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Theory and Modern Applications

Existence results of Brezis-Browder type for systems of Fredholm integral equations

Abstract

In this article, we consider the following systems of Fredholm integral equations:

u i ( t ) = h i ( t ) + 0 T g i ( t , s ) f i ( s , u 1 ( s ) , u 2 ( s ) , , u n ( s ) ) d s , t [ 0 , T ] , 1 i n , u i ( t ) = h i ( t ) + 0 g i ( t , s ) f i ( s , u 1 ( s ) , u 2 ( s ) , , u n ( s ) ) d s , t [ 0 , ) , 1 i n .

Using an argument originating from Brezis and Browder [Bull. Am. Math. Soc. 81, 73-78 (1975)] and a fixed point theorem, we establish the existence of solutions of the first system in (C[0, T])n, whereas for the second system, the existence criteria are developed separately in (C l [0,∞))nas well as in (BC[0,∞))n. For both systems, we further seek the existence of constant-sign solutions, which include positive solutions (the usual consideration) as a special case. Several examples are also included to illustrate the results obtained.

2010 Mathematics Subject Classification: 45B05; 45G15; 45M20.

1 Introduction

In this article, we shall consider the system of Fredholm integral equations:

u i ( t ) = h i ( t ) + 0 T g i ( t , s ) f i ( s , u 1 ( s ) , u 2 ( s ) , , u n ( s ) ) d s , t [ 0 , T ] , 1 i n
(1.1)

where 0 < T <∞, and also the following system on the half-line

u i ( t ) = h i ( t ) + 0 g i ( t , s ) f i ( s , u 1 ( s ) , u 2 ( s ) , , u n ( s ) ) d s , t [ 0 , ) , 1 i n .
(1.2)

Throughout, let u = (u 1, u 2,..., u n ). We are interested in establishing the existence of solutions u of the system (1.1) in (C[0, T]) n = C[0, T] × C[0, T] × × C[0, T] (n times), whereas for the system (1.2), we shall seek a solution in (C l [0, ∞))nas well as in (BC[0, ∞))n. Here, BC[0, ∞) denotes the space of functions that are bounded and continuous on [0, ∞) and C l [0, ∞) = {x BC[0, ∞) : lim t→∞ x(t) exists}.

We shall also tackle the existence of constant-sign solutions of (1.1) and (1.2). A solution u of (1.1) (or (1.2)) is said to be of constant sign if for each 1 ≤ in, we have θ i u i (t) ≥ 0 for all t [0, T] (or t [0,∞)), where θ i {-1, 1} is fixed. Note that when θ i = 1 for all 1 ≤ in, a constant-sign solution reduces to a positive solution, which is the usual consideration in the literature.

In the literature, there is a vast amount of research on the existence of positive solutions of the nonlinear Fredholm integral equations:

y ( t ) = h ( t ) + 0 T g ( t , s ) f ( y ( s ) ) d s , t [ 0 , T ]
(1.3)

and

y ( t ) = h ( t ) + 0 g ( t , s ) f ( y ( s ) ) d s , t [ 0 , ) .
(1.4)

Particular cases of (1.3) are also considered in [13]. The reader is referred to the monographs [[4, 5], and the references cited therein] for the related literature. Recently, a generalization of (1.3) and (1.4) to systems similar to (1.1) and (1.2) have been made, and the existence of single and multiple constant-sign solutions has been established for these systems in [610].

The technique used in these articles has relied heavily on various fixed point results such as Krasnosel'skii's fixed point theorem in a cone, Leray-Schauder alternative, Leggett-Williams' fixed point theorem, five-functional fixed point theorem, Schauder fixed point theorem, and Schauder-Tychonoff fixed point theorem. In the current study, we will make use of an argument that originates from Brezis and Browder [11]; therefore, the technique is different from those of [610] and the results subsequently obtained are also different. The present article also extends, improves, and complements the studies of [5, 1223]. Indeed, we have generalized the problems to (i) systems; (ii) more general form of nonlinearities f i , 1 ≤ in,; and (iii) existence of constant-sign solutions.

The outline of the article is as follows. In Section 2, we shall state the necessary fixed point theorem and compactness criterion, which are used later. In Section 3, we tackle the existence of solutions of system (1.1) in (C[0, T]) n , while Sections 4 and 5 deal with the existence of solutions of system (1.2) in (C l [0, ∞)) n and (BC[0, ∞)) n , respectively. In Section 6, we seek the existence of constant-sign solutions of (1.1) and (1.2) in (C[0, T]) n , (C l [0, ∞)) n and (BC[0, ∞)) n . Finally, several examples are presented in Section 7 to illustrate the results obtained.

2 Preliminaries

In this section, we shall state the theorems that are used later to develop the existence criteria--Theorem 2.1 [24] is Schauder's nonlinear alternative for continuous and compact maps, whereas Theorem 2.2 is the criterion of compactness on C l [0, ∞) [[16], p. 62].

Theorem 2.1[24]Let B be a Banach space with E B closed and convex. Assume U is a relatively open subset of E with 0 U and S: U ¯ E is a continuous and compact map. Then either

  1. (a)

    S has a fixed point in U ¯ , or

  2. (b)

    there exist u U and λ (0, 1) such that u = λSu.

Theorem 2.2 [[16], p. 62] Let P C l [0, ∞). Then P is compact in C l [0, ∞) if the following hold:

  1. (a)

    P is bounded in C l [0, ∞).

  2. (b)

    Any y P is equicontinuous on any compact interval of [0, ∞).

  3. (c)

    P is equiconvergent, i.e., given ε > 0, there exists T(ε) > 0 such that |y(t) - y(∞)| < ε for any tT(ε) and y P.

3 Existence results for (1.1) in (C[0, T]) n

Let the Banach space B = (C[0, T]) n be equipped with the norm:

u = max 1 i n sup t [ 0 , T ] u i ( t ) = max 1 i n u i 0

where we let |u i | 0 = sup t[0,T]|u i (t)|, 1 ≤ in. Throughout, for u B and t [0, T], we shall denote

u ( t ) = max 1 i n u i ( t ) .

Moreover, for each 1 ≤ in, let 1 ≤ p i be an integer and q i be such that 1 p i + 1 q i =1. For x L p i [ 0 , T ] , we shall define

x p i = 0 T x ( s ) p i d s 1 p i , 1 p i < ess sup s [ 0 , T ] x ( s ) , p i = .

Our first existence result uses Theorem 2.1.

Theorem 3.1 For each 1 ≤ in, assume (C1)- (C4) hold where

(C1) h i C[0, T], denote H i ≡ sup t [0, T]|h i (t)|,

(C2) f i : [0, T] × n is a L q i -Carathéodory function:

  1. (i)

    the map u α f i (t, u) is continuous for almost all t [0, T],;

  2. (ii)

    the map t α f i (t, u) is measurable for all u n ;

  3. (iii)

    for any r > 0, there exists μ r , i L q i [ 0 , T ] such that |u| ≤ r implies |f i (t, u)| ≤ μ r,i (t) for almost all t [0, T];

(C3) g i t ( s ) = g i ( t , s ) L p i [ 0 , T ] for each t [0, T];

(C4) the map t g i t is continuous from [0, T] to L p i [ 0 , T ] .

In addition, suppose there is a constant M > 0, independent of λ, with ||u|| ≠ M for any solution u (C[0, T])nto

u i ( t ) = λ h i ( t ) + 0 T g i ( t , s ) f i ( s , u ( s ) ) d s , t [ 0 , T ] , 1 i n ( 3 . 1 ) λ

for each λ (0, 1). Then, (1.1) has at least one solution in (C[0, T]) n .

Proof Let the operator S be defined by

S u ( t ) = ( S 1 u ( t ) , S 2 u ( t ) , , S n u ( t ) ) , t [ 0 , T ]
(3.2)

where

S i u ( t ) = h i ( t ) + 0 T g i ( t , s ) f i ( s , u ( s ) ) d s , t [ 0 , T ] , 1 i n .
(3.3)

Clearly, the system (1.1) is equivalent to u = Su, and (3.1) λ is the same as u = λSu.

Note that S maps (C[0, T])ninto (C[0, T]) n , i.e., S i : (C[0, T]) nC[0, T], 1 ≤ in. To see this, note that for any u (C[0, T]) n , there exits r > 0 such that ||u|| < r. Since f i is a L q i -Carathéodory function, there exists μ r , i L q i [ 0 , T ] such that |f i (s, u)| ≤ μ r,i (s) for almost all s [0, T]. Hence, for any t 1, t 2 [0, T], we find for 1 ≤ in,

S i u ( t 1 ) - S i u ( t 2 ) h i ( t 1 ) - h i ( t 2 ) + 0 T g i t 1 ( s ) - g i t 2 ( s ) p i d s 1 p i μ r , i q i 0
(3.4)

as t 1t 2, where we have used (C1) and (C3). This shows that S : (C[0, T]) n → (C[0, T]) n .

Next, we shall prove that S : (C[0, T])n→ (C[0, T])nis continuous. Let u m = ( u 1 m , u 2 m , , u n m ) u in (C[0, T]) n , i.e., u i m u i in C[0, T], 1 ≤ in. We need to show that Su mSu in (C[0, T]) n , or equivalently S i u mS i u in C[0, T], 1 ≤ in. There exists r > 0 such that ||u m ||, ||u|| < r. Since f i is a L q i -Carathéodory function, there exists μ r , i L q i [ 0 , T ] such that |f i (s, u m )|, |f i (s, u)| ≤ μ r,i (s) for almost all s [0, T]. Using a similar argument as in (3.4), we get for any t 1, t 2 [0, T] and 1 ≤ in:

S i u m ( t 1 ) - S i u m ( t 2 ) 0 and S i u ( t 1 ) - S i u ( t 2 ) 0
(3.5)

as t 1t 2. Furthermore, S i u m (t) → S i u(t) pointwise on [0, T], since, by the Lebesgue-dominated convergence theorem,

S i u m ( t ) - S i u ( t ) sup t [ 0 , T ] g i t p i 0 T f i ( s , u m ( s ) ) - f i ( s , u ( s ) ) q i d s 1 q i 0
(3.6)

as m → ∞. Combining (3.5) and (3.6) and using the fact that [0, T] is compact, gives for all t [0, T],

S i u m ( t ) - S i u ( t ) S i u m ( t ) - S i u m ( t 1 ) + S i u m ( t 1 ) - S i u ( t 1 ) + S i u ( t 1 ) - S i u ( t ) 0
(3.7)

as m → ∞. Hence, we have proved that S : (C[0, T])n→ (C[0, T])nis continuous.

Finally, we shall show that S : (C[0, T])n→ (C[0, T])nis completely continuous. Let Ω be a bounded set in (C[0, T])nwith ||u|| ≤ r for all u Ω. We need to show that S i Ω is relatively compact for 1 ≤ in. Clearly, S i Ω is uniformly bounded, since there exists μ r , i L q i [ 0 , T ] such that |f i (s, u)| ≤ μ r,i (s) for all u Ω and a.e. s [0, T], and hence

S i u 0 H i + sup t [ 0 , T ] g i t p i μ r , i q i K i , u Ω .
(3.8)

Further, using a similar argument as in (3.4), we see that S i Ω is equicontinuous. It follows from the Arzéla-Ascoli theorem [[5], Theorem 1.2.4] that S i Ω is relatively compact.

We now apply Theorem 2.1 with U = {u (C[0, T])n: ||u|| < M} and B = E = (C[0, T])nto obtain the conclusion of the theorem. □

Our subsequent results will apply Theorem 3.1. To do so, we shall show that any solution u of (3.1) λ is bounded above. This is achieved by bounding the integral of |f i (t,u(t))| (or f i ( t , u ( t ) ) ρ i ) on two complementary subsets of [0, T], namely {t [0, T] : ||u(t)|| ≤ r} and {t [0, T] : ||u(t)|| > r}, where ρ i and r are some constants--this technique originates from the study of Brezis and Browder [11]. In the next four theorems (Theorems 3.2-3.5), we shall apply Theorem 3.1 to the case p i = ∞ and q i = 1, 1 ≤ in.

