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Solutions and nonnegative solutions for a weighted variable exponent impulsive integro-differential system with multi-point and integral mixed boundary value problems

Rong Dong and Qihu Zhang*

Author Affiliations

Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan, 450002, China

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Boundary Value Problems 2013, 2013:161  doi:10.1186/1687-2770-2013-161


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/161


Received:19 March 2013
Accepted:18 June 2013
Published:5 July 2013

© 2013 Dong and Zhang; 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

This paper investigates the existence of solutions for a weighted p ( t ) -Laplacian impulsive integro-differential system with multi-point and integral mixed boundary value problems via Leray-Schauder’s degree; sufficient conditions for the existence of solutions are given. Moreover, we get the existence of nonnegative solutions.

MSC: 34B37.

Keywords:
weighted p ( t ) -Laplacian; impulsive integro-differential system; Leray-Schauder’s degree

1 Introduction

In this paper, we consider the existence of solutions and nonnegative solutions for the following weighted p ( t ) -Laplacian integro-differential system:

p ( t ) u + f ( t , u , ( w ( t ) ) 1 p ( t ) 1 u , S ( u ) , T ( u ) ) = 0 , t ( 0 , 1 ) , t t i , (1)

where u : [ 0 , 1 ] R N , f ( , , , , ) : [ 0 , 1 ] × R N × R N × R N × R N R N , t i ( 0 , 1 ) , i = 1 , , k , with the following impulsive boundary value conditions:

lim t t i + u ( t ) lim t t i u ( t ) = A i ( lim t t i u ( t ) , lim t t i ( w ( t ) ) 1 p ( t ) 1 u ( t ) ) , i = 1 , , k , (2)

lim t t i + w ( t ) | u | p ( t ) 2 u ( t ) lim t t i w ( t ) | u | p ( t ) 2 u ( t ) = B i ( lim t t i u ( t ) , lim t t i ( w ( t ) ) 1 p ( t ) 1 u ( t ) ) , i = 1 , , k , (3)

u ( 0 ) = 0 1 g ( t ) u ( t ) d t , u ( 1 ) = = 1 m 2 α u ( ξ ) 0 1 h ( t ) u ( t ) d t , (4)

where p C ( [ 0 , 1 ] , R ) and p ( t ) > 1 , p ( t ) u : = ( w ( t ) | u | p ( t ) 2 u ) is called the weighted p ( t ) -Laplacian; 0 < t 1 < t 2 < < t k < 1 , 0 < ξ 1 < < ξ m 2 < 1 ; α 0 ( = 1 , , m 2 ); g L 1 [ 0 , 1 ] is nonnegative, 0 1 g ( t ) d t = σ [ 0 , 1 ] ; h L 1 [ 0 , 1 ] , 0 1 h ( t ) d t = δ ; A i , B i C ( R N × R N , R N ) ; T and S are linear operators defined by ( S u ) ( t ) = 0 1 h ( t , s ) u ( s ) d s , ( T u ) ( t ) = 0 t k ( t , s ) u ( s ) d s , t [ 0 , 1 ] , where k , h C ( [ 0 , 1 ] × [ 0 , 1 ] , R ) .

If σ < 1 and = 1 m 2 α δ 1 , we say the problem is nonresonant, but if σ = 1 or = 1 m 2 α δ = 1 , we say the problem is resonant.

Throughout the paper, o ( 1 ) means functions which are uniformly convergent to 0 (as n + ); for any v R N , v j will denote the jth component of v; the inner product in R N will be denoted by , , | | will denote the absolute value and the Euclidean norm on R N . Denote J = [ 0 , 1 ] , J = ( 0 , 1 ) { t 1 , , t k } , J 0 = [ t 0 , t 1 ] , J i = ( t i , t i + 1 ] , i = 1 , , k , where t 0 = 0 , t k + 1 = 1 . Denote by J i o the interior of J i , i = 0 , 1 , , k . Let

P C ( J , R N ) = { x : J R N | x C ( J i , R N ) , i = 0 , 1 , , k and  lim t t i + x ( t )  exists for  i = 1 , , k } ,

w P C ( J , R ) satisfy 0 < w ( t ) , t ( 0 , 1 ) { t 1 , , t k } , and ( w ( t ) ) 1 p ( t ) 1 L 1 ( 0 , 1 ) ,

P C 1 ( J , R N ) = { x P C ( J , R N ) | x C ( J i o , R N ) , lim t t i + ( w ( t ) ) 1 p ( t ) 1 x ( t ) and  lim t t i + 1 ( w ( t ) ) 1 p ( t ) 1 x ( t )  exists for  i = 0 , 1 , , k } .

For any x = ( x 1 , , x N ) P C ( J , R N ) , denote | x i | 0 = sup { | x i ( t ) | t J } .

Obviously, P C ( J , R N ) is a Banach space with the norm x 0 = ( i = 1 N | x i | 0 2 ) 1 2 , and P C 1 ( J , R N ) is a Banach space with the norm x 1 = x 0 + ( w ( t ) ) 1 p ( t ) 1 x 0 . Denote L 1 = L 1 ( J , R N ) with the norm

x L 1 = ( i = 1 N | x i | L 1 2 ) 1 2 , x L 1 ,  where  | x i | L 1 = 0 1 | x i ( t ) | d t .

