Ny, where θ denotes the zero element of E, and the smallest N is called the normal constant of P. If x ≤ y and x = y, then we write x < y. Let P+ = P\{θ }, that is, P+ = {x ∈ P : x > θ }. For details on cone theory, see []. Consider the infinite boundary value problem (IBVP) for third-order impulsive singular integro-differential equation of mixed type on the half-line in E: ⎧ ⎪ u (t) = f (t, u(t), u (t), u (t), (Tu)(t), (Su)(t)), ∀t ∈ J+ , ⎪ ⎪ ⎪  – ⎪ ⎪ ⎨u|t=tk = Ik (u (tk )) (k = , , , . . .), u |t=tk = I¯k (u (tk– )) (k = , , , . . .), ⎪ ⎪ ⎪u |t=t = I˜k (u (t – )) (k = , , , . . .), ⎪ k k ⎪ ⎪ ⎩u() = θ , u (∞) = βu (), u () = θ ,

()

where J = [, ∞), J+ = (, ∞),  < t < · · · < tk < · · · , tk → ∞, J+ = J+ \{t , . . . , tk , . . .}, f ∈ C[J+ × P+ × P+ × P+ × P × P, P], Ik , I¯k , I˜k ∈ C[P+ , P] (k = , , , . . .), β > , u (∞) = limt→∞ u (t), and  (Tu)(t) =



t

K(t, s)u(s) ds,

(Su)(t) =



∞

()

H(t, s)u(s) ds, 

K ∈ C[D, J], D = {(t, s) ∈ J × J : t ≥ s}, H ∈ C[J × J, J]. u|t=tk , u |t=tk , and u |t=tk denote the jumps of u(t), u (t), and u (t) at t = tk , respectively, that is,     u|t=tk = u tk+ – u tk– ,

    u |t=tk = u tk+ – u tk– ,

    u |t=tk = u tk+ – u tk– ,

where u(tk+ ) and u(tk– ) represent the right and left limits of u(t) at t = tk , respectively, and u (tk+ ) (u (tk+ )) and u (tk– ) (u (tk– )) represent the right and left limits of u (t) (u (t)) at t = tk , respectively. In the following, we always assume that lim f (t, u, v, w, y, z) = ∞,

t→+

∀u, v, w ∈ P+ , y, z ∈ P,

()

lim

f (t, u, v, w, y, z) = ∞,

∀t ∈ J+ , v, w ∈ P+ , y, z ∈ P,

()

lim

f (t, u, v, w, y, z) = ∞,

∀t ∈ J+ , u, w ∈ P+ , y, z ∈ P,

()

∀t ∈ J+ , u, v ∈ P+ , y, z ∈ P,

()

u∈P+ ,u→

v∈P+ ,v→

and lim

f (t, u, v, w, y, z) = ∞,

w∈P+ ,w→

that is, f (t, u, v, w, y, z) is singular at t = , u = θ , v = θ , and w = θ . We also assume that lim

Ik (w) = ∞ (k = , , , . . .),

()

lim

¯Ik (w) = ∞ (k = , , , . . .),

()

w∈P+ ,w→

w∈P+ ,w→

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and lim

˜Ik (w) = ∞ (k = , , , . . .),

w∈P+ ,w→

()

that is, Ik (w), I¯k (w), and I˜k (w) (k = , , , . . .) are singular at w = θ . Let PC[J, E] = {u : u is a map from J into E such that u(t) is continuous at t = tk , left-continuous at t = tk , and u(tk+ ) exists, k = , , , . . .} and PC  [J, E] = {u ∈ PC[J, E] : u (t) is continuous at t = tk , and u (tk+ ) and u (tk– ) exist for k = , , , . . .}. Let u ∈ PC  [J, E]. For  < h < tk – tk– , by the mean value theorem ([], Theorem ..) we have

 u(tk ) – u(tk – h) ∈ hco u (t) : tk – h < t < tk ;

()

hence, it is easy to see that the left derivative of u(t) at t = tk , which is denoted by u– (tk ), exists, and u– (tk ) = lim+ h→

  u(tk ) – u(tk – h) = u tk– . h

()

In what follows, it is understood that u (tk ) = u– (tk ). So, for u ∈ PC  [J, E], we have u ∈ PC[J, E]. Let PC  [J, E] = {u ∈ PC[J, E] : u (t) is continuous at t = tk , and u (tk+ ) and u (tk– ) exist for k = , , , . . .}. For u ∈ PC  [J, E], we have u (tk – h) = u (t) +



tk –h

u (s) ds,

∀tk– < t < tk – h < tk (h > ),

t

so, observing the existence of u (tk– ) and taking limits as h → + in this equality, we see that u (tk– ) exists and   u tk– = u (t) +



tk

u (s) ds,

∀tk– < t < tk .

t

Similarly, we can show that u (tk+ ) exists. Hence, u ∈ PC  [J, E]. Consequently, PC  [J, E] ⊂ PC  [J, E]. For u ∈ PC  [J, E], by using u (t) and u (t) instead of u(t) and u (t) in () and () we get the conclusion: the left derivative of u (t) at t = tk , which is denoted by u– (tk ), exists, and u– (tk ) = u (tk– ). In what follows, it is understood that u (tk ) = u– (tk ). Hence, for u ∈ PC  [J, E], we have u ∈ PC  [J, E] and u ∈ PC[J, E]. A map u ∈ PC  [J, E] ∩ C  [J+ , E] is called a positive solution of IBVP () if u(t) > θ for t ∈ J+ and u(t) satisfies (). Now, we need to introduce a new space DPC  [J, E] and a new cone Q in it. Let

 u(t) u (t)  < ∞, sup u (t) < ∞ . < ∞, sup DPC [J, E] = u ∈ PC [J, E] : sup t t t∈J+ t∈J+ t∈J 

It is easy to see that DPC  [J, E] is a Banach space with norm 

uD = max uS , u T , u B ,

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where uS = sup t∈J+

u(t) , t

  u = sup u (t) , T t t∈J+

 u = sup u (t) . B t∈J

Let W = {u ∈ DPC  [J, E] : u(t) ≥ θ , u (t) ≥ θ , u (t) ≥ θ , ∀t ∈ J} and u (t) u(t) u(s) u (s) ≥ β – , ∀t, s ∈ J+ ; Q = u ∈ W :  ≥ (β – )–  , ∀t, s ∈ J+ ; t s t s

u (t) ≥ β – u (s), ∀t, s ∈ J . Obviously, W and Q are two cones in the space DPC  [J, E], and Q ⊂ W . Let Q+ = {u ∈ Q : uD > } and Qpq = {u ∈ Q : p ≤ uD ≤ q} for q > p > .

2 Several lemmas In the following, we always assume that the cone P is normal with normal constant N . Remark  For u ∈ DPC  [J, E], we have u() = θ and u () = θ . This is clear since u() = θ implies sup t∈J+

u(t) = ∞, t

and u () = θ implies sup t∈J+

u (t) = ∞. t

Lemma  For u ∈ Q, we have u(t)  – – ≤ uD , ∀t ∈ J+ , N β (β – )– uD ≤  t u (t) ≤ uD , ∀t ∈ J+ , N – β – β  uD ≤ t

() ()

and N – β – β  uD ≤ u (t) ≤ uD ,

∀t ∈ J,

()

where

 β  = min (β – )– ,  .

()

Proof The method of the proof is similar to that of Lemma  in [], but it is more complicate. For u ∈ Q, we need to establish six inequalities:  uS ≥ N – β – u B , 

 u ≥ N – β – (β – )– uS , uS ≥  N – β – u , B T 

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 u ≥ N – β – u , T B

 u ≥ N – β – (β – )– uS , T

 u ≥ N – β – u . B T

For example, we establish the first one. By Remark , u() = θ and u () = θ , so u(t ) – = t t



t



u 

 (s) ds = t–



t

s

ds 

u (r) dr.