Theorem 3.2. Let the following conditions be satisfied for each 1 ≤ in : (C1)-(C4) with p i =and q i = 1, (C5) and (C6) where

(C5) there exist B i > 0 such that for any u (C[0, T]) n ,

0 T f i ( t , u ( t ) ) 0 T g i ( t , s ) f i ( s , u ( s ) ) d s d t B i ,

(C6) there exist r > 0 and α i > 0 with i > H i such that for any u (C[0, T])n,

u i ( t ) f i ( t , u ( t ) ) r α i f i ( t , u ( t ) ) f o r u ( t ) > r and a . e . t [ 0 , T ] .

Then, (1.1) has at least one solution in (C[0, T]) n .

Proof We shall employ Theorem 3.1, and so let u = (u 1, u 2, l...., u n ) (C[0, T]) n be any solution of (3.1) λ where λ (0, 1).

Define

I = { t [ 0 , T ] : u ( t ) r } and J = { t [ 0 , T ] : u ( t ) > r } .
(3.9)

Clearly, [0, T] = I J, and hence 0 T = I + J .

Let 1 ≤ in. If t I, then by (C2), there exists μ r,i L 1[0, T] such that |f i (t, u(t))| ≤ μ r,i (t). Thus, we get

I f i ( t , u ( t ) ) d t I μ r , i ( t ) d t 0 T μ r , i ( t ) d t = μ r , i 1 .
(3.10)

On the other hand, if t J, then it is clear from (C6) that u i (t)f i (t, u(t)) ≥ 0 for a.e. t [0, T]. It follows that

J u i ( t ) f i ( t , u ( t ) ) d t r α i J f i ( t , u ( t ) ) d t .
(3.11)

We now multiply (3.1) λ by f i (t, u(t)), then integrate from 0 to T to get

0 T u i ( t ) f i ( t , u ( t ) ) d t = λ 0 T h i ( t ) f i ( t , u ( t ) ) d t + λ 0 T f i ( t , u ( t ) ) 0 T g i ( t , s ) f i ( s , u ( s ) ) d s d t .
(3.12)

Using (C5) in (3.12) yields

0 T u i ( t ) f i ( t , u ( t ) ) d t H i 0 T f i ( t , u ( t ) ) d t + B i .
(3.13)

Splitting the integrals in (3.13) and applying (3.11), we get

I u i ( t ) f i ( t , u ( t ) ) d t + r α i J f i ( t , u ( t ) ) d t H i I f i ( t , u ( t ) ) d t + H i J f i ( t , u ( t ) ) d t + B i

or

( r α i - H i ) J f i ( t , u ( t ) ) d t H i I f i ( t , u ( t ) ) d t + I u i ( t ) f i ( t , u ( t ) ) d t + B i ( H i + r ) μ r , i 1 + B i

where we have used (3.10) in the last inequality. It follows that

J f i ( t , u ( t ) ) d t ( H i + r ) μ r , i 1 + B i r α i - H i k i .
(3.14)

Finally, it is clear from (3.1) λ that for t [0, T] and 1 ≤ in,

u i ( t ) H i + 0 T g i ( t , s ) f i ( s , u ( s ) ) d s = H i + I + J g i ( t , s ) f i ( s , u ( s ) ) d s H i + sup t [ 0 , T ] g i t ( μ r , i 1 + k i ) l i
(3.15)

where we have applied (3.10) and (3.14) in the last inequality. Thus, |u i |0l i for 1 ≤ in and ||u|| ≤ max1≤in l i L. It follows from Theorem 3.1 (with M = L + 1) that (1.1) has a solution u* (C[0, T])n. □

Theorem 3.3 Let the following conditions be satisfied for each 1 ≤ in : (C1)-(C4) with p i = ∞ and q i = 1, (C7) and (C8) where

(C7) there exist constants a i ≥ 0 and b i such that for any u (C[0, T])n,

0 T f i ( t , u ( t ) ) 0 T g i ( t , s ) f i ( s , u ( s ) ) d s d t a i 0 T f i ( t , u ( t ) ) d t + b i ,

(C8) there exist r > 0 and α i > 0 with rα i > H i + a i such that for any u (C[0, T])n,

u i ( t ) f i ( t , u ( t ) ) r α i f i ( t , u ( t ) ) f o r u ( t ) > r and a . e . t [ 0 , T ] .

Then, (1.1) has at least one solution in (C[0, T]) n .

Proof The proof follows that of Theorem 3.2 until (3.12). Let 1 ≤ in. We use (C7) in (3.12) to get

0 T u i ( t ) f i ( t , u ( t ) ) d t 0 T h i ( t ) f i ( t , u ( t ) ) d t + λ 0 T f i ( t , u ( t ) ) 0 T g i ( t , s ) f i ( s , u ( s ) ) d s d t ( H i + a i ) 0 T f i ( t , u ( t ) ) d t + b i .
(3.16)

Splitting the integrals in (3.16) and applying (3.11) gives

( r α i - H i - a i ) J f i ( t , u ( t ) ) d t ( H i + a i ) I f i ( t , u ( t ) ) d t + I u i ( t ) f i ( t , u ( t ) ) d t + b i ( H i + a i + r ) μ r , i 1 + b i

where we have also used (3.10) in the last inequality. It follows that

J f i ( t , u ( t ) ) d t ( H i + a i + r ) μ r , i 1 + b i r α i - H i - a i k i .
(3.17)

The rest of the proof follows that of Theorem 3.2. □

Theorem 3.4 Let the following conditions be satisfied for each 1 ≤ in : (C1)-(C4) with p i =and q i = 1, (C9) and (C10) where

(C9) there exist constants a i ≥ 0, 0 < τ i ≤ 1 and b i such that for any u (C[0, T])n,

0 T f i ( t , u ( t ) ) 0 T g i ( t , s ) f i ( s , u ( s ) ) d s d t a i 0 T f i ( t , u ( t ) ) d t τ i + b i ,

(C10) there exist r > 0 and β i > 0 such that for any u (C[0, T])n,

u i ( t ) f i ( t , u ( t ) ) β i u ( t ) f i ( t , u ( t ) ) f o r u ( t ) > r and a . e . t [ 0 , T ] .

Then, (1.1) has at least one solution in (C[0, T]) n .

Proof Let u = (u 1, u 2,..., u n ) (C[0, T]) n be any solution of (3.1) λ where λ (0, 1). Define

r 0 = max r , max 1 i n H i + a i 2 τ i + 1 β i , I 0 = { t [ 0 , T ] : u ( t ) r 0 } and J 0 = { t [ 0 , T ] : u ( t ) > r 0 } .
(3.18)

Clearly, [0, T] = I 0 J 0 and hence 0 T = I 0 + J 0 .

Let 1 ≤ in. If t I 0, then by (C2) there exists μ r 0 , i L 1 [ 0 , T ] such that f i ( t , u ( t ) ) μ r 0 , i ( t ) and

I 0 f i ( t , u ( t ) ) d t I 0 μ r 0 , i ( t ) d t 0 T μ r 0 , i ( t ) d t = μ r 0 , i 1 .
(3.19)

Further, if t J 0, then by (C10) we have

J 0 u i ( t ) f i ( t , u ( t ) ) d t β i J 0 u ( t ) f i ( t , u ( t ) ) d t β i r 0 J 0 f i ( t , u ( t ) ) d t .
(3.20)

Now, using (3.20) and (C9) in (3.12) gives

β i r 0 J 0 f i ( t , u ( t ) ) d t I 0 u i ( t ) f i ( t , u ( t ) ) d t + 0 T h i ( t ) f i ( t , u ( t ) ) d t + a i 0 T f i ( t , u ( t ) ) d t τ i + b i I 0 u i ( t ) f i ( t , u ( t ) ) d t + 0 T h i ( t ) f i ( t , u ( t ) ) d t + a i 2 τ i I 0 f i ( t , u ( t ) ) d t τ i + J 0 f i ( t , u ( t ) ) d t τ i + b i
(3.21)

where in the last inequality, we have made use of the inequality:

( x + y ) α 2 α ( x α + y α ) , x , y 0 , α 0 .

Now, noting (3.19) we find that

I 0 u i ( t ) f i ( t , u ( t ) ) d t + I 0 h i ( t ) f i ( t , u ( t ) ) d t + a i 2 τ i I 0 f i ( t , u ( t ) ) d t τ i + b i ( r 0 + H i ) μ r 0 , i 1 + a i 2 τ i ( μ r 0 , i 1 ) τ i + b i k i
(3.22)

Substituting (3.22) in (3.21) then yields

β i r 0 J 0 f i ( t , u ( t ) ) d t J 0 h i ( t ) f i ( t , u ( t ) ) d t + a i 2 τ i J 0 f i ( t , u ( t ) ) d t τ i + k i H i J 0 f i ( t , u ( t ) ) d t + a i 2 τ i J 0 f i ( t , u ( t ) ) d t τ i + k i .

Since τ i ≤ 1, there exists a constant k i such that

( β i r 0 - H i - a i 2 τ i ) J 0 f i ( t , u ( t ) ) d t k i

which leads to

J 0 f i ( t , u ( t ) ) d t k i β i r 0 - H i - a i 2 τ i k i .
(3.23)

Finally, it is clear from (3.1) λ that for t [0, T] and 1 ≤ in,

u i ( t ) H i + 0 T g i ( t , s ) f i ( s , u ( s ) ) d s = H i + I 0 + J 0 g i ( t , s ) f i ( s , u ( s ) ) d s H i + sup t [ 0 , T ] g i t ( μ r 0 , i 1 + k i ) l i
(3.24)

where we have applied (3.19) and (3.23) in the last inequality. The conclusion now follows from Theorem 3.1. □

Theorem 3.5 Let the following conditions be satisfied for each 1 ≤ in : (C1), (C2)-(C4) with p i = ∞ and q i = 1, (C10), (C11) and (C12) where

(C11) there exist r > 0, η i > 0, γ i > 0 and ϕ i L γ i + 1 γ i [ 0 , T ] such that for any u (C[0, T])n,

u ( t ) η i f i ( t , u ( t ) γ i + ϕ i ( t ) f o r u ( t ) > r and a . e . t [ 0 , T ] ,

(C12) there exist a i ≥ 0, 0 < τ i < γ i + 1, b i , and ψ i L γ i + 1 γ i [ 0 , T ] with ψ i ≥ 0 almost everywhere on [0, T], such that for any u (C[0, T])n,

0 T f i ( t , u ( t ) ) 0 T g i ( t , s ) f i ( s , u ( s ) ) d s d t a i 0 T ψ i ( t ) f i ( t , u ( t ) ) d t τ i + b i .

Also, ϕ i C[0, T], h i L γ i + 1 γ i [ 0 , T ] , ψ i C[0, T] and 0 T g i ( t , s ) γ i + 1 γ i d sC [ 0 , T ] .

Then, (1.1) has at least one solution in (C[0, T]) n .

