AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 340487 10.1155/2013/340487 340487 Research Article Uniqueness and Existence of Positive Solutions for Singular Differential Systems with Coupled Integral Boundary Value Problems Cui Yujun 1,2 Liu Lishan 1 http://orcid.org/0000-0002-5832-2892 Zhang Xingqiu 3 Wu Yong Hong 1 School of Mathematical Sciences Qufu Normal University Qufu Shandong 273165 China qfnu.edu.cn 2 Department of Mathematics Shandong University of Science and Technology Qingdao 266590 China sdust.edu.cn 3 Department of Mathematics Liaocheng University Liaocheng Shandong 252059 China lcu.edu.cn 2013 20 11 2013 2013 24 07 2013 03 10 2013 2013 Copyright © 2013 Yujun Cui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper provides sufficient conditions for the existence and uniqueness of positive solutions to a singular differential system with integral boundary value. The emphasis here is that the boundary conditions are coupled and this is where the main novelty of this work lies. By mixed monotone method, the existence and uniqueness results of the problem are established. An example is given to demonstrate the main results.

1. Introduction

In recent years, differential system has been studied extensively in the literature (see, for instance,  and their references). Most of the results told us that the equations had at least single and multiple positive solutions. In papers , the authors obtained some of the newest results for differential system with four-point coupled boundary conditions. But there is no result on the uniqueness of solution in them.

In this paper, we discuss the existence and uniqueness of the positive solutions for a new class of boundary value problems of singular differential system. Precisely, we consider the following problem: (1)-x′′(t)=f(t,x(t),y(t)),t(0,1),-y′′(t)=g(t,x(t),y(t)),t(0,1),x(0)=01y(t)dα(t),y(0)=01x(t)dβ(t),x(1)=y(1)=0, where α and β are right continuous on [0,1), left continuous at t=1, and nondecreasing on [0,1], with α(0)=β(0)=0; 01u(s)dα(s) and 01u(s)dβ(s) denote the Riemann-Stieltjes integrals of u with respect to α and β, respectively; fC((0,1)×[0,+)×(0,+),[0,+)), gC((0,1)×  (0,+)×  [0,+),[0,+)); that is, f(t,x,y) may be singular at t=0, t=1, and y=0 and g(t,x,y) may be singular at t=0, t=1, and x=0. By a positive solution of the system (1), we mean that (x,y)(C[0,1]C2(0,1))×(C[0,1]C2(0,1)), (x,y) satisfies (1), and x>0 and y>0 on [0,1).

2. Preliminaries

For each uE:=C[0,1], we write u=max{|u(t)|:t[0,1]}. Clearly, (E,·) is a Banach space. Similarly, for each (x,y)E×E, we write (x,y)1=max{x,y}. Clearly, (E×E,·1) is a Banach space.

Throughout this paper, we shall use the following notation: (2)k(t,s)={t(1-s),0ts1,s(1-t),0st1. It is well known that k(t,s) is the Green function of the following second order boundary value problem: (3)-x′′(t)=0,0<t<1,x(0)=x(1)=0, and k(t,s) is nonnegative continuous function. It is easy to verify that for t,s[0,1]×[0,1], (4)k(t,t)k(s,s)=t(1-t)s(1-s)k(t,s)t(1-t)(or  s(1-s)).

We first list the following assumptions for convenience.

fC((0,1)×[0,+)×(0,+),[0,+)), f(t,x,y) is nondecreasing in x and nonincreasing in y, and there exist λ1,  μ1[0,1) such that (5)cλ1f(t,x,y)f(t,cx,y),x,y>0,c(0,1),(6)f(t,x,cy)c-μ1f(t,x,y),x,y>0,c(0,1);

gC((0,1)×(0,+)×[0,+),[0,+)), g(t,x,y) is nonincreasing in x and nondecreasing in y, and there exist λ2, μ2[0,1) such that (7)cλ2g(t,x,y)g(t,x,cy),x,y>0,c(0,1),(8)g(t,cx,y)c-μ2g(t,x,y),x,y>0,c(0,1).

0<01f(t,1,1-t)dt<+,  0<01g(t,1-t,1)dt<+.

