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Multiple positive solutions for first-order impulsive singular integro-differential equations on the half line in a Banach space

Yanlai Chen1* and Baoxia Qin2

Author Affiliations

1 School of Economics, Shandong University, Jinan, Shandong, 250100, People’s Republic of China

2 Department of Mathematics, Qilu Normal University, Jinan, Shandong, 250100, People’s Republic of China

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Boundary Value Problems 2013, 2013:69  doi:10.1186/1687-2770-2013-69


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/69


Received:22 December 2012
Accepted:18 March 2013
Published:2 April 2013

© 2013 Chen and Qin; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, the author discusses the multiple positive solutions for an infinite three-point boundary value problem of first-order impulsive superlinear singular integro-differential equations on the half line in a Banach space by means of the fixed-point theorem of cone expansion and compression with norm type.

MSC: 45J05, 34G20, 47H10.

Keywords:
impulsive singular integro-differential equation in a Banach space; infinite three-point boundary value problem; fixed-point theorem of cone expansion and compression with norm type

1 Introduction

In recent years, multiple solutions of boundary value problems for impulsive differential equations in scalar spaces had been extensively studied (see, for example, [1-3]). In recent papers [4] and [5], Professor D. Guo discussed two infinite boundary value problems for nth-order impulsive nonlinear singular integro-differential equations of mixed type on the half line in a Banach space. By constructing a bounded closed convex set, apart from the singularities, and using the Schauder fixed-point theorem, he obtained 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. Now, in this paper, we shall discuss the existence of two positive solutions for first-order superlinear singular equations by means of a different method, i.e., by using the fixed-point theorem of cone expansion and compression with norm type (see [6,7]), and the key point is to introduce a new cone Q.

Let E be a real Banach space and P be a cone in E, which 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 N y , 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 , we write x < y . Let P + = P { θ } , i.e., P + = { x P : x > θ } . For details on cone theory, see [7].

Consider the infinite three-point boundary value problem for a first-order impulsive nonlinear singular integro-differential equation of mixed type on the half line in E:

{ u ( t ) = f ( t , u ( t ) , ( T u ) ( t ) , ( S u ) ( t ) ) , t J + , Δ u | t = t k = I k ( u ( t k ) ) ( k = 1 , 2 , 3 , ) , u ( ) = γ u ( η ) + β u ( 0 ) , (1)

where J = [ 0 , ) , J + = ( 0 , ) , 0 < t 1 < < t k <  , t k , J + = J + { t 1 , , t k , } , f C [ J + × P + × P × P , P ] , I k C [ P + , P ] ( k = 1 , 2 , 3 , ), 0 γ < 1 , β + γ > 1 , t m 1 < η < t m (for some m), u ( ) = lim t u ( t ) and

( T u ) ( t ) = 0 t K ( t , s ) u ( s ) d s , ( S u ) ( t ) = 0 H ( t , s ) u ( s ) d s , (2)

K C [ D , R + ] , D = { ( t , s ) J × J : t s } , H C [ J × J , R + ] , R + denotes the set of all nonnegative numbers. Δ u | t = t k denotes the jump of u ( t ) at t = t k , i.e.,

Δ u | t = t k = u ( t k + ) u ( t k ) ,

where u ( t k + ) and u ( t k ) represent the right and left limits of u ( t ) at t = t k , respectively. In the following, we always assume that

lim t 0 + f ( t , u , v , w ) = , u P + , v , w P (3)

and

lim u θ + f ( t , u , v , w ) = , t J + , v , w P , (4)

(where u θ + means u > θ , u 0 ), i.e., f ( t , u , v , w ) is singular at t = 0 and u = θ . We also assume that

lim u θ + I k ( u ) = ( k = 1 , 2 , 3 , ) , (5)

i.e., I k ( u ) ( k = 1 , 2 , 3 , ) are singular at u = θ .

Let P C [ J , E ] = { u : u is a map from J into E such that u ( t ) is continuous at t t k , left continuous at t = t k , and u ( t k + ) exists, k = 1 , 2 , 3 , } and B P C [ J , E ] = { u P C [ J , E ] : sup t J u ( t ) < } . It is clear that B P C [ J , E ] is a Banach space with norm

u B = sup t J u ( t ) .

Let B P C [ J , P ] = { u B P C [ J , E ] : u ( t ) θ , t J } and Q = { u B P C [ J , P ] : u ( t ) β 1 ( 1 γ ) u ( s ) , t , s J } . Obviously, B P C [ J , P ] and Q are two cones in space B P C [ J , E ] and Q B P C [ J , P ] . u B P C [ J , P ] C 1 [ J + , E ] is called a positive solution of the infinite three-point boundary value problem (1) if u ( t ) > θ for t J and u ( t ) satisfies (1). Let Q + = { u Q : u B > 0 } and Q p q = { u Q : p u B q } for q > p > 0 .

2 Several lemmas

Let us list some conditions.

( H 1 ) sup t J 0 t K ( t , s ) d s < , sup t J 0 H ( t , s ) d s < and

lim t t 0 | H ( t , s ) H ( t , s ) | d s = 0 , t J .

In this case, let

k = sup t J 0 t K ( t , s ) d s , h = sup t J 0 H ( t , s ) d s .

