Positive Solutions for p-Laplacian Fourth-Order Differential System with Integral Boundary Conditions

Abstract

This paper investigates the existence of positive solutions for a class of singular p -Laplacian fourth-order differential equations with integral boundary conditions. By using the fixed point theory in cones, explicit range for λ and μ is derived such that for any λ and μ lie in their respective interval, the existence of at least one positive solution to the boundary value system is guaranteed.

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Jiqiang Jiang 1 and Lishan Liu 1,2 and Yonghong Wu 2

Recommended by Cengiz Çinar

1, School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
2, Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia

Received 25 March 2012; Accepted 15 May 2012

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.

1. Introduction

Boundary value problems for ordinary differential equations arise in different areas of applied mathematics and physics and so on. Fourth-order differential equations boundary value problems, including those with the p -Laplacian operator, have their origin in beam theory [1, 2], ice formation [3, 4], fluids on lungs [5], brain warping [6, 7], designing special curves on surfaces [8], and so forth. In beam theory, more specifically, a beam with a small deformation, a beam of a material that satisfies a nonlinear power-like stress and strain law, and a beam with two-sided links that satisfies a nonlinear powerlike elasticity law can be described by fourth order differential equations along with their boundary value conditions. For more background and applications, we refer the reader to the work by Timoshenko [9] on elasticity, the monograph by Soedel [10] on deformation of structure, and the work by Dulcska [11] on the effects of soil settlement. Due to their wide applications, the existence and multiplicity of positive solutions for fourth-order (including p -Laplacian operator) boundary value problems has also attracted increasing attention over the last decades; see [12-33] and references therein. In [28], Zhang and Liu studied the following singular fourth-order four-point boundary value problem [figure omitted; refer to PDF] where _[varphi]p (x)=^|x|p-2 x , p>1 , 0<ξ , η<1 , 0...4;a , b<1 , f∈C((0,1)×(0,∞),(0,∞)) , f(t,x) may be singular at t=0 and/or t=1 and x=0 . The authors gave sufficient conditions for the existence of one positive solution by using the upper and lower solution method, fixed point theorems, and the properties of the Green function.

In [32], Zhang et al. discussed the existence and nonexistence of symmetric positive solutions of the following fourth-order boundary value problem with integral boundary conditions: [figure omitted; refer to PDF] where _[varphi]p (x)=^|x|p-2 x , p>1 , w∈^L1 [0,1] is nonnegative, symmetric on the interval [0,1],f:[0,1]×[0,+∞)[arrow right][0,+∞) is continuous, f(1-t,x)=f(t,x) for all (t,x)∈[0,1]×[0,+∞) , and g,h∈^L1 [0,1] are nonnegative, symmetric on [0,1] .

Motivated by the work of the above papers, in this paper, we study the existence of positive solutions of the following singular fourth-order boundary value system with integral boundary conditions: [figure omitted; refer to PDF] where λ and μ are positive parameters, _[varphi]pi (x)=^{|x|_pi -2} x , _pi >1 , _[varphi]qi =^{[varphi]_pi -1} , 1/_pi +1/_qi =1 , _ξi ,_ηi :[0,1][arrow right]^...+ (i=1,2) are nondecreasing functions of bounded variation, and the integrals in (1.3) are Riemann-Stieltjes integrals, _f1 :[0,1]×^...0+ ×^...+ [arrow right]^...+ and _f2 :[0,1]×^...+ ×^...0+ [arrow right]^...+ are two continuous functions, and _f1 (t,x,y) may be singular at x=0 while _f2 (t,x,y) may be singular at y=0 ; _a1 , _a2 :(0,1)[arrow right]^...+ are continuous and may be singular at t=0 and/or t=1 , in which ^...+ =[0,+∞) , ^...0+ =(0,+∞) .

