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Agarwal et al Advances in Difference Equations 2011, 2011:43 http://www.advancesindifferenceequations.com/content/2011/1/43 RESEARCH Open Access Existence results of Brezis-Browder type for systems of Fredholm integral equations Ravi P Agarwal1,2*, Donal O’Regan3 and Patricia JY Wong4 * Correspondence: Agarwal@tamuk.edu Department of Mathematics, Texas A&M University - Kingsville, Kingsville, TX 78363, USA Full list of author information is available at the end of the article Abstract In this article, we consider the following systems of Fredholm integral equations: T ui (t) = hi (t) + gi (t, s)fi (s, u1 (s), u2 (s), , un (s))ds, ∞ ui (t) = hi (t) + gi (t, s)fi (s, u1 (s), u2 (s), , un (s))ds, t ∈ [0, T], ≤ i ≤ n, t ∈ [0, ∞), ≤ i ≤ n Using an argument originating from Brezis and Browder [Bull Am Math Soc 81, 7378 (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 (Cl[0,∞))n as 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 Keywords: system of Fredholm integral equations, Brezis-Browder arguments, constant-sign solutions Introduction In this article, we shall consider the system of Fredholm integral equations: T gi (t, s)fi (s, u1 (s), u2 (s), , un (s))ds, ui (t) = hi (t) + t ∈ [0, T], ≤ i ≤ n (1:1) where < T 0, there exists T(ε) >0 such that |y(t) - y(∞)| 0, there exists μr,i ∈ Lqi [0, T]such that |u| ≤ r implies |fi(t, u)| ≤ μr,i (t) for almost all t Ỵ [0, T]; (C3) git (s) = gi (t, s) ∈ Lpi [0, T]for each t Ỵ [0, T]; (C4) the map t → gitis continuous from [0, T] to Lpi [0, T] In addition, suppose there is a constant M > 0, independent of l, with ||u|| ≠ M for any solution u Ỵ (C[0, T])n to ⎛ ⎞ T ui (t) = λ ⎝hi (t) + gi (t, s)fi (s, u(s))ds⎠ , t ∈ [0, T], ≤ i ≤ n (3.1)λ for each l Ỵ (0, 1) Then, (1.1) has at least one solution in (C[0, T])n Proof Let the operator S be defined by Su(t) = (S1 u(t), S2 u(t), , Sn u(t)), t ∈ [0, T] (3:2) Agarwal et al Advances in Difference Equations 2011, 2011:43 http://www.advancesindifferenceequations.com/content/2011/1/43 Page of 35 where T Si u(t) = hi (t) + gi (t, s)fi (s, u(s))ds, t ∈ [0, T], ≤ i ≤ n (3:3) Clearly, the system (1.1) is equivalent to u = Su, and (3.1)l is the same as u = lSu Note that S maps (C[0, T])n into (C[0, T])n, i.e., Si : (C[0, T])n ® C[0, T], ≤ i ≤ n To see this, note that for any u Ỵ (C[0, T])n, there exits r > such that ||u|| such that ||um||, ||u|| such that for any u Ỵ (C[0, T])n, T T 0 ∫ fi (t, u(t)) ∫ gi (t, s)fi (s, u(s))ds dt ≤ Bi , (C6) there exist r > and > with rai >Hi such that for any u Ỵ (C[0, T])n, ui (t)fi (t, u(t)) ≥ rαi |fi (t, u(t))| for ||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 = (u1, u2, l , un) Ỵ (C[0, T])n be any solution of (3.1)l where l Ỵ (0, 1) Define I = {t ∈ [0, T] : ||u(t)|| ≤ r} J = {t ∈ [0, T] : ||u(t)|| > r} and Clearly, [0, T] = I ∪ J, and hence T = I (3:9) + J Let ≤ i ≤ n If t Ỵ I, then by (C2), there exists μr,i Ỵ L1[0, T] such that |fi(t, u(t))| ≤ μr,i(t) Thus, we get T |fi (t, u(t))|dt ≤ I μr,i (t)dt ≤ I μr,i (t)dt = ||μr,i ||1 (3:10) On the other hand, if t Ỵ J, then it is clear from (C6) that ui(t)fi(t, u(t)) ≥ for a.e t Ỵ [0, T] It follows that ui (t)fi (t, u(t))dt ≥ rαi J |fi (t, u(t))|dt (3:11) J We now multiply (3.1)l by fi(t, u(t)), then integrate from to T to get T T ui (t)fi (t, u(t))dt = λ T T ⎣fi (t, u(t)) hi (t)fi (t, u(t))dt + λ ⎡ ⎤ gi (t, s)fi (s, u(s))ds⎦ dt (3:12) Using (C5) in (3.12) yields T T ui (t)fi (t, u(t))dt ≤ Hi |fi (t, u(t))|dt + Bi (3:13) Agarwal et al Advances in Difference Equations 2011, 2011:43 http://www.advancesindifferenceequations.com/content/2011/1/43 Page of 35 Splitting the integrals in (3.13) and applying (3.11), we get |fi (t, u(t))|dt ≤ Hi ui (t)fi (t, u(t))dt + rαi I |fi (t, u(t))|dt + Hi J I |fi (t, u(t))|dt ≤ Hi |fi (t, u(t))|dt + |fi (t, u(t))|dt + Bi J or (rαi − Hi ) J I |ui (t)fi (t, u(t))|dt + Bi I ≤ (Hi + r)||μr,i ||1 + Bi where we have used (3.10) in the last inequality It follows that |fi (t, u(t))|dt ≤ J (Hi + r)||μr,i ||1 + Bi ≡ ki rαi − Hi (3:14) Finally, it is clear from (3.1)l that for t Î [0, T] and ≤ i ≤ n, T ui (t) ≤ Hi + |gi (t, s)fi (s, u(s))|ds (3:15) = Hi + ∫ + ∫ |gi (t, s)fi (s, u(s))|ds I ≤ Hi + J sup ||git ||∞ (||μr,i ||1 + ki ) ≡ li t∈[0,T] where we have applied (3.10) and (3.14) in the last inequality Thus, |ui|0 ≤ li for ≤ i ≤ n and ||u|| ≤ max1≤i≤n li ≡ L It follows from Theorem 3.1 (with M = L + 1) that (1.1) has a solution u* Ỵ (C[0, T])n □ Theorem 3.3 Let the following conditions be satisfied for each ≤ i ≤ n : (C1)-(C4) with pi = ∞ and qi = 1, (C7) and (C8) where (C7) there exist constants ≥ and bi such that for any u Ỵ (C[0, T])n, ⎡ ⎤ T T T ⎣fi (t, u(t)) gi (t, s)fi (s, u(s))ds⎦ dt ≤ |fi (t, u(t))|dt + bi , (C8) there exist r >0 and >0 with rai > Hi + such that for any u Ỵ (C[0, T])n, ui (t)fi (t, u(t)) ≥ rαi |fi (t, u(t))| for ||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 ≤ i ≤ n We use (C7) in (3.12) to get T T ui (t)fi (t, u(t))dt ≤ T T ≤ (Hi + ) |fi (t, u(t))|dt + |bi | T ⎣fi (t, u(t)) |hi (t)fi (t, u(t))|dt + λ ⎡ ⎤ gi (t, s)fi (s, u(s))ds⎦ dt (3:16) Agarwal et al Advances in Difference Equations 2011, 2011:43 http://www.advancesindifferenceequations.com/content/2011/1/43 Page of 35 Splitting the integrals in (3.16) and applying (3.11) gives (rαi − Hi − ) |fi (t, u(t))|dt ≤ (Hi + ) J |fi (t, u(t))|dt + I |ui (t)fi (t, u(t))|dt + |bi | I ≤ (Hi + + r)||μr,i ||1 + |bi | where we have also used (3.10) in the last inequality It follows that |fi (t, u(t))|dt ≤ J (Hi + + r)||μr,i ||1 + |bi | ≡ ki rαi − Hi − (3:17) The rest of the proof follows that of Theorem 3.2 □ Theorem 3.4 Let the following conditions be satisfied for each ≤ i ≤ n : (C1)-(C4) with pi = ∞ and qi = 1, (C9) and (C10) where (C9) there exist constants ≥ 0, < τi ≤ and bi such that for any u Ỵ (C[0, T])n, T ⎡ ⎤ T ⎣fi (t, u(t)) ⎡ ⎤τi T gi (t, s)fi (s, u(s))ds⎦ dt ≤ ⎣ |fi (t, u(t))|dt⎦ + bi , (C10) there exist r > and bi > such that for any u Ỵ (C[0, T])n, ui (t)fi (t, u(t)) ≥ βi ||u(t)|| · |fi (t, u(t))| for ||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 = (u1, u2, , un) Ỵ (C[0, T])n be any solution of (3.1)l where l Ỵ (0, 1) Define Hi + 2τi + , βi r0 = max r, max 1≤i≤n I0 = {t ∈ [0, T] : ||u(t)|| ≤ r0 } (3:18) J0 = {t ∈ [0, T] : ||u(t)|| > r0 } and T Clearly, [0, T] = I0 ∪ J0 and hence = + I0 J0 Let ≤ i ≤ n If t Ỵ I , then by (C2) there exists μr0 ,i ∈ L1 [0, T] such that |fi (t, u(t))| ≤ μr0 ,i (t) and T |fi (t, u(t))|dt ≤ I0 μr0 ,i (t)dt ≤ I0 μr0 ,i (t)dt = ||μr0 ,i ||1 (3:19) Further, if t Ỵ J0, then by (C10) we have ui (t)fi (t, u(t))dt ≥ βi J0 ||u(t)|| · |fi (t, u(t))|dt ≥ βi r0 J0 |fi (t, u(t))|dt (3:20) J0 Now, using (3.20) and (C9) in (3.12) gives T βi r0 |fi (t, u(t))|dt ≤ J0 |ui (t)fi (t, u(t))|dt + I0 |hi (t)fi (t, u(t))|dt ⎡ ⎤τi T |fi (t, u(t))|dt⎦ + |bi | + ⎣ (3:21) T ≤ |hi (t)fi (t, u(t))|dt ui (t)fi (t, u(t))dt + I0 ⎧⎡ ⎨ ⎣ + ⎩ τi I0 ⎤τi ⎡ |fi (t, u(t))|dt⎦ + ⎣ J0 ⎤τi ⎫ ⎬ |fi (t, u(t))|dt ⎦ ⎭ + |bi | Agarwal et al Advances in Difference Equations 2011, 2011:43 http://www.advancesindifferenceequations.com/content/2011/1/43 Page of 35 where in the last inequality, we have made use of the inequality: (x + y)α ≤ 2α (xα + yα ), x, y ≥ 0, α ≥ Now, noting (3.19) we find that ⎤τi ⎡ |ui (t)fi (t, u(t))|dt + I0 |fi (t, u(t))|dt ⎦ + |bi | τi ⎣ |hi (t)fi (t, u(t))|dt + I0 (3:22) I0 τi τi ≤ (r0 + Hi )||μr0 ,i ||1 + (||μr0 ,i ||1 ) + |bi | ≡ k i Substituting (3.22) in (3.21) then yields ⎤τi ⎡ βi r0 |fi (t, u(t))|dt ≤ J0 |fi (t, u(t))|dt ⎦ + k i τi ⎣ |hi (t)fi (t, u(t))|dt + J0 ⎡ J0 τi ⎣ ≤ Hi |fi (t, u(t))|dt + J0 ⎤τi |fi (t, u(t))|dt⎦ + k i J0 Since τi ≤ 1, there exists a constant ki such that (βi r0 − Hi − 2τi ) |fi (t, u(t))|dt ≤ ki J0 which leads to |fi (t, u(t))|dt ≤ J0 k i ≡ ki βi r0 − Hi − 2τi (3:23) Finally, it is clear from (3.1)l that for t Ỵ [0, T] and ≤ i ≤ n, T |ui (t)| ≤ Hi + |gi (t, s)fi (s, u(s))|ds (3:24) = Hi + ∫ + ∫ |gi (t, s)fi (s, u(s))|ds I0 ≤ Hi + J0 sup ||git ||∞ (||μr0 ,i ||1 + ki ) ≡ li t∈[0,T] 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 ≤ i ≤ n : (C1), (C2)(C4) with pi = ∞ and qi = 1, (C10), (C11) and (C12) where (C11) there exist r > 0, hi > 0, gi > and φ ∈ L i T])n, γi +1 γi [0, T]such that for any u Ỵ (C[0, ||u(t)|| ≥ ηi |fi (t, u(t)|γi + φi (t) for ||u(t)|| > r and a.e t ∈ [0, T], (C12) there exist ≥ 0, 0, gi > and φi ∈ Lpi [0, T]such that for any u Ỵ (C[0, T]) n , ||u(t)|| ≥ ηi |fi (t, u(t)|γi + φi (t) for ||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 = (u1, u2, , un) Ỵ (C[0, T])n be any solution of (3.1)l where l Ỵ (0, 1) Define the sets I and J as in (3.9) Let ≤ i ≤ n If t Ỵ I, then by (C2), there exists μr,i ∈ Lqi [0, T] such that |fi(t, u(t))| ≤ μr,i(t) Consequently, we have T |fi (t, u(t))|dt ≤ I μr,i (t)dt ≤ I μr,i (t)dt ≤ T pi ||μr,i ||qi (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 |fi (t, u(t))|γi +1 dt βi ηi J φi (t)fi (t, u(t)) |dt + Hi ≤ βi J |fi (t, u(t))|dt + Bi + J J |fi (t, u(t))|dt + Bi + (r + Hi )T pi ||μr,i ||qi J φi (t)fi (t, u(t)) |dt + Hi = βi J (3:31) φi (t)fi (t, u(t)) |dt + Hi ≤ βi (|ui (t)| + Hi )|fi (t, u(t))|dt I |fi (t, u(t))|dt + B i J where (3.30) has been used in the last inequality and B ≡ Bi + (r + Hi )T pi ||μr,i ||q i i Agarwal et al Advances in Difference Equations 2011, 2011:43 http://www.advancesindifferenceequations.com/content/2011/1/43 Page 21 of 35 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 ≤ i ≤ n : (D1)-(D5), (C5)∞, and (C6)∞ where (C5)∞ there exist Bi >0 such that for any u Ỵ (Cl[0, ∞))n, ⎤ ⎡ ∞ ∞ ⎣fi (t, u(t)) gi (t, s)fi (s, u(s))ds⎦ dt ≤ Bi , (C6)∞ there exist r >0 and >0 with rai > Hi such that for any u Ỵ (Cl[0, ∞))n, ui (t)fi (t, u(t)) ≥ rαi |fi (t, u(t))| for ||u(t)|| > r and a.e t ∈ [0, ∞) Then, (1.2) has at least one solution in (Cl[0, ∞))n Proof We shall employ Theorem 4.1, so let u = (u1, u2, , un) Ỵ (Cl[0, ∞))n be any solution of (4.1)l where l Ỵ (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 ≤ i ≤ n : (D1)-(D5), (C7)∞ and (C8)∞ where (C7)∞ there exist constants ≥ and bi such that for any u Ỵ (Cl[0, ∞))n, ⎤ ⎡ ∞ ∞ ∞ ⎣fi (t, u(t)) gi (t, s)fi (s, u(s))ds⎦ dt ≤ |fi (t, u(t))|dt + bi , (C8)∞ there exist r >0 and >0 with rai > Hi + such that for any u Ỵ (Cl[0, ∞))n, ui (t)fi (t, u(t)) ≥ rαi |fi (t, u(t))| for ||u(t)|| > r and a.e t ∈ [0, ∞) Then, (1.2) has at least one solution in (Cl[0, ∞))n Theorem 4.4 Let the following conditions be satisfied for each ≤ i ≤ n : (D1)-(D5), (C9)∞ and (C10)∞ where (C9)∞ there exist constants ≥ 0, < τi ≤ and bi such that for any u Ỵ (Cl[0, ∞))n, ⎤ ⎤τi ⎡ ⎡∞ ∞ ∞ ⎣fi (t, u(t)) gi (t, s)fi (s, u(s))ds⎦ dt ≤ ⎣ |fi (t, u(t))|dt⎦ + bi , (C10)∞ there exist r >0 and bi >0 such that for any u Ỵ (Cl[0, ∞))n, ui (t)fi (t, u(t)) ≥ βi ||u(t)|| · |fi (t, u(t))| for ||u(t)|| > r and a.e t ∈ [0, ∞) Then, (1.2) has at least one solution in (Cl[0, ∞))n Theorem 4.5 Let the following conditions be satisfied for each ≤ i ≤ n : (D1)-(D5), (C10)∞, (C11)∞ and (C12)∞ where (C11)∞ there exist r >0, hi >0, gi >0 and φ ∈ L i ∞))n, γi +1 γi [0, ∞)such that for any u Ỵ (Cl[0, Agarwal et al Advances in Difference Equations 2011, 2011:43 http://www.advancesindifferenceequations.com/content/2011/1/43 Page 22 of 35 ||u(t)|| ≥ ηi |fi (t, u(t)|γi + φi (t) for ||u(t)|| > r and a.e t ∈ [0, ∞), (C12)∞ there exist ≥ 0, 0 and bi >0 such that for any u Ỵ (Cl[0, ∞))n, ui (t)fi (t, u(t)) ≥ βi |ui |0 · |fi (t, u(t))| for ||u(t)|| > r and a.e t ∈ [0, ∞), sup where we denote |ui |0 = t∈[0,∞) |ui (t)| (C11)∞ There exist r > 0, hi > 0, gi > and φ ∈ L i (Cl[0, ∞))n, γi +1 γi [0, ∞) such that for any u Ỵ |ui |0 ≥ ηi |fi (t, u(t))|γi + φi (t) for ||u(t)|| > r and a.e t ∈ [0, ∞) Existence results for (1.2) in (BC[0, ∞))n Let the Banach space B = (BC[0, ∞))n be equipped with the norm: ||u|| = max sup |ui (t)| = max |ui |0 1≤i≤n t∈[0,∞) 1≤i≤n where we let |ui|0 = suptỴ[0,∞) |ui(t)|, < i < n Throughout, for u Ỵ B and t Ỵ [0, ∞) we shall denote ||u(t)|| = max |ui (t)| 1≤i≤n Moreover, for each ≤ i ≤ n, let ≤ p i ≤ ∞ be an integer and q i be such that pi + qi = For x ∈ Lpi [0, ∞), we shall define ||x||pi as in Section Our first result is a variation of an existence principle of Lee and O’Regan [25] Theorem 5.1 For each ≤ i ≤ n, assume (D2)’-(D4)’ and (D6) hold where (D6) hi Ỵ BC[0, ∞), denote Hi ≡ suptỴ[0, ∞) |hi(t)| For each k = 1, 2, , suppose there exists uk = (uk , uk , , uk ) ∈ (C[0, k])n that satisn fies k