( H 2 ) There exist a C [ J + , R + ] and g C [ R + + × R + × R + , R + ] such that

f ( t , u , v , w ) a ( t ) g ( u , v , w ) , t J + , u P + , v , w P ,

and

a = 0 a ( t ) d t < ,

where R + + = { x R + : x > 0 } .

( H 3 ) There exist γ k 0 ( k = 1 , 2 , 3 , ) and F C [ R + + , R + ] such that

I k ( u ) γ k F ( u ) , u P + ( k = 1 , 2 , 3 , ) ,

and

γ = k = 1 γ k < .

( H 4 ) For any t J + and r > p > 0 , f ( t , P p r , P r , P r ) = { f ( t , u , v , w ) : u P p r , v , w P r } and I k ( P p r ) = { I k ( u ) : u P p r } ( k = 1 , 2 , 3 , ) are relatively compact in E, where P r = { u P : u r } and P p r = { u P : p u r } .

Remark Obviously, condition ( H 4 ) is satisfied automatically when E is finite dimensional.

Remark It is clear: If condition ( H 1 ) is satisfied, then the operators T and S defined by (2) are bounded linear operators from B P C [ J , E ] into B P C [ J , E ] and T k , S h ; moreover, we have T ( B P C [ J , P ] ) B P C [ J , P ] and S ( B P C [ J , P ] ) B P C [ J , P ] .

We shall reduce the infinite three-point boundary value problem (1) to an impulsive integral equation. To this end, we consider the operator A defined by

( A u ) ( t ) = 1 β + γ 1 { η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + ( 1 γ ) 0 η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + k = m I k ( u ( t k ) ) + ( 1 γ ) k = 1 m 1 I k ( u ( t k ) ) } + 0 t f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + 0 < t k < t I k ( u ( t k ) ) , t J . (6)

In what follows, we write J 1 = [ 0 , t 1 ] , J k = ( t k 1 , t k ] ( k = 2 , 3 , 4 , ).

Lemma 1Let conePbe normal and conditions ( H 1 )-( H 4 ) be satisfied. Then operatorAdefined by (6) is a continuous operator from Q + intoQ; moreover, for any q > p > 0 , A ( Q p q ) is relatively compact.

Proof Let u Q + and u B = r . Then r > 0 and

u ( t ) β 1 ( 1 γ ) u ( s ) θ , t , s J ,

so,

u ( t ) N 1 β 1 ( 1 γ ) u B , t J , (7)

where N denotes the normal constant of cone P, and consequently,

N 1 β 1 ( 1 γ ) r u ( t ) r , t J . (8)

By condition ( H 2 ) and (8), we have

f ( t , u ( t ) , ( T u ) ( t ) , ( S u ) ( t ) ) M a ( t ) , t J , (9)

where

M = max { g ( x , y , z ) : N 1 β 1 ( 1 γ ) r x r , 0 y k r , 0 z h r } ,

which implies the convergence of the infinite integral

0 f ( t , u ( t ) , ( T u ) ( t ) , ( S u ) ( t ) ) d t (10)

and

0 f ( t , u ( t ) , ( T u ) ( t ) , ( S u ) ( t ) ) d t 0 f ( t , u ( t ) , ( T u ) ( t ) , ( S u ) ( t ) ) d t M a . (11)

On the other hand, by condition ( H 3 ) and (8), we have

I k ( u ( t k ) ) D γ k ( k = 1 , 2 , 3 , ) , (12)

where

D = max { F ( x ) : N 1 β 1 ( 1 γ ) r x r } ,

which implies the convergence of the infinite series

k = 1 I k ( u ( t k ) ) (13)

and

k = 1 I k ( u ( t k ) ) k = 1 I k ( u ( t k ) ) D γ . (14)

It follows from (6), (11), and (14) that

( A u ) ( t ) 1 β + γ 1 { η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + ( 1 γ ) 0 η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + k = m I k ( u ( t k ) ) + ( 1 γ ) k = 1 m 1 I k ( u ( t k ) ) } + 0 t f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + 0 < t k < t I k ( u ( t k ) ) 1 β + γ 1 { 0 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + k = 1 I k ( u ( t k ) ) } + 0 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + k = 1 I k ( u ( t k ) ) = β + γ β + γ 1 { 0 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + k = 1 I k ( u ( t k ) ) } β + γ β + γ 1 ( M a + D γ ) , t J ,

which implies that A u B P C [ J , P ] and

A u B β + γ β + γ 1 ( M a + D γ ) . (15)

Moreover, by (6), we have

( A u ) ( t ) 1 β + γ 1 { η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + ( 1 γ ) 0 η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + k = m I k ( u ( t k ) ) + ( 1 γ ) k = 1 m 1 I k ( u ( t k ) ) } , t J (16)

and

( A u ) ( t ) 1 β + γ 1 { η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + ( 1 γ ) 0 η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + k = m I k ( u ( t k ) ) + ( 1 γ ) k = 1 m 1 I k ( u ( t k ) ) } + 0 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + k = 1 I k ( u ( t k ) ) , t J . (17)

