# Positive Solutions for Singularp-Laplacian Fractional Differential System with Integral Boundary Conditions

Authors: Liping Wang; Zongfu Zhou; Hui Zhou
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

This paper investigates the existence of positive solutions for a class of singularp-Laplacian
fractional differential equations with integral boundary conditions. By using the Leggett-Williams fixed
point theorem, the existence of at least three positive solutions to the boundary value system is guaranteed.


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## Body

## 1. Introduction

This paper investigates the singularp-Laplacian fractional boundary value problem:
(1)D0+β(ϕp1(D0+αu(t)))=f1(t,u(t),v(t),D0+μu(t),D0+νv(t)),0<t<1,D0+β(ϕp2(D0+αv(t)))=f2(t,u(t),v(t),D0+μu(t),D0+νv(t)),0<t<1,u(0)=u′(0)=u′′(0)=0,u′′(1)=(α-1)(α-2)∫01‍u(s)dξ1(s),D0+αu(0)=0,ϕp1(D0+αu(1))=∫01‍ϕp1(D0+αu(s))dη1(s),v(0)=v′(0)=v′′(0)=0,v′′(1)=(α-1)(α-2)∫01‍v(s)dξ2(s),D0+αv(0)=0,ϕp2(D0+αv(1))=∫01‍ϕp2(D0+αv(s))dη2(s),
where 1<β≤2, 3<α≤4 and 5<α+β≤6, 0<μ≤1,1<ν<2 are real numbers. ϕpi(x)=|x|pi-2x, pi>1, ϕqi=ϕpi-1, 1/pi+1/qi=1,andξi,ηi:[0,1]→R+ are nondecreasing functions of bounded variation i=1,2. Dα and Dβ are the standard Riemann-Liouville fractional derivatives of order α,β, respectively, and the integrals in (1) are Riemann-Stieltjes integrals. Here fi: [0,1]×Ω→ℝ+ is Carathéodory function; that is, fi satisfies the local Carathéodory condition on [0,1]×Ω, and fi(t,x1,x2,x3,x4) may be singular at the value 0 of all its space variables x1,x2,x3,x4, i=1,2, in which Ω=(0,∞)4, ℝ+=[0,∞).We say thatf satisfies the local Carathéodory condition on [0,1]×Ω,  (f∈Car([0,1]×Ω)), if(i)
f(·;x1,x2,x3,x4): [0,1]→ℝ+ is measurable for all (x1,x2,x3,x4)∈Ω;(ii)
f(t;·,·,·,·): Ω→ℝ+ is continuous for a.e. t∈[0,1];(iii)
for each compact set𝒦⊂Ω there is a function φ𝒦∈L1[0,1] such that |f(t,x1,x2,x3,x4)|≤φ𝒦(t) for a.e. t∈[0,1] and all (x1,x2,x3,x4)∈𝒦.A vector(u,v)∈(C2[0,1])×(C2[0,1]) is called positive solution of system (1) if and only if (u,v) satisfies (1) and u(t)>0,v(t)≥0 or v(t)>0,u(t)≥0 for any t∈[0,1].Fractional differential equation can describe many phenomena in various fields of science and engineering, such as control, porous media, electrochemistry, and electromagnetic. For details, see [1–8] and the references therein. There are also a large number of papers dealing with the solvability of nonlinear fractional differential equations. Papers [9–13] discuss fractional boundary value problems with nonlinearities having singularities in space variables.Paper [14] is concerned with the existence of positive solutions to the following BVP for nonlinear fractional differential equations with boundary conditions involving Riemann-Stieltjes integrals:
(2)D0+αu(t)+f(t,u(t),u′(t),…,u(n-2)(t))=0,mmmmmmmmmmmmmmmmmm0<t<1,u(0)=u′(0)=⋯=un-2(0)=0,un-2(1)=∫01‍un-2(s)dA(s),
where n-1<α≤n and n≥2 is positive integer, the integrals ∫01‍un-2(s)dA(s) are Riemann-Stieltjes integrals, and f is a Carathéodory function on [0,1]×ℝn-1→ℝ+. Some existence and multiplicity results of positive solutions are obtained by using the Krasnosel'skii fixed point theorem, the Leray-Schauder nonlinear alternative, and the Leggett-Williams fixed point theorem.In the past few decades, in order to meet the demands of research, thep-Laplacian equation is introduced in some BVP, such as [15, 16].Paper [15] investigates the existence of solutions for the BVP of fractional p-Laplacian equation with the following form:
(3)D0+β(ϕp(D0+αx(t)))=f(t,x(t),D0+αx(t)),mmmmmmmmmmmmmmmmmbt∈[0,1],D0+αx(0)=D0+αx(1)=0,
where 0<α,β≤1, and 1<α+β≤2, and p>1, ϕp(s)=|s|p-2s is a p-Laplacian operator. Dα is a Caputo fractional derivative. A new result on the existence of solutions for the above fractional boundary value problem is obtained, which generalize and enrich some known results to some extent from the literature, by using the coincidence degree theory.Paper [16] studies the existence of positive solutions of the following singular fourth-order coupled system with integral boundary conditions:
(4)(ϕp1(u′′(t)))′′=λp1a1(t)f1(t,u(t),v(t)),nnnnnnnnnnnnnnnnnnnnnnnn0<t<1,(ϕp2(v′′(t)))′′=μp2a2(t)f2(t,u(t),v(t)),bbbbbbbbbbbbbbbbbbnnnnnnnn0<t<1,u(0)=u(1)=∫01‍u(s)dξ1(s),ϕp1(u′′(0))=ϕp1(u′′(1))=∫01‍ϕp1(u′′(s))dη1(s),v(0)=v(1)=∫01‍v(s)dξ2(s),ϕp2(v′′(0))=ϕp2(v′′(1))=∫01‍ϕp2(v′′(t))dη2(s),