Since u (r) ≥ β – u (t),

∀r, t ∈ J,

we have u(t ) ≥ t– β – t



t 

  s ds u (t) = β – u (t), 

∀t ∈ J,

and hence  uS ≥ N – β – u B .  From these six inequalities it is easy to prove inequalities ()-(). For example, we prove (). We have u(t) u(s) ≥ N – (β – )–  , t s

∀t, s ∈ J+ ,

so, u(t) ≥ N – (β – )– uS , t

∀t ∈ J+ ,

and therefore u(t) ≥ t



 – – N β (β   – – N β (β 

– )– u B , – )– u T ,

∀t ∈ J+ , 

and hence, () holds. Corollary For u ∈ Q+ , we have u(t) > θ and u (t) > θ for t ∈ J+ and u (t) > θ for t ∈ J. This follows from ()-(). Let us list some conditions. t ∞ (H ) supt∈J  K(t, s)s ds < ∞, supt∈J  H(t, s)s ds < ∞, and  lim 

   H t  , s – H(t, s)s ds = ,

∞

t →t 

∀t ∈ J.

In this case, let k ∗ = sup t∈J



t

K(t, s)s ds, 

h∗ = sup t∈J



∞ 

H(t, s)s ds.

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(H ) There exist a, b ∈ C[J+ , J], g ∈ C[J+ × P+ , J], and G ∈ C[J+ × J × J, J] such that     f (t, u, v, w, y, z) ≤ a(t)g u, v + b(t)G w, y, z , ∀t ∈ J+ , u, v, w ∈ P+ , y, z ∈ P, and a∗p,q =



∞

a(t)gp,q (t) dt < ∞



for any q ≥ p > , where

 gp,q (t) = max g(s , s ) : N – β – (β – )– pt  ≤ s ≤ qt  , N – β – β  pt ≤ s ≤ qt ,  ∀t ∈ J+ , and 

b∗ =

∞

b(t) dt < ∞.



In this case, let (for q ≥ p > )

 Gp,q = max G(x , x , x ) : N – β – β  p ≤ x ≤ q,  ≤ x ≤ k ∗ q,  ≤ x ≤ h∗ q , where β  is defined by (). (H ) Ik (w) ≤ tk I¯k (w) and I¯k (w) ≤ tk I˜k (w), ∀w ∈ P+ (k = , , , . . .), and there exist γk ∈ J (k = , , , . . .) and F ∈ C[J+ , J] such that   ˜Ik (w) ≤ γk F w ,

∀w ∈ P+ (k = , , , . . .),

and γ˜ =

∞ 

tk γk < ∞,

k=

and, consequently, γ∗ =

∞ 

γk ≤ t– γ˜ < ∞,

k=

γ¯ =

∞ 

tk γk ≤ t– γ˜ < ∞.

k=

In this case, let (for q ≥ p > )

 Np,q = max F(s) : N – β – β  p ≤ s ≤ q . (H ) For any t ∈ J+ , r > p > , and q > , f (t, Ppr , Ppr , Ppr , Pq , Pq ) = {f (t, u, v, w, y, z) : u, v, w ∈ Ppr , y, z ∈ Pq }, Ik (Ppr ) = {Ik (w) : w ∈ Ppr }, I¯k (Ppr ) = {I¯k (w) : w ∈ Ppr }, and I˜k (Ppr ) = {I˜k (w) :

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w ∈ Ppr } (k = , , , . . .) are relatively compact in E, where Ppr = {w ∈ P : p ≤ w ≤ r} and Pq = {w ∈ P : w ≤ q}. (H ) There exist w ∈ P+ , c ∈ C[J+ , J], and τ ∈ C[P+ , J] such that f (t, u, v, w, y, z) ≥ c(t)τ (w)w ,

∀t ∈ J+ , u, v, w ∈ P+ , y, z ∈ P,

and τ (w) → ∞ as w ∈ P+ , w → ∞, w and c∗ =



∞

c(t) dt < ∞.



(H ) There exist w ∈ P+ , d ∈ C[J+ , J], and σ ∈ C[P+ , J] such that f (t, u, v, w, y, z) ≥ d(t)σ (w)w ,

∀t ∈ J+ , u, v, w ∈ P+ , y, z ∈ P,

and σ (w) → ∞ as w ∈ P+ , w → , and d∗ =



∞

d(t) dt < ∞.



Remark  It is clear: if condition (H ) is satisfied, then the operators T and S defined by () are bounded linear operators from DPC  [J, E] into BC[J, E] (the Banach space of all bounded continuous maps from J into E with norm uB = supt∈J u(t)), and T ≤ k ∗ , S ≤ h∗ ; moreover, we have T(DPC  [J, P]) ⊂ BC[J, P] and S(DPC  [J, P]) ⊂ BC[J, P], DPC  [J, P] = {u ∈ DPC  [J, E] : u(t) ≥ θ , ∀t ∈ J} and BP[J, P] = {u ∈ BP[J, E] : u(t) ≥ θ , ∀t ∈ J}. Remark  Condition (H ) means that the function f (t, u, v, w, y, z) is superlinear with respect to w. Remark  Condition (H ) means that the function f (t, u, v, w, y, z) is singular at w = θ , and it is stronger than (). Remark  If condition (H ) is satisfied, then () implies (), and () implies (). Remark  Condition (H ) is satisfied automatically when E is finite-dimensional. Remark  In what follows, we need the following three formulas (see [], Lemma ): (a) If u ∈ PC[J, E] ∩ C  [J+ , E], then  u(t) = u() + 

t

u (s) ds +

     u tk+ – u tk– , 
∀t ∈ J.

()

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(b) If u ∈ PC  [J, E] ∩ C  [J+ , E], then 



t

u(t) = u() + tu () +

(t – s)u (s) ds



+

           u tk+ – u tk– + (t – tk ) u tk+ – u tk– ,

∀t ∈ J.

()


(c) If u ∈ PC  [J, E] ∩ C  [J+ , E], then t  u(t) = u() + tu () + u () +   

 

t

      (t – s) u (s) ds + u tk+ – u tk–  


           + (t – tk ) u tk+ – u tk– + (t – tk ) u tk+ – u tk– , 

∀t ∈ J.

()

We shall reduce IBVP () to an impulsive integral equation. To this end, we first consider the operator A defined by   ∞ ∞      –   t   f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds + (Au)(t) = I˜k u tk (β – )  k=     t + (t – s) f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds          Ik u tk– + (t – tk )I¯k u tk– + 
    + (t – tk ) I˜k u tk– , 

∀t ∈ J.

()

In what follows, we write J = [, t ], Jk = (tk– , tk ] (k = , , , . . .). Lemma  If conditions (H )-(H ) are satisfied, then operator A defined by () is a continuous operator from Q+ into Q; moreover, for any q > p > , A(Qpq ) is relatively compact. Proof Let u ∈ Q+ and uD = r. Then r > , and, by ()-() and Remark ,  – – N β (β – )– rt  ≤ u(t) ≤ rt  , 

∀t ∈ J,

and N – β – β  rt ≤ u (t) ≤ rt,

N – β – β  r ≤ u (t) ≤ r,

∀t ∈ J,

so, conditions (H ) and (H ) imply (for k ∗ , h∗ , a(t), gp,q (t), a∗p,q , Gp,q , b(t), b∗ , see conditions (H ) and (H ))   f t, u(t), u (t), u (t), (Tu)(t), (Su)(t) ≤ a(t)gr,r (t) + Gr,r b(t),

∀t ∈ J+ ,

()

∗ ∗ where gr,r (t) and Gr,r are gp,q (t) and Gp,q for p = r and q = r, respectively. By () and con∞ dition (H ) we know that the infinite integral  f (t, u(t), u (t), u (t), (Tu)(t), (Su)(t)) dt is

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convergent and 

∞



  f t, u(t), u (t), u (t), (Tu)(t), (Su)(t) dt ≤ a∗ + Gr,r b∗ . r,r

()

On the other hand, by condition (H ) we have    –  ˜Ik u t ≤ Nr,r γk k

(k = , , , . . .),

()

where Nr,r is Np,q for p = r and q = r, which implies the convergence of the infinite series ∞ ˜  – k= Ik (u (t )) and k

∞     –  ˜Ik u t ≤ Nr,r γ ∗ . k

()

k=

From () we get  (Au)(t) ≥  t (β – )



∞ 

 ∞     –      , f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds + I˜k u tk k=

∀t ∈ J+ .