Compared to previous results, our work presented in this paper has the following new features. Firstly, our study is on singular nonlinear differential systems, that is, _a1 and _a2 in (1.3) are allowed to be singular at t=0 and/or t=1 , meanwhile _f1 (t,x,y) is allowed to be singular at x=0 while _f2 (t,x,y) is allowed to be singular at y=0 , which bring about many difficulties. Secondly, the main tools used in this paper is a fixed-point theorem in cones, and the results obtained are the conditions for the existence of solutions to the more general system (1.3). Thirdly, the techniques used in this paper are approximation methods, and a special cone has been developed to overcome the difficulties due to the singularity and to apply the fixed-point theorem. Finally, we discuss the boundary value problem with integral boundary conditions, that is, system (1.3) including fourth-order three-point, multipoint and nonlocal boundary value problems as special cases. To our knowledge, very few authors studied the existence of positive solutions for p -Laplacian fourth-order differential equation with boundary conditions involving Riemann-Stieltjes integrals. Hence we improve and generalize the results of previous papers to some degree, and so it is interesting and important to study the existence of positive solutions for system (1.3).

The rest of this paper is organized as follows. In Section 2, we present some lemmas that are used to prove our main results. In Section 3, the existence of positive solution for system (1.3) is established by using the fixed point theory in cone. Finally, in Section 4, one example is also included to illustrate the main results.

Definition 1.1.

A vector (u,v)∈(^C2 [0,1]∩^C4 (0,1))×(^C2 [0,1]∩^C4 (0,1)) is said to be a positive solution of system (1.3) if and only if (u,v) satisfies (1.3) and u(t)>0 , v(t)...5;0 or u(t)...5;0 , v(t)>0 for any t∈(0,1) .

Let K be a cone in a Banach space E . For 0<r<R<+∞ , let _Kr ={x∈K:||x||<r} , ∂_Kr ={x∈K:||x||=r} , and _K¯r,R ={x∈K:r...4;||x||...4;R} . The proof of the main theorem of this paper is based on the fixed point theory in cone. We list one lemma [34, 35] which is needed in our following argument.

Lemma 1.2.

Let K be a positive cone in real Banach space E and T:_K¯r,R [arrow right]K a completely continuous operator. If the following conditions hold

(i) ||Tx||...4;||x|| for x∈∂_KR ;

(ii) there exists e∈∂_K1 such that x...0;Tx+me for any x∈∂_Kr and m>0 . Then T has a fixed point in _K¯r,R .

Remark 1.3.

If (i) and (ii) are satisfied for x∈∂_Kr and x∈∂_KR , respectively. Then Lemma 1.2 is still true.

2. Preliminaries and Lemmas

The basic space used in this paper is E=C[0,1]×C[0,1] . Obviously, the space E is a Banach space if it is endowed with the norm as follows: [figure omitted; refer to PDF]

for any (u,v)∈E . Denote ^C+ [0,1]={u∈C[0,1]:u(t)...5;0, 0...4;t...4;1} . For convenience, we list the following assumptions:

(_H1 ) : _a1 ,_a2 :(0,1)[arrow right]^...+ are continuous and [figure omitted; refer to PDF] where e(s)=s(1-s) , s∈[0,1] .

(_H2 ) : _ξi ,_ηi :[0,1][arrow right]^...+ (i=1,2) are nondecreasing functions of bounded variation, and _αi ∈[0,1) , _βi ∈[0,1) , where [figure omitted; refer to PDF]

(_H3 ) : _f1 :[0,1]×^...0+ ×^...+ [arrow right]^...+ , _f2 :[0,1]×^...+ ×^...0+ [arrow right]^...+ are continuous and satisfy [figure omitted; refer to PDF]

where _g1 ,_h2 :[0,1]×^R0+ [arrow right]^R+ are continuous and nonincreasing in the second variable, and _g2 ,_h1 :[0,1]×^R+ [arrow right]^R+ are continuous and for any constant r>0 , [figure omitted; refer to PDF]

Similar to the proof of Lemmas 2.1 and 2.2 in [32], the following two lemmas are valid.

Lemma 2.1.

If (_H2 ) holds, then for any y∈L(0,1) , the boundary value problem [figure omitted; refer to PDF] has a unique solution [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Lemma 2.2.

If (_H2 ) holds, then for any z∈L(0,1) , the boundary value problem [figure omitted; refer to PDF] has a unique solution [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Remark 2.3.