where λ and μ are positive parameters, ϕpi(x)=|x|pi-2x, pi>1, ϕqi=ϕpi-1, 1/pi+1/qi=1, and ξi,ηi:[0,1]→R+ are nondecreasing functions of bounded variation, i=1,2, and the integrals in (4) are Riemann-Stieltjes integrals, f1: [0,1]×ℝ0+×ℝ+→ℝ+ and f2: [0,1]×ℝ+×ℝ0+→ℝ+ 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)→ℝ+ are continuous and may be singular at t=0 and/or t=1, in which ℝ+=[0,∞) and ℝ0+=(0,∞). By using the fixed point theory in cones, explicit range for λ and μ is derived such that for any λ and μ which lie in their respective interval, the existence of at least one positive solution to the boundary value system is guaranteed.Inspired by above works, our work presented in this paper has the following new features. Firstly, our study is on singular nonlinear differential systems; that is,fi(t,x1,x2,x3,x4) may be singular at the value 0 of all its space variables x1,x2,x3,x4, i=1,2, which bring about many difficulties. Secondly, 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 BVP with integral boundary conditions, that is, system (1) including multipoint and nonlocal boundary value problems as special cases. To our knowledge, very few authors studied the existence of positive solutions for p-Laplacian fractional 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).Throughout the paper,∥u∥1=∫01‍|u(t)|dt is the norm in the Banach L1[0,1] and ∥u∥∞=max⁡t∈[0,1]|u(t)| is the norm in the Banach C[0,1]. Let E=C2[0,1]; then E is a Banach space endowed with the norm ∥u∥*=max⁡{∥u∥∞,∥u′∥∞,∥u′′∥∞}. Thus, (E×E,∥·∥) is a Banach with the norm defined by ∥(u,v)∥=max⁡{∥u∥*,∥v∥*} for (u,v)∈E×E.This paper is organized as follows. In Section2, we present some results of fractional calculus theory and auxiliary technical lemmas, which are used in the next two sections. Section 3 deals with the approximate problem of (1). We induce the solvability of this problem to the existence of a fixed point of an operator Tn. By the Leggett-Williams fixed point theorem, the existence of at least three fixed points of Tn is obtained. In Section 4, we prove the existence and multiplicity of positive solutions of problem (1) by applying the results of Sections 2 and 3.


## 2. Preliminaries

For the convenience of the reader, we present here the necessary definitions and lemmas from fractional calculus theory. These definitions and lemmas can be found in the recent literatures [14, 17–19].For convenience, we list the following signs and assumptions for system (1).We note that(1)
ai=∫01‍dξi(s),  bi=∫01‍dηi(s),  Ai=∫01‍sα-1dξi(s),  Bi=∫01‍sα-1dηi(s), i=1,2;(2)
eα(s)=s(1-s)α-1,  eβ(s)=s(1-s)β-1.We assume that(H1)
0≤ai<1,  0≤bi<1, and  i=1,2.Obviously,Ai≤ai,  Bi≤bi, i=1,2.Definition 1 (see [17]).
The fractional integral of orderα>0 of a function f: (0,∞)→ℝ is defined by
(5)I0+αf(t)=1Γ(α)∫0t‍(t-s)α-1f(s)ds
provided that the right-hand side exists.Definition 2 (see [17]).
The Riemann-Liouville fractional order derivative of orderα∈(n-1,n] of a function f: (0,∞)→ℝ is defined by
(6)D0+αf(t)=1Γ(n-α)dndtn∫0t‍(t-s)n-α-1f(s)ds
provided that the right-hand side exists.Lemma 3 (see [17]).
Letα∈(n-1,n] and u∈C(0,1)∩L1(0,1). Then
(7)I0+αD0+αu(t)=u(t)+C1tα-1+C2tα-2+⋯+Cntα-n,
where Ci∈ℝ,  i=1,2,…,n.Lemma 4 (see [18]).
Suppose thatμ∈(0,1], ν∈(1,2) and u∈C2[0,1], u(0)=u′(0)=0. Then D0+μu∈C[0,1],D0+νu∈C[0,1], and
(8)D0+μu(t)=1Γ(2-μ)∫0t‍(t-s)1-μu′′(s)ds,D0+νu(t)=1Γ(2-ν)∫0t‍(t-s)1-νu′′(s)ds.Lemma 5.
If (H1) holds, then for any y∈L1(0,1) and α∈(3,4] the boundary value problem
(9)-D0+αx(t)=ϕqi(y(t)),0<t<1,x(0)=x′(0)=x′′(0)=0,x′′(1)=(α-1)(α-2)∫01‍x(s)dξi(s)
has a unique solution
(10)x(t)=∫01‍Hi(t,s)ϕqi(y(s))ds,i=1,2,
where
(11)Hi(t,s)=Gα(t,s)+tα-11-Ai∫01‍Gα(τ,s)dξi(τ),i=1,2,(12)Gα(t,s)=1Γ(α){tα-1(1-s)α-3-(t-s)α-1,0≤s≤t≤1,tα-1(1-s)α-3,0≤t≤s≤1.Proof.
By Lemma3, we can see that
(13)x(t)=-I0+α(ϕqi(y(t)))+C1tα-1+C2tα-2+C3tα-3+C4tα-4.
Considering that solutions satisfy x(0)=x′(0)=x′′(0)=0 and  x′′(1)=(α-1)(α-2)∫01‍x(s)dξi(s), we can get C2=C3=C4=0 and C1=(∫01‍(1-s)α-3ϕqi(y(s))ds-∫01‍∫0s‍(s-τ)α-1ϕqi(y(τ))dτdξi(s))/Γ(α)(1-Ai). As a result,
(14)x(t)=tα-1Γ(α)(1-Ai)[∫01(1-s)α-3ϕqi(y(s))ds-∫01‍∫0s‍(s-τ)α-1ϕqi(y(τ))dτdξi(s)]-1Γ(α)∫0t‍(t-s)α-1ϕqi(y(s))ds=∫01‍Gα(t,s)ϕqi(y(s))ds+tα-11-Ai∬01Gα(τ,s)ϕqi(y(s))dξi(τ)ds=∫01‍Hi(t,s)ϕqi(y(s))ds.Lemma 6 (see [14]).