κ1>0,  κ2>0,  κ>0, where (9)κ1=01(1-t)dα(t),κ2=01(1-t)dβ(t),κ=1-κ1κ2.

Remark 1.

By (H1) and (H2), we can get (10)0<01f(t,1-t,1)dt<+,0<01g(t,1,1-t)dt<+.

Remark 2.

(i) (5) and (6) imply that (11)f(t,cx,y)cλ1f(t,x,y),x,y>0,c>1,(12)f(t,x,y)cμ1f(t,x,cy),x,y>0,c>1. Conversely, (11) implies (5) and (12) implies (6).

(ii) (7) and (8) implies that (13)g(t,cx,y)cλ2g(t,x,y),x,y>0,c>1,(14)g(t,x,y)cμ2g(t,x,cy),x,y>0,c>1. Conversely, (13) implies (7) and (14) implies (8).

Lemma 3.

Assume that (H3) holds. Let u,vL[0,1]C(0,1); then the system of BVPs (15)-x′′(t)=u(t),-y′′(t)=v(t),t(0,1),x(0)=01y(t)dα(t),y(0)=01x(t)dβ(t),x(1)=y(1)=0has integral representation (16)x(t)=01G1(t,s)u(s)ds+01H1(t,s)v(s)ds,y(t)=01G2(t,s)v(s)ds+01H2(t,s)u(s)ds, where (17)G1(t,s)=k1(1-t)κ01k(s,τ)dβ(τ)+k(t,s),H1(t,s)=1-tκ01k(s,τ)dα(τ),G2(t,s)=k2(1-t)κ01k(s,τ)dα(τ)+k(t,s),H2(t,s)=1-tκ01k(s,τ)dβ(τ).

Proof.

It is easy to see that (15) is equivalent to the system of integral equations (18)x(t)=x(0)(1-t)+01k(t,s)u(s)ds,t[0,1],(19)y(t)=y(0)(1-t)+01k(t,s)v(s)ds,t[0,1]. Integrating (18) and (19) with respect to dβ(t) and dα(t), respectively, on [0,1] gives (20)01x(t)dβ(t)=x(0)01(1-t)dβ(t)+01k(t,s)u(s)dsdβ(t),01y(t)dα(t)=y(0)01(1-t)dα(t)+01k(t,s)v(s)dsdα(t). Therefore, (21)(-κ211-κ1)(x(0)y(0))=(01k(t,s)u(s)dsdβ(t)01k(t,s)v(s)dsdα(t)), and so (22)(x(0)y(0))=1κ(-κ111-κ2)(01k(t,s)u(s)dsdβ(t)01k(t,s)v(s)dsdα(t)). Substituting (22) into (18) and (19), we have (23)x(t)=κ1(1-t)κ01k(t,s)u(s)dsdβ(t)+1-tκ01k(t,s)v(s)dsdα(t)+01k(t,s)u(s)ds,y(t)=1-tκ01k(t,s)u(s)dsdβ(t)+κ2(1-t)κ01k(t,s)v(s)dsdα(t)+01k(t,s)v(s)ds, which is equivalent to the system (16).

Remark 4.

From (4) and (H3), for t[0,1], we have (24)Gi(t,s)ρs(1-s)(or  ρ(1-t)),Hi(t,s)ρs(1-s)(or  ρ(1-t)),i=1,2,Gi(t,s)ν(1-t)s(1-s),Hi(t,s)ν(1-t)s(1-s),i=1,2, where (25)ρ=max{κ1κ01dβ(τ)+1,κ2κ01dα(τ)+1,1κ01dβ(τ),1κ01dα(τ)},ν=min{κ1κ01τ(1-τ)dβ(τ),κ2κ01τ(1-τ)dα(τ),1κ01τ(1-τ)dβ(τ),1κ01τ(1-τ)dα(τ)}.

Denote (26)P={(x,y)E×E:x(t)γ(1-t)(x,y)1,y(t)γ(1-t)(x,y)1,t[0,1]}, where γ=ν/ρ(0,1). It can be easily seen that P is a cone in E×E. For any real constant r>0, define Pr={(x,y)P:(x,y)1<r}.