For t , s∈[0,1] , we have [figure omitted; refer to PDF]

Remark 2.4.

If (_H2 ) holds, it is easy to testify _Hi (t,s) defined by (2.8) that: [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Remark 2.5.

From (2.11), we can prove that the properties of _Ki (t,s) (i=1,2) are similar to those of _Hi (t,s) (i=1,2) .

Lemma 2.6.

For x>0 , y>0 , we have [figure omitted; refer to PDF]

Proof.

The proof of this lemma is easy, and we omit it.

Let [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

It is easy to see that K is a cone of E . For any 0<r<R , let _Kr,R ={(u,v)∈K:r<||u||<R, r<||v||<R} .

Remark 2.7.

By the definition of _ρi , _σi , _γi , _νi (i=1,2) , we have 0<Λ<1 .

To overcome singularity, we consider the following approximate problem of (1.3): [figure omitted; refer to PDF] where n is a positive integer and [figure omitted; refer to PDF]

Clearly, _fin ∈C([0,1]×^...+ ×^...+ ,^...+ ) (i=1,2) .

By Lemmas 2.1 and 2.2, for each n∈... , λ>0 , μ>0 , let us define operators ^Anλ :K[arrow right]C[0,1] , ^Bnμ :K[arrow right]C[0,1] , and _Tn :K[arrow right]E by [figure omitted; refer to PDF] and _Tn (u,v)=(^Anλ (u,v),^Bnμ (u,v)) , respectively.

Lemma 2.8.

Assume that (_H1 ) - (_H3 ) hold, then for each λ>0 , μ>0 , n∈... , _Tn :_K¯r,R [arrow right]K is a completely continuous operator.

Proof.

Let λ>0 , μ>0 , and n∈... be fixed. For any (u,v)∈K , by (2.21), we have [figure omitted; refer to PDF] which implies that ^Anλ is nonnegative and concave on [0,1] . Similarly, by (2.22) we can obtain that ^Bnμ is nonnegative and concave on [0,1] . For any (u,v)∈K and t∈[0,1] , it follows from (2.13) that [figure omitted; refer to PDF]

Thus [figure omitted; refer to PDF]

On the other hand, by (2.13) and (2.18), we have [figure omitted; refer to PDF]

This implies that [figure omitted; refer to PDF]

Similar to (2.27), we also have [figure omitted; refer to PDF]

Therefore, _Tn (K)⊂K .

Next, we prove that _Tn :_K¯r,R [arrow right]K is completely continuous. Suppose (_um ,_vm )∈_K¯r,R and (_u0 ,_v0 )∈_K¯r,R with ||(_um ,_vm )-(_u0 ,_v0 )||[arrow right]0 (m[arrow right]∞) . We notice that t∈[0,1] _fin (t,_um (t),_vm (t))-_fin (t,_u0 (t),_v0 (t))[arrow right]0 (m[arrow right]∞) . Using the Lebesgue dominated convergence theorem, we have [figure omitted; refer to PDF]

Therefore, [figure omitted; refer to PDF]

Similarly, we also have [figure omitted; refer to PDF]

So ^Anλ :_K¯r,R [arrow right]C[0,1] and ^Bnβ :_K¯r,R [arrow right]C[0,1] are continuous. Therefore, _Tn :_K¯r,R [arrow right]K is also continuous.

Let D⊂_K¯r,R be any bounded set, then for any (u,v)∈D , we have (u,v)∈K , r...4;||u||...4;R , r...4;||v||...4;R , and then 0<Λr...4;u(τ)...4;R , 0<Λr...4;v(τ)...4;R for any τ∈[0,1] . By (_H3 ) , we have [figure omitted; refer to PDF]

It is easy to show that ^Anλ (D) is uniformly bounded. In order to show that _Tn is a compact operator, we only need to show that ^Anλ (D) is equicontinuous. By the uniformly continuity of _H1 (t,s) on [0,1]×[0,1] , for all [straight epsilon]>0 , there is δ>0 such that for any _t1 , _t2 , s∈[0,1] and |_t1 -_t2 |<δ , we have [figure omitted; refer to PDF]

This together with (2.15) and (2.32) implies [figure omitted; refer to PDF]

This means that ^Anλ (D) is equicontinuous. By the Arzela-Ascoli theorem, ^Anλ (D) is a relatively compact set and that ^Anλ :_K¯r,R [arrow right]C[0,1] is a completely continuous operator.