gi (t, s)fi (s, uk (s), uk (s), , uk (s))ds, n uk (t) = hi (t) + i t ∈ [0, k], ≤ i ≤ n (5:1) Agarwal et al Advances in Difference Equations 2011, 2011:43 http://www.advancesindifferenceequations.com/content/2011/1/43 Page 23 of 35 Further, for ≤ i ≤ n and k = 1, 2, , there is a bounded set B ⊆ ℝ such that uk (t) ∈ B for each t Ỵ [0, k] Then, (1.2) has a solution u* Ỵ (BC[0, ∞))n such that for i ¯ ≤ i ≤ n, u∗ (t) ∈ B for all t Ỵ [0, ∞) i Proof First we shall show that for each ≤ i ≤ n and = 1, 2, , the sequence {uk }k≥ i is uniformly bounded and equicontinuous on [0, ] (5:2) The uniform boundedness of {uk }k≥ follows immediately from the hypotheses; therei fore, we only need to prove that {uk }k≥ is equicontinuous Let ≤ i ≤ n Since i uk (t) ∈ B for each t Ỵ [0, k], there exists μB ∈ Lqi [0, ∞) such that |fi(s,uk(s))| ≤ μB(s) i for almost every s Ỵ [0, k].Fix t, t’ Ỵ [0, l] Then, from (5.1) we find that k uk (t) i − uk (t i ) ≤ |hi (t) − hi (t )| + git (s) − git (s) · |fi (s, uk (s))|ds ∞ = |hi (t) − hi (t )| + 1[0,k] git (s) − git (s) · |fi (s, uk (s))|ds ≤ |hi (t) − hi (t )| + ||git − git ||pi · ||μB ||qi → as t ® t’ Therefore, {uk }k≥ is equicontinuous on [0, l] i Let ≤ i ≤ n Now, (5.2) and the Arzéla-Ascoli theorem yield a subsequence N1 of N = {1, 2, } and a function z1 ∈ C[0, 1] such that uk → z1 uniformly on [0,1] as k ® ∞ i i i ∗ in N1 LetN2 = N1 \ {1} Then, (5.2) and the Arzéla-Ascoli theorem yield a subsequence ∗ N2 of N2 and a function z2 ∈ C[0, 2] such that uk → z2 uniformly on [0,2] as k ® ∞ in i i i N2 Note that z2 = z1 on [0,1] since N2 ⊆ N1 Continuing this process, we obtain subsei i quences of integers N1, N2, with N1 ⊇ N2 ⊇ · · · ⊇ N ⊇ · · · , where N ⊆ { , + 1, }, (5:3) and functions zi ∈ C[0, ] such that uk → zi uniformly on [0, ] as k → ∞ in N , i and zi +1 = zi on [0, ], = 1, 2, (5:4) Let ≤ i ≤ n Define a function u∗ : [0, ∞] → R by i u∗ (t) = zi (t), t ∈ [0, ] i (5:5) ¯ Clearly, u∗ ∈ C[0, ∞) and u∗ (t) ∈ B for each t Ỵ [0, l] It remains to prove that i i ∗ , u∗ , , u∗ ) = (u1 n solves (1.2) Fix t Ỵ [0, ∞) Then, choose and fix l such that t Ỵ [0, l] Take k ≥ l Now, from (5.1) we have u∗ k uk (t) i gi (t, s)fi (s, uk (s), uk (s), , uk (s))ds, t ∈ [0, ] n = hi (t) + Agarwal et al Advances in Difference Equations 2011, 2011:43 http://www.advancesindifferenceequations.com/content/2011/1/43 Page 24 of 35 or equivalently uk (t) − hi (t) − i gi (t, s)fi (s, uk (s), uk (s), , uk (s))ds n (5:6) k gi (t, s)fi (s, uk (s), uk (s), , uk (s))ds, t ∈ [0, ] n = l Since fi is a Lqi-Carathéodory function and uk (t) ∈ B for each t Ỵ [0, k], there exists i μB ∈ Lqi [0, ∞) such that |gi (t, s)fi (s, uk (s), uk (s), , uk (s))| ≤ |git (s)|μB (s), n a.e s ∈ [0, k] and |git |μB L1 [0, ) Let k đ (k ẻ Nℓ) in (5.6) Since uk → zi uniformly on [0, ℓ], i an application of Lebesgue-dominated convergence theorem gives ∞ gi (t, s)fi (s, z1 (s), z2 (s), , zn (s))ds ≤ zi (t) − hi (t) − |git (s)|μB (s)ds, t ∈ [0, ] l or equivalently (noting (5.5)) ∞ u∗ (t) − hi (t) − i gi (t, s)fi (s, u∗ (s), u∗ (s), , u∗ (s))ds ≤ n |git (s)|μB (s)ds, t ∈ [0, ] (5:7) l Finally, letting ℓ ® ∞ in (5.7) and use the fact |git |μB ∈ L1 [0, ∞) to get ∞ u∗ (t) i gi (t, s)fi (s, u∗ (s), u∗ (s), , u∗ (s))ds = 0, t ∈ [0, ∞) n − hi (t) − Hence, u∗ = (u∗ , u∗ , , u∗ ) is a solution of (1.2) □ n 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 As such, our subsequent Theorems 5.2-5.5 will make use of Theorem 5.1 and the technique used in Section 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 ≤ i ≤ n Moreover, suppose the following conditions hold for each ≤ i ≤ n and each w Ỵ {1, 2, }: (C5)w there exist Bi > such that for any u Ỵ (C[0, w])n, w w fi (t, u(t)) gi (t, s)fi (s, u(s))ds dt ≤ Bi , (C6)w there exist r >0 and >0 with rai > Hi (Hi as in (D6)) such that for any u Ỵ (C[0, w])n, ui (t)fi (t, u(t)) ≥ rαi |fi (t, u(t))| for ||u(t)|| r and a.e t ∈ [0, w] Then, (1.2) has at least one solution in (BC[0, ∞))n Proof We shall apply Theorem 5.1 To so, for w = 1, 2, , we shall show that the system Agarwal et al Advances in Difference Equations 2011, 2011:43 http://www.advancesindifferenceequations.com/content/2011/1/43 