Define an operator T:P{θ}P by (27)T(x,y)=(T1(x,y),T2(x,y)), where operators T1,T2:P{θ}Q={uEu(t)0,t[0,1]} are defined by (28)T1(x,y)(t)=01G1(t,s)f(s,x(s),y(s))ds+01H1(t,s)g(s,x(s),y(s))ds,t[0,1],T2(x,y)(t)=01G2(t,s)g(s,x(s),y(s))ds+01H2(t,s)f(s,x(s),y(s))ds,t[0,1]. Now we claim that T(x,y) is well defined for (x,y)P{θ}. In fact, since (x,y)P{θ}, we can see that (29)γ(1-t)(x,y)1x(t),y(t)(x,y)1,t[0,1].

Let c be a positive number such that (x,y)1/c<1 and c>1. From (H1) and Remark 2, we have (30)f(t,x(t),y(t))f(t,c,γ(x,y)1(1-t))cλ1f(t,1,γ(x,y)1c(1-t))cλ1+μ1(γ(x,y)1)-μ1f(t,1,1-t),g(t,x(t),y(t))cλ2+μ2(γ(x,y)1)-μ2f(t,1,1-t). Hence, for any t[0,1], by Remark 4 and equation (30), we get (31)Ti(x,y)(t)ρ01f(s,x(s),y(s))ds+ρ01g(s,x(s),y(s))dsρcλ1+μ1(γ(x,y)1)-μ101f(s,1,1-s)ds+ρcλ2+μ2(γ(x,y)1)-μ201g(s,1-s,1)ds<+,i=1,2. Thus, T is well defined on P{θ}.

Lemma 5.

Assume that (H1), (H2), and (H3) hold. Then, for any 0<a<b<+, T:(Pb¯Pa)P is a completely continuous operator.

Proof.

Firstly, we show that T(Pb¯Pa)P. By Remark 4, for τ,t,s[0,1], we obtain (32)Gi(t,s)γ(1-t)Gi(τ,s),Hi(t,s)γ(1-t)Hi(τ,s),i=1,2,H1(t,s)γ(1-t)G2(τ,s),G1(t,s)γ(1-t)H2(τ,s),H2(t,s)γ(1-t)G1(τ,s),G2(t,s)γ(1-t)H1(τ,s). Hence, for (x,y)Pb¯Pa, t[0,1], we have (33)T1(x,y)(t)=01G1(t,s)f(s,x(s),y(s))ds+01H1(t,s)g(s,x(s),y(s))dsγ(1-t)01G1(τ,s)f(s,x(s),y(s))ds+γ(1-t)01H1(τ,s)g(s,x(s),y(s))ds=γ(1-t)T1(x,y)(τ),τ[0,1],T1(x,y)(t)=01G1(t,s)f(s,x(s),y(s))ds+01H1(t,s)g(s,x(s),y(s))dsγ(1-t)01H2(τ,s)f(s,x(s),y(s))ds+γ(1-t)01G2(τ,s)g(s,x(s),y(s))ds=γ(1-t)T2(x,y)(τ),τ[0,1]. Then T1(x,y)(t)γ(1-t)T1(x,y) and T1(x,y)(t)γ(1-t)T2(x,y); that is, T1(x,y)(t)γ(1-t)(T1(x,y),T2(x,y))1. In the same way, we can prove that T2(x,y)(t)γ(1-t)(T1(x,y),T2(x,y))1. Therefore, T(Pb¯Pa)P.

Next, we prove that T is a compact operator. That is, for any bounded subset VPb¯Pa, we show that T(V) is relatively compact in E×E. Since VPb¯Pa is a bounded subset, there exists a constant c>1 such that (x,y)1=max{x,y}c for all (x,y)V. Notice that, for any (x,y)V, we have (34)T(x,y)1=max{T1(x,y),T2(x,y)} and from (H1), (H2), Remarks 2 and 4, (16), and (18), we obtain (35)Ti(x,y)(t)ρ01f(s,x(s),y(s))ds+ρ01g(s,x(s),y(s))dsρcλ1+μ1(γa)-μ101f(s,1,1-s)ds+ρcλ2+μ2(γa)-μ201g(s,1-s,1)ds<+,i=1,2. Therefore, T(V) is uniformly bounded.