In the following, P C ( J , R N ) and P C 1 ( J , R N ) will be simply denoted by PC and P C 1 , respectively. We denote

u ( t i + ) = lim t t i + u ( t ) , u ( t i ) = lim t t i u ( t ) , w ( 0 ) | u | p ( 0 ) 2 u ( 0 ) = lim t 0 + w ( t ) | u | p ( t ) 2 u ( t ) , w ( 1 ) | u | p ( 1 ) 2 u ( 1 ) = lim t 1 w ( t ) | u | p ( t ) 2 u ( t ) , A i = A i ( lim t t i u ( t ) , lim t t i ( w ( t ) ) 1 p ( t ) 1 u ( t ) ) , i = 1 , , k , B i = B i ( lim t t i u ( t ) , lim t t i ( w ( t ) ) 1 p ( t ) 1 u ( t ) ) , i = 1 , , k .

The study of differential equations and variational problems with nonstandard p ( t ) -growth conditions has attracted more and more interest in recent years (see [1-4]). The applied background of these kinds of problems includes nonlinear elasticity theory [4], electro-rheological fluids [1,3], and image processing [2]. Many results have been obtained on these kinds of problems; see, for example, [5-15]. Recently, the applications of variable exponent analysis in image restoration have attracted more and more attention [16-19]. If p ( t ) p (a constant), (1)-(4) becomes the well-known p-Laplacian problem. If p ( t ) is a general function, one can see easily p ( t ) c u c p ( t ) 1 ( p ( t ) u ) in general, but p c u = c p 1 ( p u ) , so p ( t ) represents a non-homogeneity and possesses more nonlinearity, thus p ( t ) is more complicated than p . For example:

(a) If Ω R N is a bounded domain, the Rayleigh quotient

λ p ( x ) = inf u W 0 1 , p ( x ) ( Ω ) { 0 } Ω 1 p ( x ) | u | p ( x ) d x Ω 1 p ( x ) | u | p ( x ) d x

is zero in general, and only under some special conditions λ p ( x ) > 0 (see [9]), when Ω R ( N = 1 ) is an interval, the results show that λ p ( x ) > 0 if and only if p ( x ) is monotone. But the property of λ p > 0 is very important in the study of p-Laplacian problems, for example, in [20], the authors use this property to deal with the existence of solutions.

(b) If w ( t ) 1 and p ( t ) p (a constant) and p u > 0 , then u is concave, this property is used extensively in the study of one-dimensional p-Laplacian problems (see [21]), but it is invalid for p ( t ) . It is another difference between p and p ( t ) .

In recent years, many results have been devoted to the existence of solutions for the Laplacian impulsive differential equation boundary value problems; see, for example, [22-29]. There are some methods to deal with these problems, for example, sub-super-solution method, fixed point theorem, monotone iterative method, coincidence degree. Because of the nonlinear property of p , results on the existence of solutions for p-Laplacian impulsive differential equation boundary value problems are rare (see [30-33]). In [34], using the coincidence degree method, the present author investigates the existence of solutions for p ( r ) -Laplacian impulsive differential equation with multi-point boundary value conditions, when the problem is nonresonant. Integral boundary conditions for evolution problems have various applications in chemical engineering, thermo-elasticity, underground water flow and population dynamics. There are many papers on the differential equations with integral boundary value problems; see, for example, [35-38].

In this paper, when p ( t ) is a general function, we investigate the existence of solutions and nonnegative solutions for the weighted p ( t ) -Laplacian impulsive integro-differential system with integral and multi-point boundary value conditions. Results on these kinds of problems are rare. Our results contain both of the cases of resonance and nonresonance. Our method is based upon Leray-Schauder’s degree. The homotopy transformation used in [34] is unsuitable for this paper. Moreover, this paper will consider the existence of (1) with (2), (4) and the following impulsive condition:

lim t t i + ( w ( t ) ) 1 p ( t ) 1 u ( t ) lim t t i ( w ( t ) ) 1 p ( t ) 1 u ( t ) = D i ( lim t t i u ( t ) , lim t t i ( w ( t ) ) 1 p ( t ) 1 u ( t ) ) , i = 1 , , k , (5)

where D i C ( R N × R N , R N ) , the impulsive condition (5) is called a linear impulsive condition (LI for short), and (3) is called a nonlinear impulsive condition (NLI for short). In general, p-Laplacian impulsive problems have two kinds of impulsive conditions, including LI and NLI; but Laplacian impulsive problems only have LI in general. It is another difference between p-Laplacian impulsive problems and Laplacian impulsive problems. Moreover, since the Rayleigh quotient λ p ( x ) = 0 in general and the p ( t ) -Laplacian is non-homogeneity, when we deal with the existence of solutions of variable exponent impulsive problems like (1)-(4), we usually need the nonlinear term that satisfies the sub- ( p 1 ) growth condition, but for the p-Laplacian impulsive problems, the nonlinear term only needs to satisfy the sub- ( p 1 ) growth condition.

Let N 1 , the function f : J × R N × R N × R N × R N R N is assumed to be Caratheodory, by which we mean:

(i) For almost every t J , the function f ( t , , , , ) is continuous;

(ii) For each ( x , y , s , z ) R N × R N × R N × R N , the function f ( , x , y , s , z ) is measurable on J;

(iii) For each R > 0 , there is a α R L 1 ( J , R ) such that, for almost every t J and every ( x , y , s , z ) R N × R N × R N × R N with | x | R , | y | R , | s | R , | z | R , one has

| f ( t , x , y , s , z ) | α R ( t ) .

We say a function u : J R N is a solution of (1) if u P C 1 with w ( t ) | u | p ( t ) 2 u absolutely continuous on J i o , i = 0 , 1 , , k , which satisfies (1) a.e. on J.

