Existence of symmetric positive solutions for a singular system with coupled integral boundary

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Existence of symmetric positive solutions for a singular system with coupled integral boundary

conditions

Jiqiang Jiang

^B^1,2

, Lishan Liu

^1,3

and Yonghong Wu

³

1School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China

2School of statistics, Qufu Normal University, Qufu 273165, Shandong, China

3Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia

Received 9 May 2018, appeared 3 December 2018 Communicated by Jeff R. L. Webb

Abstract. In this paper, we study a class of nonlinear singular system with coupled integral boundary condition. Based on the Guo–Krasnosel’skii fixed point theorem, some new results on the existence of symmetric positive solutions for the coupled singular system are obtained. The impact of the two different parameters on the existence of symmetric positive solutions is also investigated. Finally, an example is then given to demonstrate the applicability of our results.

Keywords: coupled singular system, symmetric positive solutions, fixed point theorem in cones, coupled integral boundary conditions.

2010 Mathematics Subject Classification: 34B15, 34B18.

1 Introduction

This paper is concerned with the existence of symmetric positive solutions for the following singular fourth-order boundary value system with coupled integral boundary conditions (BCs)











(φp(u⁰⁰(t)))⁰⁰ =λ₁a₁(t)f₁(t,u(t),v(t)), −1<t <1, (φp(v⁰⁰(t)))⁰⁰ =λ2a2(t)f2(t,u(t),v(t)), −1<t <1, u(−1) =u(1) =

Z ₁

−1v(s)dA₁(s), v(−1) =v(1) =

Z ₁

−1u(s)dA₂(s), φ_p(u⁰⁰(−1)) =φ_p(u⁰⁰(1)) =

Z ₁

−1φ_p(v⁰⁰(s))dB₁(s), φ_p(v⁰⁰(−1)) =φ_p(v⁰⁰(1)) =

Z ₁

−1φ_p(u⁰⁰(s))dB₂(s),

(1.1)

BCorresponding author. Email: qfjjq@163.com, qfjjq@qfnu.edu.cn

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where λ₁ and λ₂ are positive parameters, φ_p(x) = |x|^p⁻²x,p > 1,φ_q = φ⁻_p¹,¹_p + ¹_q = 1, f₁,f₂ :[−_{1, 1}]×[_0,_∞)×[_0,_∞)→[_0,_∞)are continuous and f₁(·_,x,y)_,f₂(·_,x,y)are symmetric on [−1, 1]for all x,y ∈ [0,∞), a₁,a₂ : (−1, 1) → [0,∞)are symmetric on (−1, 1)and may be singular att = −1 and/ort = 1, and the integrals from (1.1) are Riemann–Stieltjes integrals with a signed measure, that is, A_i,B_i (i=_{1, 2})are functions of boundary variation on[−_{1, 1}]_. By a symmetric positive solution of the system (1.1), we mean a pair of functions (u,v) ∈ (C²[−1, 1]∩C⁴(−1, 1))×(C²[−1, 1]∩C⁴(−1, 1))satisfying (1.1),u,vare symmetric andu(t)>

0,v(t)>_{0 for all}t ∈[−_{1, 1}]_.

Coupled BCs arise from the study of reaction–diffusion equations, Sturm–Liouville problems, mathematical biology and so on. In [1], Asif and Khan studied the following a coupled singular system subject to four-point coupled BCs of the type











−x⁰⁰(t) = f(t,x(t),y(t)), t∈ (0, 1),

−y⁰⁰(t) =g(t,x(t)_,y(t))_, t ∈(_{0, 1})_, x(0) =0, x(1) =αy(ξ),

y(0) =0, y(1) = βx(η),

(1.2)

where the parametersα,β,ξ,η satisfy ξ,η ∈ (0, 1)and 0 < αβξη < 1, f,g : (0, 1)×[0,∞)× [0,∞)→[0,∞)are continuous and singular att =0,t =1. The authors obtained at least one positive solution to the system (1.2) by using the Guo–Krasnosel’skii fixed-point theorem. For other recent results concerning the Coupled BCs, we refer the reader to [2,5,6,8,15].

We notice that a type of symmetric problem has received much attention, for instance, [3,7,9,11–14,16] and the references therein. At the same time, a class of boundary value problems with integral BCs appeared in heat conduction, chemical engineering, underground water flow, thermoelasticity, and plasma physics. For earlier contributions on problems with Lebesgue integral BCs, we refer the reader to [3,13,14,16] and the more general nonlocal Riemann-Stieltjes integral BCs, we refer the reader to [2,5,6,10] and references therein, such integral BCs are a general type of nonlocal boundary conditions and cover multi-point and integral BCs as special cases. Infante, Minhós, Pietramala [5] gave a general method for dealing with these problems in the important case whenp=2. Ma [14] studied the existence of a symmetric positive solution for the following singular fourth-order nonlocal boundary value problem











u⁰⁰⁰⁰(t) =h(t)f(t,u(t)), 0<t<1, u(0) =u(1) =

Z ₁

0

p(s)u(s)ds, u⁰⁰(0) =u⁰⁰(1) =

Z ₁

0 q(s)u⁰⁰(s)ds,

where p,q ∈ L¹[0, 1], h : (0, 1) → [0,+_∞) is continuous, symmetric on (0, 1) and may be singular at t = _{0 and} t = _1, f : [_{0, 1}]×[_0,+_∞) → [_0,+_∞) is continuous and f(·_,x) _is symmetric on [0, 1] for all x ∈ [0,+_∞). The existence of at least one symmetric positive solution was obtained by the application of the fixed point index in cones.