In the following, we shall show that T(V) is equicontinuous on [0,1].

For (x,y)V, t[0,1], using Lemma 3, we have (36)T1(x,y)(t)=01G1(t,s)f(s,x(s),y(s))ds+01H1(t,s)g(s,x(s),y(s))ds=κ1(1-t)κ01(01k(s,τ)dβ(τ))f(s,x(s),y(s))ds+0ts(1-t)f(s,x(s),y(s))ds+t1t(1-s)f(s,x(s),y(s))ds+1-tκ01(01k(s,τ)dα(τ))g(s,x(s),y(s))ds. Differentiating with respect to t and combining (H1) and (H2), we obtain (37)|T1(x,y)(t)|=|-κ1κ01(01k(s,τ)dβ(τ))f(s,x(s),y(s))ds-0tsf(s,x(s),y(s))ds+t1(1-s)f(s,x(s),y(s))ds-1κ01(01k(s,τ)dα(τ))g(s,x(s),y(s))ds|κ1κ01(01k(s,τ)dβ(τ))f(s,x(s),y(s))ds+0tsf(s,x(s),y(s))ds+t1(1-s)f(s,x(s),y(s))ds+1κ01(01k(s,τ)dα(τ))g(s,x(s),y(s))dscλ1+μ1(γa)-μ1(ρκ1κ01f(s,1,1-s)ds+0tsf(s,1,1-s)ds+t1(1-s)f(s,1,1-s)ds)+ρκcλ2+μ2(γa)-μ201g(s,1-s,1)ds=:K(t). Exchanging the integral order, we have (38)01K(t)dt=cλ1+μ1(γa)-μ1(ρκ1κ01f(s,1,1-s)ds+201s(1-s)f(s,1,1-s)ds)+ρκcλ2+μ2(γa)-μ201g(s,1-s,s)ds<+. From the absolute continuity of the integral, we know that T1(V) is equicontinuous on [0,1]. Thus, according to the Ascoli-Arzela theorem, T1(V) is a relatively compact set. In the same way, we can prove that T2(V) is relatively compact. Therefore, T(V) is relatively compact.

Finally, it remains to show that T is continuous. We need to prove only that T1,T2:Pb¯PaQ are continuous. Suppose that (xm,ym),(x0,y0)Pb¯Pa, and (xm,ym)-(x0,y0)10    (m). Let L=sup{(xm,ym)1,  m=0,1,2,}. Then we may still choose positive constants c such that L/c<1 and c>1. From (H1) and Remark 2, we get (39)f(t,xm(t),ym(t))cλ1+μ1(γa)-μ1f(t,1,1-t),m=0,1,2,g(t,xm(t),ym(t))cλ2+μ2(γa)-μ2g(t,1-t,1),m=0,1,2,|T1(xm,ym)(t)-T1(x0,y0)(t)|ρ01|f(s,xm(s),ym(s))-f(s,x0(s),y0(s))|ds+ρ01|g(s,xm(s),ym(s))-g(s,x0(s),y0(s))|ds.