In this paper, we always use C i to denote positive constants, if it cannot lead to confusion. Denote

z = inf t J z ( t ) , z + = sup t J z ( t ) for any  z P C ( J , R ) .

We say f satisfies the sub- ( p 1 ) growth condition if f satisfies

lim | u | + | v | + | s | + | z | + f ( t , u , v , s , z ) ( | u | + | v | + | s | + | z | ) q ( t ) 1 = 0 for  t J  uniformly,

where q ( t ) P C ( J , R ) and 1 < q q + < p .

We will discuss the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) in the following three cases:

Case (i): σ < 1 , = 1 m 2 α δ = 1 ;

Case (ii): σ = 1 , = 1 m 2 α δ 1 ;

Case (iii): σ < 1 , = 1 m 2 α δ < 1 .

This paper is organized as five sections. In Section 2, we present some preliminaries and give the operator equation which has the same solutions of (1)-(4) in the three cases, respectively. In Section 3, we give the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) when σ < 1 , = 1 m 2 α δ = 1 . In Section 4, we give the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) when σ = 1 , = 1 m 2 α δ 1 . Finally, in Section 5, we give the existence of solutions and nonnegative solutions for system (1)-(4) or (1) with (2), (4) and (5) when σ < 1 , = 1 m 2 α δ < 1 .

2 Preliminary

For any ( t , x ) J × R N , denote φ ( t , x ) = | x | p ( t ) 2 x . Obviously, φ has the following properties.

Lemma 2.1 (see [34])

φis a continuous function and satisfies:

(i) For any t [ 0 , 1 ] , φ ( t , ) is strictly monotone, i.e.,

φ ( t , x 1 ) φ ( t , x 2 ) , x 1 x 2 > 0 for any   x 1 , x 2 R N , x 1 x 2 .

(ii) There exists a function α : [ 0 , + ) [ 0 , + ) , α ( s ) + as s + such that

φ ( t , x ) , x α ( | x | ) | x | for all   x R N .

It is well known that φ ( t , ) is a homeomorphism from R N to R N for any fixed t J . Denote

φ 1 ( t , x ) = | x | 2 p ( t ) p ( t ) 1 x for  x R N { 0 } , φ 1 ( t , 0 ) = 0 , t J .

It is clear that φ 1 ( t , ) is continuous and sends bounded sets to bounded sets.

In this section, we will do some preparation and give the operator equation which has the same solutions of (1)-(4) in three cases, respectively. At first, let us now consider the following simple impulsive problem with boundary value condition (4):

( w ( t ) φ ( t , u ( t ) ) ) = f ( t ) , t ( 0 , 1 ) , t t i , lim t t i + u ( t ) lim t t i u ( t ) = a i , i = 1 , , k , lim t t i + w ( t ) | u | p ( t ) 2 u ( t ) lim t t i w ( t ) | u | p ( t ) 2 u ( t ) = b i , i = 1 , , k , } (6)

where a i , b i R N ; f L 1 .

Denote a = ( a 1 , , a k ) , b = ( b 1 , , b k ) . Obviously, a , b R k N .

We will discuss it in three cases, respectively.

2.1 Case (i)

Suppose that σ < 1 and = 1 m 2 α δ = 1 . If u is a solution of (6) with (4), we have

w ( t ) φ ( t , u ( t ) ) = w ( 0 ) φ ( 0 , u ( 0 ) ) + t i < t b i + 0 t f ( s ) d s , t J . (7)

Denote ρ 1 = w ( 0 ) φ ( 0 , u ( 0 ) ) . It is easy to see that ρ 1 is dependent on a, b and f ( ) . Define the operator F : L 1 P C as

F ( f ) ( t ) = 0 t f ( s ) d s , t J , f L 1 .

By solving for u in (7) and integrating, we find

u ( t ) = u ( 0 ) + t i < t a i + F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 1 + t i < t b i + F ( f ) ( t ) ) ] } ( t ) , t J ,

which together with boundary value condition (4) implies

u ( 0 ) = 1 ( 1 σ ) 0 1 g ( t ) ( F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 1 + t i < t b i + F ( f ) ( t ) ) ] } ( t ) + t i < t a i ) d t ,

and

= 1 m 2 α { t i < ξ a i + 0 ξ φ 1 [ t , ( w ( t ) ) 1 ( ρ 1 + t i < t b i + F ( f ) ( t ) ) ] d t } i = 1 k a i 0 1 φ 1 [ t , ( w ( t ) ) 1 ( ρ 1 + t i < t b i + F ( f ) ( t ) ) ] d t 0 1 h ( t ) ( F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 1 + t i < t b i + F ( f ) ( t ) ) ] } ( t ) + t i < t a i ) d t = 0 .

Denote W = R 2 k N × L 1 with the norm

ω = i = 1 k | a i | + i = 1 k | b i | + ϑ L 1 , ω = ( a , b , ϑ ) W ,

then W is a Banach space.

For any ω W , we denote

Λ ω ( ρ 1 ) = = 1 m 2 α { t i < ξ a i + 0 ξ φ 1 [ t , ( w ( t ) ) 1 ( ρ 1 + t i < t b i + F ( ϑ ) ( t ) ) ] d t } i = 1 k a i 0 1 φ 1 [ t , ( w ( t ) ) 1 ( ρ 1 + t i < t b i + F ( ϑ ) ( t ) ) ] d t 0 1 h ( t ) ( F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 1 + t i < t b i + F ( ϑ ) ( t ) ) ] } ( t ) + t i < t a i ) d t .