In [16], Zhang, Feng, Ge studied fourth-order boundary value problem with integral BCs:











(φp(u⁰⁰(t)))⁰⁰ =ω(t)f(t,u(t)), 0< t<1, u(0) =u(1) =

Z ₁

0 g(s)u(s)ds, φ_p(_u⁰⁰(₀)) =φ_p(_u⁰⁰(₁)) =

Z ₁

0 h(_s)φ_p(_u⁰⁰(_s))_ds,

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whereφ_p(t) =|t|^p⁻²t,p>1,φ_q= φ⁻_p¹,¹_p+¹_q =1,ω,g,h∈ L¹[0, 1]are nonnegative, symmetric on [_{0, 1}]_. f : [_{0, 1}]×[_0,_∞) → [_0,_∞)is continuous, f(₁−t,x) = f(t,x)_{for all} (t,x) ∈ [_{0, 1}]× [0,∞). By using of fixed point theorem in cones, the existence and multiplicity of symmetric positive solutions were obtained, and the nonexistence of a positive solution was also studied.

Inspired and motivated by the above mentioned work and wide applications of coupled BCs in various fields of sciences and engineering, we study the existence of symmetric positive solutions to a singular system (1.1). Of necessity u⁰(0) = 0, v⁰(0) = 0, u⁰⁰⁰(0) = 0 and v⁰⁰⁰(0) =0 for symmetric functions (u,v)∈ (C²[−1, 1]∩C⁴(−1, 1))×(C²[−1, 1]∩C⁴(−1, 1)), so the problem can be handled by considering the simpler problem (2.1) on[0, 1], then using symmetryu(−t) =u(t),v(−t) =v(t)to extend the solution to[−1, 1].

Our work presented in this paper has the following new features. First of all, we discuss the system (1.1) subject to coupled BCs withp-Laplacian operators, Riemann–Stieltjes integral BCs are a general type of nonlocal boundary conditions and cover multi-point and integral BCs as special cases, these are different from [3,7,11–14,16]. The second new feature is that the system (1.1) possesses singularity, that is, the nonlinear terms may be singular att =−1, 1.

Thirdly, we involve the parameterλ_i(i=1, 2)in the model and obtain the sufficient conditions for the existence of symmetric positive solutions of system (1.1) within certain interval of λ_i (i = 1, 2). To the best knowledge of the authors, there is no earlier literature studying the existence of symmetric positive solutions for boundary value system with coupled integral BCs.

The rest of the paper is organized as follows. In Section 2, we present a positive cone, a fixed point theorem which will be used to prove existence of symmetric positive solutions, Green’s function for the modified system and some related lemmas. In Section 3, we present main results of the paper and in Section 4 an example is given to illustrate the application of our main results.

2 Preliminaries and lemmas

We recall that the functionωis said to be concave on[a,b]_if

ω(τt₁+ (₁−τ)t₂)≥τω(t₁) + (₁−τ)ω(t₂)_, t₁,t₂ ∈[a,b]_,τ∈(_{0, 1})_, and the functionωis said to be symmetric on[−_{1, 1}]_ifω(−t) =ω(t)_,t∈[−_{1, 1}]_.

Remark 2.1. If (u,v) ∈ (C²[−1, 1]∩C⁴(−1, 1))×(C²[−1, 1]∩C⁴(−1, 1))is a symmetric positive solution of the singular system (1.1), obviously, u⁰(₀) = _0, v⁰(₀) = _0, u⁰⁰⁰(₀) = _{0 and} v⁰⁰⁰(0) = 0 are necessary. So the problem (1.1) can be handled by considering the following simpler problem











(φp(u⁰⁰(t)))⁰⁰ =λ₁a₁(t)f₁(t,u(t),v(t)), 0< t<1, (φp(v⁰⁰(t)))⁰⁰ =λ2a2(t)f2(t,u(t),v(t)), 0< t<1, u⁰(0) =0, u(1) =

Z ₁

0 v(s)dA₁(s), v⁰(0) =0, v(1) =

Z ₁

0 u(s)dA₂(s), φ_p(u⁰⁰⁰(₀)) =_0, φ_p(u⁰⁰(₁)) =

Z ₁

0

φ_p(v⁰⁰(s))dB₁(s)_, φp(v⁰⁰⁰(0)) =0, φp(v⁰⁰(1)) =

Z ₁

0 φp(u⁰⁰(s))dB2(s),

(2.1)

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on [0, 1], then using symmetryu(−t) = u(t)to extend the solution to [−1, 1]. In view of the above, we will concentrate our study on the system (2.1).