For any ϵ>0, by (H2), there exists a positive number δ(0,1/2) such that (40)[0,δ][1-δ,1]ρcλ1+μ1(γa)-μ1f(s,1,1-s)ds<ɛ4,[0,δ][1-δ,1]ρcλ2+μ2(γa)-μ2g(s,1-s,1)ds<ɛ4. On the other hand, for (x,y)Pb¯Pa and t[δ,1-δ], we have (41)aγδx(t),y(t)b. Since f(t,x,y) and g(t,x,y) are uniformly continuous in [δ,1-δ]×[aγδ,b]×[aγδ,b], we have (42)limm+|f(t,xm(t),ym(t))-f(t,x0(t),y0(t))|=limm+|g(t,xm(t),ym(t))-g(t,x0(t),y0(t))|=0 holds uniformly on t[δ,1-δ]. Then the Lebesgue dominated convergence theorem yields that (43)δ1-δ|f(s,xm(s),ym(s))-f(s,x0(s),y0(s))|ds0δ1-δ|g(s,xm(s),ym(s))-g(s,x0(s),y0(s))|ds0,m+. Thus, for above ɛ>0, there exists a natural number N, for m>N; we have (44)δ1-δρ|f(s,xm(s),ym(s))-f(s,x0(s),y0(s))|ds<ɛ4,δ1-δρ|g(s,xm(s),ym(s))-g(s,x0(s),y0(s))|ds<ɛ4. It follows from (39)–(44) that when m>N(45)T1(xm,ym)-T1(x0,y0)ρ01|f(s,xm(s),ym(s))-f(s,x0(s),y0(s))|ds+ρ01|g(s,xm(s),ym(s))-g(s,x0(s),y0(s))|ds[0,δ][1-δ,1]ρcλ1+μ1(γa)-μ1f(s,1,1-s)ds+[0,δ][1-δ,1]ρcλ2+μ2(γa)-μ2g(s,1-s,1)ds+δ1-δρ|f(s,xm(s),ym(s))-g(s,x0(s),y0(s))|ds+δ1-δρ|g(s,xm(s),ym(s))-g(s,x0(s),y0(s))|ds<ɛ. This implies that T1:Pb¯PaQ is continuous. Similarly, we can prove that T2:Pb¯PaQ is continuous. So, T:Pb¯PaP is continuous. Summing up, T:Pb¯PaP is completely continuous.

Our main tool of this paper is the following cone compression and expansion fixed point theorem.

Lemma 6 (see [<xref ref-type="bibr" rid="B10">18</xref>]).

Let E be a Banach space and P a cone in E. Suppose that Ω1 and Ω2 are two bounded open subsets of E with θΩ1, Ω1¯Ω2. If  T:P(Ω2¯Ω1)P is a completely continuous operator satisfying (46)Txx,for  xPΩ1,Txx,for  xPΩ2, then T has a fixed point in P(Ω2¯Ω1).

3. Main Results

In this section, we present our main results.

Theorem 7.

Suppose that conditions (H1), (H2), and (H3) hold. Then, if λ1+μ1<1 and λ2+μ2<1, the differential system (1) has a unique positive solution (x*,y*).

Proof.

We divide the rather long proof into three steps.

(i) The differential system (1) has at least one positive solution (x*,y*).

Choose r, R such that (47)0<rmin{×f(s,1-s,1)dsν4γλ101)1/1-λ1(ν4γλ101s(1-s)×f(s,1-s,1)dsν4γλ101)1/(1-λ1),12},Rmax{+ρ01g(s,1-s,1)ds)1/1-max{λ1,λ2}1γ(ρ01f(s,1,1-s)ds+ρ01g(s,1-s,1)ds)1/(1-max{λ1,λ2}),1γ,2}.

Clearly 0<r<1<R. By Lemma 5, T:PR¯PrP is completely continuous.

Extend T (denote T yet) to T:PR¯P which is completely continuous.

Then, for (x,y)Pr, we have (48)rγ(1-t)x(t),y(t)r,t[0,1]. By Remarks 1 and 2, (H1), and (H2), we get (49)Ti(x,y)(t)ν401s(1-s)f(s,γr(1-s),r)ds  ν401s(1-s)f(s,γr(1-s),1)dsν4γλ1rλ101s(1-s)f(s,1-s,1)dsr=(x,y)1,i=1,2,t[0,34]. This guarantees that (50)T(x,y)1(x,y)1,(x,y)Pr. On the other hand, for (x,y)PR, we have (51)Rγ(1-t)x(t),y(t)R,t[0,1]. Therefore, (52)Ti(x,y)(t)ρ01f(s,R,γR(1-s))ds+ρ01g(s,γR(1-s),R)dsρ01f(s,R,1-s)ds+ρ01g(s,1-s,R)dsρRλ101f(s,1,1-s)ds+ρRλ201g(s,1-s,1)dsρRmax{λ1,λ2}(01f(s,1,1-s)ds+01g(s,1-s,1)ds)R=(x,y)1,i=1,2,t[0,1].This guarantees that (53)T(x,y)1(x,y)1,(x,y)PR. By the complete continuity of T, (50) and (53), and Lemma 6, we obtain that T has a fixed point (x*,y*) in PR¯Pr. Consequently, (1) has a positive solution (x*,y*) in PR¯Pr.