Proof Let u = (u 1, u 2,..., u n ) (C[0, T]) n be any solution of (3.1) λ where λ (0, 1). Define the sets I and J as in (3.9). Let 1 ≤ in. Applying (C10) and (C11), we get

J u i ( t ) f i ( t , u ( t ) ) d t β i J u ( t ) f i ( t , u ( t ) ) d t β i η i J f i ( t , u ( t ) ) γ i + 1 d t + β i J ϕ i ( t ) f i ( t , u ( t ) ) d t .
(3.25)

Using (3.25) and (C12) in (3.12), we obtain

β i η i J f i ( t , u ( t ) ) γ i + 1 d t I u i ( t ) f i ( t , u ( t ) ) d t + β i J ϕ i ( t ) f i ( t , u ( t ) ) d t + 0 T h i ( t ) f i ( t , u ( t ) ) d t + a i 0 T ψ i ( t ) f i ( t , u ( t ) ) d t τ i + b i I u i ( t ) f i ( t , u ( t ) ) d t + β i J ϕ i ( t ) f i ( t , u ( t ) ) d t + 0 T h i ( t ) f i ( t , u ( t ) ) d t + a i 2 τ i I ψ i ( t ) f i ( t , u ( t ) ) d t τ i + J ψ i ( t ) f i ( t , u ( t ) ) d t τ i + b i .
(3.26)

Now, in view of (3.10) and (C12), we have

I u i ( t ) f i ( t , u ( t ) ) d t + I h i ( t ) f i ( t , u ( t ) ) d t + a i 2 τ i I ψ i ( t ) f i ( t , u ( t ) ) d t τ i + b i ( r + H i ) μ r , i 1 + a i 2 τ i I ψ i ( t ) μ r , i ( t ) d t τ i + b i k ¯ i .
(3.27)

Substituting (3.27) into (3.26) and using Hölder's inequality, we find

β i η i J f i ( t , u ( t ) ) γ i + 1 d t β i J ϕ i ( t ) f i ( t , u ( t ) ) d t + J h i ( t ) f i ( t , u ( t ) ) d t + a i 2 τ i J ψ i ( t ) f i ( t , u ( t ) ) d t τ i + k ¯ i β i 0 T ϕ i ( t ) γ i + 1 γ i d t γ i + 1 γ i J f i ( t , u ( t ) ) γ i + 1 d t 1 γ i + 1 + 0 T h i ( t ) γ i + 1 γ i d t γ i γ i + 1 J f i ( t , u ( t ) ) γ i + 1 d t 1 γ i + 1 + a i 2 τ i 0 T ψ i ( t ) γ i + 1 γ i d t τ i γ i γ i + 1 J f i ( t , u ( t ) ) γ i + 1 d t τ i γ i + 1 + k ¯ i .

Since 1 γ i + 1 <1 and τ i γ i + 1 <1, there exists a constant k i such that

J f i ( t , u ( t ) ) γ i + 1 d t k i .
(3.28)

Finally, it is clear from (3.1) λ that for t [0, T] and 1 ≤ in,

u i ( t ) H i + I + J g i ( t , s ) f i ( s , u ( s ) ) d s H i + sup t [ 0 , T ] g i t μ r , i 1 + sup t [ 0 , T ] 0 T g i ( t , s ) γ i + 1 γ i d s γ i γ i + 1 J f i ( t , u ( t ) ) γ i + 1 d t 1 γ i + 1 l i
(3.29)

where we have used (3.28) and (C12) in the last inequality, and l i is some constant. The conclusion is now immediate by Theorem 3.1. □

In the next six results (Theorem 3.6-3.11), we shall apply Theorem 3.1 for general p i and q i .

Theorem 3.6 Let the following conditions be satisfied for each 1 ≤ in : (C1)-(C4), (C5), (C10) and (C13) where

(C13) there exist r > 0, η i > 0, γ i > 0 and ϕ i L p i [ 0 , T ] such that for any u (C[0, T])n,

u ( t ) η i f i ( t , u ( t ) γ i + ϕ i ( t ) f o r u ( t ) > r and a . e . t [ 0 , T ] .

Then, (1.1) has at least one solution in (C[0, T]) n .

Proof Let u = (u 1, u 2,..., u n ) (C[0, T]) n be any solution of (3.1) λ where λ (0, 1). Define the sets I and J as in (3.9). Let 1 ≤ in. If t I, then by (C2), there exists μ r , i L q i [ 0 , T ] such that |f i (t, u(t))| ≤ μ r,i (t). Consequently, we have

I f i ( t , u ( t ) ) d t I μ r , i ( t ) d t 0 T μ r , i ( t ) d t T 1 p i μ r , i q i .
(3.30)

On the other hand, using (C10) and (C13), we derive at (3.25).

Next, applying (C5) in (3.12) leads to (3.13). Splitting the integrals in (3.13) and using (3.25), we find that

β i η i J f i ( t , u ( t ) ) γ i + 1 d t β i J ϕ i ( t ) f i ( t , u ( t ) ) d t + H i J f i ( t , u ( t ) ) d t + B i + I ( u i ( t ) + H i ) f i ( t , u ( t ) ) d t β i J ϕ i ( t ) f i ( t , u ( t ) ) d t + H i J f i ( t , u ( t ) ) d t + B i + ( r + H i ) T 1 p i μ r , i q i = β i J ϕ i ( t ) f i ( t , u ( t ) ) d t + H i J f i ( t , u ( t ) ) d t + B i
(3.31)

where (3.30) has been used in the last inequality and B i B i + ( r + H i ) T 1 p i μ r , i q i .

Now, an application of Hölder's inequality gives

J ϕ i ( t ) f i ( t , u ( t ) ) d t 0 T ϕ i ( t ) γ i + 1 γ i d t γ i γ i + 1 J f i ( t , u ( t ) ) γ i + 1 d t 1 γ i + 1 .
(3.32)

Another application of Hölder's inequality yields

0 T ϕ i ( t ) γ i + 1 γ i d t T γ i p i - γ i - 1 p i γ i 0 T ϕ i ( t ) p i d t γ i + 1 γ i p i .
(3.33)

Substituting (3.33) into (3.32) then leads to

J ϕ i ( t ) f i ( t , u ( t ) ) d t T γ i p i - γ i - 1 p i ( γ i + 1 ) ϕ i p i J f i ( t , u ( t ) ) γ i + 1 d t 1 γ i + 1 .
(3.34)

Further, using Hölder's inequality again, we get

J f i ( t , u ( t ) ) d t T γ i γ i + 1 J f i ( t , u ( t ) ) γ i + 1 d t 1 γ i + 1 .
(3.35)

Substituting (3.34) and (3.35) into (3.31), we obtain

β i η i J f i ( t , u ( t ) ) γ i + 1 d t A i J f i ( t , u ( t ) ) γ i + 1 d t 1 γ i + 1 + B i
(3.36)

where A i T γ i p i - γ i - 1 p i ( γ i + 1 ) β i ϕ i p i + H i T γ i γ i + 1 . Since 1 γ i + 1 <1, from (3.36), there exists a constant k i such that

J f i ( t , u ( t ) ) γ i + 1 d t k i .
(3.37)

Finally, it is clear from (3.1) λ that for t [0, T] and 1 ≤ in,

u i ( t ) H i + ( I + J ) | g i ( t , s ) f i ( s , u ( s ) ) | d s H i + ( sup t [ 0 , T ] g i t p i ) μ r , i q i + T γ i p i γ i 1 p i ( γ i + 1 ) ( sup t [ 0 , T ] g i t p i ) [ J f i ( s , u ( s ) ) γ i + 1 d s ] 1 γ i + 1 l i ( a constant ) ,
(3.38)

where in the second last inequality a similar argument as in (3.34) is used, and in the last inequality we have used (3.37). An application of Theorem 3.1 completes the proof. □

Theorem 3.7 Let the following conditions be satisfied for each 1 ≤ in : (C1)-(C4), (C7), (C10) and (C13). Then, (1.1) has at least one solution in (C[0, T]) n .

Proof Let u = (u 1, u 2,..., u n ) (C[0, T]) n be any solution of (3.1) λ where λ (0, 1). Define the sets I and J as in (3.9). Let 1 ≤ in. As in the proof of Theorems 3.3 and 3.6, respectively, (C7) leads to (3.16), whereas (C10) and (C13) yield (3.25).

Splitting the integrals in (3.16) and applying (3.25), we find that

β i η i J f i ( t , u ( t ) ) γ i + 1 d t β i J ϕ i ( t ) f i ( t , u ( t ) ) d t + ( H i + a i ) J f i ( t , u ( t ) ) d t + b i + I ( u i ( t ) + H i + a i ) f i ( t , u ( t ) ) d t β i J ϕ i ( t ) f i ( t , u ( t ) ) d t + ( H i + a i ) J f i ( t , u ( t ) ) d t + b i + ( r + H i + a i ) T 1 p i μ r , i q i = β i J J ϕ i ( t ) f i ( t , u ( t ) ) d t + ( H i + a i ) J f i ( t , u ( t ) ) d t + B i
(3.39)

where B i b i + ( r + H i + a i ) T 1 p i μ r , i q i . Substituting (3.34) and (3.35) into (3.39) then leads to

β i η i J f i ( t , u ( t ) ) γ i + 1 d t A i J f i ( t , u ( t ) ) γ i + 1 d t 1 γ i + 1 + B i
(3.40)

where A i T γ i p i - γ i - 1 p i ( γ i + 1 ) β i ϕ i p i + ( H i + a i ) T γ i γ i + 1 . Since 1 γ i + 1 <1, from (3.40), we can obtain (3.37) where k i is some constant. The rest of the proof proceeds as that of Theorem 3.6. □

Theorem 3.8 Let the following conditions be satisfied for each 1 ≤ in : (C1)-(C4), (C10), (C13), and (C14) where

(C14) there exist constants a i ≥ 0, 0 < τ i < γ i + 1 and b i such that for any u (C[0, T])n,

0 T f i ( t , u ( t ) ) 0 T g i ( t , s ) f i ( s , u ( s ) ) d s d t a i 0 T f i ( t , u ( t ) ) d t τ i + b i .

Then, (1.1) has at least one solution in (C[0, T])n.

Proof Let u = (u 1, u 2,..., u n ) (C[0, T]) n be any solution of (3.1) λ where λ (0, 1). Define the sets I and J as in (3.9). Let 1 ≤ in. From the proof of Theorem 3.6, we see that (C10) and (C13) lead to (3.25).

Using (3.25) and (C14) in (3.12), we obtain

β i η i J f i ( t , u ( t ) ) γ i + 1 d t I u i ( t ) f i ( t , u ( t ) ) d t + β i J ϕ i ( t ) f i ( t , u ( t ) ) d t + 0 T h i ( t ) f i ( t , u ( t ) ) d t + a i 0 T f i ( t , u ( t ) ) d t τ i + b i I u i ( t ) f i ( t , u ( t ) ) d t + β i J ϕ i ( t ) f i ( t , u ( t ) ) d t + 0 T h i ( t ) f i ( t , u ( t ) ) d t + a i 2 τ i I f i ( t , u ( t ) ) d t τ i + J f i ( t , u ( t ) ) d t τ i + b i .
(3.41)

Note that

I u i ( t ) f i ( t , u ( t ) ) d t + I h i ( t ) f i ( t , u ( t ) ) d t + a i 2 τ i I f i ( t , u ( t ) ) d t τ i + b i ( r + H i ) I f i ( t , u ( t ) ) d t + a i 2 τ i I f i ( t , u ( t ) ) d t τ i + b i ( r + H i ) T 1 p i μ r , i q i + a i 2 τ i T τ i p i ( μ r , i q i ) τ i + b i k i
(3.42)

where we have used (3.30) in the last inequality. Substituting (3.42) into (3.41) and using (3.34) and (3.35) then provides

β i η i J f i ( t , u ( t ) ) γ i + 1 d t β i J ϕ i ( t ) f i ( t , u ( t ) ) d t + J h i ( t ) f i ( t , u ( t ) ) d t + a i 2 τ i J f i ( t , u ( t ) ) d t τ i + k i β i T γ i p i - γ i - 1 p i ( γ i + 1 ) ϕ i p i J f i ( t , u ( t ) ) γ i + 1 d t 1 γ i + 1 + H i T γ i γ i + 1 J f i ( t , u ( t ) ) γ i + 1 d t 1 γ i + 1 + a i 2 τ i T τ i γ i γ i + 1 J f i ( t , u ( t ) ) γ i + 1 d t τ i γ i + 1 + k i .
(3.43)

Since 1 γ i + 1 <1 and τ i γ i + 1 <1, there exists a constant k i such that (3.37) holds. The rest of the proof is similar to that of Theorem 3.6. □

Theorem 3.9 Let the following conditions be satisfied for each 1 ≤ in : (C1)-(C4), (C10), (C13), and (C15) where

(C15) there exist constants d i ≥ 0, 0 < τ i < γ i + 1 and e i such that for any u (C[0, T])n,

0 T f i ( t , u ( t ) ) 0 T g i ( t , s ) f i ( s , u ( s ) ) d s d t d i 0 T f i ( t , u ( t ) ) q i d t τ i q i + e i .