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:

k(u,v)k₁ :=max{kuk,kvk}, kuk= max

0≤t≤1|u(t)|, kvk= max

0≤t≤1|v(t)|

for any(u,v)∈ E.

Set

G(t,s) =

(1−t, 0≤s ≤t≤1, 1−s, 0≤t ≤s≤1, κ_i =

Z ₁

0 dA_i(t), $_i =

Z ₁

0 G(t,t)dA_i(t), G_i(s) =

Z ₁

0 G(t,s)dA_i(t), i=1, 2,

˜ κ_i =

Z ₁

0 dB_i(t), $˜_i =

Z ₁

0 G(t,t)dB_i(t), Ge_i(s) =

Z ₁

0 G(t,s)dB_i(t), i=1, 2.

(2.2) Obviously,

G(t,t)G(s,s)≤ G(t,s)≤ G(s,s)orG(t,t), ∀t,s∈ [0, 1]. (2.3) In the rest of the paper, we make the following assumptions:

(_H₁) A_i,B_i (i=_{1, 2})are functions of boundary variation on[−_{1, 1}]_,_∆₁=₁−κ₁κ₂, and κ_i >0, κ˜_i >0, 0<κ₁κ₂<1, 0<κ˜₁κ˜₂<1, i=1, 2.

(_H₂) a₁,a₂:(−1, 1)→[0,∞)are continuous, symmetric on(−1, 1)and 0<

Z ₁

0 G(s,s)a_i(s)ds<_∞, i=1, 2.

(H₃) f₁, f₂:[−1, 1]×[0,∞)×[0,∞)→[0,∞)are continuous and f₁(·,x,y), f₂(·,x,y)are symmetric on[−1, 1]for all x,y∈[0,∞), i.e., f_i satisfy f_i(−t,x,y) = f_i(t,x,y) (i=1, 2). Lemma 2.2. Assume that (H₁)holds. Then for any x,y ∈ L¹(0, 1)∩C(0, 1), the system of BVPs consisting of the equations

−u⁰⁰(t) =φ⁻_p¹(x(t)), −v⁰⁰(t) =φ⁻_p¹(y(t)), t ∈(0, 1) (2.4) and the BCs

u⁰(0) =0, u(1) =

Z ₁

0 v(t)dA₁(t), v⁰(0) =0, v(1) =

Z ₁

0 u(t)dA2(t) (2.5) has a unique integral representation

u(t) =

Z ₁

0 H₁(t,s)φ⁻_p¹(x(s))ds+

Z ₁

0 K₁(s)φ⁻_p¹(y(s))ds, (2.6) v(t) =

Z ₁

0 H₂(t,s)φ⁻_p¹(y(s))ds+

Z ₁

0 K₂(s)φ⁻_p¹(x(s))ds, (2.7) where

H₁(t,s) =G(t,s) + ^κ¹

∆1

G₂(s), K₁(s) = ¹

∆1

G₁(s), (2.8)

H2(t,s) =G(t,s) + ^κ²

∆1

G₁(s), K2(s) = ¹

∆1

G₂(s). (2.9)

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Proof. Let

u(t) =

Z ₁

0 G(t,s)φ⁻_p¹(x(s))ds+c₁+c₃(t−1), (2.10) v(t) =

Z ₁

0 G(t,s)φ⁻_p¹(y(s))ds+c₂+c₄(t−1), (2.11) where c₁,c₂,c₃ andc₄ are constants to be determined. Clearly, u(t) andv(t)satisfy (2.4). In the following, we determine c_i (1 ≤ i ≤ 4) so that u(t) and v(t) satisfy (2.5). Substituting (2.10) and (2.11) into (2.5), we obtainc3 =_c₄=_{0 and}

c₁−κ₁c₂ =

Z ₁

0

G₁(s)φ⁻_p¹(y(s))ds, (2.12)

−κ₂c₁+c₂ =

Z ₁

0

G₂(s)φ⁻_p¹(x(s))ds. (2.13) Note that

1 −κ₁

−κ₂ 1

=1−κ₁κ2=_∆₁6=0.

Thus, the system (2.12)–(2.13) has a unique solution forc_i (1 ≤ i≤ 2). By the Cramer’s rule and simple calculations, it follows that

c₁ = ¹

∆1

Z ₁

0

G₁(s)φ⁻_p¹(y(s))ds+κ₁ Z ₁

0

G₂(s)φ⁻_p¹(x(s))ds

, c2 = ¹

∆1

Z ₁

0

G₂(s)φ⁻_p¹(x(s))ds+κ2

Z ₁

0

G₁(s)φ⁻_p¹(y(s))ds

. Then from (2.10) and (2.11), it is obvious that (2.6) and (2.7) hold.