In the same way, we can show that ^Bnμ :_K¯r,R [arrow right]C[0,1] is also completely continuous, and so _Tn :_K¯r,R [arrow right]K is completely continuous. Now since λ , μ , and n are given arbitrarily, the conclusion of this lemma is valid.

3. Main Results

For notational convenience, we denote by [figure omitted; refer to PDF] where α denotes 0 or ∞ . The main results of this paper are the following.

Theorem 3.1.

Assume that (_H1 ) - (_H3 ) hold. Then we have:

(_C1 ) : If ^f10 , _f1∞ , ^f20 ∈(0,∞) and _M1 /_f1∞ <_N1 /^f10 , then for each λ∈(_M1 /_f1∞ ,_N1 /^f10 ) , μ∈(0,_N2 /^f20 ) , the system (1.3) has at least one positive solution.

(_C2 ) : If ^f10 , ^f20 , _f2∞ ∈(0,∞) and _M2 /_f2∞ <_N2 /^f20 , then for each λ∈(0,_N1 /^f10 ) , μ∈(_M2 /_f2∞ ,_N2 /^f20 ) , the system (1.3) has at least one positive solution.

(_C3 ) : If ^f10 =0 , _f1∞ =∞ , 0<^f20 <∞ , then for each λ∈(0,∞) , μ∈(0,_N2 /^f20 ) , the system (1.3) has at least one positive solution.

(_C4 ) : If 0<^f10 <∞ , ^f20 =0 , _f2∞ =∞ , then for each λ∈(0,_N1 /^f10 ) , μ∈(0,∞) , the system (1.3) has at least one positive solution.

(_C5 ) : If ^fi0 =0 , _fi∞ =∞ (i=1,2) , then for each λ∈(0,∞) , μ∈(0,∞) , the system (1.3) has at least one positive solution.

(_C6 ) : If 0<^f10 <∞ , _f1∞ =∞ or _f2∞ =∞ , 0<^f20 <∞ , then for each λ∈(0,_N1 /^f10 ) , μ∈(0,_N2 /^f20 ) , the system (1.3) has at least one positive solution.

(_C7 ) : If ^f10 =0 , 0<_f1∞ <∞ , and ^f10 =0 , 0<_f2∞ <∞ , then for each λ∈(_M1 /_f1∞ ,∞) , μ∈(0,∞) or λ∈(0,∞) , μ∈(_M2 /_f2∞ ,∞) , the system (1.3) has at least one positive solution.

Proof.

We only prove the condition in which (_C1 ) holds. The other cases can be proved similarly.

Let λ∈(_M1 /_f1∞ ,_N1 /^f10 ) , μ∈(0,_N2 /(^f20 )) , choose _{[straight epsilon]1} >0 such that _f1∞ -_{[straight epsilon]1} >0 and [figure omitted; refer to PDF]

It follows from ^fi0 ∈(0,∞) of (_C1 ) that there exists _r1 >0 such that for any t∈[0,1] , [figure omitted; refer to PDF]

Let _Kr1 ={(u,v)∈K:||u||<_r1 ,||v||<_r1 } . For any (u,v)∈∂_Kr1 , n>1/_r1 , by (2.13), (3.3), we have [figure omitted; refer to PDF]

Similarly, we also have [figure omitted; refer to PDF]

Therefore, we have [figure omitted; refer to PDF]

On the other hand, by _f1∞ >_f1∞ -_{[straight epsilon]1} >0 , there exists _R0 >0 such that [figure omitted; refer to PDF]