(ii) Suppose that (x,y) is a positive solution of the differential system (1).

Then there exist real numbers 0<m<1 such that (54)m(1-t)x(t)1m(1-t),m(1-t)y(t)1m(1-t),t[0,1]. From Lemma 5, we know that (x,y)P{θ}. So, we have (55)γ(x,y)1(1-t)x(t),y(t)(x,y)1. Let c be a constant such that (x,y)1/c<1 and c>1/γ. By Lemma 3, we get (56)x(t)ρ(1-t)01f(s,c,γ(x,y)1c(1-s))ds+ρ(1-t)01g(s,γ(x,y)1c(1-s),c)dscλ1+μ1(γ(x,y)1)-μ1ρ(1-t)01f(s,1,1-s)ds+cλ2+μ2(γ(x,y)1)-μ2ρ(1-t)01g(s,1-s,1)ds=:C(1-t),t[0,1]. In the same way, we can prove that y(t)C(1-t), t[0,1]. Then we may pick out m such that m=min{γ(x,y)1,1/C,1/2}, which implies that (54) holds.

(iii) The differential system (1) has a unique positive solution (x*,y*).

Assuming the contrary, we find that the differential system (1) has a positive solution (x*,y*) different from (x*,y*). By (54), there exist δ1,δ2>0, such that (57)δ1(1-t)x*(t),y*(t)1δ1(1-t),t[0,1],δ2(1-t)x*(t),y*(t)1δ2(1-t),t[0,1]. Hence, we have (58)δ1δ2x*(t)x*(t)1δ1δ2x*(t),δ1δ2y*(t)y*(t)1δ1δ2y*(t),t[0,1]. Clearly, δ1δ21. Put (59)δ*=sup{δδx*(t)x*(t)1δx*(t),δy*(t)y*(t)1δy*(t),t[0,1]}. It is easy to see that 1>δ*δ1δ2>0, and (60)δ*x*(t)x*(t)1δ*x*(t),δ*y*(t)y*(t)1δ*y*(t),t[0,1]. So, by (H1), we have (61)f(t,x*(t),y*(t))f(t,δ*x*(t),1δ*y*(t))δ*λ1+μ1f(t,x*(t),y*(t))δ*σf(t,x*(t),y*(t)),g(t,x*(t),y*(t))δ*λ2+μ2g(t,x*(t),y*(t))δ*σg(t,x*(t),y*(t)), where σ=max{λ1+μ1,λ2+μ2} such that σ<1. Therefore, we have (62)x*(t)=T1(x*,y*)(t)=01G1(t,s)f(s,x*(s),y*(s))ds+01H1(t,s)g(s,x*(s),y*(s))dsδ*σ01G1(t,s)f(s,x*(s),y*(s))ds+01H1(t,s)g(s,x*(s),y*(s))ds=δ*σT1(x*,y*)(t)=δ*σx*(t). Similarly, we can get (63)y*(t)δ*σy*(t),x*(t)δ*σx*(t),y*(t)δ*σy*(t). Noticing that 0<δ*, σ<1, we get to a contradiction with the maximality of δ*. Thus, the differential system (1) has a unique positive solution (x*,y*). This completes the proof of Theorem 7.

4. An Example

In this section, we give an example to illustrate the usefulness of our main results. Let us consider the singular differential system with couple boundary value problem (64)-x′′=xyt(1-t)3,-y′′=y3x,x(1)=y(1)=0,x(0)=y(13)+y(12),y(0)=01x(t)dt2.

Let (65)f(t,x,y)=xyt(1-t)3,g(t,x,y)=y3x,α(t)={0,t[0,13),1,t[13,12),2,t[12,1],β(t)=t2,λ1=μ2=12,λ2=μ1=13; then (66)κ1=76,κ2=13,κ=1-κ1κ2=1118,01f(s,1,1-s)ds=B(23,16),01g(s,1-s,1)ds=B(1,12). So all conditions of Theorem 7 are satisfied for (64), and our conclusion follows from Theorem 7.

Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original paper. This project is supported by the National Natural Science Foundation of China (11371221, 11071141), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001), the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province, the Postdoctoral Science Foundation of Shandong Province, and Foundation of SDUST.