Similar to the proof of Lemma2.2, we have

Lemma 2.3. Assume that(H₁)–(H₃)hold. Then for any u,v∈ C[0, 1], the system











−x⁰⁰(t) =λ₁a₁(t)f₁(t,u(t),v(t)), t∈(0, 1),

−y⁰⁰(t) =λ₂a₂(t)f₂(t,u(t),v(t)), t ∈(0, 1), x⁰(0) =0, x(1) =

Z ₁

0

y(t)dB₁(t), y⁰(0) =0, y(1) =

Z ₁

0 x(t)dB2(t), has a unique integral representation

x(t) =λ₁ Z ₁

0

He₁(t,s)a₁(s)f₁(s,u(s)_,v(s))ds+λ₂ Z ₁

0

Ke₁(s)a₂(s)f₂(s,u(s)_,v(s))ds, y(t) =λ₂

Z ₁

0

He₂(t,s)a₂(s)f₂(s,u(s),v(s))ds+λ₁ Z ₁

0

Ke₂(s)a₁(s)f₁(s,u(s),v(s))ds, where∆2 =1−κ˜₁κ˜₂, andKe₁(s) = _∆¹

2Ge₁(s),Ke₂(s) = _∆¹

2Ge₂(s), He₁(t,s) =G(t,s) + ^κ^˜¹

∆2

Ge₂(s), He₂(t,s) =G(t,s) + ^κ^˜²

∆2

Ge₁(s).

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Lemma 2.4. Assume that(H₁)holds. Then the functions H_i(t,s), He_i(t,s), K_i(s), Ke_i(s) (i = 1, 2) are continuous and

H_i(t,s)>0, He_i(t,s)>0, K_i(s)>0, Ke_i(s)>0, for t,s ∈[0, 1), i=1, 2, H_i(t,s)≥0, He_i(t,s)≥0, K_i(s)≥0, Ke_i(s)≥0, for t,s ∈[0, 1], i=1, 2.

Proof. It follows from (2.3), Lemmas2.2and2.3that the results of Lemma2.4are true.

Lemma 2.5. Assume that(H₁)holds. For all t,s∈ [0, 1],we have κ_i$_j

∆1

G(s,s)≤H_i(t,s)≤ ¹

∆1

G(s,s), i=1, 2, i+j=3, (2.14)

$_i

∆1

G(s,s)≤ K_i(s)≤ ^κⁱ

∆1

G(s,s), i=1, 2, (2.15)

˜ κ_i$˜_j

∆2

G(s,s)≤He_i(t,s)≤ ¹

∆2

G(s,s), i=1, 2, i+j=3, (2.16)

˜

$_i

∆2

G(s,s)≤ Ke_i(s)≤ ^κ^˜ⁱ

∆2

G(s,s), i=1, 2, (2.17) whereκ_i,κ˜_i,$_i and$˜_i(i=1, 2)are defined by(2.2).

Proof. First, we will show that (2.14) is true. By (2.3), the first equalities of (2.8) and (2.9), we obtain

H_i(t,s) =G(t,s) + ^κⁱ

∆1

G_j(s)≤ G(s,s) + ^κⁱ

∆1

Z ₁

0 G(t,s)dA_j(t)

≤G(s,s) + ^κⁱ

∆1

Z ₁

0 dA_j(t)·G(s,s) = ^∆¹+κ_iκ_j

∆1

G(s,s)

= ¹

∆1

G(s,s), t,s∈[0, 1], i=1, 2,i+j=3.

On the other hand, by (2.3), the first equalities of (2.8) and (2.9), we also have H_i(t,s) =G(t,s) + ^κⁱ

∆1

G_j(s)≥ ^κⁱ

∆1

G_j(s)

≥ ^κⁱ

∆1

Z ₁

0 G(t,t)G(s,s)dA_j(t)

= ^κⁱ^$^j

∆1

G(s,s), t,s∈[0, 1], i=1, 2, i+j=3.

Next we show that (2.15) holds. In fact, using (2.3), the second equalities of (2.8) and (2.9), we get

K_i(s) = ¹

∆1

Z ₁

0 G(t,s)dA_i(t)≤ ¹

∆1

Z ₁

0 G(s,s)dA_i(t) = ^κⁱ

∆1

G(s,s), s∈ [0, 1], i=1, 2.

On the other hand, by (2.3), the second equalities of (2.8) and (2.9), we also have K_i(s) = ¹

∆1

Z ₁

0 G(t,s)dA_i(t)≥ ¹

∆1

Z ₁

0 G(t,t)G(s,s)dA_i(t) = ^$ⁱ

∆1

G(s,s), s∈[0, 1], i=1, 2.

Similar to the proof of (2.14) and (2.15), we obtain (2.16) and (2.17).

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Remark 2.6. From Lemma2.5, fort,s ∈[0, 1]we have

νG(s,s)≤ H_i(t,s)≤µG(s,s)_, νG(s,s)≤K_i(s)≤µG(s,s)_, i=_{1, 2,}

˜

νG(s,s)≤ He_i(t,s)≤µG˜ (s,s), νG˜ (s,s)≤Ke_i(s)≤µG˜ (s,s), i=1, 2, where

µ= ^max{1,κ₁,κ2}

∆1

, ν= ^min{κ₁$2,κ2$₁,$₁,$2}

∆1

,

˜

µ= ^max{1, ˜κ₁, ˜κ₂}

∆2

, ν˜ = ^min{κ˜₁$˜₂, ˜κ₂$˜₁, ˜$₁, ˜$₂}

∆2

.