If (H1) holds, then for any z∈L1(0,1) and β∈(1,2] the boundary value problem
(15)-D0+βx(t)=z(t),0<t<1,y(0)=0,x(1)=∫01‍y(s)dηi(s)
has a unique solution
(16)y(t)=∫01‍Ki(t,s)z(s)ds,i=1,2,
where
(17)Ki(t,s)=Gβ(t,s)+tβ-11-Bi∫01‍Gβ(τ,s)dηi(τ),i=1,2,(18)Gβ(t,s)=1Γ(β){tβ-1(1-s)β-1-(t-s)β-1,0≤s≤t≤1,tβ-1(1-s)β-1,0≤t≤s≤1.Lemma 7 (see [18]).
LetGα and Gβ be as defined in (12) and (18). Then(1)
Gα(t,s)∈C([0,1]×[0,1]) and Gα(t,s)>0 on (0,1)×(0,1),(2)
eα(1-t)eα(s)/Γ(α)≤Gα(t,s)≤1/Γ(α) for (t,s)∈[0,1]×[0,1],(3)
(∂/∂t)Gα(t,s)∈C([0,1]×[0,1]) and (∂/∂t)Gα(t,s)>0 on (0,1)×(0,1),(4)
(∂/∂t)Gα(t,s)≤1/Γ(α-1) for (t,s)∈[0,1]×[0,1],(5)
(∂2/∂t2)Gα(t,s)∈C([0,1]×[0,1]) and (∂2/∂t2)Gα(t,s)>0 on (0,1)×(0,1),(6)
(∂2/∂t2)Gα(t,s)≤1/Γ(α-2) for (t,s)∈[0,1]×[0,1],(7)
Gβ(t,s)∈C([0,1]×[0,1]) and Gβ(t,s)>0 on (0,1)×(0,1),(8)
eβ(1-t)eβ(s)/Γ(β)≤Gβ(t,s)≤1/Γ(β) for (t,s)∈[0,1]×[0,1].Proof.
For notational convenience, we denote by(19)Gα*(t,s)=1Γ(α){tα-1(1-s)α-1-(t-s)α-1,0≤s≤t≤1,tα-1(1-s)α-1,0≤t≤s≤1.
From paper [19], (7) and (8) hold. Likewise, we have eα(1-t)eα(s)/Γ(α)≤Gα*(t,s), since (1-s)α-1≤(1-s)α-3; for s∈[0,1], we have Gα(t,s)≤Gα*(t,s), so we have eα(1-t)eα(s)/Γ(α)≤Gα(t,s), and from [18], (1)–(6) hold.Lemma 8.
LetHi, Ki be as defined in (11) and (17), i=1,2. Then(1)
Hi(t,s)∈C([0,1]×[0,1]) and Hi(t,s)>0 on (0,1)×(0,1),(2)
eα(1-t)eα(s)/Γ(α)≤Hi(t,s)≤1/(Γ(α)(1-ai)) for (t,s)∈[0,1]×[0,1],(3)
(∂/∂t)Hi(t,s)∈C([0,1]×[0,1]) and (∂/∂t)Hi(t,s)>0 on (0,1)×(0,1),(4)
(∂/∂t)Hi(t,s)≤1/(Γ(α-1)(1-ai)) for (t,s)∈[0,1]×[0,1],(5)
(∂2/∂t2)Hi(t,s)∈C([0,1]×[0,1]) and (∂2/∂t2)Hi(t,s)>0 on (0,1)×(0,1),(6)
(∂2/∂t2)Hi(t,s)≤1/(Γ(α-2)(1-ai)) for (t,s)∈[0,1]×[0,1],(7)
Ki(t,s)∈C([0,1]×[0,1]) and Ki(t,s)>0 on (0,1)×(0,1),(8)
eβ(1-t)eβ(s)/Γ(β)≤Ki(t,s)≤1/(Γ(β)(1-bi)) for (t,s)∈[0,1]×[0,1].Proof .
By (H1), (11), and Lemma 7, we have Gα(t,s)≤Hi(t,s)≤(1/Γ(α))(1+ai/(1-Ai))≤1/(Γ(α)(1-ai)), so (2) holds; likewise, (1) and (3)–(8) hold.


## 3. Auxiliary Regular Problem

To overcome singularity, we consider the following approximate problem of (1):
(20)D0+β(ϕp1(D0+αu(t)))=f1n(t,u(t),v(t),D0+μu(t),D0+νv(t)),0<t<1,D0+β(ϕp2(D0+αv(t)))=f2n(t,u(t),v(t),D0+μu(t),D0+νv(t)),0<t<1,u(0)=u′(0)=u′′(0)=0,u′′(1)=(α-1)(α-2)∫01‍u(s)dξ1(s),D0+αu(0)=0,ϕp1(D0+αu(1))=∫01‍ϕp1(D0+αu(s))dη1(s),v(0)=v′(0)=v′′(0)=0,v′′(1)=(α-1)(α-2)∫01‍v(s)dξ2(s),D0+αv(0)=0,ϕp2(D0+αv(1))=∫01‍ϕp2(D0+αv(s))dη2(s),
where n is a positive integer and
(21)fin(t,x1,x2,x3,x4)=fi(t,max⁡{x1,n-1},max⁡{x2,n-1},max⁡{x3,n-1},max⁡{x4,n-1}).
Clearly, fin∈Car([0,1]×ℝ+×ℝ+×ℝ+×ℝ+,ℝ+), i=1,2.Define a coneP in E as
(22)P={u∈E:u(0)=u′(0)=0,u(t)≥0,u′(t)≥0,u′′(t)≥0,t∈[0,1]}.