()

Moreover, by condition (H ) we have           Ik u tk– + (t – tk )I¯k u tk– + (t – tk ) I˜k u tk–     –      ≤ t I¯k u tk + (t – tk ) I˜k u tk–     –      ≤ ttk I˜k u tk + (t – tk ) I˜k u tk–          = t  + tk I˜k u tk– ≤ t  I˜k u tk– , ∀ < tk < t,  so, () gives  (Au)(t) ≤ t (β – )



∞

  f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds



  ∞     –    ∞  ˜ + f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds Ik u t k +   k=

+

∞     I˜k u tk– k=

 ≤ (β – )  + 

∞



∞ 

 ∞     –      f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds + I˜k u tk

  f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds +

k= ∞  k=

   I˜k u tk–

Guo Boundary Value Problems (2016) 2016:70

β –  = (β – ) +

∞ 

Page 10 of 31



∞

  f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds



    –  I˜k u tk ,

∀t ∈ J+ .

()

k=

On the other hand, by () we have   ∞ ∞      –   t   f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds + (Au) (t) = I˜k u tk β –  k=  t       + (t – s)f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds + I¯k u tk– 




   + (t – tk )I˜k u tk– ,

∀t ∈ J,

()

so, (Au) (t)  ≥ t β –



∞



 ∞     –  , f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds + I˜k u tk 







k=

∀t ∈ J+ ,

()

and, by condition (H ),  (Au) (t) ≤ t β – 

 

∞

+ 

β = β –

∞

∞     f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds + I˜k u tk–











k=

∞       f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds + I˜k u tk–



k=

∞



 ∞     –  ˜ , f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds + Ik u t k 







k=

∀t ∈ J+ .

()

In addition, () gives   ∞ ∞      –      (Au) (t) = f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds + I˜k u tk β –  k=  t       + f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds + I˜k u tk– , 




∀t ∈ J,

()

so,  (Au) (t) ≥ β – 

∀t ∈ J,

 

∞

 ∞     –  I˜k u tk , f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds + 







k=

()

Guo Boundary Value Problems (2016) 2016:70

Page 11 of 31

and  (Au) (t) ≤ β – 





∞



∞

+

∞     f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds + I˜k u tk–









k=

  f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds +



β = β –

 



∞ 

   I˜k u tk–

k=

∞

 ∞     , f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds + I˜k u tk– 



k=

∀t ∈ J.

()

It follows from (), (), (), (), and () that Au ∈ DPC  [J, E] and  N(β – )  ∗ a + Gr,r b∗ + Nr,r γ ∗ , (β – ) r,r   (Au) ≤ Nβ a∗ + Gr,r b∗ + Nr,r γ ∗ , r,r T β –   (Au) ≤ Nβ a∗ + Gr,r b∗ + Nr,r γ ∗ . r,r B β –

AuS ≤

() () ()

Moreover, (), (), (), (), (), and () imply (Au) (s) (Au) (t) ≥ β – , t s

(Au)(t) (Au)(s) ≥ (β – )– , t s

∀t, s ∈ J+ ,

and (Au) (t) ≥ β – (Au) (s),

∀t, s ∈ J,

and hence, Au ∈ Q. Thus, we have proved that A maps Q+ into Q. ¯ D→ Now, we are going to show that A is continuous. Let un , u¯ ∈ Q+ and un – u ¯ D = ¯r (¯r > ), and we may assume that r¯ ≤ un D ≤ ¯r (n = , , , . . .). (n → ∞). Write u So, ()-() imply  – – N β (β – )– r¯ t  ≤ un (t) ≤ ¯r t  ,  ∀t ∈ J (n = , , , . . .), N – β – β  r¯ t ≤ un (t) ≤ ¯rt,

 – – ¯ ≤ ¯rt  , N β (β – )– r¯ t  ≤ u(t)  ()

N – β – β  r¯ t ≤ u¯  (t) ≤ ¯rt,

∀t ∈ J (n = , , , . . .),

()

and N – β – β  r¯ ≤ un (t) ≤ ¯r, ∀t ∈ J (n = , , , . . .).

N – β – β  r¯ ≤ u¯  (t) ≤ ¯r, ()

Guo Boundary Value Problems (2016) 2016:70

Page 12 of 31

By () we have ¯ (Aun )(t) – (Au)(t) t  ∞    f s, un (s), u (s), u (s), (Tun )(s), (Sun )(s) ≤ n (β – ) 

 ∞     –     –      ˜Ik u t ds + ¯ ¯ ¯ u¯ (s), u¯ (s), (T u)(s), – I˜k u¯ tk (Su)(s) – f s, u(s), n k 

k= t

   f (s, un (s), u (s), u (s), Tun (s), (Sun )(s) n n     ds ¯ ¯ ¯ u¯  (s), u¯  (s), (T u)(s), (T u)(s) – f s, u(s), +

+

               Ik u t – – Ik u¯  t – +  ¯Ik u t – – I¯k u¯  t – n k k n k k t  
k

        ˜Ik u t – – I˜k u¯  t – , + n k k  
∀t ∈ J+ (n = , , , . . .).

()

k

When  < t ≤ t , we have        ¯Ik u t – – I¯k u¯  t – = , n k k

       Ik u t – – Ik u¯  t – = , n k k 

so, sup t∈J+

               Ik u t – – Ik u¯  t – = sup  Ik u t – – Ik u¯  t – n k k n k k  t  
k

≤

 t

∞ 

   –     Ik u t – Ik u¯  tk– n k

()

k=

and sup t∈J+

              ¯Ik u t – – I¯k u¯  t – = sup  ¯Ik u t – – I¯k u¯  t – n k k n k k t 
≤

∞         ¯Ik u t – – I¯k u¯  t – . n k k t k=

It follows from ()-() that ¯ S = sup Aun – Au t∈J+

∞       ¯   (Aun )(t) – (Au)(t) Ik u t – – Ik u¯  t – ≤ n k k   t t k=

∞         ¯Ik u t – – I¯k u¯  t – n k k t k=  ∞  β f s, un (s), u (s), u (s), + n n (β – ) 

+

()

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Page 13 of 31

   ds ¯ u¯  (s), u¯  (s), (T u)(s), ¯ ¯ (Tun )(s), (Sun )(s) – f s, u(s), (Su)(s)  ∞     –     –  ˜ ˜ – Ik u¯ t (n = , , , . . .). Ik u t + n

k

k

()

k=

It is clear that     ¯ u¯  (t), u¯  (t), (T u)(t), ¯ ¯ (Su)(t) f t, un (t), un (t), un (t), (Tun )(t), (Sun )(t) → f t, u(t), as n → ∞, ∀t ∈ J+ , and, similarly to () and observing ()-(), we have     f t, un (t), u (t), u (t), (Tun )(t), (Sun )(t) – f t, u(t), ¯ u¯  (t), u¯  (t), (T u)(t), ¯ ¯ (Su)(t) n n   ≤  a(t)gr¯,¯r (t) + Gr¯,¯r b(t) = σ (t), ∀t ∈ J+ (n = , , , . . .), and by condition (H ) we see that theorem implies  lim

∞ 

σ (t) dt < ∞. Hence, the dominated convergence

∞

n→∞ 

  f t, un (t), u (t), u (t), (Tun )(t), (Sun )(t) n n

  dt = . ¯ ¯ ¯ u¯  (t), u¯  (t), (T u)(t), (Su)(t) – f t, u(t),

()

On the other hand, similarly to () and observing (), we have    –  ˜Ik u¯ t ≤ Nr¯,¯r γk k

   –  ˜Ik u t ≤ Nr¯,¯r γk , n k

(k, n = , , , . . .).