Page 25 of 35 w gi (t, s)fi (s, u(s))ds, t ∈ [0, w], ≤ i ≤ n ui (t) = hi (t) + (5:8) has a solution in (C[0, w])n Obviously, (5.8) is just (1.1) with T = w Let w Î {1, 2, } be fixed Let u = (u1, u2, , un) Ỵ (C[0,w])n be any solution of (3.1)l (with T = w) where l Ỵ (0, 1) We shall model after the proof of Theorem 3.2 with T = w and Hi given in (D6) As in (3.9), define I = {t ∈ [0, w] : ||u(t)|| ≤ r} and J = {t ∈ [0, w] : ||u(t)|| > r} Let ≤ i ≤ n If t Î I, then by (D2) there exists μr,i Î L1[0, ∞) such that ∞ |fi (t, u(t))|dt ≤ I μr,i (t)dt ≤ μr,i (t)dt = ||μr,i ||1 I [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 |fi (t, u(t))|dt ≤ J (Hi + r)||μr,i ||1 + Bi ≡ ki rαi − Hi (independent of w) Further, the analog of (3.15) appears as |ui (t)| ≤ sup |hi (t)| + t∈[0,w] ≤ Hi + sup ess sup |gi (t, s)| (||μr,i ||1 + ki ) t∈[0,w] sup ess t∈[0,∞) s∈[0,w] (5:9) sup |gi (t, s)| (||μr,i ||1 + ki ) ≡ li (independent of w), t ∈ [0, w] s∈[0,∞) Hence, ||u|| ≤ max1≤i≤n li = 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∗ (t)| ≤ lifor each t Ỵ [0, w] All the conditions of Theorem 5.1 are now satisfied, it i 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 ≤ i ≤ n Moreover, suppose the following conditions hold for each ≤ i ≤ n and each w Ỵ {1, 2, } : (C7)w there exist constants ≥ and bi such that for any u Ỵ (C[0, w])n, ⎤ ⎡ w w w ⎣fi (t, u(t)) gi (t, s)fi (s, u(s))ds⎦ dt ≤ fi (t, u(t)) dt + bi , (C8)w there exist r >0 and > with rai > Hi + (Hi as in (D6)) such that for any u Ỵ (C[0, w])n, ui (t)fi (t, u(t)) ≥ rαi |fi (t, u(t))| for ||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 ≤ i ≤ n Moreover, suppose the following conditions hold for each ≤ i ≤ n and each w Ỵ {1, 2, } : Agarwal et al Advances in Difference Equations 2011, 2011:43 http://www.advancesindifferenceequations.com/content/2011/1/43 Page 26 of 35 (C9)w there exist constants ≥ 0, 0 and bi >0 such that for any u Î (C[0, w])n, ui (t)fi (t, u(t)) ≥ βi ||u(t)|| · ||fi (t, u(t))| for ||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 ≤ i ≤ n Moreover, suppose the following conditions hold for each ≤ i ≤ n and each w Ỵ {1, 2, } : (C10)w, γi + such that for any u Ỵ (C[0, (C11)w there exist r >0, hi >0, gi >0 and φi ∈ L γi [0, w] w])n, ||u(t)|| ≥ ηi |fi (t, u(t)|γi + φi (t) for ||u(t)|| > r and a.e t ∈ [0, w], γi + with ψi ≥ almost ψi ∈ L γi [0, w] everywhere on [0, w], such that for any u Î (C[0, w])n, ⎤ ⎤τi ⎡ ⎡ w w w (C12)w there exist ≥ 0, < τi < gi + 1, bi, and ⎣fi (t, u(t)) gi (t, s)fi (s, u(s))ds⎦ dt ≤ ⎣ Also, ji Ỵ C[0, w], ψi (t)|fi (t, u(t))|dt ⎦ + bi γi + , ψi Ỵ C[0, w] and hi ∈ L γi [0, w] w γi + gi (t, s) γi ds Ỵ 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 (C10)w There exist r >0 and bi >0 such that for any u Ỵ (C[0, w])n, ui (t)fi (t, u(t)) ≥ βi |ui |0 · |fi (t, u(t))| for ||u(t)|| > r and a.e t ∈ [0, w], sup where we denote |ui |0 = t∈[0,w] |ui (t)| (C11)wThere exist r >0, hi >0, gi >0 and γi + such that for any u Ỵ (C φi ∈ L γi [0, w] [0, w])n, |ui |0 ≥ ηi |fi (t, u(t)|γi + φi (t) for ||u(t)|| > r and a.e t ∈ [0, w] 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, (Cl[0, ∞))n and (BC[0, ∞))n Once again, we shall employ an argument originated from Brezis and Browder [11] Throughout, let θi Ỵ {-1, 1}, ≤ i ≤ n be fixed For each ≤ j ≤ n, we define [0, ∞)j = [0, ∞), θj = (−∞, 0], θj = −1 Agarwal et al Advances in Difference Equations 2011, 2011:43 http://www.advancesindifferenceequations.com/content/2011/1/43 Page 27 of 35 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 ≤ i ≤ n : (C1), (C2)(C4) with pi = ∞ and qi = 1, (C5), (C6) and (E1)-(E3) where (E1) θihi(t) ≥ for t Ỵ [0, T], (E2) gi(t, s) ≥ for s, t Ỵ [0, T], n (E3) θi fi(t, u) ≥ for (t, u) ∈ [0, T] × j=1 [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 T gi (t, s)fi∗ (s, u(s))ds, t ∈ [0, T], ≤ i ≤ n ui (t) = hi (t) + (6:1) has a solution in (C[0, T])n, where, fi∗ (t, u1 , , un ) = fi (t, v1 , , ), t ∈ [0, T], ≤ i ≤ n (6:2) where for ≤ j ≤ n, vj = uj , θj uj ≥ 0, θj uj ≤ Clearly, fi∗ (t, u) : [0, T] × Rn → R and fi∗satisfies (C2) We shall employ Theorem 3.1, so let