Let _R1 >max {2_r1 ,^Λ-1_R0 } , _KR1 ={(u,v)∈K:||u||<_R1 ,||v||<_R1 } . Next, we take (_{[straight phi]1} ,_{[straight phi]2} )=(1,1)∈∂_K1 , and for any (u,v)∈∂_KR1 , m>0 , n∈... , we will show [figure omitted; refer to PDF]

Otherwise, there exist (_u0 ,_v0 )∈∂_KR1 and _m0 >0 such that [figure omitted; refer to PDF]

From (_u0 ,_v0 )∈∂_KR1 , we know that ||_u0 ||=_R1 or ||_v0 ||=_R1 . Without loss of generality, we may suppose that ||_u0 ||=_R1 , then _u0 (τ)...5;Λ||_u0 ||=Λ_R1 >_R0 for any τ∈[0,1] . So, by (2.13), (3.8), we have [figure omitted; refer to PDF]

This implies that _R1 >_R1 , which is a contradiction. This yields that (3.9) holds. By (3.7), (3.9), and Lemma 1.2, for any n>1/_r1 and λ∈(_M1 /_f1∞ ,_N1 /^f10 ) , μ∈(0,_N2 /^f20 ) , we obtain that _Tn has a fixed point (_un ,_vn ) in _{K¯r1 ,R1} satisfying _r1 <||_un ||<_R1 , _r1 <||_vn ||<_R1 .

Let _{{(un ,vn )}n...5;n1} be the sequence of solutions of boundary value problems (2.19), where _n1 >1/_r1 is a fixed integer. It is easy to see that they are uniformly bounded. Next we show that _{{un }n...5;n1} are equicontinuous on [0,1] . From (_un ,_vn )∈_{K¯r1 ,R1} , we know that _R1 ...5;_un (τ)...5;Λ||_un ||...5;Λ_r1 , _R1 ...5;_vn (τ)...5;Λ||_vn ||...5;Λ_r1 , τ∈[0,1] . For any [straight epsilon]>0 , by the continuous of _H1 (t,s) in [0,1]×[0,1] , there exists _δ1 >0 such that for any _t1 , _t2 , s∈[0,1] and |_t1 -_t2 |<_δ1 , we have [figure omitted; refer to PDF]

This combining with (2.15), (2.32) implies that for any _t1 , _t2 ∈[0,1] and |_t1 -_t2 |<_δ1 , we have [figure omitted; refer to PDF]

Similarly, _{{vn }n...5;n1} are also equicontinuous on [0,1] . By the Ascoli-Arzela theorem, the sequence _{{(un ,vn )}n...5;n1} has a subsequence being uniformly convergent on [0,1] . From Lemma 2.2, we know that [figure omitted; refer to PDF]

Since the properties of _Ki (t,s) (i=1,2) are similar to those of _Hi (t,s) (i=1,2) , so (^{un[variant prime][variant prime]} ,^{vn[variant prime][variant prime]} ) have the similar properties of (_un ,_vn ) , that is, (^{un[variant prime][variant prime]} ,^{vn[variant prime][variant prime]} ) also has a subsequence being uniformly convergent on [0,1] . Without loss of generality, we still assume that _{{(un ,vn )}n...5;n1} itself uniformly converges to (u,v) on [0,1] and _{{(^{un[variant prime][variant prime]} ,^{vn[variant prime][variant prime]} )}n...5;n1} itself uniformly converges to (^{u[variant prime][variant prime]} ,^{v[variant prime][variant prime]} ) on [0,1] , respectively. Since _{{(un ,vn )}n...5;n1} ∈_{K¯r1 ,R1} ⊂K , so we have _un ...5;0 , _vn ...5;0 . By (2.19), we have [figure omitted; refer to PDF]