By Lemma 4, we can obtain that
(23)D0+μu∈C[0,1],D0+νu∈C[0,1],D0+μu(t)≥0,D0+νu(t)≥0,
for u∈P and t∈[0,1].For eachn∈ℕ+, let us define operators T1n:P→E, T2n:P→E and Tn:P×P→E×E by
(24)Tin(u,v)(t)=∫01‍Hi(t,s)ϕqi×(∫01‍Ki(s,τ)fin×(τ,u(τ),v(τ),D0+μu(τ),D0+νv(τ))dτ∫01‍)ds,i=1,2,
and Tn(u,v)=(T1n(u,v),T2n(u,v)). By Lemmas 5 and 6, we know that fixed points of Tn are positive solution of the system (20).Lemma 9.
T
n
:
P
×
P
→
P
×
P is a completely continuous operator.Proof.
We divide the proof into three steps.
Step
1. We prove that Tn:P×P→P×P is well defined.
For all(u,v)∈P×P, let ρi(t)=fin(t,u(t),v(t),D0+μu(t),D0+νv(t)). Then by (21) and (23), we have ρi∈L1[0,1], i=1,2. It follows from Lemma 8 that Hi, (∂/∂t)Hi, (∂2/∂t2)Hi, Ki are nonnegative and continuous on [0,1]×[0,1]. Therefore, we get Tin(u,v)∈C2[0,1], Tin(u,v)(0)=Tin′(u,v)(0)=0 and Tin(u,v)(t)≥0, Tin′(u,v)(t)≥0, and Tin′′(u,v)(t)≥0 on [0,1]. As a result, Tin(u,v)∈P; then we can get Tn(u,v)∈P×P.
Step
2. We prove that Tn: P×P→P×P is continuous.
Let{(um,vm)}⊂P×P be a convergent sequence. Suppose that lim⁡m→∞(um,vm)=(u,v). That is, lim⁡m→∞um=u and lim⁡m→∞vm=v. Then lim⁡m→∞um(j)(t)=u(t) and lim⁡m→∞vm(j)(t)=v(t) uniformly on [0,1], where j=0,1,2. Since, by Lemma 4, for μ∈(0,1] and ν∈(1,2),
(25)|D0+μum(t)-D0+μu(t)|≤1Γ(2-μ)∫0t‍(t-s)1-μ|um′′(s)-u′′(s)|ds≤∥um′′-u′′∥∞Γ(3-μ),|D0+νvm(t)-D0+νv(t)|≤1Γ(2-ν)∫0t‍(t-s)1-ν|vm′′(s)-v′′(s)|ds≤∥vm′′-v′′∥∞Γ(3-ν),
we have lim⁡m→∞D0+μum(t)=D0+μu(t) and lim⁡m→∞D0+νvm(t)=D0+νv(t) uniformly on [0,1]. In addition, it follows from (8), for μ∈(0,1] and ν∈(1,2), that
(26)∥D0+μum∥∞≤∥um′′∥∞Γ(2-μ)∫0t‍(t-s)1-μds≤∥um′′∥∞Γ(3-μ),∥D0+νvm∥∞≤∥vm′′∥∞Γ(2-ν)∫0t‍(t-s)1-νds≤∥vm′′∥∞Γ(3-ν).
Let(27)ρim(t)=fin(t,um(t),vm(t),D0+μum(t),D0+νvm(t)).
Sincelim⁡m→∞um=u, lim⁡m→∞vm=v, and lim⁡m→∞D0+μum(t)=D0+μu(t) and lim⁡m→∞⁡D0+νvm(t)=D0+νv(t) uniformly on [0,1], we have lim⁡m→∞ρim(t)=ρm(t), for t∈[0,1]. Since {um} and {vm} are bounded in C2[0,1] and fin∈Car([0,1]×ℝ+×ℝ+×ℝ+×ℝ+,ℝ+), inequalities (26) imply that {D0+μum} and {D0+νvm} are bounded in C[0,1]. As a result, there exists φi∈L1[0,1] such that ρim(t)≤φi(t), for t∈[0,1] and all m∈ℕ+; from (8) in Lemma 8, we have Ki(s,τ)ρim(τ)≤φi(τ)/Γ(β)(1-bi); for s,τ∈[0,1] and all m∈ℕ+. From the Lebesgue dominated convergence theorem, we can obtain
(28)lim⁡m→∞∫01‍Ki(s,τ)ρim(τ)dτ=∫01‍Ki(s,τ)ρi(τ)dτ,hhhhhhhhhhhhhhhhhhhhhhhhhhhhfors∈[0,1].
Since ϕqi(x)=x1/(pi-1) is continuous on [0,∞), we have ϕqi is uniformly continuous on [0,∥φi∥1/(1-bi)Γ(β)]; that is, for all ϵ>0, ∃δ>0, such that for all x1, x2∈[0,∞) and |x1-x2|<δ, we have
(29)|ϕqi(x1)-ϕqi(x2)|<ϵ.
For (28) we can say that for above δ>0 and ∃N>0, for each m>N, we have
(30)|∫01‍Ki(s,τ)ρim(τ)dτ-∫01‍Ki(s,τ)ρi(τ)dτ|<δ.
Since Lemma 8 and ρim(t)≤φi(t), for t∈[0,1], all m∈ℕ+, we have
(31)0≤∫01‍Ki(s,τ)ρim(τ)dτ≤∥φi∥1(1-bi)Γ(β),0≤∫01‍Ki(s,τ)ρim(τ)dτ≤∥φi∥1(1-bi)Γ(β).
From (29), (30), and (31), we can get
(32)|ϕqi(∫01‍Ki(s,τ)ρim(τ)dτ)-ϕqi(∫01‍Ki(s,τ)ρi(τ)dτ)|<ϵ,
so
(33)|Tin(um,vm)(t)-Tin(u,v)(t)|≤1Γ(α)(1-ai)∫01‍|ϕqi(∫01‍Ki(s,τ)ρim(τ)dτ)-ϕqi(∫01‍Ki(s,τ)ρi(τ)dτ)|ds<ϵΓ(α)(1-ai),
and likewise,
(34)|Tin′(um,vm)(t)-Tin′(u,v)(t)|<ϵΓ(α-1)(1-ai),|Tin′′(um,vm)(t)-Tin′′(u,v)(t)|<ϵΓ(α-2)(1-ai).