Denote ξ m 1 = 1 . Then

Λ ω ( ρ 1 ) = = 1 m 2 α { ξ t i a i + ξ 1 φ 1 [ t , ( w ( t ) ) 1 ( ρ 1 + t i < t b i + F ( ϑ ) ( t ) ) ] d t } + 0 1 h ( t ) ( t 1 φ 1 [ t , ( w ( t ) ) 1 ( ρ 1 + t i < t b i + F ( ϑ ) ( t ) ) ] d t + t i t a i ) d t = = 1 m 2 ( α ξ ξ + 1 h ( t ) d t ) ξ 1 φ 1 [ t , ( w ( t ) ) 1 ( ρ 1 + t i < t b i + F ( ϑ ) ( t ) ) ] d t = 1 m 2 ξ ξ + 1 h ( t ) ξ t φ 1 [ s , ( w ( s ) ) 1 ( ρ 1 + s i < s b i + F ( ϑ ) ( s ) ) ] d s d t + 0 ξ 1 h ( t ) t 1 φ 1 [ s , ( w ( s ) ) 1 ( ρ 1 + s i < s b i + F ( ϑ ) ( s ) ) ] d s d t = 1 m 2 α ξ t i a i + 0 1 h ( t ) t i t a i d t .

Throughout the paper, we denote

Lemma 2.2Suppose that h ( t ) 0 on [ ξ 1 , 1 ] , α ξ ξ + 1 h ( t ) d t ( = 1 , , m 2 ) and h ( t ) 0 on [ 0 , ξ 1 ] . Then the function Λ ω ( ) has the following properties:

(i) For any fixed ω W , the equation

Λ ω ( ρ 1 ) = 0 (8)

has a unique solution ρ 1 ˜ ( ω ) R N .

(ii) The function ρ 1 ˜ : W R N , defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, for any ω = ( a , b , ϑ ) W , we have

| ρ 1 ˜ ( ω ) | 3 N [ ( 2 N ) p + ( δ E + 1 E i = 1 k | a i | ) p # 1 + i = 1 k | b i | + ϑ L 1 ] ,

where the notation M p # 1 means

M p # 1 = { M p + 1 , M > 1 , M p 1 , M 1 .

Proof (i) From Lemma 2.1, it is immediate that

Λ ω ( x 1 ) Λ ω ( x 2 ) , x 1 x 2 < 0 for  x 1 x 2 , x 1 , x 2 R N ,

and hence, if (8) has a solution, then it is unique.

Set R 0 = 3 N [ ( 2 N ) p + ( δ E + 1 E i = 1 k | a i | ) p # 1 + i = 1 k | b i | + ϑ L 1 ] .

Suppose that | ρ 1 | > R 0 , it is easy to see that there exists some j 0 { 1 , , N } such that the absolute value of the j 0 th component of ρ 1 satisfies

Thus the j 0 th component of ρ 1 + t i < t b i + F ( ϑ ) ( t ) keeps sign on J, namely, for any t J , we have

Obviously, we have

then it is easy to see that the j 0 th component of Λ ω ( ρ 1 ) keeps the same sign of . Thus,

Λ ω ( ρ 1 ) 0 .

Let us consider the equation

λ Λ ω ( ρ 1 ) + ( 1 λ ) ρ 1 = 0 , λ [ 0 , 1 ] . (9)

According to the preceding discussion, all the solutions of (9) belong to b ( R 0 + 1 ) = { x R N | x | < R 0 + 1 } . Therefore

d B [ Λ ω ( ρ 1 ) , b ( R 0 + 1 ) , 0 ] = d B [ I , b ( R 0 + 1 ) , 0 ] 0 ,

it means the existence of solutions of Λ ω ( ρ 1 ) = 0 .

In this way, we define a function ρ 1 ˜ ( ω ) : W R N , which satisfies Λ ω ( ρ 1 ˜ ( ω ) ) = 0 .

(ii) By the proof of (i), we also obtain ρ 1 ˜ sends bounded sets to bounded sets, and

| ρ 1 ˜ ( ω ) | 3 N [ ( 2 N ) p + ( δ E + 1 E i = 1 k | a i | ) p # 1 + i = 1 k | b i | + ϑ L 1 ] .

It only remains to prove the continuity of ρ 1 ˜ . Let { ω n } be a convergent sequence in W and ω n ω , as n + . Since { ρ 1 ˜ ( ω n ) } is a bounded sequence, it contains a convergent subsequence { ρ 1 ˜ ( ω n j ) } . Suppose that ρ 1 ˜ ( ω n j ) ρ 0 as j + . Since Λ ω n j ( ρ 1 ˜ ( ω n j ) ) = 0 , letting j + , we have Λ ω ( ρ 0 ) = 0 , which together with (i) implies ρ 0 = ρ 1 ˜ ( ω ) , it means ρ 1 ˜ is continuous. This completes the proof. □

Now we denote by N f ( u ) : [ 0 , 1 ] × P C 1 L 1 the Nemytskii operator associated to f defined by

N f ( u ) ( t ) = f ( t , u ( t ) , ( w ( t ) ) 1 p ( t ) 1 u ( t ) , S ( u ) , T ( u ) ) on  J . (10)

We define ρ 1 : P C 1 R N as

ρ 1 ( u ) = ρ 1 ˜ ( A , B , N f ) ( u ) , (11)

where A = ( A 1 , , A k ) , B = ( B 1 , , B k ) .

It is clear that ρ 1 ( ) is continuous and sends bounded sets of P C 1 to bounded sets of R N , and hence it is compact continuous.