DenoteC⁺[0, 1] ={u∈C[0, 1]:u(t)≥0, 0≤t≤1}. Let

K=

(u,v)∈C⁺[_{0, 1}]×C⁺[_{0, 1}]_:u,vare concave on [_{0, 1}]_, min

t∈[0,1]u(t)≥γk(u,v)k₁, min

t∈[0,1]v(t)≥γk(u,v)k₁

,

where γ := ^νφ

−1 p (ν˜)

µφ⁻p¹(µ˜). Clearly 0 < γ < 1. It is easy to see that K is a cone of E. For any real constant r > _{0, define}K_r = {(u,v) _: (u,v) ∈ K,k(u,v)k₁ < r} _and ∂K_r = {(u,v) _: (u,v) ∈ K,k(u,v)k₁=r}.

Employing Lemmas2.2and2.3, the system (2.1) can be expressed as











u(t) =

Z ₁

0 H₁(t,s)φ⁻_p¹

λ₁ Z ₁

0

He₁(s,τ)a₁(τ)f₁(τ,u(τ),v(τ))dτ +λ₂

Z ₁

0

Ke₁(τ)a₂(τ)f₂(_τ,u(τ)_,v(τ))dτ

ds +

Z ₁

0 K₁(s)φ⁻_p¹

λ₂ Z ₁

0

He₂(s,τ)a₂(τ)f₂(τ,u(τ),v(τ))dτ +λ₁

Z ₁

0

Ke₂(τ)a₁(τ)f₁(τ,u(τ),v(τ))dτ

ds, v(t) =

Z ₁

0 H2(t,s)φ⁻_p¹

λ2

Z ₁

0

He2(s,τ)a2(τ)f2(τ,u(τ),v(τ))dτ +λ₁

Z ₁

0

Ke₂(τ)a₁(τ)f₁(τ,u(τ),v(τ))dτ

ds +

Z ₁

0 K₂(s)φ⁻_p¹

λ₁ Z ₁

0

He₁(s,τ)a₁(τ)f₁(τ,u(τ),v(τ))dτ +λ2

Z ₁

0

Ke₁(τ)a2(τ)f2(τ,u(τ),v(τ))dτ

ds.

(2.18)

By a solution of the system (2.1), we mean a solution of the corresponding system of integral equations (2.18). Defined an operator T : K → K by T(u,v) = (T₁(u,v)_,T₂(u,v))_{, where}

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operatorsT_i :K→Kare defined by T_i(u,v)(t) =

Z ₁

0 H_i(t,s)φ⁻_p¹

λ_i Z ₁

0

He_i(s,τ)a_i(τ)f_i(τ,u(τ),v(τ))dτ +λ_j

Z ₁

0

Ke_i(τ)a_j(τ)f_j(τ,u(τ),v(τ))dτ

ds +

Z ₁

0

K_i(s)φ⁻_p¹

λ_j Z ₁

0

He_j(s,τ)a_j(τ)f_j(_τ,u(τ)_,v(τ))dτ +λ_i

Z ₁

0

Ke_j(τ)a_i(τ)f_i(τ,u(τ),v(τ))dτ

ds, t∈ [0, 1],i=1, 2,i+j=3.

(2.19)

Clearly,(u,v)∈K is a fixed point ofTif and only if(u,v)is a solution of system (2.1).

Lemma 2.7. Assume that(H₁)–(H3)hold. Then T:K→K is well defined. Furthermore, T:K →K is a completely continuous operator.

Proof. For any fixed(u,v)∈ K, there exists a constant r >0 such that k(u,v)k ≤ r. Thus, for anyt ∈[_{0, 1}], it follows from (2.19) and Remark2.6that

T_i(u,v)(t) =

Z ₁

0 H_i(t,s)φ⁻_p¹

λ_i Z ₁

0

He_i(s,τ)a_i(τ)f_i(τ,u(τ),v(τ))dτ +λ_j

Z ₁

0

Ke_i(τ)a_j(τ)f_j(τ,u(τ),v(τ))dτ

ds +

Z ₁

0 K_i(s)φ⁻_p¹

λ_j Z ₁

0

He_j(s,τ)a_j(τ)f_j(τ,u(τ),v(τ))dτ +λ_i

Z ₁

0

Ke_j(τ)a_i(τ)f_i(τ,u(τ),v(τ))dτ

ds

≤µ Z ₁

0 G(s,s)φ⁻_p¹

λ_iµ˜ Z ₁

0 G(τ,τ)a_i(τ)f_i(τ,u(τ),v(τ))dτ +λ_jµ˜

Z ₁

0 G(τ,τ)a_j(τ)f_j(τ,u(τ),v(τ))dτ

ds +µ

Z ₁

0

G(s,s)φ⁻_p¹

λ_jµ˜ Z ₁

0

G(_τ,τ)a_j(τ)f_j(_τ,u(τ)_,v(τ))dτ +λ_iµ˜

Z ₁

0 G(τ,τ)a_i(τ)f_i(τ,u(τ),v(τ))dτ

ds

≤2µφ⁻_p¹(µM˜ )

Z ₁

0 G(s,s)φ⁻_p¹

λ₁ Z ₁

0 G(τ,τ)a₁(τ)dτ+λ₂ Z ₁

0 G(τ,τ)a₂(τ)dτ

<_∞, i=1, 2, i+j=3, where

M= max

(t,x,y)∈[0,1]×[0,r]×[0,r]f₁(t,x,y) + max

(t,x,y)∈[0,1]×[0,r]×[0,r] f₂(t,x,y). ThusT: K→Kis well defined.