Then
(35)∥Tin(um,vm)-Tin(u,v)∥*<ϵΓ(α-2)(1-ai).
That is, lim⁡m→∞Tin(um,vm)=Tin(u,v), so lim⁡m→∞Tn(um,vm)=Tn(u,v).
We prove thatTn: P×P→P×P is continuous.
Step
3. We prove that Tn:P×P→P×P is compact; that is, for all 𝒟∈P×P, let 𝒟 be a bounded set, and Tn(𝒟) is relatively compact in E×E. For notational convenience, we denote by Li=[∥φi∥1/(1-bi)Γ(β)]1/(pi-1),i=1,2.
(I) We proveTn(𝒟) is uniformly bounded in E×E.
For all𝒟∈P×P, 𝒟 is a bounded set; then ∃R>0, such that for all (u,v)∈𝒟, we have ∥(u,v)∥≤R; that is, ∥u∥*≤R, ∥v∥*≤R, and from (26), for μ∈(0,1] and ν∈(1,2), we have ∥D0+μu∥∞≤R/Γ(3-μ) and ∥D0+νv∥∞≤R/Γ(3-ν). Since fin∈Car([0,1]×ℝ+×ℝ+×ℝ+×ℝ+,ℝ+), for some φi∈L1[0,1], such that fin(t,u(t),v(t),D0+μu(t),D0+νv(t))≤φi(t), for t∈[0,1] and all m∈ℕ+, then
(36)0≤Tin(u,v)(t)=∫01‍Hi(t,s)ϕqi×(∫01‍Ki(s,τ)fin(τ,u(τ),v(τ),D0+μu(τ),D0+νv(τ)D0+μu(τ))dτ∫01‍)ds≤1Γ(α)(1-ai)∫01‍ϕqi(1Γ(β)(1-bi)∫01‍φi(τ)dτ)ds≤1Γ(α)(1-ai)[∥φi∥1(1-bi)Γ(β)]1/(pi-1)=LiΓ(α)(1-ai),0≤Tin′(u,v)(t)=∫01‍∂∂tHi(t,s)ϕqi×(∫01‍Ki(s,τ)fin(τ,u(τ),v(τ),D0+μu(τ),D0+νv(τ)D0+μu(τ))dτ∫01‍)ds≤LiΓ(α-1)(1-ai),0≤Tin′′(u,v)(t)=∫01‍∂2∂t2Hi(t,s)ϕqi×(∫01‍Ki(s,τ)fin(τ,u(τ),v(τ),D0+μu(τ),D0+νv(τ)D0+μu(τ))dτ∫01‍)ds≤LiΓ(α-2)(1-ai),
and then ∥Tin(u,v)∥*≤Li/Γ(α-2)(1-ai); that is Tin(𝒟) is uniformly bounded in E×E, so Tn(𝒟) is also uniformly bounded in E×E.
(II) We proveTn(j)(𝒟) is equicontinuous on [0,1], where j=1,2.
From Lemma8, we can get Hi(t,s), (∂/∂t)Hi(t,s), and (∂2/∂t2)Hi(t,s) is uniformly continuous on [0,1]; that is, for all ϵ>0, ∃δ>0 such that for all t1, t2, s∈[0,1] and |t1-t2|<δ, we have |Hi(t1,s)-Hi(t2,s)|<ϵ/Li,|(∂/∂t)Hi(t1,s)-(∂/∂t)Hi(t2,s)|<ϵ/Li, and  |(∂2/∂t2)Hi(t1,s)-(∂2/∂t2)Hi(t2,s)|<ϵ/Li.
For all(u,v)∈𝒟, we have
(37)|Tin(u,v)(t1)-Tin(u,v)(t2)|≤∫01‍|Hi(t1,s)-Hi(t2,s)|ϕqi×(∫01‍Ki(s,τ)fin(D0+μu(τ),D0+νv(τ)τ,u(τ),v(τ),D0+μu(τ),D0+νv(τ))dτ∫01‍)ds<ϵLi·Li=ϵ;
likewise
(38)|Tin′(u,v)(t1)-Tin′(u,v)(t2)|<ϵ,|Tin′′(u,v)(t1)-Tin′′(u,v)(t2)|<ϵ.
That is, Tin(j)(𝒟) is equicontinuous on [0,1], so Tn(j)(𝒟) is also equicontinuous on [0,1], j=1,2.
Applying the Arzelà-Ascoli theorem,Tn(𝒟) is relatively compact in E×E, so we prove that Tn:P×P→P×P is compact.
From Steps 1–3, we can getTn:P×P→P×P is a completely continuous operator.
The proof of Lemma9 is completed.Lemma 10 (see [20, 21]).
LetW be a cone in a real Banach space F, Wc={x∈W:∥x∥<c}, let θ be a nonnegative continuous concave functional on W such that θ(x)≤∥x∥ for all x∈Wc¯, and W(θ,b,d)={x∈W:θ(x)≥b,∥x∥≤d}. Suppose that  A: Wc¯→Wc¯ is completely continuous and there exist positive constants 0<a<b<d≤c such that(C1)
{x∈W(θ,b,d):θ(x)>b}≠∅ and θ(Ax)>b for x∈W(θ,b,d),(C2)
∥Ax∥<a for x∈Wa¯,(C3)
θ(Ax)>b for x∈W(θ,b,c) with ∥Ax∥>d.Then A has at least three fixed points x1, x2, and x3 satisfying
(39)∥x1∥<a,b<θ(x2),∥x3∥>a,θ(x3)<b.Remark 11.