If u is a solution of (6) with (4), we have

u ( t ) = u ( 0 ) + t i < t a i + F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 1 ˜ ( ω ) + t i < t b i + F ( f ) ( t ) ) ] } ( t ) , t [ 0 , 1 ] .

For fixed a , b R k N , we denote K ( a , b ) : L 1 P C 1 as

K ( a , b ) ( ϑ ) ( t ) = F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 1 ˜ ( a , b , ϑ ) + t i < t b i + F ( ϑ ) ( t ) ) ] } ( t ) , t J .

Define K 1 : P C 1 P C 1 as

K 1 ( u ) ( t ) = F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 1 ( u ) + t i < t B i + F ( N f ( u ) ) ( t ) ) ] } ( t ) , t J .

Lemma 2.3 (i) The operator K ( a , b ) is continuous and sends equi-integrable sets in L 1 to relatively compact sets in P C 1 .

(ii) The operator K 1 is continuous and sends bounded sets in P C 1 to relatively compact sets in P C 1 .

Proof (i) It is easy to check that K ( a , b ) ( ϑ ) ( ) P C 1 , ϑ L 1 , a , b R k N . Since and

K ( a , b ) ( ϑ ) ( t ) = φ 1 [ t , ( w ( t ) ) 1 ( ρ 1 ˜ ( a , b , ϑ ) + t i < t b i + F ( ϑ ) ) ] , t [ 0 , 1 ] ,

it is easy to check that K ( a , b ) ( ) is a continuous operator from L 1 to P C 1 .

Let now U be an equi-integrable set in L 1 , then there exists α L 1 such that

| u ( t ) | α ( t ) a.e. in  J  for any  u L 1 .

We want to show that K ( a , b ) ( U ) ¯ P C 1 is a compact set.

Let { u n } be a sequence in K ( a , b ) ( U ) , then there exists a sequence { ϑ n } U such that u n = K ( a , b ) ( ϑ n ) . For any t 1 , t 2 J , we have

| F ( ϑ n ) ( t 1 ) F ( ϑ n ) ( t 2 ) | = | 0 t 1 ϑ n ( t ) d t 0 t 2 ϑ n ( t ) d t | = | t 1 t 2 ϑ n ( t ) d t | | t 1 t 2 α ( t ) d t | .

Hence the sequence { F ( ϑ n ) } is uniformly bounded and equi-continuous. By the Ascoli-Arzela theorem, there exists a subsequence of { F ( ϑ n ) } (which we rename the same) which is convergent in PC. According to the bounded continuity of the operator ρ 1 ˜ , we can choose a subsequence of { ρ 1 ˜ ( a , b , ϑ n ) + F ( ϑ n ) } (which we still denote { ρ 1 ˜ ( a , b , ϑ n ) + F ( ϑ n ) } ) which is convergent in PC, then is convergent in PC.

Since

K ( a , b ) ( ϑ n ) ( t ) = F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 1 ˜ ( a , b , ϑ n ) + t i < t b i + F ( ϑ n ) ) ] } ( t ) , t [ 0 , 1 ] ,

it follows from the continuity of φ 1 and the integrability of in L 1 that K ( a , b ) ( ϑ n ) is convergent in PC. Thus { u n } is convergent in P C 1 .

(ii) It is easy to see from (i) and Lemma 2.2.

This completes the proof. □

Let us define P 1 : P C 1 P C 1 as

P 1 ( u ) = 0 1 g ( t ) [ K 1 ( u ) ( t ) + t i < t A i ] d t 1 σ .

It is easy to see that P 1 is compact continuous.

Lemma 2.4Suppose that σ < 1 , = 1 m 2 α δ = 1 ; h ( t ) 0 on [ ξ 1 , 1 ] , α ξ ξ + 1 h ( t ) d t ( = 1 , , m 2 ) and h ( t ) 0 on [ 0 , ξ 1 ] . Thenuis a solution of (1)-(4) if and only ifuis a solution of the following abstract operator equation:

u = P 1 ( u ) + t i < t A i + K 1 ( u ) . (12)

Proof Suppose that u is a solution of (1)-(4). By integrating (1) from 0 to t, we find that

w ( t ) φ ( t , u ( t ) ) = ρ 1 ( u ) + t i < t B i + F ( N f ( u ) ) ( t ) , t ( 0 , 1 ) , t t 1 , , t k . (13)

It follows from (13) and (4) that

u ( t ) = u ( 0 ) + t i < t A i u ( t ) = + F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 1 ( u ) + t i < t B i + F ( N f ( u ) ) ) ] } ( t ) , t [ 0 , 1 ] , u ( 0 ) = 1 ( 1 σ ) u ( 0 ) = × 0 1 g ( t ) ( F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 1 ( u ) + t i < t B i + F ( N f ( u ) ) ) ] } ( t ) + t i < t A i ) d t u ( 0 ) = 0 1 g ( t ) [ K 1 ( u ) ( t ) + t i < t A i ] d t 1 σ = P 1 ( u ) . (14)

Combining the definition of ρ 1 , we can see

u = P 1 ( u ) + t i < t A i + K 1 ( u ) .

Conversely, if u is a solution of (12), then (2) is satisfied. It is easy to check that

u ( 0 ) = P 1 ( u ) = 0 1 g ( t ) [ K 1 ( u ) ( t ) + t i < t A i ] d t 1 σ , u ( 0 ) = σ u ( 0 ) + 0 1 g ( t ) [ K 1 ( u ) ( t ) + t i < t A i ] d t = 0 1 g ( t ) u ( t ) d t , (15)

and

u ( 1 ) = P 1 ( u ) + i = 1 k A i + K 1 ( u ) ( 1 ) .