()

By () and condition (H ), using a similar method in the proof of Lemma  of [], we can prove

n→∞

∞     –     Ik u t – Ik u¯  tk– = , n k

lim

∞     –     ˜Ik u t – I˜k u¯  tk– = . n k

lim

k=

lim

n→∞

∞     –     ¯Ik u t – I¯k u¯  tk– =  () n k k=

and

n→∞

()

k=

It follows from (), (), (), and () that ¯ S = . lim Aun – Au

()

n→∞

On the other hand, similarly to () and observing () and (), we easily get  ∞ ∞      –     –  β    (Aun ) – (Au) ¯Ik u t f s, un (s), ¯ T ≤ – I¯k u¯ tk + n k t β –  k=

un (s), un (s), (Tun )(s), (Sun )(s)



Guo Boundary Value Problems (2016) 2016:70

Page 14 of 31

  ds ¯ u¯  (s), u¯  (s), (T u)(s), ¯ ¯ – t s, u(s), (Su)(s)  ∞     –     –  ˜ ˜ – Ik u¯ t (n = , , , . . .), Ik u t + n

k

k

()

k=

and (Aun ) – (Au) ¯  B ≤

 ∞   β f s, un (s), u (s), u (s), (Tun )(s), (Sun )(s) n n β –    ds ¯ ¯ ¯ u¯  (s), u¯  (s), (T u)(s), (Su)(s) – f s, u(s),  ∞     –     –  ˜ – I˜k u¯ t + Ik u t . n

k

k

()

k=

So, (), () and (), (), () imply ¯  T =  lim (Aun ) – (Au)

()

¯  B = . lim (Aun ) – (Au)

()

n→∞

and

n→∞

¯ D →  as n → ∞, and the continuity It follows from (), (), and () that Aun – Au of A is proved. Finally, we prove that A(Qpq ) is relatively compact, where q > p >  are arbitrarily given. Let u¯ n ∈ Qpq (n = , , , . . .). Then, by ()-() and Remark , we have  – – N β (β – )– pt  ≤ u¯ n (t) ≤ qt  , N – β – β  pt ≤ u¯ n (t) ≤ qt,  N – β – β  p ≤ u¯  (t) ≤ q, ∀t ∈ J (n = , , , . . .).

()

n

Similarly to (), (), ()-() and observing (), we get   f t, u¯ n (t), u¯  (t), u¯  (t), (T u¯ n )(t), (Su¯ n )(t) ≤ a(t)gp,q (t) + Gp,q b(t), n n ∀t ∈ J+ (n = , , , . . .),    –  ˜Ik u¯ t ≤ Np,q γk (k, n = , , , . . .), n k

() ()

 N(β – )  ∗ ap,q + Gp,q b∗ + Np,q γ ∗ (n = , , , . . .), (β – )   (Au¯ n ) ≤ Nβ a∗ + Gp,q b∗ + Np,q γ ∗ (n = , , , . . .), p,q T β – Au¯ n S ≤

() ()

and   (Au¯ n ) ≤ Nβ a∗ + Gp,q b∗ + Np,q γ ∗ p,q B β –

(n = , , , . . .).

()

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From () we see that the functions {(Au¯ n )(t)} (n = , , , . . .) are uniformly bounded on [, r] for any r > . Consider Ji = (ti– , ti ] for any fixed i. By () we have   (Au¯ n ) t  – (Au¯ n )(t)   ∞ ∞     –    (t  ) – t    = f (s, u¯ n (s), u¯ n (s), u¯ n (s), (T u¯ n (s), Su¯ n (s) ds + I˜k u¯ n tk (β – )  k=      t    + t – s – (t – s) f s, u¯ n (s), u¯ n (s), u¯ n (s), (T u¯ n )(s), (Su¯ n )(s) ds       t     t – s f s, u¯ n (s), u¯ n (s), u¯ n (s), (T u¯ n )(s), (Su¯ n )(s) ds +  t

i–       –        –      ˜ ¯ , t – tk – (t – tk ) Ik u¯ tk t – t Ik u¯ n tk + +  k=

∀t, t  ∈ Ji , t  > t (n = , , , . . .).

()

Since 

t – s



        – (t – s) = t  – t t  + t – s ≤ t  – t t  + t = t  – t  ,

∀t, t  ∈ Ji , t  > t,  ≤ s ≤ t and, similarly, 

t  – tk



  – (t – tk ) ≤ t  – t  ,

∀t, t  ∈ Ji , t  > t (k = , , . . . , i – ),

() implies   θ ≤ (Au¯ n ) t  – (Au¯ n )(t)   ∞ ∞       –  (t  ) – t    ˜ ≤ f s, u¯ n (s), u¯ n (s), u¯ n (s), (T u¯ n )(s), (Su¯ n )(s) ds + Ik u¯ n tk (β – )  k=            ∞  t –t + t –t + f s, u¯ n (s), u¯ n (s), u¯ n (s), (T u¯ n )(s), (Su¯ n )(s) ds   ∞ ∞    –               ¯ t –t Ik u¯ n tk + I˜k u¯ n tk– , + t –t  k=



k=



∀t, t ∈ Ji , t > t (n = , , , . . .),

()

and, consequently, by (), (), condition (H ), and () we get   (Au¯ n ) t  – (Au¯ n )(t) ≤

 N         ∗  N[(t  ) – t  ]  ∗ ap,q + Gp,q b∗ + Np,q γ ∗ + t –t + t –t ap,q + Gp,q b∗ (β – )    N      t – t Np,q γ ∗ , ∀t, t  ∈ Ji , t  > t (n = , , , . . .). () + N t – t Np,q γ¯ + 

Guo Boundary Value Problems (2016) 2016:70

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From () we know that the maps {wn (t)} (n = , , , . . .) defined by  (Au¯ n )(t), ∀t ∈ Ji = (ti– , ti ], wn (t) = + (Au¯ n )(ti– ), ∀t = ti–

(n = , , , . . .)

()

+ ) denotes the right limit of (Au¯ n )(t) at t = ti– ) are equicontinuous on J¯i = [ti– , ti ]. ((Au¯ n )(ti– On the other hand, for any  > , choose a sufficiently large τ >  and a sufficiently large positive integer j such that





∞

a(t)gp,q (t) dt + Gp,q τ

∞

b(t) dt < , τ

Np,q

∞ 

γk < .