It is clear,

0 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + k = 1 I k ( u ( t k ) ) 1 1 γ { η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + ( 1 γ ) 0 η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + k = m I k ( u ( t k ) ) + ( 1 γ ) k = 1 m 1 I k ( u ( t k ) ) } , (18)

so, (17) and (18) imply

( A u ) ( t ) { 1 β + γ 1 + 1 1 γ } { η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + ( 1 γ ) 0 η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + k = m I k ( u ( t k ) ) + ( 1 γ ) k = 1 m 1 I k ( u ( t k ) ) } , t J . (19)

It follows from (16) and (19) that

( A u ) ( t ) 1 β + γ 1 ( 1 β + γ 1 + 1 1 γ ) 1 ( A u ) ( s ) = β 1 ( 1 γ ) ( A u ) ( s ) , t , s J . (20)

Hence, A u Q , i.e., A maps Q + into Q.

Now, we are going to show that A is continuous. Let u n , u ¯ Q + , u n u ¯ B 0 ( n ). Write u ¯ B = 2 r ¯ ( r ¯ > 0 ) and we may assume that

r ¯ u n B 3 r ¯ ( n = 1 , 2 , 3 , ) .

So, by (7),

N 1 β 1 ( 1 γ ) r ¯ u n ( t ) 3 r ¯ , t J ( n = 1 , 2 , 3 , ) (21)

and

N 1 β 1 ( 1 γ ) r ¯ < 2 N 1 β 1 ( 1 γ ) r ¯ u ¯ ( t ) 2 r ¯ < 3 r ¯ , t J . (22)

Similar to (15), it is easy to get

A u n A u ¯ B β + γ β + γ 1 { 0 f ( s , u n ( s ) , ( T u n ) ( s ) , ( S u n ) ( s ) ) f ( s , u ¯ ( s ) , ( T u ¯ ) ( s ) , ( S u ¯ ) ( s ) ) d s + k = 1 I k ( u n ( t k ) ) I k ( u ¯ ( t k ) ) } ( n = 1 , 2 , 3 , ) . (23)

It is clear that

f ( t , u n ( t ) , ( T u n ) ( t ) , ( S u n ) ( t ) ) f ( t , u ¯ ( t ) , ( T u ¯ ) ( t ) , ( S u ¯ ) ( t ) ) as  n , t J , (24)

and, similar to (9) and observing (21) and (22), we have

f ( t , u n ( t ) , ( T u n ) ( t ) , ( S u n ) ( t ) ) f ( t , u ¯ ( t ) , ( T u ¯ ) ( t ) , ( S u ¯ ) ( t ) ) 2 M ¯ a ( t ) = d ( t ) , t J ( n = 1 , 2 , 3 , ) ; d L [ J , R + ] , (25)

where

M ¯ = max { g ( x , y , z ) : N 1 β 1 ( 1 γ ) r ¯ x 3 r ¯ , 0 y 3 k r ¯ , 0 z 3 h r ¯ } .

It follows from (24), (25), and the dominated convergence theorem that

lim n 0 f ( t , u n ( t ) , ( T u n ) ( t ) , ( S u n ) ( t ) ) f ( t , u ¯ ( t ) , ( T u ¯ ) ( t ) , ( S u ¯ ) ( t ) ) d t = 0 . (26)

On the other hand, for any ϵ > 0 , we can choose a positive integer j such that

D ¯ k = j + 1 γ k < ϵ , (27)

where

D ¯ = max { F ( x ) : N 1 β 1 ( 1 γ ) r ¯ x 3 r ¯ } .

And then, choose an positive integer n 0 such that

k = 1 j I k ( u n ( t k ) ) I k ( u ¯ ( t k ) ) < ϵ , n > n 0 . (28)

From (27), (28), and observing condition ( H 3 ) and (21), (22), we get

k = 1 I k ( u n ( t k ) ) I k ( u ¯ ( t k ) ) < ϵ + 2 D ¯ k = j + 1 γ k < 3 ϵ , n > n 0 ,

hence,

lim n k = 1 I k ( u n ( t k ) ) I k ( u ¯ ( t k ) ) = 0 . (29)

It follows from (23), (26), and (29) that A u n A u ¯ B 0 as n , and the continuity of A is proved.

Finally, we prove that A ( Q p q ) is relatively compact, where q > p > 0 are arbitrarily given. Let v n A ( Q p q ) ( n = 1 , 2 , 3 , ). Then, by (7),

N 1 β 1 ( 1 γ ) p v n ( t ) q , t J ( n = 1 , 2 , 3 , ) . (30)

Similar to (9), (12), (15), and observing (30), we have

f ( t , v n ( t ) , ( T v n ) ( t ) , ( S v n ) ( t ) ) M 1 a ( t ) , t J + ( n = 1 , 2 , 3 , ) , (31)

I k ( v n ( t k ) ) D 1 γ k ( k , n = 1 , 2 , 3 , ) (32)

and

A v n B β + γ β + γ 1 ( M 1 a + D 1 γ ) ( n = 1 , 2 , 3 , ) , (33)

where

M 1 = max { g ( x , y , z ) : N 1 β 1 ( 1 γ ) p x q , 0 y k q , 0 z h q }

and

D 1 = max { F ( x ) : N 1 β 1 ( 1 γ ) p x q } .