By the condition of the mapping ρ 1 , we have

= 1 m 2 α { t i < ξ A i + 0 ξ φ 1 [ t , ( w ( t ) ) 1 ( ρ 1 + t i < t B i + F ( N f ( u ) ) ( t ) ) ] d t } i = 1 k A i 0 1 φ 1 [ t , ( w ( t ) ) 1 ( ρ 1 + t i < t B i + F ( N f ( u ) ) ( t ) ) ] d t 0 1 h ( t ) ( F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 1 + t i < t B i + F ( N f ( u ) ) ( t ) ) ] } ( t ) + t i < t A i ) d t = 0 .

Thus

u ( 1 ) = = 1 m 2 α u ( ξ ) 0 1 h ( t ) u ( t ) d t . (16)

It follows from (15) and (16) that (4) is satisfied.

From (12), we have

w ( t ) φ ( t , u ( t ) ) = ρ 1 ( u ) + t i < t B i + F ( N f ( u ) ) ( t ) , t ( 0 , 1 ) , t t i , ( w ( t ) φ ( t , u ) ) = N f ( u ) ( t ) , t ( 0 , 1 ) , t t i . (17)

It follows from (17) that (3) is satisfied.

Hence u is a solution of (1)-(4). This completes the proof. □

2.2 Case (ii)

Suppose that σ = 1 and = 1 m 2 α δ 1 . If u is a solution of (6) with (4), we have

w ( t ) φ ( t , u ( t ) ) = w ( 0 ) φ ( 0 , u ( 0 ) ) + t i < t b i + 0 t f ( s ) d s , t J .

Denote ρ 2 = w ( 0 ) φ ( 0 , u ( 0 ) ) . It is easy to see that ρ 2 is dependent on a, b and f ( ) . Boundary value condition (4) implies that

0 1 g ( t ) ( F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 2 + t i < t b i + F ( f ) ( t ) ) ] } ( t ) + t i < t a i ) d t = 0 , u ( 0 ) = = 1 m 2 α { t i < ξ a i + 0 ξ φ 1 [ t , ( w ( t ) ) 1 ( ρ 2 + t i < t b i + F ( f ) ( t ) ) ] d t } 1 i = 1 m 2 α + δ u ( 0 ) = i = 1 k a i + 0 1 φ 1 [ t , ( w ( t ) ) 1 ( ρ 2 + t i < t b i + F ( f ) ( t ) ) ] d t 1 = 1 m 2 α + δ u ( 0 ) = 0 1 h ( t ) ( F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 2 + t i < t b i + F ( f ) ( t ) ) ] } ( t ) + t i < t a i ) d t 1 = 1 m 2 α + δ .

For any ω W , we denote

Γ ω ( ρ 2 ) = 0 1 g ( t ) ( F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 2 + t i < t b i + F ( ϑ ) ( t ) ) ] } ( t ) + t i < t a i ) d t .

Throughout the paper, we denote .

Lemma 2.5The function Γ ω ( ) has the following properties:

(i) For any fixed ω W , the equation Γ ω ( ρ 2 ) = 0 has a unique solution ρ 2 ˜ ( ω ) R N .

(ii) The function ρ 2 ˜ : W R N , defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, for any ω = ( a , b , ϑ ) W , we have

| ρ 2 ˜ ( ω ) | 3 N [ ( 2 N ) p + ( E 1 + 1 E 1 i = 1 k | a i | ) p # 1 + i = 1 k | b i | + ϑ L 1 ] ,

where the notation M p # 1 means

M p # 1 = { M p + 1 , M > 1 , M p 1 , M 1 .

Proof Similar to the proof of Lemma 2.2, we omit it here. □

We define ρ 2 : P C 1 R N as ρ 2 ( u ) = ρ 2 ˜ ( A , B , N f ) ( u ) , where A = ( A 1 , , A k ) , B = ( B 1 , , B k ) .

It is clear that ρ 2 ( ) is continuous and sends bounded sets of P C 1 to bounded sets of R N , and hence it is compact continuous.

For fixed a , b R k N , we denote K ( a , b ) : L 1 P C 1 as

K ( a , b ) ( ϑ ) ( t ) = F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 2 ˜ ( a , b , ϑ ) + t i < t b i + F ( ϑ ) ( t ) ) ] } ( t ) , t J .

Define K 2 : P C 1 P C 1 as

K 2 ( u ) ( t ) = F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 2 ( u ) + t i < t B i + F ( N f ( u ) ) ( t ) ) ] } ( t ) , t J .

Similar to the proof of Lemma 2.3, we have the following.

Lemma 2.6 (i) The operator K ( a , b ) is continuous and sends equi-integrable sets in L 1 to relatively compact sets in P C 1 .

(ii) The operator K 2 is continuous and sends bounded sets in P C 1 to relatively compact sets in P C 1 .

Let us define P 2 : P C 1 P C 1 as

P 2 ( u ) = = 1 m 2 α [ t i < ξ A i + K 2 ( u ) ( ξ ) ] i = 1 k A i 1 = 1 m 2 α + δ K 2 ( u ) ( 1 ) + 0 1 h ( t ) [ K 2 ( u ) ( t ) + t i < t A i ] d t 1 = 1 m 2 α + δ .

It is easy to see that P 2 is compact continuous.

Lemma 2.7Suppose that σ = 1 , = 1 m 2 α δ 1 , thenuis a solution of (1)-(4) if and only ifuis a solution of the following abstract operator equation:

u = P 2 ( u ) + t i < t A i + K 2 ( u ) .