u = (u1, u2, , un) Ỵ (C[0, T])n be any solution of ⎛ ⎞ T ui (t) = λ ⎝hi (t) + gi (t, s)fi∗ (s, u(s))ds⎠ , t ∈ [0, T], ≤ i ≤ n (6.3)λ where l Ỵ (0, 1) Using (E1)-(E3), we have for t Ỵ [0, T] and ≤ i ≤ n, ⎛ ⎞ T θi ui (t) = λ ⎝θi hi (t) + gi (t, s)θi fi∗ (s, u(s))ds⎠ ≥ 0 Hence, u is a constant-sign solution of (6.3)l, and it follows that fi∗ (t, u(t)) = fi (t, u(t)), t ∈ [0, T], ≤ i ≤ n (6:4) Noting (6.4), we see that (6.3)l is the same as (3.1)l Therefore, using a similar technique as in the proof of Theorem 3.2, we obtain (3.15) and subsequently ||u|| ≤ max1≤i≤n li ≡ L It now follows from Theorem 3.1 (with M = L + 1) that (6.1) has a solution u* Ỵ (C[0, T])n Noting (E1)-(E3), we have for t Ỵ [0, T] and ≤ i ≤ n, θi u∗ (t) = θi hi (t) + i T gi (t, s)θi fi∗ (s, u∗ (s))ds ≥ Thus, u* is of constant sign From (6.2), it is then clear that fi∗ (t, u∗ (t)) = fi (t, u∗ (t)), t ∈ [0, T], ≤ i ≤ n Hence, u* is actually a solution of (1.1) This completes the proof of the theorem □ Agarwal et al Advances in Difference Equations 2011, 2011:43 http://www.advancesindifferenceequations.com/content/2011/1/43 Based on the proof of Theorem 6.1, we can develop parallel results to Theorems 3.33.11 as follows Theorem 6.2 Let the following conditions be satisfied for each ≤ i ≤ n : (C1), (C2)(C4) with pi = ∞ and qi = 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 ≤ i ≤ n : (C1), (C2)(C4) with pi = ∞ and qi = 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 ≤ i ≤ n : (C1), (C2)(C4) with pi = ∞ and qi = 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 ≤ 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 ≤ 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 ≤ 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 ≤ 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 ≤ 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 ≤ 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 (Cl[0, ∞))n The first result is “parallel” to Theorem 4.2 Theorem 6.11 Let the following conditions be satisfied for each ≤ i ≤ n : (D1)-(D5), (C5)∞, (C6)∞ and (E1)∞-(E3)∞ where (E1)∞ θihi(t) ≥ for t Ỵ [0, ∞), (E2)∞ gi(t, s) ≥ for s, t Ỵ [0, ∞), n (E3)θifi(t,u) ≥ for (t, u) ∈ [0, ∞) × j=1 [0, ∞)j Then, (1.2) has at least one constant-sign solution in (Cl[0, ∞))n Page 28 of 35 Agarwal et al Advances in Difference Equations 2011, 2011:43 http://www.advancesindifferenceequations.com/content/2011/1/43 Page 29 of 35 Proof First, we shall show that the system ∞ gi (t, s)fi∗ (s, u(s))ds, t ∈ [0, ∞), ≤ i ≤ n ui (t) = hi (t) + (6:5) has a solution in (Cl[0, ∞))n Here, fi∗ (t, u1 , , un ) = fi (t, v1 , , ), t ∈ [0, ∞), ≤ i ≤ n (6:6) where vj = uj , θj uj ≥ 0, θj uj ≤ Clearly, fi∗ (t, u) : [0, ∞] × Rn → R and fi∗ satisfies (D2) We shall employ Theorem 4.1, so let u = (u1, u2, , un) Ỵ (Cl[0, ∞))n be any solution of ⎛ ∞ ui (t) = λ ⎝hi (t) + ⎞ gi (t, s)fi∗ (s, u(s))ds⎠ , t ∈ [0, ∞), ≤ i ≤ n (6.7)λ where l Ỵ (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* Ỵ (Cl [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 ≤ i ≤ n : (D1)-(D5), (C7)∞, (C8)∞ and (E1)∞-(E3)∞ Then, (1.2) has at least one constant-sign solution in (Cl [0, ∞))n Theorem 6.13 Let the following conditions be satisfied for each ≤ i ≤ n : (D1)-(D5), (C9)∞, (C10)∞ and (E1)∞-(E3)∞ Then, (1.2) has at least one constant-sign solution in (Cl[0, ∞))n Theorem 6.14 Let the following conditions be satisfied for each ≤ i ≤ n : (D1)-(D5), (C10)∞-(C12)∞ and (E1)∞-(E3)∞ Then, (1.2) has at least one constant-sign solution in (Cl[0, ∞))n Remark 6.3 Similar to Remark 4.2, in Theorem 6.14 the conditions (C10) ∞ and (C11)∞ can be replaced by (C10)∞ and (C11)∞ 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 ≤ i ≤ n, assume (D2)’-(D4)’ and (D6) For each k = 1, 2, , suppose there exists a constant-sign uk = (uk , uk , , uk ) ∈ (C[0, k])nthat satisfies n k uk (t) i gi (t, s)fi (s, uk (s), uk (s), , uk (s))ds, t ∈ [0, k], ≤ i ≤ n (6:8) n = hi (t) + Agarwal et al Advances in Difference Equations 2011, 2011:43 http://www.advancesindifferenceequations.com/content/2011/1/43 Further, for ≤ i ≤ n and k = 1, 2, , there is a bounded set B ⊆ ℝ such that