From (3.15) and (3.16), we know that _{{^{un [variant prime]} (1/2)}n...5;n1} , _{{^{vn [variant prime]} (1/2)}n...5;n1} , _{{^{un [variant prime][variant prime]} (1/2)}n...5;n1} , _{{^{vn [variant prime][variant prime]} (1/2)}n...5;n1} , _{{^{un [variant prime][variant prime][variant prime]} (1/2)}n...5;n1} , _{{^{vn [variant prime][variant prime][variant prime]} (1/2)}n...5;n1} are bounded sets. Without loss of generality, we may assume (^{un [variant prime]} (1/2),^{vn [variant prime]} (1/2))[arrow right](_c1 ,_d1 ),(^{un [variant prime][variant prime]} (1/2),^{vn [variant prime][variant prime]} (1/2)) [arrow right](_c2 ,_d2 ),(^{un [variant prime][variant prime][variant prime]} (1/2),^{vn [variant prime][variant prime][variant prime]} (1/2))[arrow right](_c3 ,_d3 ) as n[arrow right]∞ . Then by (3.15), (3.16), and the Lebesgue dominated convergence theorem, we have [figure omitted; refer to PDF]

By (3.17) and (3.18), direct computation shows that [figure omitted; refer to PDF]

On the other hand, (u,v) satisfies the boundary condition of (1.3). In fact, _un (0)=_un (1)=^∫01 ..._un (s)d_ξ1 (s) , _vn (0)=_vn (1)=^∫01 ..._vn (s)d_ξ2 (s) , _[varphi]p1 (^{un[variant prime][variant prime]} (0))=_[varphi]p1 (^{un[variant prime][variant prime]} (1))=^∫01 ..._[varphi]p1 (^{un[variant prime][variant prime]} (s))d_η1 (s) , _[varphi]p2 (^{vn[variant prime][variant prime]} (0))=_[varphi]p2 (^{vn[variant prime][variant prime]} (1))=^∫01 ..._[varphi]p2 (^{vn[variant prime][variant prime]} (s))d_η2 (s) , and so the conclusion holds by letting n[arrow right]∞ .

Theorem 3.2.

Assume that (_H1 ) - (_H3 ) hold. Then we have:

(_D1 ) : If _f10 , ^f1∞ , ^f2∞ ∈(0,∞) and _M1 /_f10 <_N1 /^f1∞ , then for each λ∈(_M1 /_f10 ,_N1 /^f1∞ ) , μ∈(0,_N2 /^f2∞ ) , the system (1.3) has at least one positive solution.

(_D2 ) : If ^f1∞ , _f20 , ^f2∞ ∈(0,∞) and _M2 /_f20 <_N2 /^f2∞ , then for each λ∈(0,_N1 /^f1∞ ) , μ∈(_M2 /_f20 ,_N2 /^f2∞ ) , the system (1.3) has at least one positive solution.

(_D3 ) : If _f10 =∞ , ^f1∞ =0 , 0<^f2∞ <∞ , then for each λ∈(0,∞) , μ∈(0,_N2 /(^f2∞ )) , the system (1.3) has at least one positive solution.

(_D4 ) : If 0<^f1∞ <∞ , _f20 =∞ , ^f2∞ =0 , then for each λ∈(0,_N1 /^f1∞ ) , μ∈(0,∞) , the system (1.3) has at least one positive solution.

(_D5 ) : If _fi0 =∞ , ^fi∞ =0 (i=1,2) , then for each λ∈(0,∞) , μ∈(0,∞) , the system (1.3) has at least one positive solution.

(_D6 ) : If 0<^f1∞ <∞ , _f10 =∞ or _f20 =∞ , 0<^f2∞ <∞ , then for each λ∈(0,_N1 /^f1∞ ) , μ∈(0,_N2 /^f2∞ ) , the system (1.3) has at least one positive solution.

(_D7 ) : If ^f1∞ =0 , 0<_f10 <∞ , and ^f2∞ =0 , 0<_f20 <∞ , then for each λ∈(_M1 /_f10 ,∞) , μ∈(0,∞) or λ∈(0,∞) , μ∈(_M2 /_f20 ,∞) , the system (1.3) has at least one positive solution.

Proof.

We may suppose that condition (_D1 ) holds. Similarly, we can prove the other cases.