Proof Similar to the proof of Lemma 2.4, we omit it here. □

2.3 Case (iii)

Suppose that σ < 1 and = 1 m 2 α δ < 1 . If u is a solution of (6) with (4), we have

w ( t ) φ ( t , u ( t ) ) = w ( 0 ) φ ( 0 , u ( 0 ) ) + t i < t b i + 0 t f ( s ) d s , t J .

Denote ρ 3 = w ( 0 ) φ ( 0 , u ( 0 ) ) . It is easy to see that ρ 3 is dependent on a, b and f ( ) .

From u ( 0 ) = 0 1 g ( t ) u ( t ) d t , we have

u ( 0 ) = 1 ( 1 σ ) × 0 1 g ( t ) ( F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 3 + t i < t b i + F ( f ) ( t ) ) ] } ( t ) + t i < t a i ) d t . (18)

From u ( 1 ) = = 1 m 2 α u ( ξ ) 0 1 h ( t ) u ( t ) d t , we obtain

u ( 0 ) = = 1 m 2 α { t i < ξ a i + 0 ξ φ 1 [ t , ( w ( t ) ) 1 ( ρ 3 + t i < t b i + F ( f ) ( t ) ) ] d t } 1 = 1 m 2 α + δ i = 1 k a i + 0 1 φ 1 [ t , ( w ( t ) ) 1 ( ρ 3 + t i < t b i + F ( f ) ( t ) ) ] d t 1 = 1 m 2 α + δ 0 1 h ( t ) ( F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 3 + t i < t b i + F ( f ) ( t ) ) ] } ( t ) + t i < t a i ) d t 1 = 1 m 2 α + δ . (19)

For fixed ω W , we denote

ϒ ω ( ρ 3 ) = 1 ( 1 σ ) 0 1 g ( t ) ( F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 3 + t i < t b i + F ( ϑ ) ( t ) ) ] } ( t ) + t i < t a i ) d t = 1 m 2 α { t i < ξ a i + 0 ξ φ 1 [ t , ( w ( t ) ) 1 ( ρ 3 + t i < t b i + F ( ϑ ) ( t ) ) ] d t } 1 = 1 m 2 α + δ + i = 1 k a i + 0 1 φ 1 [ t , ( w ( t ) ) 1 ( ρ 3 + t i < t b i + F ( ϑ ) ( t ) ) ] d t 1 = 1 m 2 α + δ + 0 1 h ( t ) ( F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 3 + t i < t b i + F ( ϑ ) ( t ) ) ] } ( t ) + t i < t a i ) d t 1 = 1 m 2 α + δ , ρ 3 R N .

From (18) and (19), we have ϒ ω ( ρ 3 ) = 0 .

Obviously, ϒ ω ( ρ 3 ) can be rewritten as

ϒ ω ( ρ 3 ) = 1 ( 1 σ ) 0 1 g ( t ) ( F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 3 + t i < t b i + F ( ϑ ) ( t ) ) ] } ( t ) + t i < t a i ) d t + = 1 m 2 α { ξ t i a i + ξ 1 φ 1 [ t , ( w ( t ) ) 1 ( ρ 3 + t i < t b i + F ( ϑ ) ( t ) ) ] d t } 1 = 1 m 2 α + δ + ( 1 = 1 m 2 α ) 0 1 φ 1 [ t , ( w ( t ) ) 1 ( ρ 3 + t i < t b i + F ( ϑ ) ( t ) ) ] d t 1 = 1 m 2 α + δ + i = 1 k a i ( 1 = 1 m 2 α ) 1 = 1 m 2 α + δ + 0 1 h ( t ) ( F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 3 + t i < t b i + F ( ϑ ) ( t ) ) ] } ( t ) + t i < t a i ) d t 1 = 1 m 2 α + δ .

Denote ξ m 1 = 1 . Moreover, we also have

ϒ ω ( ρ 3 ) = 1 ( 1 σ ) 0 1 g ( t ) ( F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 3 + t i < t b i + F ( ϑ ) ( t ) ) ] } ( t ) + t i < t a i ) d t + = 1 m 2 α ξ t i a i 1 = 1 m 2 α + δ + = 1 m 2 ( α ξ ξ + 1 h ( t ) d t ) ξ 1 φ 1 [ t , ( w ( t ) ) 1 ( ρ 3 + t i < t b i + F ( ϑ ) ( t ) ) ] d t 1 = 1 m 2 α + δ + = 1 m 2 ξ ξ + 1 h ( t ) ξ t φ 1 [ s , ( w ( s ) ) 1 ( ρ 3 + s i < s b i + F ( ϑ ) ( s ) ) ] d s d t 1 = 1 m 2 α + δ 0 ξ 1 h ( t ) t 1 φ 1 [ s , ( w ( s ) ) 1 ( ρ 3 + s i < s b i + F ( ϑ ) ( s ) ) ] d s d t + 0 1 h ( t ) t i t a i d t 1 = 1 m 2 α + δ + 0 1 φ 1 [ t , ( w ( t ) ) 1 ( ρ 3 + t i < t b i + F ( ϑ ) ( t ) ) ] d t + i = 1 k a i .

Lemma 2.8Suppose that α , g, hsatisfy one of the following:

(10) = 1 m 2 α 1 , g ( t ) ( 1 = 1 m 2 α + δ ) + h ( t ) ( 1 σ ) 0 ;

(20) h ( t ) 0 on [ ξ 1 , 1 ] , α ξ ξ + 1 h ( t ) d t ( = 1 , , m 2 ) and h ( t ) 0 on [ 0 , ξ 1 ] .