Ifd=c, then condition (C1) of Lemma 10 implies condition (C3) of Lemma 10.We noteW=P×P, Wc={(u,v)∈W:∥(u,v)∥<c}, and a nonnegative continuous concave functional θ on the cone W defined by
(40)θ(u,v)=min⁡t∈[ζ,ω]|u(t)+v(t)|,
in which Γ(3-μ)/2<ζ<ω<1. Since 0<μ≤1, we get 0<Γ(3-μ)/2<1, and θ is well defined.For notational convenience, we introduce the following constants:(41)hα=min⁡t∈[ζ,ω]eα(1-t),hβ=min⁡t∈[ζ,ω]eβ(1-t),(42)mi=∫01‍eα(s)ϕqi(∫ζω‍eβ(τ)dτ)ds,vvvvvvvvvvvini=(hβΓ(β))1/(pi-1),vvvvvvvvvvvvvvvvvvvvvvi=1,2,(43)Ji=Γ(α-2)(1-ai)[Γ(β)(1-bi)]1/(pi-1),wwwwwwiΛ=[(hβΓ(β))(m1n1+m2n2)]-1,wwwwwwwwwwwwwwwwwwwwwi=1,2.We work with the following conditions onfi in (1), i=1,2.∃
a,  b,  c and 2<a<b<c, such that fi satisfies the following conditions:(H2)
fi(t,x1,x2,x3,x4) < (Jia)pi-1, for (t,x1,x2,x3,x4)∈[0,1] × (0,a] × (0,a] × (0,/(0,a/Γ(3-μ)] × (0,a/Γ(3-ν)],(H3)
fi(t,x1,x2,x3,x4) ≤ (Jic)pi-1, for (t,x1,x2,x3,x4)∈[0,1] × (0,c] × (0,c] × (0,c/Γ(3-μ)] × (0,c/Γ(3-ν)],(H4)
fi(t,x1,x2,x3,x4) > (Λc)pi-1, for (t,x1,x2,x3,x4)∈[ζ,ω] × [b,c] × [b,c] × [(ζ/Γ(3-μ))b,(ω/Γ(3-μ))c] × [(ζ/Γ(3-ν))b,(ω/Γ(3-ν))c], where Ji, Λ are defined by (43).Sincefin(t,x1,x2,x3,x4)=fi(t,max⁡{x1,n-1},max⁡{x2,n-1},max⁡{x3,n-1},max⁡{x4,n-1}), we can get fin satisfies the following conditions:(H2)′
fin(t,x1,x2,x3,x4) < (Jia)pi-1, for (t,x1,x2,x3,x4)∈[0,1] × [0,a] × [0,a] × [0,a/Γ(3-μ)] × [0,a/Γ(3-ν)],(H3)′
fin(t,x1,x2,x3,x4) ≤ (Jic)pi-1, for (t,x1,x2,x3,x4)∈[0,1] × [0,c] × [0,c] × [0,c/Γ(3-μ)] × [0,c/Γ(3-ν)],(H4)′
fin(t,x1,x2,x3,x4) > (Λc)pi-1, for (t,x1,x2,x3,x4)∈[ζ,ω] × [b,c] × [b,c] × [(ζ/Γ(3-μ))b,(ω/Γ(3-μ))c] × [(ζ/Γ(3-ν))b,(ω/Γ(3-ν))c].Theorem 12.
Assume that (H1)–(H4) hold. Then, for n∈ℕ+, BVP (20) have at least three positive solutions (u1n,v1n), (u2n,v2n), and (u3n,v3n) in Wc¯, satisfying
(44)∥(u1n,v1n)∥<a,b<min⁡t∈[ζ,ω]|u2n(t)+v2n(t)|,a<∥(u3n,v3n)∥,min⁡t∈[ζ,ω]|u3n(t)+v3n(t)|<b.Proof.
We will show that all conditions of Lemma10 are satisfied.
Firstly, we proveTn: Wc¯→Wc¯ is completely continuous.
For all(u,v)∈Wc¯, then ∥(u,v)∥≤c. So we have 0≤u(j)(t)≤c, 0≤v(j)(t)≤c, and t∈[0,1], j=0,1,2. By (26) we have 0≤D0+μu(t)≤c/Γ(3-μ) and 0≤D0+νv(t)≤c/Γ(3-ν). By condition (H3), we can get (H3)′ hold, so it follows from condition (H3) that
(45)fin(τ,u(τ),v(τ),D0+μu(τ),D0+νv(τ))≤(Jic)pi-1,wwwwwwwwwwwwwwwwwwwwwwiτ∈[0,1].
Thus, for any (u,v)∈Wc¯, by (45), we have
(46)|Tin(u,v)(t)|=∫01‍Hi(t,s)ϕqi(∫01‍Ki(s,τ)fin(τ,u(τ),v(τ),D0+μu(τ),D0+νv(τ)D0+μu(τ))dτ∫01‍)ds≤1Γ(α)(1-ai)×∫01‍ϕqi(∫01‍[1(1-bi)Γ(β)](Jic)pi-1dτ)ds=1Γ(α)(1-ai)[1(1-bi)Γ(β)]1/(pi-1)·Jic<JicJi=c,|Tin′(u,v)(t)|=∫01‍∂∂tHi(t,s)ϕqi×(∫01‍Ki(s,τ)fin(τ,u(τ),v(τ),D0+μu(τ),D0+νv(τ)D0+μu(τ))dτ∫01‍)ds≤1Γ(α-1)(1-ai)×∫01‍ϕqi(∫01‍[1(1-bi)Γ(β)](Jic)pi-1dτ)ds=1Γ(α-1)(1-ai)[1(1-bi)Γ(β)]1/(pi-1)·Jic<JicJi=c,|Tin′′(u,v)(t)|=∫01‍∂2∂t2Hi(t,s)ϕqi×(∫01‍Ki(s,τ)fin(τ,u(τ),v(τ),D0+μu(τ),D0+νv(τ)D0+μu(τ))dτ∫01‍)ds≤1Γ(α-2)(1-ai)×∫01‍ϕqi(∫01‍[1(1-bi)Γ(β)](Jic)pi-1dτ)ds=1Γ(α-2)(1-ai)[1(1-bi)Γ(β)]1/(pi-1)·Jic<JicJi=c,
which means that ∥Tin(u,v)∥*≤c, so we have ∥Tn(u,v)∥≤c and (u,v)∈Wc¯. Therefore, Tn: Wc¯→Wc¯. By Lemma 9, we know that Tn: Wc¯→Wc¯ is completely continuous.