For all(u,v)∈ K, by (2.19) we have (T_i(u,v))⁰⁰(t) =−φ⁻_p¹(λ_i

Z ₁

0

He_i(s,τ)a_i(τ)f_i(_τ,u(τ)_,v(τ))dτ +λ_j

Z ₁

0

Ke_i(τ)a_j(τ)f_j(τ,u(τ),v(τ))dτ≤0,

(9)

which implies that T_i(u,v) is concave on [0, 1]. Further, by (2.19), Lemmas 2.2 and 2.3 we obtainT_i(u,v)(0)≥0, T_i(u,v)(1)≥0. It follows thatT_i(u,v)(t)≥0 fort∈[0, 1].

On the other hand, for(u,v)∈K,t ∈[0, 1], using (2.19) and Remark2.6, we obtain T_i(u,v)(t)≤µ

Z ₁

0 G(s,s)φ⁻_p¹

λ_iµ˜ Z ₁

0 G(τ,τ)a_i(τ)f_i(τ,u(τ),v(τ))dτ +λ_jµ˜

Z ₁

ds +µ

Z ₁

0 G(s,s)φ⁻_p¹

λ_jµ˜ Z ₁

0 G(τ,τ)a_j(τ)f_j(τ,u(τ),v(τ))dτ +λ_iµ˜

Z ₁

0 G(τ,τ)a_i(τ)f_i(τ,u(τ),v(τ))dτ

ds

≤2µφ⁻_p¹(µ˜)

Z ₁

0 G(s,s)φ⁻_p¹

λ_i Z ₁

0 G(τ,τ)a_i(τ)f_i(τ,u(τ),v(τ))dτ +λ_j

Z ₁

0

G(_τ,τ)a_j(τ)f_j(_τ,u(τ)_,v(τ))dτ

ds, i=_{1, 2,} i+j=_3, which implies that

kT_i(u,v)k ≤2µφ⁻_p¹(µ˜)

Z ₁

0 G(s,s)φ⁻_p¹

λ_i Z ₁

Z ₁

ds, i=1, 2, i+j=3.

(2.20)

Also, for(u,v)∈K,t∈[_{0, 1}], using Remark2.6, (2.19) and (2.20), we have T₁(u,v)(t) =

Z ₁

0 H₁(t,s)φ⁻_p¹

λ₁ Z ₁

0

He₁(s,τ)a₁(τ)f₁(τ,u(τ),v(τ))dτ +λ₂

Z ₁

0

Ke₁(τ)a₂(τ)f₂(τ,u(τ),v(τ))dτ

ds +

Z ₁

0 K₁(s)φ⁻_p¹

λ₂ Z ₁

0

He₂(s,τ)a₂(τ)f₂(τ,u(τ),v(τ))dτ +λ₁

Z ₁

0

Ke2(τ)a₁(τ)f₁(τ,u(τ),v(τ))dτ

ds

≥ ν Z ₁

0 G(s,s)φ⁻_p¹

λ₁ν˜ Z ₁

0 G(τ,τ)a₁(τ)f₁(τ,u(τ),v(τ))dτ +λ₂ν˜

Z ₁

0 G(τ,τ)a₂(τ)f₂(τ,u(τ),v(τ))dτ

ds +ν

Z ₁

0 G(s,s)φ⁻_p¹

λ₂ν˜ Z ₁

0 G(τ,τ)a₂(τ)f₂(τ,u(τ),v(τ))dτ +λ₁ν˜

Z ₁

0

G(_τ,τ)a₁(τ)f₁(_τ,u(τ)_,v(τ))dτ

ds

≥2νφ⁻_p¹(ν˜)

Z ₁

0 G(s,s)φ⁻_p¹

λ_i Z ₁

Z ₁

ds

≥ γkT_i(u,v)k, i=1, 2, i+j=3.

(10)

This implies that

tmin∈[0,1]T₁(u,v)(t)≥γk(T₁(u,v),T2(u,v))k₁. In the same way as above, we can prove that

min

t∈[0,1]T₂(u,v)(t)≥γk(T₁(u,v),T₂(u,v))k₁. Hence,T(K)⊂K.