Let λ∈(_M1 /_f10 ,_N1 /^f1∞ ) , μ∈(0,_N2 /^f2∞ ) . We can choose _{[straight epsilon]2} >0 such that _N1 -_{[straight epsilon]2} >0 , _N2 -_{[straight epsilon]2} >0 and [figure omitted; refer to PDF]

It follows from (_D1 ) and (2.16) that there exists ^R2* >0 such that for any t∈[0,1] [figure omitted; refer to PDF]

Let _R2 =^Λ-1R2* , _KR2 ={(u,v)∈K:||u||<_R2 , ||v||<_R2 } . For any (u,v)∈∂_KR2 , n∈... , by (2.13), (3.21), we have [figure omitted; refer to PDF]

Similarly, by (3.22) we have ||^Bnμ (u,v)||<_R2 . Therefore, [figure omitted; refer to PDF]

On the other hand, choose _{[straight epsilon]3} >0 such that _M1 +_{[straight epsilon]3} <λ_f10 . By the condition _f10 ∈(0,∞) of (_D1 ) and (2.16), there exists ^r2* >0 such that [figure omitted; refer to PDF]

Let 0<_r2 <min {_R2 ,^r2* } , _Kr2 ={(u,v)∈K:||u||<_r2 , ||v||<_r2 } . Next, we take (_{[straight phi]1} ,_{[straight phi]2} )=(1,1)∈∂_K1 , n>1/_r2 , and for any (u,v)∈∂_Kr2 , m>0 , we will show [figure omitted; refer to PDF]

Otherwise, there exist (_u0 ,_v0 )∈∂_Kr2 and _m0 >0 such that [figure omitted; refer to PDF]

From (_u0 ,_v0 )∈∂_Kr2 , we know that ||_u0 ||=_r2 or ||_v0 ||=_r2 . Without loss of generality, we may suppose that ||_u0 ||=_r2 , then _u0 (τ)...5;Λ||_u0 ||...5;Λ_r2 for any τ∈[0,1] . So, we have [figure omitted; refer to PDF]

This implies that _r2 >_r2 , which is a contradiction. This yields that (3.26) holds. By (3.24), (3.26), and Lemma 1.2, for any n>1/_r2 and λ∈(_M1 /_f10 ,_N1 /^f1∞ ) , μ∈(0,_N2 /^f2∞ ) , we obtain that _Tn has a fixed point (_un ,_vn ) in _{K¯r2 ,R2} and _r2 <||_un ||<_R2 , _r2 <||_vn ||<_R2 . The rest of proof is similar to Theorem 3.1.

4. An Example

Example 4.1.

We consider system (1.3) with _p1 =3/2 , _p2 =7/3 , _a1 (t)=1/(t(1-t) , _a2 (t)=1/((1-t) t) , [figure omitted; refer to PDF]

Obviously, _a1 , _a2 are singular at t=0 and t=1 , _f1 (t,u,v) is singular at u=0 and _f2 (t,u,v) is singular at v=0 . Choose _g1 (t,u)=(^t2 +1)/u , _h1 (t,v)=1+sin (^v2 +v+t) , _g2 (t,u)=2+sin (u+ln (t+1)) , and _h2 (t,v)=(^t4 +t+3)/[radical]v . Let [figure omitted; refer to PDF]

By direct calculation, we have _α1 =1/4 , _α2 =1/3 , _β1 =3/5 , _β2 =4/7 , ^∫01 ...e(s)_ai (s)ds=(2/3) (i=1,2) . It is easy to check that _f10 =_f20 =∞ , ^f1∞ =^f2∞ =0 , and the conditions (_H1 ) - (_H3 ) and (_D5 ) are satisfied. By Theorem 3.2, system (1.3) has at least one positive solution provided λ , μ∈(0,+∞) .

Remark 4.2.

Example 4.1 not only implies that _f1 (t,u,v) , _f2 (t,u,v) can be singular at u=0 and v=0 , respectively, but also indicates that there is a large number of functions that satisfy the conditions of Theorem 3.2. In addition, the condition (_D5 ) is also easy to check.

Acknowledgments

The first and second authors were supported financially by the National Natural Science Foundation of China (11071141, 11126231) and the Natural Science Foundation of Shandong Province of China (ZR2009AL014, ZR2011AQ008). The third author was supported financially by the Australia Research Council through an ARC Discovery Project Grant.

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