Consider J i = ( t i 1 , t i ] for any fixed i. By (6) and (31), we have

( A v n ) ( t ) ( A v n ) ( t ) t t f ( s , v n ( s ) , ( T v n ) ( s ) , ( S v n ) ( s ) ) d s M 1 t t a ( s ) d s , t , t J i , t > t ( n = 1 , 2 , 3 , ) , (34)

which implies that the functions { w n ( t ) } ( n = 1 , 2 , 3 , ) defined by

w n ( t ) = { ( A v n ) ( t ) , t J i = ( t i 1 , t i ] , ( A v n ) ( t i 1 + ) , t = t i 1 ( n = 1 , 2 , 3 , ) (35)

( ( A u n ) ( t i 1 + ) denotes the right limit of ( A u n ) ( t ) at t = t i 1 ) are equicontinuous on J ¯ i = [ t i 1 , t i ] . On the other hand, for any ϵ > 0 , choose a sufficiently large τ > η and a sufficiently large positive integer j > m such that

M 1 τ a ( s ) d s < ϵ , D 1 k = j + 1 γ k < ϵ . (36)

We have, by (35), (6), (31), (32), and (36),

w n ( t ) = 1 β + γ 1 { η τ f ( s , v n ( s ) , ( T v n ) ( s ) , ( S v n ) ( s ) ) d s + τ f ( s , v n ( s ) , ( T v n ) ( s ) , ( S v n ) ( s ) ) d s + ( 1 γ ) 0 η f ( s , v n ( s ) , ( T v n ) ( s ) , ( S v n ) ( s ) ) d s + k = m j I k ( v n ( t k ) ) + k = j + 1 I k ( v n ( t k ) ) + ( 1 γ ) k = 1 m 1 I k ( v n ( t k ) ) } + 0 t f ( s , v n ( s ) , ( T v n ) ( s ) , ( S v n ) ( s ) ) d s + k = 1 i 1 I k ( v n ( t k ) ) , t J ¯ i ( n = 1 , 2 , 3 , ) (37)

and

τ f ( s , v n ( s ) , ( T v n ) ( s ) , ( S v n ) ( s ) ) d s < ϵ ( n = 1 , 2 , 3 , ) , (38)

k = j + 1 I k ( v n ( t k ) ) < ϵ ( n = 1 , 2 , 3 , ) . (39)

It follows from (37), (38), (39), and ([8], Theorem 1.2.3) that

α ( W ( t ) ) 1 β + γ 1 { 2 η τ α ( f ( s , V ( s ) , ( T V ) ( s ) , ( S V ) ( s ) ) ) d s + 2 ϵ + 2 ( 1 γ ) 0 η α ( f ( s , V ( s ) , ( T V ) ( s ) , ( S V ) ( s ) ) ) d s + k = m j α ( I k ( V ( t k ) ) ) + 2 ϵ + ( 1 γ ) k = 1 m 1 α ( I k ( V ( t k ) ) ) } + 2 0 t α ( f ( s , V ( s ) , ( T V ) ( s ) , ( S V ) ( s ) ) ) d s + k = 1 i 1 α ( I k ( V ( t k ) ) ) , t J ¯ i , (40)

where W ( t ) = { w n ( t ) : n = 1 , 2 , 3 , } , V ( s ) = { v n ( s ) : n = 1 , 2 , 3 , } , ( T V ) ( s ) = { ( T v n ) ( s ) : n = 1 , 2 , 3 , } , ( S V ) ( s ) = { ( S v n ) ( s ) : n = 1 , 2 , 3 , } and α ( U ) denotes the Kuratowski measure of noncompactness of bounded set U E (see [[8], Section 1.2]). Since V ( s ) P p q and ( T V ) ( s ) , ( S V ) ( s ) P q for s J , where p = N 1 β 1 ( 1 γ ) p and q = max { q , k q , h q } , we see that, by condition ( H 4 ),

α ( f ( s , V ( s ) , ( T V ) ( s ) , ( S V ) ( s ) ) ) = 0 , s J (41)

and

α ( I k ( V ( t k ) ) ) = 0 ( k = 1 , 2 , 3 , ) . (42)

It follows from (40) to (42) that

α ( W ( t ) ) 4 ϵ β + γ 1 , t J ¯ i ,

which implies by virtue of the arbitrariness of ϵ that α ( W ( t ) ) = 0 for t J ¯ i .

By the Ascoli-Arzela theorem (see [[8], Theorem 1.2.5]), we conclude that W = { w n : n = 1 , 2 , 3 , } is relatively compact in C [ J ¯ i , E ] , hence, { w n ( t ) } has a subsequence which is convergent uniformly on J ¯ i , so, { ( A v n ) ( t ) } has a subsequence which is convergent uniformly on J i . Since i may be any positive integer, so, by diagonal method, we can choose a subsequence { ( A v n i ) ( t ) } of { ( A v n ) ( t ) } such that { ( A v n i ) ( t ) } is convergent uniformly on each J k ( k = 1 , 2 , 3 , ). Let

lim i ( A v n i ) ( t ) = w ( t ) , t J .