()

k=j+

We have, by (), (), (), (), and (), t wn (t) = (β – )



τ 

  f s, u¯ n (s), u¯ n (s), u¯ n (s), (T u¯ n ), (Su¯ n )(s) ds +

u¯ n (s), (T u¯ n )(s), (Su¯ n )(s)



+

+

 

t



i–  k=

∞ τ

j ∞         ds + I˜k u¯ n tk– + I˜k u¯ n tk– k=





 f s, u¯ n (s), u¯ n (s),



k=j+

  (t – s) f s, u¯ n (s), u¯ n (s), u¯  (s), (T u¯ n )(s), (Su¯ n )(s) ds

          Ik u¯ n tk– + (t – tk )I¯k u¯ n tk– + (t – tk ) I˜k u¯  tk– , 

∀t ∈ J¯i (n = , , , . . .),

()

and  ∞     f s, u¯ n (s), u¯ n (s), u¯ n (s), (T u¯ n )(s), (Su¯ n )(s) ds < τ ∞      – I˜k u¯ n tk <  (n = , , , . . .).

(n = , , , . . .),

() ()

k=j+

It follows from ()-() and [], Theorem .., that   α W (t) ≤

  τ    t  α f s, V (s), V  (s), V  (s), (TV )(s), (SV )(s) ds +  (β – )   j      –  +  α I˜k V tk + k=



t

+

   (t – s) α f s, V (s), V  (s), V  (s), (TV )(s), (SV )(s) ds

 i–          α Ik V  tk– + (t – tk )α I¯k V  tk– + k=

     , + (t – tk ) α I˜k V  tk– 

∀t ∈ J¯i ,

()

Guo Boundary Value Problems (2016) 2016:70

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where W (t) = {wn (t) : n = , , , . . .}, V (s) = {u¯ n (s) : n = , , , . . .}, V  (s) = {u¯ n (s) : n = , , , . . .}, V  (s) = {u¯ n (s) : n = , , , . . .}, (TV )(s) = {(T u¯ n )(s) : n = , , , . . .}, (SV )(s) = {(Su¯ n )(s) : n = , , , . . .}, and α(U) denotes the Kuratowski measure of noncompactness of a bounded set U ⊂ E (see [], Section .). Since V (s), V  (s), V  (s) ⊂ Pp∗ r∗ for s ∈ J+ and (TV )(s) ⊂ Pq∗ , (SV )(s) ⊂ Pq∗ for s ∈ J, where p∗ = min{  N – β – (β – )– ps , N – β – β  ps, N – β – β  p}, r∗ = max{qs , qs, q}, and q∗ = max{k ∗ q, h∗ q}, we see that, by condition (H ),    α f s, V (s), V  (s), V  (s), (TV )(s), (SV )(s) = ,

∀s ∈ J+ ,

()

and     α Ik V  tk– = ,

    α I¯k V  tk– = ,

    α I˜k V  tk– =  (k = , , , . . .).

()

It follows from ()-() that   t   , α W (t) ≤ β –

∀t ∈ J¯i ,

which implies by virtue of the arbitrariness of  that α(W (t)) =  for t ∈ J¯i . Hence, by Ascoli-Arzela theorem (see [], Theorem ..) we conclude that W = {wn : n = , , , . . .} is relatively compact in C[J¯i , E], and therefore, {wn (t)} (n = , , , . . .) has a subsequence that is convergent uniformly on J¯i , so, {(Au¯ n )(t)} (n = , , , . . .) has a subsequence that is convergent uniformly on Ji . Since i may be any positive integer, by the diagonal method, we can choose a subsequence {(Au¯ nj )(t)} (j = , , , . . .) of {(Au¯ n )(t)} (n = , , , . . .) such that {(Au¯ nj )(t)} (j = , , , . . .) is convergent uniformly on each Ji (i = , , , . . .). Let ¯ lim (Au¯ nj )(t) = w(t),

j→∞

∀t ∈ J.

()

By a similar method, we can prove that {(Au¯ nj ) (t)} (j = , , , . . .) has a subsequence that is convergent uniformly on each Ji (i = , , , . . .). For simplicity of notation, we may assume that {(Au¯ nj ) (t)} (j = , , , . . .) itself converges uniformly on each Ji (i = , , , . . .). Let lim (Au¯ nj ) (t) = y¯ (t),

j→∞

∀t ∈ J.

()

Again by a similar method, we can prove that {(Au¯ nj ) (t)} (j = , , , . . .) has a subsequence that is convergent uniformly on each Ji (i = , , , . . .). Again for simplicity of notation, we may assume that {(Au¯ nj ) (t)} (j = , , , . . .) itself converges uniformly on each Ji (i = , , , . . .). Let lim (Au¯ nj ) (t) = z¯ (t).

()

j→∞

By ()-() and the uniformity of convergence we have ¯  (t) = y¯ (t), w

y¯  (t) = z¯ (t),

∀t ∈ J,

()

Guo Boundary Value Problems (2016) 2016:70

Page 18 of 31

¯ ∈ PC  [J, E]. From ()-() we get and so, w ¯ S≤ w

 N(β – )  ∗ a + Gp,q b∗ + Np,q γ ∗ (β – ) p,q

and   Nβ  ∗ w ¯ T ≤ a + Gp,q b∗ + Np,q γ ∗ , β –  p,q   Nβ  ∗ w ¯ B ≤ ap,q + Gp,q b∗ + Np,q γ ∗ . β – ¯ ∈ DPC  [J, E], and Consequently, w ¯ D≤ w

 Nβ  ∗ ap,q + Gp,q b∗ + Np,q γ ∗ . β –

Let  >  be arbitrarily given. Choose a sufficiently large positive number η such that 



∞

∞

a(t)gp,q (t) dt + Gp,q

b(t) dt + Np,q

η

η



γk < .

()

tk ≥η

For any η < t < ∞, we have, by (), (Au¯ nj ) (t) – (Au¯ nj ) (η) =



t

η

+

  f s, u¯ nj (s), u¯ nj (s), u¯ nj (s), (T u¯ nj )(s), (Su¯ nj )(s) ds



   I˜k u¯ nj tk–

(j = , , , . . .),

η≤tk
so, from () and () we get (Au¯ n ) (t) – (Au¯ n ) (η) ≤ j j





t

a(s)gp,q (s) ds + Gp,q η

+ Np,q



t

b(s) ds η

γk

(j = , , , . . .),

η≤tk
which implies by () that (Au¯ n ) (t) – (Au¯ n ) (η) < , j j

∀t > η (j = , , , . . .),

and, letting j → ∞ and observing () and (), we get  w ¯ (t) – w ¯  (η) ≤ ,

∀t > η.

¯  (t) on [, η] as j → ∞, On the other hand, since {(Au¯ nj ) (t)} converges uniformly to w there exists a positive integer j such that (Au¯ n ) (t) – w ¯  (t) < , j

∀t ∈ [, η], j > j ,

Guo Boundary Value Problems (2016) 2016:70

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and hence (Au¯ n ) (t) – w ¯  (t) ≤ (Au¯ nj ) (t) – (Au¯ nj ) (η) + (Au¯ nj ) (η) – w ¯  (η) j  ¯  (t) < , ∀t > η, j > j . ¯ (η) – w + w Consequently, (Au¯ n ) – w ¯  B ≤ , j

∀j > j ,

and hence ¯  B = . lim (Au¯ nj ) – w

()

j→∞

It is clear that () and () imply        (Au¯ nj ) tk+ – (Au¯ nj ) tk– = Ik u¯ nj tk–

(k, j = , , , . . .)

()

and        (Au¯ nj ) tk+ – (Au¯ nj ) tk– = I¯k u¯ nj tk–

(k, j = , , , . . .).