Then, (1.1) has at least one solution in (C[0, T]) n .

Proof Let u = (u 1, u 2,..., u n ) (C[0, T]) n be any solution of (3.1) λ where λ (0, 1). Define the sets I and J as in (3.9). Let 1 ≤ in. As before, we see that (C10) and (C13) lead to (3.25).

Using (3.25) and (C15) in (3.12), we obtain

β i η i J f i ( t , u ( t ) ) γ i + 1 d t β i J ϕ i ( t ) f i ( t , u ( t ) ) d t + I u i ( t ) f i ( t , u ( t ) ) d t + 0 T h i ( t ) f i ( t , u ( t ) ) d t + d i 0 T f i ( t , u ( t ) ) q i d t τ i q i + e i β i J ϕ i ( t ) f i ( t , u ( t ) ) d t + I u i ( t ) f i ( t , u ( t ) ) d t + H i 0 T f i ( t , u ( t ) ) d t + d i 2 τ i q i I f i ( t , u ( t ) ) q i d t τ i q i + I f i ( t , u ( t ) ) q i d t τ i q i + e i .
(3.44)

Now, it is clear that

I u i ( t ) f i ( t , u ( t ) ) d t + H i I f i ( t , u ( t ) ) d t + d i 2 τ i q i I f i ( t , u ( t ) ) q i d t τ i q i + e i ( r + H i ) I μ r , i ( t ) d t + d i 2 τ i q i I ( μ r , i ( t ) ) q i d t τ i q i + e i ( r + H i ) 0 T μ r , i ( t ) d t + d i 2 τ i q i 0 T ( μ r , i ( t ) ) q i d t τ i q i + e i k ^ i .
(3.45)

Moreover, an application of Hölder's inequality gives

J f i ( t , u ( t ) ) q i d t T γ i + 1 - q i γ i + 1 J f i ( t , u ( t ) ) γ i + 1 d t q i γ i + 1 .
(3.46)

Substituting (3.45) into (3.44) and using (3.34), (3.35) and (3.46) then leads to

β i η i J f i ( t , u ( t ) ) γ i + 1 d t β i J ϕ i ( t ) f i ( t , u ( t ) ) d t + H i J f i ( t , u ( t ) ) d t + d i 2 τ i q i J f i ( t , u ( t ) ) q i d t τ i q i + k ^ i β i T γ i p i - γ i - 1 p i ( γ i + 1 ) ϕ i p i J f i ( t , u ( t ) ) γ i + 1 d t 1 γ i + 1 + H i T γ i γ i + 1 J f i ( t , u ( t ) ) γ i + 1 d t 1 γ i + 1 + d i 2 τ i q i T τ i ( γ i + 1 - q i ) q i ( γ i + 1 ) J f i ( t , u ( t ) ) γ i + 1 d t τ i γ i + 1 + k ^ i .
(3.47)

Noting 1 γ i + 1 <1 and τ i γ i + 1 <1, there exists a constant k i such that (3.37) holds. The rest of the proof follows that of Theorem 3.6. □

Theorem 3.10 Let the following conditions be satisfied for each 1 ≤ in : (C1)-(C4), (C10), (C13) and (C16) where

(C16) there exist constants c i ≥ 0, d i ≥ 0, 0 < τ i < γ i + 1 and e i with β i η i >2 c i ( 2 T ) γ i + 1 - q i q i such that for any u (C[0, T])n,

0 T f i ( t , u ( t ) ) 0 T g i ( t , s ) f i ( s , u ( s ) ) d s d t c i 0 T f i ( t , u ( t ) ) q i d t γ i + 1 q i + d i 0 T f i ( t , u ( t ) ) q i d t τ i q i + e i .

Then, (1.1) has at least one solution in (C[0, T]) n .

Proof Let u = (u 1, u 2,..., u n ) (C[0, T]) n be any solution of (3.1) λ where λ (0, 1). Define the sets I and J as in (3.9). Let 1 ≤ in. As before, we see that (C10) and (C13) lead to (3.25).

Using (3.25) and (C16) in (3.12) gives

β i η i J f i ( t , u ( t ) ) γ i + 1 d t β i J ϕ i ( t ) f i ( t , u ( t ) ) d t + I u i ( t ) f i ( t , u ( t ) ) d t + 0 T h i ( t ) f i ( t , u ( t ) ) d t + c i 0 T f i ( t , u ( t ) ) q i d t γ i + 1 q i + d i 0 T f i ( t , u ( t ) ) q i d t τ i q i + e i β i J ϕ i ( t ) f i ( t , u ( t ) ) d t + I u i ( t ) f i ( t , u ( t ) ) d t + H i 0 T f i ( t , u ( t ) ) d t + c i 2 γ i + 1 q i I f i ( t , u ( t ) ) q i d t γ i + 1 q i + J f i ( t , u ( t ) ) q i d t γ i + 1 q i + d i 2 τ i q i I f i ( t , u ( t ) ) q i d t τ i q i + J f i ( t , u ( t ) ) q i d t τ i q i + e i .
(3.48)

Now, it is clear that

I u i ( t ) f i ( t , u ( t ) ) d t + H i I f i ( t , u ( t ) ) d t + c i 2 γ i + 1 q i I f i ( t , u ( t ) ) q i d t γ i + 1 q i + d i 2 τ i q i I f i ( t , u ( t ) ) q i d t τ i q i + e i ( r + H i ) I μ r , i ( t ) d t + c i 2 γ i + 1 q i I ( μ r , i ( t ) ) q i d t γ i + 1 q i + d i 2 τ i q i I ( μ r , i ( t ) ) q i d t τ i q i + e i ( r + H i ) 0 T μ r , i ( t ) d t + c i 2 γ i + 1 q i 0 T ( μ r , i ( t ) ) q i d t γ i + 1 q i + d i 2 τ i q i 0 T ( μ r , i ( t ) ) q i d t τ i q i + e i k i .
(3.49)

Substituting (3.49) into (3.48) and then using (3.34), (3.35) and (3.46) leads to

β i η i J f i ( t , u ( t ) ) γ i + 1 d t β i T γ i p i - γ i - 1 p i ( γ i + 1 ) ϕ i p i J f i ( t , u ( t ) ) γ i + 1 d t 1 γ i + 1 + H i T γ i γ i + 1 J f i ( t , u ( t ) ) γ i + 1 d t 1 γ i + 1 + c i 2 γ i + 1 q i T γ i + 1 - q i q i J f i ( t , u ( t ) ) γ i + 1 d t + d i 2 τ i q i T τ i ( γ i + 1 - q i ) q i ( γ i + 1 ) J f i ( t , u ( t ) ) γ i + 1 d t τ i γ i + 1 + k i .
(3.50)

Noting 1 γ i + 1 <1, τ i γ i + 1 <1 as well as β i η i >2 c i ( 2 T ) γ i + 1 - q i q i , from (3.50) there exists a constant k i such that (3.37) holds. The rest of the proof proceeds as that of Theorem 3.6. □

Theorem 3.11 Let the following conditions be satisfied for each 1 ≤ in : (C1)-(C4), (C10), (C13) and (C17) where

(C17) there exist a i ≥ 0, 0 i < γ i + 1, b i , and ψ i L p i [ 0 , T ] with ψ i ≥ 0 almost everywhere on [0, T], such that for any u (C[0, T])n,

0 T f i ( t , u ( t ) ) 0 T g i ( t , s ) f i ( s , u ( s ) ) d s d t a i 0 T ψ i ( t ) f i ( t , u ( t ) ) d t τ i + b i .

Then, (1.1) has at least one solution in (C[0, T]) n .

Proof Let u = (u 1, u 2,..., u n ) (C[0, T]) n be any solution of (3.1) λ where λ (0, 1). Define the sets I and J as in (3.9). Let 1 ≤ in. Once again, conditions (C10) and (C13) give rise to (3.25).

Similar to the proof of Theorem 3.5, we apply (3.25) and (C17) in (3.12) to get (3.26). Next, using (3.30) and Hölder's inequality, we find that

I u i ( t ) f i ( t , u ( t ) ) d t + I h i ( t ) f i ( t , u ( t ) ) d t + a i 2 τ i I ψ i ( t ) f i ( t , u ( t ) ) d t τ i + b i ( r + H i ) T 1 p i μ r , i q i + a i 2 τ i I ψ i ( t ) μ r , i ( t ) d t τ i + b i ( r + H i ) T 1 p i μ r , i q i + a i 2 τ i ( ψ i p i μ r , i q i ) τ i + b i k i .
(3.51)

Substituting (3.51) into (3.26) and applying (3.34) and (3.35), we find that

β i η i J f i ( t , u ( t ) ) γ i + 1 d t β i J ϕ i ( t ) f i ( t , u ( t ) ) d t + J h i ( t ) f i ( t , u ( t ) ) d t + a i 2 τ i J ψ i ( t ) f i ( t , u ( t ) ) d t τ i + k i β i T γ i p i - γ i - 1 p i ( γ i + 1 ) ϕ i p i J f i ( t , u ( t ) ) γ i + 1 d t 1 γ i + 1 + H i T γ i γ i + 1 J f i ( t , u ( t ) ) γ i + 1 d t 1 γ i + 1 + a i 2 τ i T τ i ( γ i p i - γ i - 1 ) p i ( γ i + 1 ) ( ψ i p i ) τ i J f i ( t , u ( t ) ) γ i + 1 d t τ i γ i + 1 + k i .
(3.52)

Since 1 γ i + 1 <1 and τ i γ i + 1 <1, from (3.52), there exists a constant k i such that (3.37) holds. The rest of the proof proceeds as that of Theorem 3.6. □

Remark 3.1 In Theorem 3.5, the conditions (C10) and (C11) can be replaced by the following, which is evident from the proof.

(C10)' There exist r > 0 and β i > 0 such that for any u (C[0, T]) n ,

u i ( t ) f i ( t , u ( t ) ) β i u i 0 f i ( t , u ( t ) ) for u ( t ) > r and a . e . t [ 0 , T ] ,

where we denote u i 0 = sup t [ 0 , T ] u i ( t ) .

(C11)' There exist r > 0, η i > 0, γ i > 0 and ϕ i L γ i + 1 γ i [ 0 , T ] such that for any u (C[0, T]) n ,

u i 0 η i f i ( t , u ( t ) γ i + ϕ i ( t ) for u ( t ) > r and a . e . t [ 0 , T ] .

Remark 3.2 In Theorems 3.6-3.11, the conditions (C10) and (C13) can be replaced by (C10)' and (C13)' below, and the proof will be similar.

(C13)' There exist r > 0, η i > 0, γ i > 0, and ϕ i L p i [ 0 , T ] such that for any u (C[0, T]) n ,

u i 0 η i f i ( t , u ( t ) ) γ i + ϕ i ( t ) for u ( t ) > r and a . e . t [ 0 , T ] .

4 Existence results for (1.2) in (C l [0, ∞)) n

Let the Banach space B = (C l [0, ∞)) n be equipped with the norm:

u = max 1 i n sup t [ 0 , ) u i ( t ) = max 1 i n u i 0

where we let |u i |0 = sup t[0,∞)|u i (t)|, 1 ≤ in. Throughout, for u B and t [0, ∞), we shall denote that

u ( t ) = max 1 i n u i ( t ) .

Moreover, for each 1 ≤ in, let 1 ≤ p i ≤ ∞ be an integer and q i be such that 1 p i + 1 q i =1. For x L p i [ 0 , ) , we shall define that

x p i = 0 x ( s ) p i d s 1 p i , 1 p i < ess sup s [ 0 , ) x ( s ) , p i = .

We shall apply Theorem 2.1 to obtain the first existence result for (1.2) in (C l [0, ∞)) n .

Theorem 4.1 For each 1 ≤ i ≤ n, assume (D1)-(D5) hold where

(D1) h i C l [0, ∞), denote H i ≡ sup t[0,∞)|h i (t)|,

(D2) f i : [0, ∞) × n is a L 1-Carathéodory function, i.e.,

  1. (i)

    the map u α f i (t, u) is continuous for almost all t [0, ∞),

  2. (ii)

    the map t α f i (t, u) is measurable for all u n ,

  3. (iii)

    for any r > 0, there exists μ r,i L 1[0, ∞) such that |u|r implies |f i (t, u)| ≤ μ r,i (t) for almost all t [0, ∞).