It is clear that w P C [ J , P ] . By (33), we have

A v n i B β + γ β + γ 1 ( M 1 a + D 1 γ ) ( i = 1 , 2 , 3 , ) ,

which implies that w B P C [ J , P ] and

w B β + γ β + γ 1 ( M 1 a + D 1 γ ) .

Let ϵ > 0 be arbitrarily given and choose a sufficiently large positive number τ such that

M 1 τ a ( s ) d s + D 1 t k τ γ k < ϵ . (43)

For any τ < t < , we have, by (6),

( A v n i ) ( t ) ( A v n i ) ( τ ) = τ t f ( s , v n i ( s ) , ( T v n i ) ( s ) , ( S v n i ) ( s ) ) d s + τ t k < t I k ( v n i ( t ) ) ( i = 1 , 2 , 3 , ) ,

which implies by virtue of (31), (32), and (43) that

( A v n i ) ( t ) ( A v n i ) ( τ ) M 1 τ t a ( s ) d s + D 1 τ t k < t γ k < ϵ ( i = 1 , 2 , 3 , ) . (44)

Letting i in (44), we get

w ( t ) w ( τ ) ϵ , t > τ . (45)

On the other hand, since { ( A v n i ) ( t ) } converges uniformly to w ( t ) on [ 0 , τ ] as i , there exists a positive integer i 0 such that

( A v n i ) ( t ) w ( t ) < ϵ , t [ 0 , τ ] , i > i 0 . (46)

It follows from (44) to (46) that

( A v n i ) ( t ) w ( t ) ( A v n i ) ( t ) ( A v n i ) ( τ ) + ( A v n i ) ( τ ) w ( τ ) + w ( τ ) w ( t ) < 3 ϵ , t > τ , i > i 0 . (47)

By (46) and (47), we have

A v n i w B 3 ϵ , i > i 0 ,

hence, A v n i w B 0 as i , and the relative compactness of A ( Q p q ) is proved. □

Lemma 2Let conePbe normal and conditions ( H 1 )-( H 4 ) be satisfied. Then u Q + C 1 [ J + , E ] is a positive solution of the infinite three-point boundary value problem (1) if and only if u Q + is a solution of the following impulsive integral equation:

u ( t ) = 1 β + γ 1 { η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + ( 1 γ ) 0 η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + k = m I k ( u ( t k ) ) + ( 1 γ ) k = 1 m 1 I k ( u ( t k ) ) } + 0 t f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + 0 < t k < t I k ( u ( t k ) ) , t J , (48)

i.e., uis a fixed point of operatorAdefined by (6) in Q + .

Proof For u P C [ J , E ] C 1 [ J + , E ] , it is easy to get the following formula:

u ( t ) = u ( 0 ) + 0 t u ( s ) d s + 0 < t k < t [ u ( t k + ) u ( t k ) ] , t J . (49)

Let u Q + C 1 [ J + , E ] be a positive solution of the infinite three-point boundary value problem (1). By (1) and (49), we have

u ( t ) = u ( 0 ) + 0 t f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + 0 < t k < t I k ( u ( t k ) ) , t J . (50)

We have shown in the proof of Lemma 1 that the infinite integral (10) and the infinite series (13) are convergent, so, by taking limits as t in both sides of (50), we get

u ( ) = u ( 0 ) + 0 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + k = 1 I k ( u ( t k ) ) . (51)

On the other hand, by (1) and (50), we have

u ( ) = γ u ( η ) + β u ( 0 ) (52)

and

u ( η ) = u ( 0 ) + 0 η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + k = 1 m 1 I k ( u ( t k ) ) . (53)

It follows from (51) to (53) that

u ( 0 ) = 1 β + γ 1 { η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + ( 1 γ ) 0 η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + k = m I k ( u ( t k ) ) + ( 1 γ ) k = 1 m 1 I k ( u ( t k ) ) } ,

and, substituting it into (50), we see that u ( t ) satisfies equation (48), i.e., u = A u .

Conversely, assume that u Q + is a solution of Equation (48). We have, by (48),

u ( 0 ) = 1 β + γ 1 { η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + ( 1 γ ) 0 η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + k = m I k ( u ( t k ) ) + ( 1 γ ) k = 1 m 1 I k ( u ( t k ) ) } (54)

and

u ( η ) = 1 β + γ 1 { η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + ( 1 γ ) 0 η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + k = m I k ( u ( t k ) ) + ( 1 γ ) k = 1 m 1 I k ( u ( t k ) ) } + 0 η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + k = 1 m 1 I k ( u ( t k ) ) . (55)

Moreover, by taking limits as t in (48), we see that u ( ) exists and

u ( ) = 1 β + γ 1 { η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + ( 1 γ ) 0 η f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + k = m I k ( u ( t k ) ) + ( 1 γ ) k = 1 m 1 I k ( u ( t k ) ) } + 0 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + k = 1 I k ( u ( t k ) ) . (56)

It follows from (54) to (56) that

γ u ( η ) + β u ( 0 ) = u ( ) .