()

By the uniformity of convergence of {(Au¯ nj )(t)} and {(Au¯ nj ) (t)} we see that     ¯ tk– , lim (Au¯ nj ) tk– = w

    ¯ tk+ lim (Au¯ ni ) tk+ = w

j→∞

j→∞

(k = , , , . . .)

and     ¯  tk– , lim (Au¯ nj ) tk– = w

    ¯  tk+ lim (Au¯ nj ) tk+ = w

j→∞

j→∞

(k = , , , . . .),

so, () and () imply that limits limj→∞ Ik (u¯ nj (tk– )) (k = , , , . . .) and limj→∞ I¯k (u¯ nj (tk– )) (k = , , , . . .) exist and        ¯ tk– = lim Ik u¯ nj tk– , ¯ tk+ – w w

       ¯  tk+ – w ¯  tk– = lim I¯k u¯ nj tk– w

j→∞

j→∞

(k = , , , . . .).

Let    lim Ik u¯ nj tk– = zk ,

j→∞

   lim I¯k u¯ nj tk– = z¯ k

j→∞

(k = , , , . . .).

Then zk ≥ θ , z¯ k ≥ θ (k = , , , . . .), and     ¯ tk– = zk , ¯ tk+ – w w

    ¯  tk– = z¯ k ¯  tk+ – w w

(k = , , , . . .).

()

By () and condition (H ) we have    –  Ik u¯ t ≤ NNp,q t  γk , nj k k

   –  ¯Ik u¯ t ≤ NNp,q tk γk nj k

(k, j = , , , . . .),

()

Guo Boundary Value Problems (2016) 2016:70

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so, zk  ≤ NNp,q tk γk ,

¯zk  ≤ NNp,q tk γk

(k = , , , . . .).

()

For any given  > , by condition (H ) we can choose a sufficiently large positive integer k such that Np,q

∞ 

tk γk < ,

∞ 

Np,q

k=k +

tk γk < ,

()

k=k +

and then, choose another sufficiently large positive integer j such that     –  Ik u¯ t – zk < , nj k k

    –  ¯Ik u¯ t – z¯ k < , nj k k

∀j > j (k = , , . . . , k ).

()

It follows from ()-() that k ∞     –      –  Ik u¯ t Ik u¯ t – z – zk ≤ k nj k nj k k=

k=

+

∞ ∞     –   Ik u¯ t + zk  < , nj k k=k +

∀j > j ,

k=k +

and k ∞     –      –  ¯Ik u¯ t ¯Ik u¯ t ¯ – zk ≤ – z¯ k nj k nj k k=

k=

+

∞ ∞     –   ¯Ik u¯ t + ¯zk  < , nj k k=k +

∀j > j ,

k=k +

and hence lim

j→∞

∞     –  Ik u¯ t – zk = , nj k

lim

j→∞

k=

∞     –  ¯Ik u¯ t – z¯ k = . nj k k=

By (), (), and ()-() we have (Au¯ nj ) (t) = ¯  (t) = w



t



t



(Au¯ nj ) (s) ds +

   I¯k u¯ nj tk– ,

∀t ∈ J (j = , , , . . .),


¯  (s) ds + w







z¯ k ,

∀t ∈ J,


and  (Au¯ nj )(t) =



t

(t – s)(Au¯ nj ) (s) ds +

∀t ∈ J (j = , , , . . .),

       Ik u¯ nj tk– + (t – tk )I¯k u¯ nj tk– , 
()

Guo Boundary Value Problems (2016) 2016:70

 ¯ = w(t)

t

Page 21 of 31

¯  (s) ds + (t – s)w





 zk + (t – tk )¯zk ,

∀t ∈ J,


which imply (Au¯ n ) (t) – w ¯  (t) ≤ t (Au¯ nj ) – w ¯  B j     ¯Ik u¯  t – – z¯ k , + nj k

∀t ∈ J (j = , , , . . .),

()


and      Ik u¯  t – – zk (Au¯ n )(t) – w(t) ≤ t  (Au¯ n ) – w ¯ ¯ + nj k j j B 
    + t ¯Ik u¯ nj tk– – z¯ k ,

∀t ∈ J.

()

Since     ¯Ik u¯  t – – z¯ k = , nj k

    Ik u¯  t – – zk = , nj k 
∀ < t ≤ t (j = , , , . . .),


() and () imply ∞     –  (Au¯ n ) – w ¯Ik u¯ t ¯  T ≤ (Au¯ nj ) – w ¯  B + t– – z¯ k (j = , , , . . .) nj k j

()

k=

and ∞     –  Ik u¯ t ¯ S ≤ (Au¯ nj ) – w ¯  B + t– Au¯ nj – w – zk nj k k=

+ t–

∞ 

   –  ¯Ik u¯ t – z¯ k nj k

(j = , , , . . .).

()

k=

By (), (), (), and () we get ¯  T = , lim (Au¯ nj ) – w

j→∞

¯ S = . lim Au¯ nj – w

j→∞

()

¯ D →  as j → ∞, and the relative compactness It follows from () and () that Au¯ nj – w  of A(Qpq ) is proved. Lemma  Let conditions (H )-(H ) be satisfied. Then u ∈ Q+ ∩ C  [J+ , E] is a positive solution of IBVP () if and only if u ∈ Q+ is a solution of the following impulsive integral equation:   ∞ ∞       –  t   ˜ u(t) = f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds + Ik u tk (β – )  k=     t + (t – s) f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds  

Guo Boundary Value Problems (2016) 2016:70

Page 22 of 31

       + Ik u tk– + (t – tk )I¯k u tk– 
   –   ˜ + (t – tk ) Ik u tk , 

∀t ∈ J,

()

that is, u is a fixed point of the operator A defined by (). Proof The method of the proof is similar to that of Lemma  in [], the difference is in using formula () instead of formula () with discussion in a Banach space. We omit the proof.  Lemma  (Fixed point theorem of cone expansion and compression with norm type; see [], Corollary , or [], Theorem , or [], Theorem ..; see also [, ]) Let P be a cone in a real Banach space E, and  ,  be two bounded open sets in E such that ¯  ⊂  , where θ denotes the zero element of E, and  ¯ i denotes the closure of i θ ∈  ,  ¯  \ ) → P be completely continuous (i.e., continuous (i = , ). Let an operator A : P ∩ ( and compact). Suppose that one of the following two conditions is satisfied: (a) Ax ≤ x,

∀x ∈ P ∩ ∂ ;

Ax ≥ x,

∀x ∈ P ∩ ∂ ,

where ∂i denotes the boundary of i (i = , ); (b) Ax ≥ x,

∀x ∈ P ∩ ∂ ;

Ax ≤ x,

∀x ∈ P ∩ ∂ .

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

3 Main theorem Theorem Let conditions (H )-(H ) be satisfied. Assume that there exists r >  such that  Nβ  ∗ a + Gr,r b∗ + Nr,r γ ∗ < r, β –  r,r

()

where a∗r,r , Gr,r , and Nr,r are a∗p,q , Gp,q , and Np,q for p = r and q = r, respectively (for a∗p,q , Gp,q , Np,q , b∗ , and γ ∗ ; see conditions (H ) and (H )). Then IBVP () has at least two positive solutions u∗ , u∗∗ ∈ Q+ ∩ C  [J+ , E] such that  < inf

t∈J+

u∗ (t) u∗ (t) ≤ sup < r, t t t∈J+

(u∗ ) (t) (u∗ ) (t) ≤ sup < r, t∈J+ t t t∈J+      < inf  u∗ (t) ≤ sup  u∗ (t) < r,  < inf

t∈J

t∈J

 – – u∗∗ (t) u∗∗ (t) ≤ sup < ∞, N β (β – )– r < inf  t∈J+  t t t∈J+ N – β – β  r < inf

t∈J+

(u∗∗ ) (t) (u∗∗ ) (t) ≤ sup < ∞, t t t∈J+

Guo Boundary Value Problems (2016) 2016:70

Page 23 of 31

and     N – β – β  r < inf u∗∗ (t) ≤ sup u∗∗ (t) < ∞. t∈J

t∈J

Proof By Lemma  and Lemma  the operator A defined by () is continuous from Q+ into Q and we need to prove that A has two fixed points u∗ and u∗∗ in Q+ such that  < u∗ D < r < u∗∗ D . By condition (H ) there exists r >  such that f (t, u, v, w, y, z) ≥

N  β  (β – ) c(t)ww , c∗ w 

∀t ∈ J+ , u, v, w ∈ P+ , w ≥ r , y, z ∈ P.