Then the function ϒ ω ( ) has the following properties:

(i) For any fixed ω W , the equation ϒ ω ( ρ 3 ) = 0 has a unique solution ρ 3 ˜ ( ω ) R N .

(ii) The function ρ 3 ˜ : W R N , defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, for any ω = ( a , b , ϑ ) W , we have

| ρ 3 ˜ ( ω ) | 3 N { ( 2 N ) p + [ ( E 1 + 1 ( 1 σ ) E 1 + ( δ + 1 ) E + 1 ( 1 = 1 m 2 α + δ ) E ) i = 1 k | a i | ] p # 1 + i = 1 k | b i | + ϑ L 1 } ,

where the notation M p # 1 means

M p # 1 = { M p + 1 , M > 1 , M p 1 , M 1 .

Proof Similar to the proof of Lemma 2.2, we omit it here. □

We define ρ 3 : P C 1 R N as ρ 3 ( u ) = ρ 3 ˜ ( A , B , N f ) ( u ) , where A = ( A 1 , , A k ) , B = ( B 1 , , B k ) .

It is clear that ρ 3 ( ) is continuous and sends bounded sets of P C 1 to bounded sets of R N , and hence it is compact continuous.

For fixed a , b R k N , we denote K ( a , b ) : L 1 P C 1 as

K ( a , b ) ( ϑ ) ( t ) = F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 3 ˜ ( a , b , ϑ ) + t i < t b i + F ( ϑ ) ( t ) ) ] } ( t ) , t J .

Define K 3 : P C 1 P C 1 as

K 3 ( u ) ( t ) = F { φ 1 [ t , ( w ( t ) ) 1 ( ρ 3 ( u ) + t i < t B i + F ( N f ( u ) ) ( t ) ) ] } ( t ) , t J .

Similar to the proof of Lemma 2.3, we have

Lemma 2.9 (i) The operator K ( a , b ) is continuous and sends equi-integrable sets in L 1 to relatively compact sets in P C 1 .

(ii) The operator K 3 is continuous and sends bounded sets in P C 1 to relatively compact sets in P C 1 .

Let us define P 3 : P C 1 P C 1 as

P 3 ( u ) = 0 1 g ( t ) [ K 3 ( u ) ( t ) + t i < t A i ] d t 1 σ .

It is easy to see that P 3 is compact continuous.

Lemma 2.10Suppose that σ < 1 , = 1 m 2 α δ < 1 and α , g, hsatisfy one of the following:

(10) = 1 m 2 α 1 , g ( t ) ( 1 = 1 m 2 α + δ ) + h ( t ) ( 1 σ ) 0 ;

(20) h ( t ) 0 on [ ξ 1 , 1 ] , α ξ ξ + 1 h ( t ) d t ( = 1 , , m 2 ) and h ( t ) 0 on [ 0 , ξ 1 ] .

Thenuis a solution of (1)-(4) if and only ifuis a solution of the following abstract operator equation:

u = P 3 ( u ) + t i < t A i + K 3 ( u ) .

Proof Similar to the proof of Lemma 2.4, we omit it here. □

3 Existence of solutions in Case (i)

In this section, we apply Leray-Schauder’s degree to deal with the existence of solutions for system (1)-(4) or (1) with (2), (4) and (5) when σ < 1 , = 1 m 2 α δ = 1 .

When f satisfies the sub- ( p 1 ) growth condition, we have the following theorem.

Theorem 3.1Suppose that σ < 1 , = 1 m 2 α δ = 1 ; h ( t ) 0 on [ ξ 1 , 1 ] , α ξ ξ + 1 h ( t ) d t ( = 1 , , m 2 ) and h ( t ) 0 on [ 0 , ξ 1 ] ; fsatisfies the sub- ( p 1 ) growth condition; and operatorsAandBsatisfy the following conditions:

i = 1 k | A i ( u , v ) | C 1 ( 1 + | u | + | v | ) q + 1 p + 1 , i = 1 k | B i ( u , v ) | C 2 ( 1 + | u | + | v | ) q + 1 , ( u , v ) R N × R N , (20)

then problem (1)-(4) has at least a solution.

Proof First we consider the following problem:

( S 1 ) { p ( t ) u = λ N f ( u ) ( t ) , t ( 0 , 1 ) , t t i , lim t t i + u ( t ) lim t t i u ( t ) = λ A i ( lim t t i u ( t ) , lim t t i ( w ( t ) ) 1 p ( t ) 1 u ( t ) ) , i = 1 , , k , lim t t i + w ( t ) | u | p ( t ) 2 u ( t ) lim t t i w ( t ) | u | p ( t ) 2 u ( t ) = λ B i ( lim t t i u ( t ) , lim t t i ( w ( t ) ) 1 p ( t ) 1 u ( t ) ) , i = 1 , , k , u ( 0 ) = 0 1 g ( t ) u ( t ) d t , u ( 1 ) = = 1 m 2 α u ( ξ ) 0 1 h ( t ) u ( t ) d t .

Denote

where N f ( u ) is defined in (10).

Obviously, ( S 1 ) has the same solution as the following operator equation when λ = 1 :

u = Ψ f ( u , λ ) . (21)

It is easy to see that the operator is compact continuous for any λ [ 0 , 1 ] . It follows from Lemma 2.2 and Lemma 2.3 that Ψ f ( , λ ) is compact continuous from P C 1 to P C 1 for any λ [ 0 , 1 ] .

We claim that all the solutions of (21) are uniformly bounded for λ [ 0 , 1 ] . In fact, if it is false, we can find a sequence of solutions { ( u <