On the other hand, direct differentiation of (48) gives

u ( t ) = f ( t , u ( t ) , ( T u ) ( t ) , ( S u ) ( t ) ) , t J + ,

and, it is clear, by (48),

Δ u | t = t k = I k ( u ( t k ) ) ( k = 1 , 2 , 3 , ) .

Hence, u C 1 [ J + , E ] and u ( t ) satisfies (1). Since u Q + , so (7) holds and u B > 0 , hence u ( t ) > θ for t J . Consequently, u ( t ) is a positive solution of the infinite three-point boundary value problem (1). □

Lemma 3 (The fixed-point theorem of cone expansion and compression with norm type; see [[6], Theorem 3] or [[7], Theorem 2.3.4])

LetPbe a cone in real Banach spaceEand Ω 1 , Ω 2 be two bounded open sets inEsuch that θ Ω 1 , Ω ¯ 1 Ω 2 , and operator A : P ( Ω ¯ 2 Ω 1 ) P be completely continuous, whereθdenotes the zero element ofEand Ω ¯ i denotes the closure of Ω i ( i = 1 , 2 ). Suppose that one of the following two conditions is satisfied:

( a ) A x x , x P Ω 1 ; A x x , x P Ω 2 ,

where Ω i denotes the boundary of Ω i ( i = 1 , 2 ).

( b ) A x x , x P Ω 1 ; A x x , x P Ω 2 .

ThenAhas at least one fixed point in P ( Ω ¯ 2 Ω 1 ) .

Remark 1 Lemma 3 is different from the Krasnoselskii fixed-point theorem of cone expansion and compression (see [[9], Theorem 44.1]). In Krasnoselskii’s theorem, the condition corresponding to (a) is

( a ) A x ≱ x , x P Ω 1 , A x ≰ x , x P Ω 2 .

It is clear, conditions (a) and (a′) are independent each other. On the other hand, in Krasnoselskii’s theorem, Ω 1 and Ω 2 are balls with center θ.

3 Main theorems

Let us list more conditions.

( H 5 ) There exist u 0 P + , b C [ J + , R + + ] and τ C [ P + , R + ] such that

f ( t , u , v , w ) b ( t ) τ ( u ) u 0 , t J + , u P + , v , w P ,

and

τ ( u ) u as  u P + , u ,

and

b = 0 b ( t ) d t < .

Remark 2 Condition ( H 5 ) means that f ( t , u , v , w ) is superlinear with respect to u.

( H 6 ) There exist u 1 P + , c C [ J + , R + + ] and σ C [ P + , R + ] such that

f ( t , u , v , w ) c ( t ) σ ( u ) u 1 , t J + , u P + , v , w P ,

and

σ ( u ) as  u P + , u 0 ,

and

c = 0 c ( t ) d t < .

Theorem 1Let conePbe normal and conditions ( H 1 )-( H 6 ) be satisfied. Assume that there exists a ξ > 0 such that

N ( β + γ ) β + γ 1 ( M ξ a + D ξ γ ) < ξ , (57)

whereNdenotes the normal constant ofP, and

M ξ = max { g ( x , y , z ) : N 1 β 1 ( 1 γ ) ξ x ξ , 0 y k ξ , 0 z h ξ } , (58)

D ξ = max { F ( x ) : N 1 β 1 ( 1 γ ) ξ x ξ } (59)

(for g ( x , y , z ) , F ( x ) , a and γ ; see conditions ( H 2 ) and ( H 3 )). Then the infinite three-point boundary value problem (1) has at least two positive solutions u , u Q + C 1 [ J + , E ] such that 0 < u B < ξ < u B .

Proof By Lemma 1 and Lemma 2, operator A defined by (6) is continuous from Q + into Q and we need to prove that A has two fixed points u and u in Q + such that 0 < u B < ξ < u B .

By condition ( H 5 ), there exists a r 1 > 0 such that

τ ( u ) β ( β + γ 1 ) N 2 ( 1 γ ) 2 b u 0 u , u P + , u r 1 , (60)

so,

f ( t , u , v , w ) β ( β + γ 1 ) N 2 u ( 1 γ ) 2 b u 0 b ( t ) u 0 , t J + , u P + , v , w P , u r 1 . (61)

Choose

r 2 > max { N β ( 1 γ ) 1 r 1 , ξ } . (62)

For u Q , u B = r 2 , we have by (7) and (62),

u ( t ) N 1 β 1 ( 1 γ ) u B = N 1 β 1 ( 1 γ ) r 2 > r 1 , t J , (63)

so, (6), (63), (61), and (7) imply

( A u ) ( t ) 1 γ β + γ 1 ( 0 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s ) β N 2 ( 1 γ ) b u 0 ( 0 u ( s ) b ( s ) d s ) u 0 N u B b u 0 ( 0 b ( s ) d s ) u 0 = N u B u 0 u 0 , t J , (64)

and consequently,

A u B u B , u Q , u B = r 2 . (65)

By condition ( H 6 ), there exists r 3 > 0 such that

σ ( u ) ( β + γ 1 ) N ξ ( 1 γ ) c u 1 , u P + , 0 < u < r 3 , (66)

so,

f ( t , u , v , w ) ( β + γ 1 ) N ξ ( 1 γ ) c u 1 c ( t ) u 1 , t J + , u P + , v , w P , 0 < u < r 3 . (67)