()

Choose  –

 r > max N  β  β  r , r .

()

For u ∈ Q, uD = r , we have, by () and (), u (t) ≥ N – β – β  r > r , ∀t ∈ J, so, () and () imply  (Au) (t) ≥ β – 



∞

  f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds



  ∞  N  β  (β  )– u (s) ds w c(s) ≥ c∗ w     ∞ Nr Nr ≥ ∗ c(s) ds w = w , c w   w 

∀t ∈ J,

and, consequently, (Au) (t) ≥ r , ∀t ∈ J; hence, AuD ≥ uD ,

∀u ∈ Q, uD = r .

()

By condition (H ) there exists r >  such that f (t, u, v, w, y, z) ≥

N(β – )r d(t)w , d∗ w 

∀t ∈ J+ , u, v, w ∈ P+ ,  < w < r , y, z ∈ P.

()

Choose  < r < min{r , r}.

()

For u ∈ Q, uD = r , we have, by () and (), r > r ≥ u (t) ≥ N – β – β  r > , so, we get, by () and (),  (Au) (t) ≥ β – 

≥



∞

  f s, u(s), u (s), u (s), (Tu)(s), (Su)(s) ds



Nr d∗ w 



∞ 

 Nr w , d(s) ds w = w 

∀t ∈ J,

Guo Boundary Value Problems (2016) 2016:70

Page 24 of 31

and, consequently, (Au) (t) ≥ r > r , ∀t ∈ J; hence, AuD > uD ,

∀u ∈ Q, uD = r .

()

On the other hand, for u ∈ Q, uD = r, ()-() imply AuD ≤

 Nβ  ∗ a + Gr,r b∗ + Nr,r γ ∗ , β –  r,r

so, () implies AuD < uD ,

∀u ∈ Q, uD = r.

()

By () and () we know that  < r < r < r , and, by Lemma  the operator A is completely continuous from Qr r into Q; hence, (), (), (), and Lemma  imply that A has two fixed points u∗ , u∗∗ ∈ Qr r such that r < u∗ D < r < u∗∗ D ≤ r . The proof is complete.  Example Consider the infinite system of scalar third-order impulsive singular integrodifferential equations of mixed type on the half-line: ⎧  –t   un (t) = e  { ⎪  +    ⎪  ⎪  n t  [un (t)+un– (t)+ ∞ um (t)]  [un+ (t)+ ∞ ⎪ m= m= um (t)] ⎪  ⎪   ⎪    +  [un+ (t) + ∞ } ⎪ m= um (t)] +  ∞ ⎪ m= um (t) ⎪  ⎪ –t t –(t+)s e ⎪  ⎪ + un (s) ds)  {(  e ⎪ ⎪ t  n ⎪  ⎪ ∞ u (s) ds ⎨ n +( ) }, ∀ < t < ∞, t = k (k = , , , . . . ; n = , , , . . .), (+t+s)



⎪ un |t=k = n– –k–  –  √ ∞  – (k = , , , . . . ; n = , , , . . .), ⎪ ⎪ [un (k )] + ⎪ m= um (k ) ⎪ ⎪   – – –k– ⎪ √ | = n k  (k = , , , . . . ; n = , , , . . .), u ⎪ ∞  – t=k n ⎪ [un (k – )] + ⎪ m= um (k ) ⎪ ⎪ ⎪ ⎪ un |t=k = n– k – –k–  –  √∞  – (k = , , , . . . ; n = , , , . . .), ⎪ ⎪ un (k )] + m= um (k ) ⎪ ⎩ un () = , un () = , un (∞) = un () (n = , , , . . .).

()

Conclusion The infinite system () has at least two positive solutions {u∗n (t)} (n = , , , . . .) and {u∗∗ n (t)} (n = , , , . . .) such that ∞  < inf


 < inf

∗ m= um (t) t

∞

∗  m= (um ) (t)

t


 < inf

≤t<∞

≤ sup

∞   m=

∞


≤ sup

∗ m= um (t) t

< ,

∞


∗  m= (um ) (t)

t

< ,

∞    ∗  u∗m (t) ≤ sup um (t) < ,

∞

≤t<∞ m=

∞ ∗∗ ∗∗  m= um (t) m= um (t) < inf ≤ sup < ∞,  
Guo Boundary Value Problems (2016) 2016:70

Page 25 of 31

and ∞ ∞    ∗∗   ∗∗   < inf um (t) ≤ sup um (t) < ∞.  ≤t<∞ m= ≤t<∞ m=

 ∞ Proof Let E = l = {u = (u , . . . , un , . . .) : ∞ n= |un | < ∞} with norm u = n= |un | and P = {u = (u , . . . , un , . . .) ∈ E : un ≥ , n = , , , . . .}. Then P is a normal cone in E with normal constant N = , and the infinite system () can be regarded as an IBVP of the form (). In this situation, u = (u , . . . , un , . . .), v = (v , . . . , vn , . . .), w = (w , . . . , wn , . . .), y = (y , . . . , yn , . . .), z = (z , . . . , zn , . . .), tk = k (k = , , , . . .), K(t, s) = e–(t+)s , H(t, s) = ( + t + s)– , β = , β  =  , f = (f , . . . , fn , . . .), Ik = (Ik , . . . , Ikn , . . .), I¯k = (I¯k , . . . , I¯kn , . . .), and I˜k = (I˜k , . . . , I˜kn , . . .), in which   –  –   ∞ ∞       fn (t, u, v, w, y, z) = u v + u + u + + v n n– m n+ m   n t   m= m=       – ∞ ∞     wn+ + + wm + wm   m= m= e–t

e–t 

+

  

  yn + zn ,

n t   – ∞   Ikn (w) = n– –k– wn +  wm ,

∀t ∈ J+ , u, v, w ∈ P+ , y, z ∈ P,

()

∀w ∈ P+ (k = , , , . . . ; n = , , , . . .), ()

m=

 I¯kn (w) = n– k – –k–

 – ∞   wn +  wm , m=

∀w ∈ P+ (k = , , , . . . ; n = , , , . . .),

()

and  I˜kn (w) = n– k – –k–

 – ∞   wn +  wm , m=

∀w ∈ P+ (k = , , , . . . ; n = , , , . . .).

()

It is easy to see that f ∈ C[J+ × P+ × P+ × P+ × P × P, P], Ik , I¯k , I˜k ∈ C[P+ , P] (k = , , , . . .), and condition (H ) is satisfied with k ∗ ≤ e– (by using the fact that the function φ(s) = s e–s ( ≤ s < ∞) attains its maximum at s = ) and h∗ ≤ . We have by ()       –  –  – u v w w + + +     n t    –t  –t  e  e        ≤ + u–  + v  + w + w–  y + z     n t  n t        y + z , ∀t ∈ J+ , u, v, w ∈ P+ , y, z ∈ P, + ()  

 ≤ fn (t, u, v, w, y, z) ≤

e–t



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so, observing the inequality

∞

 n= n

< , we get

 ∞  e–t      f (t, u, v, w, z) = u–  + v–  + w fn (t, u, v, w, z) <      t n=     w– + y + z , ∀t ∈ J+ , u, v, w ∈ P+ , y, z ∈ P, +    which implies that condition (H ) is satisfied for a(t) = b(t) =

e–t  

t

,

g(s , s ) =

 –   –  s + s   

and G(x , x , x ) =

   –     x + x + x + x        

()

with (for q ≥ p > )

 –   –  pt  pt s + s : ≤ s ≤ qt  , ≤ s ≤ qt gp,q (t) = max                 t–  + t –  , ∀t ∈ J+ , =  p  q        ∞    ∞ e–t    ∞ e–t a(t)gp,q (t) dt = dt + () a∗p,q =   dt < ∞,  p  q    t t and b∗ =



∞ 

e–t 

t

dt < ∞.