uk (t) ∈ Bfor each t Ỵ [0, k] Then, (1.2) has a constant-sign solution u*Ỵ (BC[0, ∞))n i ¯ such that for ≤ i ≤ n, u∗ (t) ∈ Bfor all t Ỵ [0, ∞) i Proof Using a similar technique as in the proof of Theorem 5.1, we can show that (5.2) holds Let ≤ i ≤ n Together with the Arzéla-Ascoli theorem, we obtain subsequences of integers N1, N2, satisfying (5.3), and functions zi ∈ C[0, ] such that (5.4) holds Define a function u∗ : [0, ∞) → R by (5.5), i.e., i u∗ (t) = zi (t), t ∈ [0, ] i Since θi uk ≥ 0, we have θi zi ≥ and so θi u∗ ≥ Hence, u∗ is of constant sign The i i i 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 ≤ i ≤ n Moreover, suppose the following conditions hold for each ≤ i ≤ n and each w Ỵ {1, 2, ,} : (C5)w, (C6)w and (E1)w - (E3)w where (E1)w θihi(t) ≥ for t Ỵ [0, w], (E2)w gi(t, s) ≥ for s, t Ỵ [0, w], n (E3)w θifi(t,u) ≥ for (t, u) ∈ [0, w] × j=1 [0, ∞)j Then, (1.2) has at least one constant-sign solution in (BC[0, ∞))n Proof We shall apply Theorem 6.15 To 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∗ (t)| ≤ li i for each t Ỵ [0, w] and ≤ 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 ≤ i ≤ n Moreover, suppose the following conditions hold for each ≤ i ≤ n and each w Ỵ {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 ≤ i ≤ n Moreover, suppose the following conditions hold for each ≤ i ≤ n and each w Ỵ {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 ≤ i ≤ n Moreover, suppose the following conditions hold for each ≤ i ≤ n and each w Ỵ {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 (C10)w and (C11)w Examples We shall now illustrate the results obtained through some examples Page 30 of 35 Agarwal et al Advances in Difference Equations 2011, 2011:43 http://www.advancesindifferenceequations.com/content/2011/1/43 Page 31 of 35 Example 7.1 In system (1.1), consider the following fi, ≤ i ≤ n : fi (t, u) = κi (t, u), u ∈ P 0, otherwise (7:1) Here, P = {u ∈ (C[0, T])n : u1 (t), u2 (t), , un (t) > c for all t ∈ [0, T]} where c >0 is a given constant, and i is such that (a) the map u a fi(t, u) is continuous for almost all t Ỵ [0, T]; (b) the map t a fi(t, u) is measurable for all u Ỵ ℝn; (c) for any r >0, there exists μr,i Ỵ L1[0, T] such that |u| ≤ r implies |i (t, u)| ≤ μr,i (t) for almost all t Ỵ [0, T]; (d) for any u Ỵ P, ui(t)i(t, u(t)) ≥ for all t Ỵ [0, T] Next, suppose for each ≤ i ≤ n, hi ∈ C[0, T] with Hi ≡ sup |hi (t)| < c (7:2) t∈[0,T] Clearly, conditions (C1) and (C2) with qi = are fulfilled We shall check that condition (C6) is satisfied Pick r > c and αi = c, ≤ i ≤ n Then, from (7.2) we have rai = c r > Hi Let u Ỵ P Then, from (7.1) we have fi(t, u) = i(t, u) Consider ||u(t)|| > r where t Ỵ [0, T] If ||u(t)|| = |ui(t)|, then noting (d) we have ui (t)fi (t, u(t)) = |ui (t)| · |fi (t, u(t))| = ||u(t)|| · |fi (t, u(t))| > r|fi (t, u(t))| c > r · · |fi (t, u(t))| r = rαi |fi (t, u(t))| (7:3) If ||u(t)|| = |uk(t)| for some k ≠ i, then ui (t)fi (t, u(t)) = |ui (t)| · |fi (t, u(t))| = r · c · |fi (t, u(t))| r = rαi |fi (t, u(t))| >r· |ui (t)| · |fi (t, u(t))| r (7:4) Therefore, from (7.3) and (7.4) we see that condition (C6) holds for u Ỵ P For u Ỵ (C[0, T])n\P, we have fi(t, u) = and (C6) is trivially true Hence, we have shown that condition (C6) is satisfied The next example considers a convolution kernel gi(t, s) which arises in nonlinear diffusion and percolation problems; the particular case when n = has been investigated by Bushell and Okrasiński [26] Example 7.2 Consider system (1.1) with (7.1), (7.2), and for ≤ i ≤ n, gi (t, s) = (t − s)γi −1 where gi > (7:5) Agarwal et al Advances in Difference Equations 2011, 2011:43 http://www.advancesindifferenceequations.com/content/2011/1/43 Page 32 of 35 Clearly, gi satisfies (C3) and (C4) with pi = ∞ Next, we shall check condition (C5) For u Ỵ P (P is given in Example 7.1), we have T ⎡ T gi (t, s)fi (s, u(s))ds⎦ dt = ⎣fi (t, u(t)) ⎤ T ⎡ ⎣κi (t, u(t)) T ≤T ⎤ T ⎡ γi −1 (t − s) ⎤ T ⎣κi (t, u(t)) γi −1 κi (s, u(s))ds⎦ dt (7:6) κi (s, u(s))ds⎦ dt ≤ Bi since i(t, u) satisfies (c) (note (c) is stated in Example 7.1) This shows that condition (C5) holds for u Î P