Next, we prove that T : K → K is completely continuous. For any natural number n, we set

a_in(t) =











a_i(t), 0≤t≤ ⁿ−1 n , inf

1−¹_n≤s<t

a_i(s), 1− ¹

n ≤t≤1,

i = _{1, 2. Then} a_1n,a_2n : [_{0, 1}] → [_0,+_∞) are continuous and a_1n(t) ≤ a₁(t)_,a_2n(t) ≤ a₂(t)_, t∈ (0, 1). Let

T_in(u,v)(t) =

Z ₁

0 H_i(t,s)φ⁻_p¹

λ_i Z ₁

0

He_i(s,τ)a_in(τ)f_i(τ,u(τ),v(τ))dτ +λ_j

Z ₁

0

Ke_i(τ)a_jn(τ)f_j(τ,u(τ),v(τ))dτ

ds +

Z ₁

0 K_i(s)φ⁻_p¹

λ_j Z ₁

0

He_j(s,τ)a_jn(τ)f_j(τ,u(τ),v(τ))dτ +λ_i

Z ₁

0

Ke_j(τ)a_in(τ)f_i(τ,u(τ),v(τ))dτ

ds,

i = 1, 2, i+j = 3. Similar to [14], by the approximating theorem of completely continuous operators, we can proveT:K→K is a completely continuous operator.

Lemma 2.8 ([4]). Let X be a real Banach space, P be a cone in X. Assume thatΩ1 andΩ2 are two bounded open sets of X with θ ∈ _Ω₁ and Ω1 ⊂ _Ω₂. Let T : P∩(_Ω₂\_Ω₁) → P be a completely continuous operator such that either

(i) kTxk ≤ kxk, x ∈P∩_∂Ω₁andkTxk ≥ kxk, x ∈P∩_∂Ω₂,or (ii) kTxk ≥ kxk, x ∈P∩_∂Ω₁andkTxk ≤ kxk, x ∈P∩_∂Ω₂. Then T has at least one fixed point in P∩(_Ω₂\_Ω₁).

3 Main results

Denote

L₁ =max

2µφ⁻_p¹

˜ µ

Z ₁

0 G(τ,τ)a₁(τ)dτ

, 2µφ⁻_p¹

˜ µ

Z ₁

, L₂ =min

2νφ⁻_p¹

˜ ν

Z ₁

0 G(τ,τ)a₁(τ)dτ

, 2νφ⁻_p¹

˜ ν

Z ₁

, M_i = max

t∈[0,1] f_i(t, 1, 1), m_i = min

t∈[0,1]f_i(t, 1, 1), i=1, 2.

And, we also suppose

(11)

(H₄) f_i(t,x,y)is nondecreasing inxand nonincreasing iny, and there existξ_i,η_i ∈[0, 1)such that

c^ξⁱf_i(t,x,y)≤ f_i(t,cx,y), f_i(t,x,cy)≤ c⁻^ηⁱf_i(t,x,y), ∀x,y>0, c∈(0, 1), i=1, 2.

(H₅) f_i(t,x,y)is nonincreasing inxand nondecreasing iny, and there existξ_i,η_i ∈[0, 1)such that

c^ξⁱf_i(t,x,y)≤ f_i(t,x,cy), f_i(t,cx,y)≤c⁻^ηⁱf_i(t,x,y), ∀x,y >0,c∈ (0, 1), i=1, 2.

Remark 3.1. (H₄)implies that

f_i(t,cx,y)≤c^ξⁱf_i(t,x,y), f_i(t,x,cy)≤ c^ηⁱf_i(t,x,y), ∀x,y>0, c>1, i=1, 2.

Remark 3.2. (H₅)implies that

f_i(t,x,cy)≤c^ξⁱf_i(t,x,y), f_i(t,x,y)≤ c^ηⁱf_i(t,cx,y), ∀x,y>0, c>1, i=1, 2.

Theorem 3.3. Assume that f_i(t, 1, 1) 6= 0, t ∈ [0, 1]and (H₁)–(H₄)hold. Then for any0 < r₀ <

1< R₀< _∞, the system(1.1)has at least one symmetric positive solution for λ_i ∈ ^r

p−1−ξi

0

m_iγ^ξⁱφ_p(L2)^,

R₀^p⁻¹⁻^ξⁱγ^ηⁱ M_iφ_p(L₁)

!

, i=1, 2 (3.1)

provided that M_iφ_p(L₁)r₀^p⁻¹⁻^ξⁱ <m_iφ_p(L₂)R₀^p⁻¹⁻^ξⁱγ^ξⁱ⁺^ηⁱ (i=1, 2). Proof. For any(u,v)∈ ∂K_r₀, by the definition ofk · k, we have

r₀ =k(u,v)k₁≥ kuk ≥u(t)≥γk(u,v)k₁=γr₀,

r₀ =k(u,v)k₁≥ kvk ≥v(t)≥γk(u,v)k₁= γr₀. (3.2) Suppose thatλ_i satisfy (3.1). So, for any(u,v)∈∂K_r₀, by (H₄)and (3.2), we have