()

By ()-() it is obvious that  ≤ Ikn (w) ≤ k I¯kn (w),

I¯kn (w) ≤ I˜kn (w),

∀w ∈ P+ (k = , , , . . . ; n = , , , . . .),

so, Ik (w) ≤ k I¯k (w),

I¯k (w) ≤ k I˜k (w),

∀w ∈ P+ (k = , , , . . .).

Moreover, from () we get  ,  ≤ I˜kn (w) ≤ n– k – –k– √ w

∀w ∈ P+ (k = , , , . . . ; n = , , , . . .),

so, ∞   ˜Ik (w) = I˜kn (w) ≤ k – –k– √ , w n=

∀w ∈ P+ (k = , , , . . .),

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which implies that condition (H ) is satisfied for γk = k – –k– and 

F(s) = s– 

()

with γ˜ =

∞  k=

k  γk =

 , 

∞ 

γ∗ =

γk < γ˜ .

()

k=

On the other hand, () implies fn (t, u, v, w, y, z) ≥

e–t 

w ,

∀t ∈ J+ , u, v, w ∈ P+ , y, z ∈ P,



w– ,

∀t ∈ J+ , u, v, w ∈ P+ , y, z ∈ P,

n t 

and fn (t, u, v, w, y, z) ≥

e–t n t 

()

so, we see that condition (H ) is satisfied for     w = ,  , . . . ,  , . . . ,  n

c(t) =

e–t t

 

,

τ (w) =

 w , 

,

σ (w) =

 w– . 

and condition (H ) is satisfied for     w = ,  , . . . ,  , . . . ,  n

d(t) =

e–t t

 

In addition, by () we have fn (t, u, v, w, y, z) ≥

e–t    

–  u  =

n t

e–t  



  

( )n t

u–  ,

∀t ∈ J+ , u, v, w ∈ P+ , y, z ∈ P,

()

and fn (t, u, v, w, y, z) ≥

e–t   

 –  v  =

n t

e–t  



  

( )n t

v–  ,

∀t ∈ J+ , u, v, w ∈ P+ , y, z ∈ P. It follows from (), (), and () that  ∞   e–t e–t –  –  f (t, u, v, w, y, z) ≥ > ,   u   u  n (  )t  (  )t  n= ∀t ∈ J+ , u, v, w ∈ P+ , y, z ∈ P,

()

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 ∞   e–t e–t –  –  f (t, u, v, w, y, z) ≥ v > ,     v  n (  )t  (  )t  n= ∀t ∈ J+, u, v, w ∈ P+ , y, z ∈ P, and ∞    e–t e–t – – f (t, u, v, w, y, z) ≥ w >   w ,  n   t t n= ∀t ∈ J+ , u, v, w ∈ P+ , y, z ∈ P; hence, (), (), (), and () are satisfied, that is, f (t, u, v, w, y, z) is singular at t = , u = θ , v = θ , and w = θ (θ = (, , . . . , , . . .)). On the other hand, from ()-() we get   – Ikn (w) ≥ n– –k– w + w , ∀w ∈ P+ (k = , , , . . . ; n = , , , . . .),   – I¯kn (w) ≥ n– k – –k– w + w , ∀w ∈ P+ (k = , , , . . . ; n = , , , . . .), and   – I˜kn (w) ≥ n– k – –k– w + w ,

∀w ∈ P+ (k = , , , . . . ; n = , , , . . .),

so,  ∞       – – Ik (w) ≥ –k– w + w > –k– w + w ,  n n= ∀w ∈ P+ (k = , , , . . .),  ∞       – – ¯Ik (w) ≥ k – –k– w + w > k – –k– w + w ,  n n= ∀w ∈ P+ (k = , , , . . .), and  ∞       – – ˜Ik (w) ≥ k – –k– w + w > k – –k– w + w ,  n n= ∀w ∈ P+ (k = , , , . . .), which imply that (), (), and () are satisfied, that is, Ik (w), I¯k (w), and I˜k (w) are singular at w = θ . Now, we check that condition (H ) is satisfied. Let t ∈ J+ , r > p > , and q >  be (m) fixed, and {y(m) } be any sequence in f (t, Ppr , Ppr , Ppr , Pq , Pq ), where y(m) = (y(m)  , . . . , yn , . . .). Then, by () we have  ≤ y(m) n ≤

e–t 

n t 



 –    –      – p + p + r + p + q + q      

(n, m = , , , . . .).



()

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So, {y(m) n } is bounded, and, by the diagonal method we can choose a subsequence {mi } ⊂ {m} such that i) → y ¯n y(m n

as i → ∞ (n = , , , . . .),

()

which implies by () that  ≤ y¯ n ≤



e–t

 –    –      – p + p + r + p + q + q      



n t 



(n = , , , . . .).

()

Consequently, y¯ = (¯y , . . . , y¯ n , . . .) ∈ l = E. Let  >  be given. Choose a positive integer n such that e–t 

t



  ∞    –    –       –  +  + < . p p r p q q + + +  n        n=n +

()



By () we can choose a positive integer i such that   (m ) y i – y¯ n  <  , n n

∀i > i (n = , , . . . , n ).

()

It follows from ()-() that n ∞ ∞ (m )   (m )    (m )    (m )  y i – y¯ = y i – y¯ n  ≤ y i – y¯ n  + y i  n n n n=

+

n=n +

n= ∞ 

|¯yn | <

n=n +

   + + = ,   

∀i > i ;

hence, y(mi ) → y¯ in E as i → ∞. Thus, we have proved that f (t, Ppr , Ppr , Ppr , Pq , Pq ) is relatively compact in E. Similarly, we can prove that Ik (Ppr ), I¯k (Ppr ), and I˜k (Ppr ) (k = , , , . . .) are relatively compact in E. Hence, condition (H ) is satisfied. Finally, we check that inequality () is satisfied for r = , that is,    a∗, + G, b∗ + N, γ ∗ < .

()

Since     = . · · · < ,   , e– = . · · · < 

 ,  √   = . · · · < ,  

  = . · · · <

and  

∞

e–t dt < tα



 

dt + tα

 

∞

e–t dt =

  + e– , –α 

∀ < α < ,

Guo Boundary Value Problems (2016) 2016:70

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we have, by () and (), a∗,

      –    –      () + e () + e + <                     + + + <         =

,  + < . + . = . , ,

and b∗ <

   + = .   

Moreover, () implies

   –       G, ≤ max x + x + x + x : ≤ x ≤ ,  ≤ x ≤ ,  ≤ x ≤                 + + = + , ≤ +       , and () implies

√   N, = max s–  : ≤ s ≤  =   < ,  so, G, b∗ <



  +  ,



 

 =

 , + < . + . = ., , ,

and, observing (), we have N, γ ∗ <

 < .. 

Consequently,    a∗, + G, b∗ + N, γ ∗ < (. + . + .) = . < . Hence, () holds, and our conclusion follows from the theorem.



Competing interests The author declares that they have no competing interests. Received: 10 December 2015 Accepted: 13 March 2016 References 1. Lakshmikantham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) 2. Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations. World Scientific, Singapore (1995) 3. Benchohra, M, Henderson, J, Ntouyas, SK: Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York (2006)

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