Next, similar to (46), it follows from condition (H2) that if (u,v)∈Wa¯, then ∥T(u,v)∥<a. So the condition (C2) of Lemma 10 holds.
Now, we takeu0(t)=(b+c)/4 and v0(t)=(b+c)/4, where t∈[0,1]; then
(47)θ(u0,v0)=min⁡t∈[ζ,ω]|u0(t)+v0(t)|=b+c2>b,∥(u0,v0)∥=b+c4<c,
which means that (u0,v0)∈W(θ,b,c) and θ(u0,v0)>b, This proves that {(u0,v0)∈W(θ,b,c):θ(u0,v0)>b}≠∅.
On the other hand, if(u,v)∈W(θ,b,c), then b≤u(j)(t)≤c and b≤v(j)(t)≤c, where t∈[ζ,ω] and j=0,1,2. By Lemma 4, for t∈[ζ,ω], we have
(48)D0+μu(t)=1Γ(2-μ)∫0t‍(t-s)1-μu′′(s)ds≥tΓ(3-μ)b≥ζΓ(3-μ)b,D0+μu(t)=1Γ(2-μ)∫0t‍(t-s)1-μu′′(s)ds≤tΓ(3-μ)c≤ωΓ(3-μ)c,
and similarly, we can get (ζ/Γ(3-ν))b≤D0+νv(t)≤(ω/Γ(3-ν))c, where t∈[ζ,ω]. By condition (H4), we can get (H4)′ hold, so it follows from condition (H4) that
(49)fin(τ,u(τ),v(τ),D0+μu(τ),D0+νv(τ))>(Λb)pi-1,wwwwwwwwwwwwwwwwwwwwwwτ∈[ζ,ω]
since
(50)θ(Tn(u,v))=θ(T1n(u,v),T2n(u,v))=min⁡t∈[ζ,ω]|T1n(u,v)(t)+T2n(u,v)(t)|=T1n(u,v)(t0)+T2n(u,v)(t0),
in which t0∈[ζ,ω].
Hence, by(49), we have
(51)min⁡t∈[ζ,ω]Tin(u,v)(t)=min⁡t∈[ζ,ω]∫01‍Hi(t,s)ϕqi×(∫01‍Ki(s,τ)fin(τ,u(τ),v(τ),D0+μu(τ),D0+νv(τ)D0+μu(τ))dτ∫01‍)ds≥min⁡t∈[ζ,ω]∫01‍eα(1-t)eα(s)Γ(α)ϕqi×(∫01‍eβ(1-s)eβ(τ)Γ(β)×fin(τ,u(τ),v(τ),D0+μu(τ),D0+νv(τ)D0+μu(τ))dτ∫01‍)ds>min⁡t∈[ζ,ω]eα(1-t)Γ(α)∫01‍eα(s)ϕqi×(∫ζω‍hβΓ(β)eβ(τ)(Λb)pi-1dτ)ds=hαΓ(α)·hβΓ(β)1/(pi-1)∫01‍eα(s)ϕqi(∫ζω‍eβ(τ)dτ)ds·Λb=hαΓ(α)miniΛb,
so we can get
(52)Tin(u,v)(t0)≥min⁡t∈[ζ,ω]Tin(u,v)(t)>hαΓ(α)miniΛb.
Hence, by (50), (51), and (52), we have
(53)θ(Tn(u,v))=T1n(u,v)(t0)+T2n(u,v)(t0)>hβΓ(β)(m1n1+m2n2)Γb=b,
which implies that θ(Tn(u,v)), for (u,v)∈W(θ,b,c). This shows that condition (C1) of Lemma 10 is also satisfied.
By Remark11, condition (C3) of Lemma 10 holds; that is, we show that all conditions of Lemma 10 are satisfied. So we obtain that BVP (20) have at least three positive solutions (u1n,v1n), (u2n,v2n), and (u3n,v3n) in Wc¯, such that ∥(u1n,v1n)∥<a, b<θ(u2n,v2n), ∥(u3n,v3n)∥>a, and θ(u3n,v3n)<b, hence satisfying
(54)∥(u1n,v1n)∥<a,b<min⁡t∈[ζ,ω]|u2n(t)+v2n(t)|,a<∥(u3n,v3n)∥,min⁡t∈[ζ,ω]|u3n(t)+v3n(t)|<b.
The proof of Theorem12 is completed.Lemma 13.
Assume that (H1)–(H4) hold. Let (un,vn) in Wc¯ be a solution of problem (20). Then the sequence {(un,vn)} is relatively compact in E×E.Proof.
Firstly, we prove{(un,vn)} is uniformly bounded. Note that
(55)un(t)=∫01‍H1(t,s)ϕq1×(∫01‍K1(s,τ)f1n(τ,u(τ),v(τ),D0+μu(τ),D0+νv(τ)D0+μu(τ))dτ∫01‍)ds,vn(t)=∫01‍H2(t,s)ϕq2×(∫01‍K2(s,τ)f2n(τ,u(τ),v(τ),D0+μu(τ),D0+νv(τ)D0+μu(τ))dτ∫01‍)ds.
Since (un,vn) is in Wc¯, we can get {(un,vn)} is uniformly bounded.
Next, we prove{(un,vn)} is equicontinuous on [0,1]. Since (un,vn) is in Wc¯, we have ∥un∥*≤c,  ∥vn∥*≤c, and by (26), we have 0≤D0+μu(t)≤c/Γ(3-μ) and 0≤D0+νv(t)≤c/Γ(3-μ). From Lemma 8, we can get H1(t,s), (∂/∂t)H1(t,s), and (∂2/∂t2)H1(t,s) are uniformly continuous on [0,1], that is, for all ϵ>0, ∃δ>0 such that for all t1 and t2, s∈[0,1], |t1-t2|<δ, and we have
(56)|H1(t1,s)-H1(t2,s)|<ϵ,|∂∂tH1(t1,s)-∂∂tH1(t2,s)|<ϵ,|∂2∂t2H1(t1,s)-∂2∂t2H1(t2,s)|<ϵ.