T_i(u,v)(t)≥ ν Z ₁

0 G(s,s)φ⁻_p¹

λ_iν˜ Z ₁

0 G(τ,τ)a_i(τ)f_i(τ,γr₀, 1)dτ +λ_jν˜

Z ₁

0 G(τ,τ)a_j(τ)f_j(τ,γr₀, 1))dτ

ds +ν

Z ₁

0 G(s,s)φ⁻_p¹

λ_jν˜ Z ₁

0 G(τ,τ)a_j(τ)f_j(τ,γr0, 1)dτ +λ_iν˜

Z ₁

0 G(τ,τ)a_i(τ)f_i(τ,γr₀, 1)dτ

ds

≥ ν

φ⁻_p¹

λ_iγ^ξⁱr^ξ₀ⁱν˜ Z ₁

0 G(τ,τ)a_i(τ)f_i(τ, 1, 1)dτ

+φ⁻_p¹

λ_jγ^ξ^jr₀^ξ^jν˜ Z ₁

0 G(τ,τ)a_j(τ)f_j(τ, 1, 1)dτ

≥ ν

φ⁻_p¹

λ_iγ^ξⁱr^ξ₀ⁱm_iν˜ Z ₁

0

G(_τ,τ)a_i(τ)dτ

+φ⁻_p¹

λ_jγ^ξ^jr₀^ξ^jm_jν˜ Z ₁

0

G(_τ,τ)a_j(τ)dτ

≥ φ⁻_p¹(λ_iγ^ξⁱr^ξ₀ⁱm_i)^L²

2 +φ⁻_p¹(λ_jγ^ξ^jr^ξ₀^jm_j)^L²

2 ≥r₀, t ∈[0, 1], i=1, 2, i+j=3,

(12)

which implies that

kT(u,v)k₁ =max{kT₁(u,v)k,kT₂(u,v)k} ≥r₀=k(u,v)k₁, (u,v)∈∂K_r₀. (3.3) On the other hand, for any(u,v)∈ ∂K_R₀,t∈[0, 1], similar to (3.2), we have

R0 =k(u,v)k₁≥u(t)≥γk(u,v)k₁ =γR0> γ,

R0 =k(u,v)k₁ ≥v(_t)≥γk(u,v)k₁ =_γR₀> _γ. ^(3.4) By (3.1), (3.4), Remarks2.6and3.1, for anyt∈ [0, 1], we have

T_i(u,v)(t)≤ µ Z ₁

0 G(s,s)φ⁻_p¹

λ_iµ˜ Z ₁

0 G(τ,τ)a_i(τ)f_i(τ,R0,γ)dτ +λ_jµ˜

Z ₁

0 G(τ,τ)a_j(τ)f_j(τ,R₀,γ)dτ

ds +µ

Z ₁

0 G(s,s)φ⁻_p¹

λ_jµ˜ Z ₁

0 G(τ,τ)a_j(τ)f_j(τ,R₀,γ)dτ +λ_iµ˜

Z ₁

0 G(τ,τ)a_i(τ)f_i(τ,R0,γ)dτ

ds

≤ µ

φ⁻_p¹

λ_iµR˜ ^ξ₀ⁱγ⁻^ηⁱ Z ₁

0

G(τ,τ)a_i(τ)f_i(τ, 1, 1)dτ

+φ⁻_p¹

λ_jµR˜ ^ξ₀ⁱγ⁻^η^j Z ₁

0 G(τ,τ)a_j(τ)f_j(τ, 1, 1)dτ

≤ µ

φ⁻_p¹

λ_iµR˜ ^ξ₀ⁱγ⁻^ηⁱM_i Z ₁

0 G(τ,τ)a_i(τ)dτ

+φ⁻_p¹

λ_jµR˜ ^ξ₀ⁱγ⁻^η^jM_j Z ₁

0

G(_τ,τ)a_j(τ)dτ

≤ φ⁻_p¹(λ_iR₀^ξⁱγ⁻^ηⁱM_i)^L¹

2 +φ⁻_p¹(λ_jR^ξ₀ⁱγ⁻^η^jM_j)^L¹

2 ≤ R₀, i=1, 2, i+j=3.

Therefore, we have

kT(u,v)k₁=_max{kT₁(u,v)k_,kT₂(u,v)k} ≤R₀ =k(u,v)k₁_, ∀(u,v)∈∂K_R₀. (3.5) It follows from (3.3), (3.5) and Lemma2.8that for anyλ_i ∈ ^r

p−1−ξi 0

m_iγ^ξⁱφp(L₂),^R

p−1−ξi

0 γ^ηⁱ

Miφp(L1)

(i=1, 2), T has a fixed point (u₀,v₀) ∈ K_R₀ \K^¯_r₀ with r₀ ≤ k(u₀,v₀)k₁ ≤ R₀. Moreover, (u₀,v₀) is positive. In fact, fromk(u0,v0)k₁≥r0>0, by construction of the coneK, we have

tmin∈[0,1]u₀(t)≥γk(u₀,v₀)k₁>_0,

which implies thatu₀(t)>0 for allt∈[0, 1]. Similarly, we also havev₀(t)>0 for allt∈ [0, 1]. Hence,(u0,v0)is a positive solution of the system (2.1). Let

u^∗(t) =

(u₀(−t), −1≤t <0,

u₀(t), 0≤t ≤1, v^∗(t) =

(v₀(−t), −1≤ t<0, v₀(t), 0≤ t≤1.

By Remark2.1we know that(u^∗,v^∗)is the desired symmetric positive solution for the system (1.1).