(D3) g i t ( s ) = g i ( t , s ) L [ 0 , ) for each t [0, ∞),

(D4) the map t g i t is continuous from [0, ∞) to L [0, ∞),

(D5) there exists g ̃ i L [ 0 , ) such that g i t g ̃ i in L [0, ∞) as t → ∞, i.e.,

lim t g i t - g ̃ i = lim t ess sup s [ 0 , ) g i ( t , s ) - g ̃ i ( s ) =0.

In addition, suppose there is a constant M > 0, independent of λ, with ||u|| ≠ M for any solution u (C l [0, ∞)) n to

u i ( t ) = λ h i ( t ) + 0 g i ( t , s ) f i ( s , u ( s ) ) d s , t [ 0 , ) , 1 i n ( 4 . 1 ) λ

for each λ (0, 1). Then, (1.2) has at least one solution in (C l [0, ∞)) n .

Proof To begin, let the operator S be defined by

Su ( t ) = ( S 1 u ( t ) , S 2 u ( t ) , , S n u ( t ) ) ,t [ 0 , )
(4.2)

where

S i u ( t ) = h i ( t ) + 0 g i ( t , s ) f i ( s , u ( s ) ) d s , t [ 0 , ) , 1 i n .
(4.3)

Clearly, the system (1.2) is equivalent to u = Su, and (4.1) λ is the same as u = λSu.

First, we shall show that S : (C l [0, ∞)) n → (C l [0, ∞)) n , or equivalently S i : (C l [0, ∞)) nC l [0, ∞), 1 ≤ in. Let u (C l [0, ∞)) n . Then, there exists r > 0 such that ||u|| ≤ r, and from (D2) there exists μ r,i L 1[0, ∞) such that |f i (s, u)| ≤ μ r,i (s) for almost all s [0, ∞). Let t 1, t 2 [0, ∞). Together with (D1) and (D4), we find that

S i u ( t 1 ) - S i u ( t 2 ) h i ( t 1 ) - h i ( t 2 ) + 0 g i t 1 ( s ) - g i t 2 ( s ) μ r , i ( s ) d s h i ( t 1 ) - h i ( t 2 ) + g i t 1 - g i t 2 μ r , i 1 0
(4.4)

as t 1t 2. Hence, S i u C[0, ∞).

To see that S i u is bounded, we have for t [0, ∞),

S i u ( t ) H i + 0 g i ( t , s ) μ r , i ( s ) d s H i + g i t μ r , i 1 .
(4.5)

By (D5), there exists T 1 > 0 such that for t > T 1,

g i t g ̃ i +1.

On the other hand, for t [0, T 1], we have

g i t sup t [ 0 , T 1 ] g i t .

Hence,

sup t [ 0 , ) g i t max sup t [ 0 , T 1 ] g i t , g ̃ i + 1 K i .
(4.6)

It follows from (4.5) that for t [0, ∞),

S i u ( t ) H i + K i μ r , i 1 M i .
(4.7)

Hence, S i u is bounded.

It remains to check the existence of the limit lim t→∞ S i u(t). We claim that

lim t S i u ( t ) = h i ( ) + 0 g ̃ i ( s ) f i ( s , u ( s ) ) d s
(4.8)

where h i (∞) ≡ lim t→∞ h i (t). In fact, it follows from (D5) that

0 [ g i t ( s ) - g ̃ i ( s ) ] f i ( s , u ( s ) ) d s g i t - g ̃ i μ r , i 1 0

as t → ∞. This implies

lim t 0 g i t ( s ) f i ( s , u ( s ) ) d s = 0 g ̃ i ( s ) f i ( s , u ( s ) ) d s

and so (4.8) is proved. We have hence shown that S : (C l [0, ∞)) n → (C l [0, ∞)) n .

Next, we shall prove that S : (C l [0, ∞)) n → (C l [0, ∞)) n is continuous. Let {u m } be a sequence in (C l [0, ∞)) n and u m = ( u 1 m , u 2 m , , u n m ) u. In (C l [0, ∞)) n , i.e., u i m u i , in C l [0, ∞), 1 ≤ in. We need to show that Su mSu in (C l [0, ∞)) n , or equivalently S i u mS i u in C l [0, ∞), 1 ≤ in. There exists r > 0 such that ||u m ||, ||u|| < r, Noting (D2), there exists μ r,i L 1[0, ∞) such that |f i (s, u m )|, |f i (s, u)| ≤ μ r,i (s) for almost all s [0, ∞). Denote S i u(∞) ≡ lim t→∞ S i u(t) and S i u m(∞) ≡ lim t→∞ S i u m(t). In view of (4.8), we get that

S i u m ( ) - S i u ( ) 0 g ̃ i ( s ) [ f i ( s , u m ( s ) ) - f i ( s , u ( s ) ) ] d s .
(4.9)

Since

g ̃ i ( s ) [ f i ( s , u m ( s ) ) - f i ( s , u ( s ) ) ] 0 a s m for almost every s [ 0 , )

and

g ̃ i ( s ) [ f i ( s , u m ( s ) ) - f i ( s , u ( s ) ) ] 2 μ r , i ( s ) g ̃ i ( s ) L 1 [ 0 , ) ,

by the Lebesgue-dominated convergence theorem, it is clear from (4.9) that

S i u m ( ) - S i u ( ) 0 as m .
(4.10)

Further, using (4.8) again we find that

S i u ( t ) - S i u ( ) h i ( t ) - h i ( ) + 0 g i t ( s ) - g ̃ i ( s ) μ r , i ( s ) d s h i ( t ) - h i ( ) + g i t - g ̃ i μ r , i 1 0
(4.11)

as t → ∞. Similarly, we also have that

S i u m ( t ) - S i u m ( ) 0 as t .
(4.12)

Combining (4.10)-(4.12), we have

S i u m ( t ) - S i u ( t ) 0 as t and m

or equivalently, there exist T ^ >0 such that

S i u m ( t ) - S i u ( t ) 0 as m , for all t > T ^ .
(4.13)

It remains to check the convergence in [ 0 , T ^ ] . As in (4.4), we find for any t 1 , t 2 [ 0 , T ^ ] ,

S i u m ( t 1 ) - S i u m ( t 2 ) 0 and S i u ( t 1 ) - S i u ( t 2 ) 0
(4.14)

as t 1t 2. Furthermore, S i u m (t) → S i u(t) pointwise on [ 0 , T ^ ] , since, by the Lebesgue-dominated convergence theorem,

S i u m ( t ) - S i u ( t ) sup t [ 0 , T ^ ] g i t 0 f i ( s , u m ( s ) ) - f i ( s , u ( s ) ) d s 0
(4.15)

as m → ∞. Combining (4.14) and (4.15) and the fact that [ 0 , T ^ ] is compact yields

S i u m ( t ) - S i u ( t ) 0 as m , for all t [ 0 , T ^ ]
(4.16)

Coupling (4.13) and (4.16), we see that S i u mS i u in C l [0, ∞).

Finally, we shall show that S : (C l [0, ∞)) n → (C l [0, ∞)) n is completely continuous. Let Ω be a bounded set in (C l [0, ∞)) n with ||u|| ≤ r for all u Ω We need to show that S i Ω is relatively compact for 1 ≤ in. First, we see that S i Ω is bounded; in fact, this follows from an earlier argument in (4.7). Next, using a similar argument as in (4.4), we see that S i Ω is equicontinuous. Moreover, S i Ω is equiconvergent follows as in (4.11). By Theorem 2.2, we conclude that S i Ω is relatively compact. Hence, S : (C l [0, ∞)) n → (C l [0, ∞)) n is completely continuous.

We now apply Theorem 2.1 with U = {u (C l [0, ∞)) n : ||u|| < M} and B = E = (C l [0, ∞)) n to obtain the conclusion of the theorem. □

Remark 4.1 In Theorem 4.1, the conditions (D2)-(D5) can be stated in terms of general p i and q i as follows, and the proof will be similar:

(D2)' f i : [0, ∞) × n is a L q i -Carathéodory function, i.e.,

  1. (i)

    the map u α f i (t, u) is continuous for almost all t [0, ∞),

  2. (ii)

    the map t α f i (t, u) is measurable for all u n ,

  3. (iii)

    for any r > 0, there exists μ r , i L q i [ 0 , ) such that |u|r implies |f i (t, u)| ≤ μ r,i (t) for almost all t [0, ∞),

(D3)' g i t ( s ) = g i ( t , s ) L p i [ 0 , ) , for each t [0, ∞),

(D4)' the map t g i t is continuous from [0, ∞) to L p i [ 0 , ) ,

(D5)' there exists g ̃ i L p i [ 0 , ) such that g i t g ̃ i , in L p i [ 0 , ) as t → ∞, i.e.,

lim t g i t - g ̃ i p i = lim t 0 g i ( t , s ) - g ̃ i ( s ) p i d s 1 p i = 0 .

Our subsequent Theorems 4.2-4.5 use an argument originating from Brezis and Browder [11]. These results are parallel to Theorems 3.2-3.5 for system (1.1).

Theorem 4.2 Let the following conditions be satisfied for each 1 ≤ in : (D1)-(D5), (C5) , and (C6) where

(C5) there exist B i > 0 such that for any u (C l [0, ∞)) n ,

0 f i ( t , u ( t ) ) 0 g i ( t , s ) f i ( s , u ( s ) ) d s d t B i ,

(C6) there exist r > 0 and α i > 0 with rα i > H i such that for any u (C l [0, ∞)) n ,

u i ( t ) f i ( t , u ( t ) ) r α i f i ( t , u ( t ) ) f o r u ( t ) > r and a . e . t [ 0 , ) .

Then, (1.2) has at least one solution in (C l [0, ∞)) n .

Proof We shall employ Theorem 4.1, so let u = (u 1, u 2,..., u n ) (C l [0, ∞)) n be any solution of (4.1) λ where λ (0, 1). The rest of the proof is similar to that of Theorem 3.2 with the obvious modification that [0, T] be replaced by [0, ∞). Also, noting (4.6) we see that the analog of (3.15) holds. □

In view of the proof of Theorem 4.2, we see that the proof of subsequent Theorems 4.3-4.5 will also be similar to that of Theorems 3.3-3.5 with the appropriate modification. As such, we shall present the results and omit the proof.

Theorem 4.3 Let the following conditions be satisfied for each 1 ≤ in : (D1)-(D5), (C7) and (C8) where

(C7) there exist constants a i ≥ 0 and b i such that for any u (C l [0, ∞)) n ,

0 f i ( t , u ( t ) ) 0 g i ( t , s ) f i ( s , u ( s ) ) d s d t a i 0 f i ( t , u ( t ) ) d t + b i ,

(C8) there exist r > 0 and α i > 0 with rα i > H i + a i such that for any u (C l [0, ∞)) n ,

u i ( t ) f i ( t , u ( t ) ) r α i f i ( t , u ( t ) ) f o r u ( t ) > r and a . e . t [ 0 , ) .

Then, (1.2) has at least one solution in (C l [0, ∞)) n .

Theorem 4.4 Let the following conditions be satisfied for each 1 ≤ in : (D1)-(D5), (C9) and (C10) where

(C9) there exist constants a i ≥ 0, 0 < τ i 1 and b i such that for any u (C l [0, ∞)) n ,

0 f i ( t , u ( t ) ) 0 g i ( t , s ) f i ( s , u ( s ) ) d s d t a i 0 f i ( t , u ( t ) ) d t τ i + b i ,

(C10) there exist r > 0 and β i > 0 such that for any u (C l [0, ∞)) n ,

u i ( t ) f i ( t , u ( t ) ) β i u ( t ) f i ( t , u ( t ) ) f o r u ( t ) > r and a . e . t [ 0 , ) .

Then, (1.2) has at least one solution in (C l [0, ∞)) n .