Choose

0 < r 4 < min { r 3 , ξ } . (68)

For u Q , u B = r 4 , we have by (68) and (7),

r 3 > u ( t ) N 1 β 1 ( 1 γ ) u B = N 1 β 1 ( 1 γ ) r 4 > 0 , (69)

so, we get by (6), (69), and (67),

( A u ) ( t ) 1 γ β + γ 1 ( 0 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s ) N ξ c u 1 ( 0 c ( s ) d s ) u 1 = N ξ u 1 u 1 , t J ,

which implies

( A u ) ( t ) ξ > r 4 , t J ,

and consequently,

A u B > u B , u Q , u B = r 4 . (70)

On the other hand, for u Q , u B = ξ , by condition ( H 2 ), condition ( H 3 ), (58), and (59), we have

f ( t , u ( t ) , ( T u ) ( t ) , ( S u ) ( t ) ) M ξ a ( t ) , t J + (71)

and

I k ( u ( t k ) ) D ξ γ k ( k = 1 , 2 , 3 , ) . (72)

It is clear, by (17),

( A u ) ( t ) β + γ β + γ 1 ( 0 f ( s , u ( s ) , ( T u ) ( s ) , ( S u ) ( s ) ) d s + k = 1 I k ( u ( t k ) ) ) , t J . (73)

It follows from (71) to (73) that

A u B N ( β + γ ) β + γ 1 ( M ξ a + D ξ γ ) . (74)

Thus, (74) and (57) imply

A u B < u B , u Q , u B = ξ . (75)

From (62) and (68), we know 0 < r 4 < ξ < r 2 , and by Lemma 1, A : Q r 4 r 2 Q is completely continuous, where Q r 4 r 2 = { u Q : r 4 u B r 2 } , hence, (65), (70), (75), and Lemma 3 imply that A has two fixed points u , u Q + such that r 4 < u B < ξ < u B r 2 . The proof is complete. □

Theorem 2Let conePbe normal and conditions ( H 1 )-( H 4 ) and ( H 6 ) be satisfied. Assume that

g ( x , y , z ) x + y + z 0 as   x (76)

uniformly for y , z R + , and

F ( x ) x 0 as   x (77)

(for g ( x , y , z ) and F ( x ) , see conditions ( H 2 ) and ( H 3 )). Then the infinite three-point boundary value problem (1) has at least one positive solution u Q + C 1 [ J + , E ] .

Proof As in the proof of Theorem 1, we can choose r 4 > 0 such that (70) holds (in this case, we put ξ = 1 in (66) and (68)). On the other hand, by (76) and (77), there exists r 5 > 0 such that

g ( x , y , z ) ϵ 0 ( x + y + z ) , x > r 5 , y 0 , z 0 (78)

and

F ( x ) ϵ 0 x , x > r 5 , (79)

where

ϵ 0 = β + γ 1 N ( β + γ ) [ ( 1 + k + h ) a + γ ] . (80)

Choose

r 6 > max { N β ( 1 γ ) 1 r 5 , r 4 } . (81)

For u Q , u B = r 6 , we have by (7) and (81),

u ( t ) N 1 β 1 ( 1 γ ) r 6 > r 5 , t J

so, (78) and (79) imply

g ( u ( t ) , ( T u ) ( t ) , ( S u ) ( t ) ) ϵ 0 ( u ( t ) + ( T u ) ( t ) + ( S u ) ( t ) ) ϵ 0 ( 1 + k + h ) r 6 , t J (82)

and

F ( u ( t k ) ) ϵ 0 u ( t k ) ϵ 0 r 6 ( k = 1 , 2 , 3 , ) . (83)

It follows from (73), conditions ( H 2 ), condition ( H 3 ), (82), (83), and (80) that

( A u ) ( t ) N ( β + γ ) β + γ 1 { ϵ 0 ( 1 + k + h ) r 6 0 a ( s ) d s + ϵ 0 r 6 k = 1 γ k } = N ( β + γ ) ϵ 0 r 6 β + γ 1 { ( 1 + k + h ) a + γ } = r 6 , t J ,

and consequently,

A u B u B , u Q , u B = r 6 . (84)

Since r 6 > r 4 by virtue of (81), we conclude from (70), (84), and Lemma 3 that A has a fixed point u Q + such that r 4 < u B r 6 . The theorem is proved. □

Example 1

Consider the infinite system of scalar first-order impulsive singular integro-differential equations of mixed type on the half line:

{ u n ( t ) = e 2 t 20 n 2 t { 1 8 ( u n + 1 ( t ) + m = 1 u m ( t ) ) 2 + 1 9 ( m = 1 u m ( t ) ) 1 } u n ( t ) = + e 3 t 18 n 3 t { ( 0 t e ( t + 1 ) s u n ( s ) d s ) 2 + 1 2 ( 0 u n + 2 ( s ) d s ( 1 + t + s ) 2 ) 3 } , 0 < t < , t k ( k = 1 , 2 , 3 , ; n = 1 , 2 , 3 , ) , Δ u n | t = k = 5 k 16 n 2 { u 2 n ( k ) + 1 3 ( m = 1 u m ( k ) ) 1 2 } ( k = 1 , 2 , 3 , ; n = 1 , 2 , 3 , ) , u n ( ) = 1 2 u n ( 9 2 ) + 6 u n ( 0 ) ( n = 1 , 2 , 3 , ) . (85)

Conclusion Infinite system (85) has at least two positive solutions { u n ( t ) } ( n = 1 , 2 , 3 , ) and { u n ( t ) } ( n = 1 , 2 , 3 , ) such that

0 < inf 0 t < n = 1 u n ( t ) sup 0 t < n = 1 u n ( t ) < 1 < sup 0 t < n = 1 u n ( t ) , inf 0 t < n = 1 u n ( t ) > 0 .