By (H2) and Lemma 8, we have
(57)|un(t1)-un(t2)|≤∫01‍|H1(t1,s)-H1(t2,s)|ϕq1×(∫01‍K1(s,τ)f1n(τ,u(τ),v(τ),D0+μu(τ),D0+νv(τ)D0+μu(τ))dτ∫01‍)ds<ϵϕq1(1(1-b1)Γ(β)(J1a)p1-1)=J1a[(1-b1)Γ(β)]1/(p1-1)ϵ,
and similar to (57), we get
(58)|un′(t1)-un′(t2)|<J1a[(1-b1)Γ(β)]1/(p1-1)ϵ,|un′′(t1)-un′′(t2)|<J1a[(1-b1)Γ(β)]1/(p1-1)ϵ.
That is, {un(j)} is equicontinuous on [0,1]andj=1,2. Likewise, we can get {vn(j)} is equicontinuous on [0,1]. That is, {(un(j),vn(j))} is equicontinuous on [0,1]andj=1,2.
Applying the Arzelà-Ascoli theorem, sequence{(un,vn)} is relatively compact in E×E.
The proof of Lemma13 is completed.


## 4. Main Results

Theorem 14.
Assume that (H1)–(H4) hold. Then, for n∈ℕ+, BVP (1) have at least three positive solutions (u1,v1), (u2,v2), and (u3,v3) in Wc¯, satisfying
(59)∥(u1,v1)∥≤a,b≤min⁡t∈[ζ,ω]|u2(t)+v2(t)|,a≤∥(u3,v3)∥,min⁡t∈[ζ,ω]|u3(t)+v3(t)|≤b.Proof.
If (H1)–(H4) hold, by Theorem 12 and Lemma 13, we get BVP (20) have at least three positive solutions (u1n,v1n), (u2n,v2n), and (u3n,v3n) in Wc¯, satisfying
(60)∥(u1n,v1n)∥<a,b<min⁡t∈[ζ,ω]|u2n(t)+v2n(t)|,a<∥(u3n,v3n)∥,min⁡t∈[ζ,ω]|u3n(t)+v3n(t)|<b,
for n∈ℕ+ and for t∈[0,1] and the sequence (u1n,v1n), (u2n,v2n), and (u3n,v3n) are relatively compact in E×E.
We first consider the situation(u1n,v1n); without loss of generality, assume that (u1n,v1n) is convergent in E×E and lim⁡n→∞(u1n,v1n)=(u1,v1); that is, lim⁡n→∞u1n=u1 and lim⁡n→∞v1n=v1. Similar to the proof of Lemma 9, we have lim⁡n→∞D0+μu1n(t)=D0+μu1(t) and lim⁡n→∞D0+νv1n(t)=D0+νv1(t) uniformly on [0,1]. Then
(61)lim⁡n→∞fin(t,u1n(t),v1n(t),D0+μu1n(t),D0+νv1n(t))=fi(t,u1(t),v1(t),D0+μu1(t),D0+νv1(t)).
Since fin∈Car([0,1]×ℝ+×ℝ+×ℝ+×ℝ+,ℝ+), then there exists φi∈L1[0,1], such that
(62)fin(τ,u1n(τ),v1n(τ),D0+μu1n(τ),D0+νv1n(τ))≤φi(τ),
for τ∈[0,1] and all m∈ℕ+. Hence we can get
(63)H1(t,s)ϕq1×(∫01‍K1(s,τ)f1n(τ,u(τ),v(τ),D0+μu(τ),D0+νv(τ))dτ)≤1Γ(α)(1-a1)[1(1-b1)Γ(β)]1/(p1-1)ϕq1(∥φ1∥1).
By (61), (63), and the Lebesgue dominated convergence theorem, we can obtain
(64)lim⁡n→∞∫01‍K1(s,τ)f1n(τ,u1n(τ),v1n(τ),D0+μu1n(τ),D0+νv1n(τ)D0+μu(τ))dτ=∫01‍K1(s,τ)f1(t,u1(τ),v1(τ),D0+μu1(τ),D0+νv1(τ)D0+μu(τ))dτu1(t)=lim⁡n→∞u1n(t)=lim⁡n→∞∫01‍H1(t,s)ϕq1×(∫01‍K1(s,τ)f1n(τ,u(τ),v(τ),D0+μu(τ),D0+νv(τ)D0+μu(τ))dτ∫01‍)ds=∫01‍H1(t,s)lim⁡n→∞ϕq1×(∫01‍K1(s,τ)f1n(τ,u(τ),v(τ),D0+μu(τ),D0+νv(τ)D0+μu(τ))dτ∫01‍)ds=∫01‍H1(t,s)ϕq1×(∫01‍K1(s,τ)f1(t,u1(τ),v1(τ),D0+μu1(τ),D0+νv1(τ)D0+μu(τ))dτ∫01‍)ds.
Similarly, we can also get
(65)v1n(t)=∫01‍H2(t,s)ϕq2×(∫01‍K2(s,τ)f2(t,u1(τ),v1(τ),D0+μu1(τ),D0+νv1(τ)D0+μu(τ))dτ∫01‍)ds.
By Lemmas 5 and 6, we can obtain (u1,v1) is positive solution of BVP (1).
Similarly, we can also obtain(u2,v2) and (u3,v3) are positive solutions of BVP (1).
Since(66)∥(u1n,v1n)∥<a,b<min⁡t∈[ζ,ω]|u2n(t)+v2n(t)|,a<∥(u3n,v3n)∥,min⁡t∈[ζ,ω]|u3n(t)+v3n(t)|<b,
for n∈ℕ+ and for t∈[0,1], now, we pass to the limit as n→∞ in (66). Hence, we have
(67)∥(u1,v1)∥≤a,b≤min⁡t∈[ζ,ω]|u2(t)+v2(t)|,a≤∥(u3,v3)∥,min⁡t∈[ζ,ω]|u3(t)+v3(t)|≤b.
The proof of Theorem14 is completed.



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*Source: 984875-2014-04-27.xml*