Theorem 4.5 Let the following conditions be satisfied for each 1 ≤ in : (D1)-(D5), (C10) , (C11) and (C12) where

(C11) there exist r > 0, η i > 0, γ i > 0 and ϕ i L γ i + 1 γ i [ 0 , ) such that for any u (C l [0, ∞)) n ,

u ( t ) η i f i ( t , u ( t ) γ i + ϕ i ( t ) f o r u ( t ) > r and a . e . t [ 0 , ) ,

(C12) there exist a i ≥ 0, 0 < τ i < γ i + 1, b i , and ψ i L γ i + 1 γ i [ 0 , ) with ψ i ≥ 0 almost everywhere on [0, ∞), such that for any u (C l [0, ∞))n,

0 f i ( t , u ( t ) ) 0 g i ( t , s ) f i ( s , u ( s ) ) d s d t a i 0 ψ i ( t ) f i ( t , u ( t ) ) d t τ i + b i .

Also, ϕ i BC[0, ∞), h i L γ i + 1 γ i [ 0 , ) , ψ i BC[0, ∞) and 0 g i ( t , s ) γ i + 1 γ i d sBC [ 0 , ) .

Then, (1.2) has at least one solution in (C l [0, ∞)) n .

We also have a remark similar to Remark 3.1.

Remark 4.2 In Theorem 4.5 the conditions (C10) and (C11) can be replaced by the following; this is evident from the proof.

( C 1 0 ) There exist r > 0 and β i > 0 such that for any u (C l [0, ∞)) n ,

u i ( t ) f i ( t , u ( t ) ) β i u i 0 f i ( t , u ( t ) ) for u ( t ) > r and a . e . t [ 0 , ) ,

where we denote u i 0 = sup t [ 0 , ) u i ( t ) .

( C 1 1 ) There exist r > 0, η i > 0, γ i > 0 and ϕ i L γ i + 1 γ i [ 0 , ) such that for any u (C l [0, ∞)) n ,

u i 0 η i f i ( t , u ( t ) ) γ i + ϕ i ( t ) for u ( t ) > r and a . e . t [ 0 , ) .

5 Existence results for (1.2) in (BC[0, ∞)) n

Let the Banach space B = (BC[0, ∞)) n be equipped with the norm:

u = max 1 i n sup t 0 , u i ( t ) = max 1 i n u i 0

where we let |u i | 0 = sup t[0,∞)|u i (t)|, 1 < i < n. Throughout, for u B and t [0, ∞) we shall denote

u ( t ) = max 1 i n u i ( t ) .

Moreover, for each 1 ≤ in, let 1 ≤ p i ≤ ∞ be an integer and q i be such that 1 p i + 1 q i =1. For x L p i [ 0 , ) , we shall define x p i as in Section 4.

Our first result is a variation of an existence principle of Lee and O'Regan [25].

Theorem 5.1 For each 1 ≤ in, assume (D2)'-(D4)' and (D6) hold where

(D6) h i BC[0, ∞), denote H i ≡ sup t[0, ∞)|h i (t)|.

For each k = 1, 2,..., suppose there exists u k = ( u 1 k , u 2 k , , u n k ) ( C [ 0 , k ] ) n that satisfies

u i k ( t ) = h i ( t ) + 0 k g i ( t , s ) f i ( s , u 1 k ( s ) , u 2 k ( s ) , , u n k ( s ) ) d s , t [ 0 , k ] , 1 i n .
(5.1)

Further, for 1 ≤ in and k = 1, 2,..., there is a bounded set B such that u i k ( t ) B for each t [0, k]. Then, (1.2) has a solution u* (BC[0, ∞)) n such that for 1 ≤ in, u i * ( t ) B ̄ for all t [0, ∞).

Proof First we shall show that

for each 1 i n and = 1 , 2 , , the sequence { u i k } k is uniformly bounded and equicontinuous on [ 0 , ] .
(5.2)

The uniform boundedness of { u i k } k follows immediately from the hypotheses; therefore, we only need to prove that { u i k } k is equicontinuous. Let 1 ≤ in. Since u i k ( t ) B for each t [0, k], there exists μ B L q i [ 0 , ) such that |f i (s,u k (s))| ≤ μ B (s) for almost every s [0, k].Fix t, t' [0, λ]. Then, from (5.1) we find that

u i k ( t ) - u i k ( t ) h i ( t ) - h i ( t ) + 0 k g i t ( s ) - g i t ( s ) f i ( s , u k ( s ) ) d s = h i ( t ) - h i ( t ) + 0 1 [ 0 , k ] g i t ( s ) - g i t ( s ) f i ( s , u k ( s ) ) d s h i ( t ) - h i ( t ) + g i t - g i t p i μ B q i 0

as tt'. Therefore, { u i k } k is equicontinuous on [0, λ].

Let 1 ≤ in. Now, (5.2) and the Arzéla-Ascoli theorem yield a subsequence N 1 of = {1, 2,...} and a function z i 1 C [ 0 , 1 ] such that u i k z i 1 uniformly on [0,1] as k → ∞ in N 1. Let N 2 * = N 1 \ 1 . Then, (5.2) and the Arzéla-Ascoli theorem yield a subsequence N 2 of N 2 * and a function z i 2 C [ 0 , 2 ] such that u i k z i 2 uniformly on [0,2] as k → ∞ in N 2. Note that z i 2 = z i 1 on [0,1] since N 2 N 1. Continuing this process, we obtain subsequences of integers N 1, N 2,... with

N 1 N 2 N , where N { , + 1 , } ,
(5.3)

and functions z i C [ 0 , ] such that

u i k z i uniformly on [ 0 , ] as k in N , and z i + 1 = z i on [ 0 , ] , = 1 , 2 , .
(5.4)

Let 1 ≤ in. Define a function u i * : [ 0 , ] by

u i * ( t ) = z i ( t ) , t [ 0 , ] .
(5.5)

Clearly, u i * C [ 0 , ) and u i * ( t ) B ̄ for each t [0, λ]. It remains to prove that u * = ( u 1 * , u 2 * , , u n * ) solves (1.2). Fix t [0, ∞). Then, choose and fix λ such that t [0, λ]. Take kλ. Now, from (5.1) we have

u i k ( t ) = h i ( t ) + 0 k g i ( t , s ) f i ( s , u 1 k ( s ) , u 2 k ( s ) , , u n k ( s ) ) d s , t [ 0 , ]

or equivalently

u i k ( t ) - h i ( t ) - 0 g i ( t , s ) f i ( s , u 1 k ( s ) , u 2 k ( s ) , , u n k ( s ) ) d s = l k g i ( t , s ) f i ( s , u 1 k ( s ) , u 2 k ( s ) , , u n k ( s ) ) d s , t [ 0 , ] .
(5.6)

Since f i is a L q i -Carathéodory function and u i k ( t ) B for each t [0, k], there exists μ B L q i [ 0 , ) such that

g i ( t , s ) f i ( s , u 1 k ( s ) , u 2 k ( s ) , , u n k ( s ) ) g i t ( s ) μ B ( s ) , a . e . s [ 0 , k ]

and g i t μ B L 1 [ 0 , ) . Let k → ∞ (k N ) in (5.6). Since u i k z i uniformly on [0, ], an application of Lebesgue-dominated convergence theorem gives

z i ( t ) - h i ( t ) - 0 g i ( t , s ) f i ( s , z 1 ( s ) , z 2 ( s ) , , z n ( s ) ) d s l g i t ( s ) μ B ( s ) d s , t [ 0 , ]

or equivalently (noting (5.5))

u i * ( t ) - h i ( t ) - 0 g i ( t , s ) f i ( s , u 1 * ( s ) , u 2 * ( s ) , , u n * ( s ) ) d s l g i t ( s ) μ B ( s ) d s , t [ 0 , ] .
(5.7)

Finally, letting → ∞ in (5.7) and use the fact g i t μ B L 1 [ 0 , ) to get

u i * ( t ) - h i ( t ) - 0 g i ( t , s ) f i ( s , u 1 * ( s ) , u 2 * ( s ) , , u n * ( s ) ) d s = 0 , t [ 0 , ) .

Hence, u * = ( u 1 * , u 2 * , , u n * ) is a solution of (1.2). □

It is noted that one of the conditions in Theorem 5.1, namely, (5.1) has a solution in (C[0, k]) n , which has already been discussed in Section 3. As such, our subsequent Theorems 5.2-5.5 will make use of Theorem 5.1 and the technique used in Section 3. These results are parallel to Theorems 3.2-3.5 and 4.2-4.5.

Theorem 5.2 Let (D2)-(D4) and (D6) be satisfied for each 1 ≤ in. Moreover, suppose the following conditions hold for each 1 ≤ in and each w {1, 2,...}:

(C5) w there exist B i > 0 such that for any u (C[0, w]) n ,

0 w f i ( t , u ( t ) ) 0 w g i ( t , s ) f i ( s , u ( s ) ) d s d t B i ,

(C6) w there exist r > 0 and α i > 0 with rα i > H i (H i as in (D6)) such that for any u (C[0, w]) n ,

u i ( t ) f i ( t , u ( t ) ) r α i f i ( t , u ( t ) ) foru ( t ) randa.e.t [ 0 , w ] .

Then, (1.2) has at least one solution in (BC[0, ∞)) n .

Proof We shall apply Theorem 5.1. To do so, for w = 1, 2,..., we shall show that the system

u i ( t ) = h i ( t ) + 0 w g i ( t , s ) f i ( s , u ( s ) ) d s , t [ 0 , w ] , 1 i n
(5.8)

has a solution in (C[0, w]) n . Obviously, (5.8) is just (1.1) with T = w. Let w {1, 2,...} be fixed.

Let u = (u 1, u 2,..., u n ) (C[0,w]) n be any solution of (3.1)λ (with T = w) where λ (0, 1). We shall model after the proof of Theorem 3.2 with T = w and H i given in (D6). As in (3.9), define

I = { t [ 0 , w ] : u ( t ) r } and J = { t [ 0 , w ] : u ( t ) > r } .

Let 1 ≤ in. If t I, then by (D2) there exists μ r,i L 1[0, ∞) such that

I f i ( t , u ( t ) ) d t I μ r , i ( t ) d t 0 μ r , i ( t )d t = μ r , i 1

[which is the analog of (3.10)]. Proceeding as in the proof of Theorem 3.2, we then obtain the analog of (3.14) as

J f i ( t , u ( t ) ) d t ( H i + r ) μ r , i 1 + B i r α i - H i k i ( independent of w ) .

Further, the analog of (3.15) appears as

u i ( t ) sup t [ 0 , w ] h i ( t ) + sup t [ 0 , w ] ess sup s [ 0 , w ] g i ( t , s ) ( μ r , i 1 + k i ) H i + sup t [ 0 , ) ess sup s [ 0 , ) g i ( t , s ) ( μ r , i 1 + k i ) l i ( independent of w ) , t [ 0 , w ] .
(5.9)

Hence, ||u|| ≤ max1≤in l i = L and we conclude from Theorem 3.1 that (5.8) has a solution u* in (C[0, w]) n . Using similar arguments as in getting (5.9), we find u i * ( t ) l i for each t [0, w]. All the conditions of Theorem 5.1 are now satisfied, it follows that (1.2) has at least one solution in (BC[0, ∞)) n . □

The proof of subsequent Theorems 5.3-5.5 will model after the proof of Theorem 5.2, and will employ similar arguments as in the proof of Theorems 3.3-3.5. As such, we shall present the results and omit the proof.

Theorem 5.3 Let (D2)-(D4) and (D6) be satisfied for each 1 ≤ i ≤ n. Moreover, suppose the following conditions hold for each 1 ≤ i ≤ n and each w {1, 2,...} :

(C7) w there exist constants a i ≥ 0 and b i such that for any u (C[0, w]) n ,

0 w [ f i ( t , u ( t ) ) 0 w g i ( t , s ) f i ( s , u ( s ))d s ] d t a i 0 w | f i ( t , u ( t ) ) | d t + b i ,

(C8) w there exist r > 0 and α i > 0 with rα i > H i + a i (H i as in (D6)) such that for any u (C[0, w]) n ,

u i ( t ) f i ( t , u ( t ) ) r α i f i ( t , u ( t ) ) f o r u ( t ) > r and a . e . t [ 0 , w ] .