Proof Let E = l 1 = { u = ( u 1 , , u n , ) : n = 1 | u n | < } with norm u = n = 1 | u n | and P = { ( u 1 , , u n , ) : u n 0 , n = 1 , 2 , 3 , } . Then P is a normal cone in E with normal constant N = 1 , and infinite system (85) can be regarded as an infinite three-point boundary value problem of form (1). In this situation, u = ( u 1 , , u n , ) , v = ( v 1 , , v n , ) , w = ( w 1 , , w n , ) , t k = k ( k = 1 , 2 , 3 , ), K ( t , s ) = e ( t + 1 ) s , H ( t , s ) = ( 1 + t + s ) 2 , η = 9 2 , γ = 1 2 , β = 6 , f = ( f 1 , , f n , ) and I k = ( I k 1 , , I k n , ) , in which

f n ( t , u , v , w ) = e 2 t 20 n 2 t { 1 8 ( u n + 1 + m = 1 u m ) 2 + 1 9 ( m = 1 u m ) 1 } + e 3 t 18 n 3 t ( v n 2 + 1 2 w n + 2 3 ) , t J + = ( 0 , ) , u P + = { u P : u > 0 } , v , w P ( n = 1 , 2 , 3 , ) (86)

and

I k n ( u ) = 5 k 16 n 2 { u 2 n + 1 3 ( m = 1 u m ) 1 2 } , u P + ( k = 1 , 2 , 3 , ; n = 1 , 2 , 3 , ) . (87)

It is easy to see that f C [ J + × P + × P × P , P ] , I k C [ P + , P ] ( k = 1 , 2 , 3 , ) and condition ( H 1 ) is satisfied and k 1 , h 1 . We have, by (86),

0 f n ( t , u , v , w ) e 2 t 20 n 2 t { 1 8 ( 2 u ) 2 + 1 9 u 1 } + e 3 t 18 n 3 t ( v 2 + 1 2 w 3 ) e 2 t n 2 t ( 1 40 u 2 + 1 180 u 1 + 1 18 v 2 + 1 36 w 3 ) , t J + , u P + , v , w P ( n = 1 , 2 , 3 , ) , (88)

so, observing the inequality n = 1 1 n 2 < 2 , we get

f ( t , u , v , w ) = n = 1 f n ( t , u , v , w ) e 2 t t ( 1 20 u 2 + 1 90 u 1 + 1 9 v 2 + 1 18 w 3 ) , t J + , u P + , v , w P ,

which implies that condition ( H 2 ) is satisfied for

a ( t ) = e 2 t t

and

g ( x , y , z ) = 1 20 x 2 + 1 90 x + 1 9 y 2 + 1 18 z 3

with

a = 0 e 2 t t d t < 0 1 d t t + 1 e 2 t d t = 2 + 1 2 e 2 < 29 14 .

By (87), we have

0 I k n ( u ) 5 k 16 n 2 ( u + 1 3 u 1 2 ) , u P + ( k = 1 , 2 , 3 , ; n = 1 , 2 , 3 , ) , (89)

so,

I k ( u ) 1 8 5 k ( u + 1 3 u 1 2 ) , u P + ( k = 1 , 2 , 3 , ) ,

which implies that condition ( H 3 ) is satisfied for γ k = 1 8 5 k ( γ = 1 32 ) and

F ( x ) = x + 1 3 x .

On the other hand, (86) implies

f n ( t , u , v , w ) e 2 t 160 n 2 t u 2 , t J + , u P + , v , w P ( n = 1 , 2 , 3 , )

and

f n ( t , u , v , w ) e 2 t 180 n 2 t u 1 , t J + , u P + , v , w P ( n = 1 , 2 , 3 , ) , (90)

so, we see that condition ( H 5 ) is satisfied for b ( t ) = e 2 t 160 t ( b < 29 2 , 240 ), τ ( u ) = u 2 and u 0 = ( 1 , , 1 n 2 , ) and condition ( H 6 ) is satisfied for c ( t ) = e 2 t 180 t ( c < 29 2 , 520 ), σ ( u ) = u 1 and u 1 = ( 1 , , 1 n 2 , ) . In addition, from (90), we have

f ( t , u , v , w ) ( n = 1 1 n 2 ) e 2 t 160 t u 1 > e 2 t 160 t u 1 , t J + , u P + , v , w P ,

which implies that (3) and (4) hold, i.e., f ( t , u , v , w ) is singular at t = 0 and u = θ