Agarwal R. P. O'Regan D. A coupled system of boundary value problems Applicable Analysis 1998 69 3-4 381 385 10.1080/00036819808840668 MR1706522 ZBL0919.34022 Agarwal R. P. O’Regan D. Singular boundary value problems Nonlinear Analysis: Theory, Methods & Applications 1996 27 645 656 10.1016/0362-546X(95)00073-5 ZBL0900.39005 Agarwal R. P. O'Regan D. Nonlinear superlinear singular and nonsingular second order boundary value problems Journal of Differential Equations 1998 143 1 60 95 10.1006/jdeq.1997.3353 MR1604959 ZBL0902.34015 Asif N. A. Khan R. A. Multiplicity results for positive solutions of a coupled system of singular boundary value problems Communications on Applied Nonlinear Analysis 2010 17 2 53 68 MR2669019 ZBL1260.34039 Asif N. A. Khan R. A. Henderson J. Existence of positive solutions to a system of singular boundary value problems Dynamic Systems and Applications 2010 19 2 395 404 MR2741930 ZBL1215.34029 Asif N. A. Khan R. A. Positive solutions to singular system with four-point coupled boundary conditions Journal of Mathematical Analysis and Applications 2012 386 2 848 861 10.1016/j.jmaa.2011.08.039 MR2834792 ZBL1232.34034 Asif N. A. Khan R. A. Positive solutions for a class of coupled system of singular three-point boundary value problems Boundary Value Problems 2009 2009 18 273063 10.1155/2009/273063 MR2525584 ZBL1181.34030 Cheng X. Zhong C. Existence of positive solutions for a second-order ordinary differential system Journal of Mathematical Analysis and Applications 2005 312 1 14 23 10.1016/j.jmaa.2005.03.016 MR2175200 ZBL1088.34016 Cui Y. Sun J. On existence of positive solutions of coupled integral boundary value problems for a nonlinear singular superlinear differential system Electronic Journal of Qualitative Theory of Differential Equations 2012 41 1 13 MR2920964 Liu B. Liu L. Wu Y. Positive solutions for a singular second-order three-point boundary value problem Applied Mathematics and Computation 2008 196 2 532 541 10.1016/j.amc.2007.06.013 MR2388709 ZBL1138.34015 Liu Y. Yan B. Multiple solutions of singular boundary value problems for differential systems Journal of Mathematical Analysis and Applications 2003 287 2 540 556 10.1016/S0022-247X(03)00568-7 MR2024339 ZBL1055.34041 H. Yu H. Liu Y. Positive solutions for singular boundary value problems of a coupled system of differential equations Journal of Mathematical Analysis and Applications 2005 302 1 14 29 10.1016/j.jmaa.2004.08.003 MR2106544 ZBL1076.34022 Ma R. Multiple nonnegative solutions of second-order systems of boundary value problems Nonlinear Analysis: Theory, Methods & Applications 2000 42 6 1003 1010 10.1016/S0362-546X(99)00152-2 MR1780450 ZBL0973.34014 Wang H. On the number of positive solutions of nonlinear systems Journal of Mathematical Analysis and Applications 2003 281 1 287 306 10.1016/S0022-247X(03)00100-8 MR1980092 ZBL1036.34032 Wei Z. Positive solution of singular Dirichlet boundary value problems for second order differential equation system Journal of Mathematical Analysis and Applications 2007 328 2 1255 1267 10.1016/j.jmaa.2006.06.053 MR2290050 ZBL1115.34025 Xu X. Existence and multiplicity of positive solutions for multi-parameter three-point differential equations system Journal of Mathematical Analysis and Applications 2006 324 1 472 490 10.1016/j.jmaa.2005.12.026 MR2262485 ZBL1113.34017 Yuan Y. Zhao C. Liu Y. Positive solutions for systems of nonlinear singular differential equations Electronic Journal of Differential Equations 2008 2008 74 1 14 MR2411070 ZBL1179.34025 Guo D. J. Lakshmikantham V. Nonlinear Problems in Abstract Cones 1988 5 Boston, Mass, USA Academic Press viii+275 MR959889