Then, (1.2) has at least one solution in (BC[0, ∞)) n .

Theorem 5.4 Let (D2)-(D4) and (D6) be satisfied for each 1 ≤ i ≤ n. Moreover, suppose the following conditions hold for each 1 ≤ i ≤ n and each w {1, 2,...} :

(C9) w there exist constants a i ≥ 0, 0 i ≤ 1 and b i such that for any u (C[0, w]) n ,

0 w f i ( t , u ( t ) ) 0 w g i ( t , s ) f i ( s , u ( s ) ) d s d t a i 0 w f i ( t , u ( t ) ) d t τ i + b i ,

(C10) w there exist r > 0 and β i > 0 such that for any u (C[0, w]) n ,

u i ( t ) f i ( t , u ( t ) ) β i u ( t ) f i ( t , u ( t ) ) f o r u ( t ) > r and a . e . t [ 0 , w ] .

Then, (1.2) has at least one solution in (BC[0, ∞)) n .

Theorem 5.5 Let (D2)-(D4) and (D6) be satisfied for each 1 ≤ i ≤ n. Moreover, suppose the following conditions hold for each 1 ≤ i ≤ n and each w {1, 2,...} : (C10) w ,

(C11) w there exist r > 0, η i > 0, γ i > 0 and ϕ i L γ i + 1 γ i [ 0 , w ] such that for any u (C[0, w]) n ,

u ( t ) η i f i ( t , u ( t ) γ i + ϕ i ( t ) f o r u ( t ) > r and a . e . t [ 0 , w ] ,

(C12) w there exist a i ≥ 0, 0 < τ i < γ i + 1, b i , and ψ i L γ i + 1 γ i [ 0 , w ] with ψ i ≥ 0 almost everywhere on [0, w], such that for any u (C[0, w]) n ,

0 w f i ( t , u ( t ) ) 0 w g i ( t , s ) f i ( s , u ( s ) ) d s d t a i 0 w ψ i ( t ) f i ( t , u ( t ) ) d t τ i + b i .

Also, ϕ i C[0, w], h i L γ i + 1 γ i [ 0 , w ] , ψ i C[0, w] and 0 w g i ( t , s ) γ i + 1 γ i d s C[0, w].

Then, (1.2) has at least one solution in (BC[0, ∞)) n .

We also have a remark similar to Remark 3.1.

Remark 5.1 In Theorem 5.5 the conditions (C10) w and (C11) w can be replaced by the following, this is evident from the proof.

( C 1 0 ) w There exist r > 0 and β i > 0 such that for any u (C[0, w]) n ,

u i ( t ) f i ( t , u ( t ) ) β i u i 0 f i ( t , u ( t ) ) for u ( t ) >r and a.e.t [ 0 , w ] ,

where we denote u i 0 = sup t [ 0 , w ] u i ( t ) .

( C 1 1 ) w There exist r > 0, η i > 0, γ i > 0 and ϕ i L γ i + 1 γ i [ 0 , w ] such that for any u (C[0, w]) n ,

u i 0 η i f i ( t , u ( t ) γ i + ϕ i ( t ) for u ( t ) > r and  a . e . t [ 0 , w ] .

6 Existence of constant-sign solutions

In this section, we shall establish the existence of constant-sign solutions of the systems (1.1) and (1.2), in (C[0, T]) n , (C l [0, ∞)) n and (BC[0, ∞)) n . Once again, we shall employ an argument originated from Brezis and Browder [11].

Throughout, let θ i {-1, 1}, 1 ≤ i ≤ n be fixed. For each 1 ≤ j ≤ n, we define

[ 0 , ) j = { [ 0 , ) , θ j = 1 ( , 0 ] , θ j = 1.

6.1 System (1.1)

Our first result is "parallel" to Theorem 3.2.

Theorem 6.1 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1), (C2)-(C4) with p i = ∞ and q i = 1, (C5), (C6) and (E1)-(E3) where

(E1) θ i h i (t) ≥ 0 for t [0, T],

(E2) g i (t, s) ≥ 0 for s, t [0, T],

(E3) θ i f i (t, u) ≥ 0 for ( t , u ) [ 0 , T ] × j = 1 n [ 0 , ) j .

Then, (1.1) has at least one constant-sign solution in (C[0, T]) n .

Proof First, we shall show that the system

u i ( t ) = h i ( t ) + 0 T g i ( t , s ) f i * ( s , u ( s ) ) d s , t [ 0 , T ] , 1 i n
(6.1)

has a solution in (C[0, T]) n , where,

f i * ( t , u 1 , , u n ) = f i ( t , v 1 , , v n ) ,t [ 0 , T ] ,1in
(6.2)

where for 1 ≤ j ≤ n,

v j = u j , θ j u j 0 0 , θ j u j 0 .

Clearly, f i * ( t , u ) : [ 0 , T ] × n and f i * satisfies (C2).

We shall employ Theorem 3.1, so let u = (u 1, u 2,..., u n ) (C[0, T]) n be any solution of

u i ( t ) = λ h i ( t ) + 0 T g i ( t , s ) f i * ( s , u ( s ) ) d s , t [ 0 , T ] , 1 i n ( 6 . 3 ) λ

where λ (0, 1). Using (E1)-(E3), we have for t [0, T] and 1 ≤ i ≤ n,

θ i u i ( t ) = λ θ i h i ( t ) + 0 T g i ( t , s ) θ i f i * ( s , u ( s ) ) d s 0 .

Hence, u is a constant-sign solution of (6.3) λ , and it follows that

f i * ( t , u ( t ) ) = f i ( t , u ( t ) ) ,t [ 0 , T ] ,1in.
(6.4)

Noting (6.4), we see that (6.3) λ is the same as (3.1) λ . Therefore, using a similar technique as in the proof of Theorem 3.2, we obtain (3.15) and subsequently ||u|| ≤ max1≤in l i L. It now follows from Theorem 3.1 (with M = L + 1) that (6.1) has a solution u* (C[0, T]) n .

Noting (E1)-(E3), we have for t [0, T] and 1 ≤ i ≤ n,

θ i u i * ( t ) = θ i h i ( t ) + 0 T g i ( t , s ) θ i f i * ( s , u * ( s ) ) d s0.

Thus, u* is of constant sign. From (6.2), it is then clear that

f i * ( t , u * ( t ) ) = f i ( t , u * ( t ) ) ,t [ 0 , T ] ,1in.

Hence, u* is actually a solution of (1.1). This completes the proof of the theorem. □

Based on the proof of Theorem 6.1, we can develop parallel results to Theorems 3.3-3.11 as follows.

Theorem 6.2 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1), (C2)-(C4) with p i = ∞ and q i = 1, (C7), (C8) and (E1)-(E3). Then, (1.1) has at least one constant-sign solution in (C[0, T]) n .

Theorem 6.3 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1), (C2)-(C4) with p i = ∞ and q i = 1, (C9), (C10) and (E1)-(E3). Then, (1.1) has at least one constant-sign solution in (C[0, T]) n .

Theorem 6.4 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1), (C2)-(C4) with p i = ∞ and q i = 1, (C10)-(C12) and (E1)-(E3). Then, (1.1) has at least one constant-sign solution in (C[0, T]) n .

Theorem 6.5 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1)-(C4), (C5), (C10), (C13) and (E1)-(E3). Then, (1.1) has at least one constant-sign solution in (C[0, T]) n .

Theorem 6.6 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1)-(C4), (C7), (C10), (C13) and (E1)-(E3). Then, (1.1) has at least one constant-sign solution in (C[0, T]) n .

Theorem 6.7 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1)-(C4), (C10), (C13), (C14) and (E1)-(E3). Then, (1.1) has at least one constant-sign solution in (C[0, T]) n .

Theorem 6.8 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1)-(C4), (C10), (C13), (C15) and (E1)-(E3). Then, (1.1) has at least one constant-sign solution in (C[0, T]) n .

Theorem 6.9 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1)-(C4), (C10), (C13), (C16) and (E1)-(E3). Then, (1.1) has at least one constant-sign solution in (C[0, T]) n .

Theorem 6.10 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (C1)-(C4), (C10), (C13), (C17) and (E1)-(E3). Then, (1.1) has at least one constant-sign solution in (C[0, T]) n .

Remark 6.1 Similar to Remarks 3.1 and 3.2, in Theorem 6.4 the conditions (C10) and (C11) can be replaced by (C10)' and (C11)'; whereas in Theorems 6.5-6.10, (C10) and (C13) can be replaced by (C10)' and (C13)'.

6.2 System (1.2)

We shall first obtain the existence of constant-sign solutions of (1.2) in (C l [0, ∞)) n . The first result is "parallel" to Theorem 4.2.

Theorem 6.11 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (D1)-(D5), (C5) , (C6) and (E1) -(E3) where

(E1) θ i h i (t) ≥ 0 for t [0, ∞),

(E2) g i (t, s) ≥ 0 for s, t [0, ∞),

(E3)θ i f i (t,u) ≥ 0 for ( t , u ) [ 0 , ) × j = 1 n [ 0 , ) j .

Then, (1.2) has at least one constant-sign solution in (C l [0, ∞)) n .

Proof First, we shall show that the system

u i ( t ) = h i ( t ) + 0 g i ( t , s ) f i * ( s , u ( s ) ) d s , t [ 0 , ) , 1 i n
(6.5)

has a solution in (C l [0, ∞)) n . Here,

f i * ( t , u 1 , , u n ) = f i ( t , v 1 , , v n ) ,t [ 0 , ) ,1in
(6.6)

where

v j = u j , θ j u j 0 0 , θ j u j 0 .

Clearly, f i * ( t , u ) : [ 0 , ] × n and f i * satisfies (D2).

We shall employ Theorem 4.1, so let u = (u 1, u 2,..., u n ) (C l [0, ∞)) n be any solution of

u i ( t ) = λ h i ( t ) + 0 g i ( t , s ) f i * ( s , u ( s ) ) d s , t [ 0 , ) , 1 i n ( 6 . 7 ) λ

where λ (0, 1). Then, using a similar technique as in the proof of Theorem 6.1 (and also Theorem 4.2), we can show that (1.2) has a constant-sign solution u* (C l [0, ∞)) n . □

Remark 6.2 Similar to Remark 4.1, in Theorem 6.11 the conditions (D2)-(D5) can be replaced by (D2)'-(D5)'.

Based on the proof of Theorem 6.11, we can develop parallel results to Theorems 4.3-4.5 as follows.

Theorem 6.12 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (D1)-(D5), (C7) , (C8) and (E1) -(E3) . Then, (1.2) has at least one constant-sign solution in (C l [0, ∞)) n .

Theorem 6.13 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (D1)-(D5), (C9) , (C10) and (E1) -(E3) . Then, (1.2) has at least one constant-sign solution in (C l [0, ∞)) n .

Theorem 6.14 Let the following conditions be satisfied for each 1 ≤ i ≤ n : (D1)-(D5), (C10) -(C12) and (E1) -(E3) . Then, (1.2) has at least one constant-sign solution in (C l [0, ∞)) n .

Remark 6.3 Similar to Remark 4.2, in Theorem 6.14 the conditions (C10) and (C11) can be replaced by ( C 1 0 ) and ( C 1 1 ) .

We shall now obtain the existence of constant-sign solutions of (1.2) in (BC[0, ∞)) n . The first result is 'parallel' to Theorem 5.1.

Theorem 6.15 For each 1 ≤ in, assume (D2)'-(D4)' and (D6). For each k = 1, 2,..., suppose there exists a constant-sign u k = ( u 1 k , u 2 k , , u n k ) ( C [ 0 , k ] ) n that satisfies

u i k ( t ) = h i ( t ) + 0 k g i ( t , s ) f i ( s , u 1 k ( s ) , u 2 k ( s ) , , u n k ( s ) ) d s , t [ 0 , k ] , 1 i n .
(6.8)

Further, for 1 ≤ i ≤ n and k = 1, 2,..., there is a bounded set B such that u i k ( t ) B for each t [0, k]. Then, (1.2) has a constant-sign solution u* (BC[0, ∞)) n such that for 1 ≤ i ≤ n, u i * ( t ) B ̄ for all t [0, ∞).