For u Î (C[0, T])n\P, we have fi(t, u) = 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 gi(t, s) of which the particular case when n = originates from the well known Emden differential equation Example 7.3 Consider system (1.1) with (7.1), (7.2), and for ≤ i ≤ n, gi (t, s) = (t − s)sγi (7:7) where gi ≥ Clearly, gi satisfies (C3) and (C4) with pi = ∞ Next, we see that condition (C5) is satisfied In fact, for u Ỵ P, corresponding to (7.6) we have T ⎡ T gi (t, s)fi (s, u(s))ds⎦ dt = ⎣fi (t, u(t)) ⎤ T ⎡ ⎤ T (t − s)sγi κi (s, u(s))ds⎦ dt ⎣κi (t, u(t)) T ⎡ T ⎣κi (t, u(t)) ≤ T γi +1 ⎤ κi (s, u(s))ds⎦ dt (7:8) ≤ Bi 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 θi = 1, ≤ i ≤ n Consider system (1.1) with (7.1), (7.2), and for ≤ i ≤ n, hi (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 gi(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 Agarwal et al Advances in Difference Equations 2011, 2011:43 http://www.advancesindifferenceequations.com/content/2011/1/43 Page 33 of 35 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 fi, ≤ i ≤ n : fi (t, u) = κi (t, u), u ∈ P∞ 0, otherwise (7:10) Here, P∞ = {u ∈ (Cl [0, ∞))n : u1 (t), u2 (t), , un (t) > c for all t ∈ [0, ∞)} where c >0 is a given constant, and i is such that (a)∞ the map u a fi(t, u) is continuous for almost all t Ỵ [0, ∞); (b)∞ the map t a fi(t, u) is measurable for all u Ỵ ℝn; (c)∞ for any r > 0, there exists μr,i Ỵ L1[0, ∞) such that |u| ≤ r implies |i(t, u)| ≤ μr,i(t) for almost all t Ỵ [0, ∞); (d)∞ for any u Ỵ P∞, ui(t) i(t, u(t)) ≥ for all t Ỵ [0, ∞) Next, suppose for each ≤ i ≤ n, hi ∈ Cl [0, ∞) with Hi ≡ sup |hi (t)| < c (7:11) t∈[0,∞) 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 ≤ i ≤ n, gi (t, s) = 1 + s + (1 + t)γi (7:12) where gi ≥ ˜ Clearly, gi satisfies (D3), (D4) and (D5) (take gi (s) = s+1) Next, we shall check condi- tion (C5)∞ For u Ỵ P∞ (P∞ is given in Example 7.5), we have ⎤ ⎡ ∞ ∞ gi (t, s)fi (s, u(s))ds⎦ dt ⎣fi (t, u(t)) ∞ ⎡ ∞ ⎣κi (t, u(t)) = ∞ ⎡ ⎤ 1 + s + (1 + t)γ i ⎤ ∞ ⎣κi (t, u(t)) ≤2 κi (s, u(s))ds⎦ dt (7:13) κi (s, u(s))ds⎦ dt ≤ Bi since i(t, u) satisfies (c)∞ (note (c)∞ is stated in Example 7.5) This shows that condition (C5)∞ holds for u Î P∞ For u Î (Cl[0, ∞))n\P∞, we have fi(t, u) = 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 (Cl[0, ∞))n Agarwal et al Advances in Difference Equations 2011, 2011:43 http://www.advancesindifferenceequations.com/content/2011/1/43 Page 34 of 35 The next example shows the existence of a positive solution in (Cl[0, ∞))n, this is the special case of constant-sign solution usually considered in the literature Example 7.7 Let θi = 1, ≤ i ≤ n Consider system (1.2) with (7.10)-(7.12), and for ≤ i ≤ n, hi (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 (Cl[0, ∞))n Acknowledgements The authors would like to thank the referee for the comments which help to improve the paper Author details Department of Mathematics, Texas A&M University - Kingsville, Kingsville, TX 78363, USA 2Department of Mathematics, Faculty of Science, King Abdulaziz University, 21589 Jeddah, Saudi Arabia 3Department of Mathematics, National University of Ireland, Galway, Ireland 4School of Electrical and Electronic Engineering, Nanyang Technological University 50 Nanyang Avenue, Singapore 639798, Singapore Authors’ contributions All authors contributed equally to the manuscript and read and approved the final draft Competing interests The authors declare that they have no competing interests Received: 18 March 2011 Accepted: 11 October 2011 Published: 11 October 2011 References Erbe, LH, Hu, S, Wang, H: Multiple positive solutions of some boundary value problems J Math Anal Appl 184, 640–648 (1994) 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Triple solutions of constant sign for a system of Fredholm integral equations Cubo 6, 1–45 (2004) Agarwal, RP, O’Regan, D, Wong, PJY: Constant-sign solutions of a system of integral equations:... Constant-sign solutions of a system of Fredholm integral equations Acta Appl Math 80, 57–94 (2004) Agarwal, RP, O’Regan, D, Wong, PJY: Eigenvalues of a system of Fredholm integral equations Math