Proof Using a similar technique as in the proof of Theorem 5.1, we can show that (5.2) holds. Let 1 ≤ i ≤ n. Together with the Arzéla-Ascoli theorem, we obtain subsequences of integers N 1, N 2,... satisfying (5.3), and functions z i C [ 0 , ] such that (5.4) holds. Define a function u i * : [ 0 , ) by (5.5), i.e.,

u i * ( t ) = z i ( t ) , t [ 0 , ] .

Since θ i u i k 0, we have θ i z i 0 and so θ i u i * 0. Hence, u i * is of constant sign. The rest of the proof is the same as that of Theorem 5.1. □

The next result is "parallel" to Theorem 5.2.

Theorem 6.16 Let (D2)-(D4) and (D6) be satisfied for each 1 ≤ i ≤ n. Moreover, suppose the following conditions hold for each 1 ≤ i ≤ n and each w {1, 2,...,} : (C5) w , (C6) w and (E1) w - (E3) w where

(E1) w θihi(t) ≥ 0 for t [0, w],

(E2) w gi(t, s) ≥ 0 for s, t [0, w],

(E3) w θ i f i (t,u) ≥ 0 for ( t , u ) [ 0 , w ] × j = 1 n [ 0 , ) j .

Then, (1.2) has at least one constant-sign solution in (BC[0, ∞)) n .

Proof We shall apply Theorem 6.15. To do so, for w = 1, 2,..., we shall show that the system (5.8) has a constant-sign solution u* in (C[0, w]) n . The proof of this is similar to that of Theorem 6.1 (with T = w) and Theorem 5.2. As in (5.9) we have u i * ( t ) l i for each t [0, w] and 1 ≤ i ≤ n. All the conditions of Theorem 6.15 are now satisfied and the conclusion is immediate. □

Based on the proof of Theorem 6.16, we can develop parallel results to Theorems 5.3-5.5 as follows:

Theorem 6.17 Let (D2)-(D4) and (D6) be satisfied for each 1 ≤ i ≤ n. Moreover, suppose the following conditions hold for each 1 ≤ i ≤ n and each w {1, 2,...} : (C7) w , (C8) w and (E1) w -(E3) w . Then, (1.2) has at least one constant-sign solution in (BC[0, ∞)) n .

Theorem 6.18 Let (D2)-(D4) and (D6) be satisfied for each 1 ≤ i ≤ n. Moreover, suppose the following conditions hold for each 1 ≤ i ≤ n and each w {1, 2,...} : (C9) w , (C10) w and (E1) w -(E3) w . Then, (1.2) has at least one constant-sign solution in (BC[0, ∞)) n .

Theorem 6.19 Let (D2)-(D4) and (D6) be satisfied for each 1 ≤ i ≤ n. Moreover, suppose the following conditions hold for each 1 ≤ i ≤ n and each w {1, 2,...} : (C11) w , (C12) w and (E1) w -(E3)w. Then, (1.2) has at least one constant-sign solution in (BC[0, ∞)) n .

Remark 6.4 Similar to Remark 5.1, in Theorem 6.19 the conditions (C10) w and (C11) w can be replaced by ( C 1 0 ) w and ( C 1 1 ) w .

7 Examples

We shall now illustrate the results obtained through some examples.

Example 7.1 In system (1.1), consider the following f i , 1 ≤ i ≤ n :

f i ( t , u ) = κ i ( t , u ) , u P 0 , otherwise .
(7.1)

Here,

P = { u ( C [ 0 , T ] ) n : u 1 ( t ) , u 2 ( t ) , , u n ( t ) > c for all  t [ 0 , T ] }

where c > 0 is a given constant, and κ i is such that

  1. (a)

    the map u α f i (t, u) is continuous for almost all t [0, T];

  2. (b)

    the map t α f i (t, u) is measurable for all u n ;

  3. (c)

    for any r > 0, there exists μ r,i L 1[0, T] such that |u| ≤ r implies |κ i (t, u)| ≤ μ r,i (t) for almost all t [0, T];

  4. (d)

    for any u P, u i(t)κ i (t, u(t)) ≥ 0 for all t [0, T].

Next, suppose for each 1 ≤ in,

h i C [ 0 , T ] with H i sup t [ 0 , T ] h i ( t ) < c .
(7.2)

Clearly, conditions (C1) and (C2) with q i = 1 are fulfilled. We shall check that condition (C6) is satisfied. Pick r > c and α i = c r , 1 ≤ in. Then, from (7.2) we have i = c > H i .

Let u P. Then, from (7.1) we have f i (t, u) = κ i (t, u). Consider ||u(t)|| > r where t [0, T]. If ||u(t)|| = |u i (t)|, then noting (d) we have

u i ( t ) f i ( t , u ( t ) ) = u i ( t ) f i ( t , u ( t ) ) = u ( t ) f i ( t , u ( t ) ) > r f i ( t , u ( t ) ) > r c r f i ( t , u ( t ) ) = r α i f i ( t , u ( t ) ) .
(7.3)

If ||u(t)|| = |u k (t)| for some ki, then

u i ( t ) f i ( t , u ( t ) ) = u i ( t ) f i ( t , u ( t ) ) = r u i ( t ) r f i ( t , u ( t ) ) > r c r f i ( t , u ( t ) ) = r α i f i ( t , u ( t ) ) .
(7.4)

Therefore, from (7.3) and (7.4) we see that condition (C6) holds for u P.

For u (C[0, T]) n \P, we have f i (t, u) = 0 and (C6) is trivially true. Hence, we have shown that condition (C6) is satisfied.

The next example considers a convolution kernel g i (t, s) which arises in nonlinear diffusion and percolation problems; the particular case when n = 1 has been investigated by Bushell and Okrasiński [26].

Example 7.2 Consider system (1.1) with (7.1), (7.2), and for 1 ≤ in,

g i ( t , s ) = ( t - s ) γ i - 1
(7.5)

where γ i > 1.

Clearly, g i satisfies (C3) and (C4) with p i = ∞. Next, we shall check condition (C5). For u P (P is given in Example 7.1), we have

0 T [ f i ( t , u ( t ) ) 0 T g i ( t , s ) f i ( s , u ( s ) ) d s ] d t = 0 T [ κ i ( t , u ( t ) ) 0 T ( t s ) γ i 1 κ i ( s , u ( s ) ) d s ] d t T γ i 1 0 T [ κ i ( t , u ( t ) ) 0 T κ i ( s , u ( s ) ) d s ] d t B i
(7.6)

since κ i (t, u) satisfies (c) (note (c) is stated in Example 7.1). This shows that condition (C5) holds for u P. For u (C[0, T]) n \P, we have f i (t, u) = 0 and (C5) is trivially true. Therefore, condition (C5) is satisfied.

It now follows from Theorem 3.2 that the system (1.1) with (7.1), (7.2) and (7.5) has at least one solution in (C[0, T]) n .

The next example considers an g i (t, s) of which the particular case when n = 1 originates from the well known Emden differential equation.

Example 7.3 Consider system (1.1) with (7.1), (7.2), and for 1 ≤ in,

g i ( t , s ) = ( t - s ) s γ i
(7.7)

where γ i ≥ 0.

Clearly, g i satisfies (C3) and (C4) with p i = ∞. Next, we see that condition (C5) is satisfied. In fact, for u P, corresponding to (7.6) we have

0 T f i ( t , u ( t ) ) 0 T g i ( t , s ) f i ( s , u ( s ) ) d s d t = 0 T κ i ( t , u ( t ) ) 0 T ( t - s ) s γ i κ i ( s , u ( s ) ) d s d t T γ i + 1 0 T κ i ( t , u ( t ) ) 0 T κ i ( s , u ( s ) ) d s d t B i .
(7.8)

Hence, by Theorem 3.2 the system (1.1) with (7.1), (7.2) and (7.7) has at least one solution in (C[0, T]) n .

Our next example illustrates the existence of a positive solution in (C[0, T]) n , this is the particular case of constant-sign solution usually considered in the literature.

Example 7.4 Let θ i = 1, 1 ≤ in. Consider system (1.1) with (7.1), (7.2), and for 1 ≤ in,

h i ( t ) 0 , t [ 0 , T ] .
(7.9)

Clearly, condition (E1) is met, and noting (d) in Example 7.1 condition (E3) is also fulfilled. Moreover, both g i (t, s) in (7.5) and (7.7) satisfy condition (E2). From Examples 7.1-7.3, we see that all the conditions of Theorem 6.1 are met. Hence, we conclude that

the system (1.1) with (7.1), (7.2), (7.9) and (7.5).

and

the system (1.1) with (7.1), (7.2), (7.9) and (7.7).

each of which has at least one positive solution in (C[0, T]) n .

Example 7.5 In system (1.2), consider the following f i , 1 ≤ in :

f i ( t , u ) = κ i ( t , u ) , u P 0 , o t h e r w i s e .
(7.10)

Here,

P = { u ( C l [ 0 , ) ) n : u 1 ( t ) , u 2 ( t ) , , u n ( t ) > c for all  t [ 0 , ) }

where c > 0 is a given constant, and κ i is such that

(a) the map u α f i (t, u) is continuous for almost all t [0, ∞);

(b) the map t α f i (t, u) is measurable for all u n ;

(c) for any r > 0, there exists μ r , i L 1[0, ∞) such that |u| ≤ r implies |κ i (t, u)| ≤ μ r,i (t) for almost all t [0, ∞);

(d) for any u P , u i (t) κ i (t, u(t)) ≥ 0 for all t [0, ∞).

Next, suppose for each 1 ≤ in,

h i C l [ 0 , ) w i t h H i sup t [ 0 , ) h i ( t ) < c .
(7.11)

Clearly, conditions (D1) and (D2) are satisfied. Moreover, using a similar technique as in Example 7.1, we see that condition (C6) is satisfied.

Example 7.6 Consider system (1.2) with (7.10), (7.11), and for 1 ≤ in,

g i ( t , s ) = 1 s + 1 + 1 ( 1 + t ) γ i
(7.12)

where γ i 1.

Clearly, g i satisfies (D3), (D4) and (D5) (take g ̃ i ( s ) = 1 s + 1 ). Next, we shall check condition (C5). For u P (P is given in Example 7.5), we have

0 f i ( t , u ( t ) ) 0 g i ( t , s ) f i ( s , u ( s ) ) d s d t = 0 κ i ( t , u ( t ) ) 0 1 s + 1 + 1 ( 1 + t ) γ i κ i ( s , u ( s ) ) d s d t 2 0 κ i ( t , u ( t ) ) 0 κ i ( s , u ( s ) ) d s d t B i
(7.13)

since κ i (t, u) satisfies (c) (note (c) is stated in Example 7.5). This shows that condition (C5) holds for u P . For u (C l [0, ∞)) n \P , we have f i (t, u) = 0 and (C5) is trivially true. Hence, condition (C5) is satisfied.

We can now conclude from Theorem 4.2 that the system (1.2) with (7.10), (7.11) and (7.12) has at least one solution in (C l [0, ∞)) n .

The next example shows the existence of a positive solution in (C l [0, ∞)) n , this is the special case of constant-sign solution usually considered in the literature.

Example 7.7 Let θ i = 1, 1 ≤ in. Consider system (1.2) with (7.10)-(7.12), and for 1 ≤ in,

h i ( t ) 0,t [ 0 , ) .
(7.14)

Clearly, conditions (E1)-(E3) are satisfied. Noting Examples 7.5 and 7.6, we see that all the conditions of Theorem 6.11 are met. Hence, the system (1.2) with (7.11)-(7.12) has at least one positive solution in (C l [0, ∞)) n .

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Agarwal, R.P., O'Regan, D. & Wong, P.J. Existence results of Brezis-Browder type for systems of Fredholm integral equations. Adv Differ Equ 2011, 43 (2011). https://doi.org/10.1186/1687-1847-2011-43

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