We are concerned with the uniqueness of solutions for a class of p-Laplacian fractional order nonlinear systems
with nonlocal boundary conditions. Based on some properties of the p-Laplacian operator, the criterion of uniqueness for solutions is established.
1. Introduction
Fractional order differential systems arise from many branches of applied mathematics and physics, such as gas dynamics, Newtonian fluid mechanics, nuclear physics, and biological process [1–12]. In the recent years, there has a significant development in fractional calculus. For example, by using the contraction mapping principle, ur Rehman and Khan [13] established the existence and uniqueness of positive solutions for the fractional order differential equation with multipoint boundary conditions:
(1)Dtαy(t)=f(t,y(t),Dtβy(t)),t∈(0,1),y(0)=0,Dtβy(1)-∑i=1m-2ζiDtβy(ξi)=y0,
where 1<α≤2, 0<β<1, ζi∈[0,+∞), and 0<ξi<1, with ∑i=1m-2ζiξi<1. In [14], by using the fixed point theorem of mixed monotone operator, Zhang et al. studied the existence and uniqueness of positive solution for the following fractional order differential systems with multipoint boundary conditions:
(2)-Dtαx(t)=f(t,x(t),Dtβx(t),y(t)),-Dtγy(t)=g(t,x(t)),t∈(0,1),Dtβx(0)=0,Dtμx(1)=∑j=1p-2ajDtμx(ξj),y(0)=0,Dtνy(1)=∑j=1p-2bjDtνy(ξj),
where 1<γ<α≤2, 1<α-β<γ, 0<β≤μ<1, 0<ν<1, and 0<ξ1<ξ2<⋯<ξp-2<1, aj,bj∈[0,+∞) with ∑j=1p-2ajξjα-μ-1<1 and ∑j=1p-2bjξjγ-1<1; Dt is the standard Riemann-Liouville derivative. Some interesting results were also obtained by Zhang et al. [1, 2, 5, 7, 9], Goodrich [15–17], and Ahmad and Nieto [18].
On the other hand, the p-Laplacian equation
(3)(φp(x′(t)))′=f(t,x(t),x′(t)),
where φp(s)=|s|p-2s, p>1, can describe the turbulent flow in a porous medium; see [19]. Recently, by using Krasnoselskii’ s fixed point theorem and the Leggett-Williams theorem, Wang et al. [20] investigated the existence of positive solutions for the nonlocal fractional order differential equation with a p-Laplacian operator:
(4)Dtα(φp(Dtβx))(t)+f(t,x(t))=0,x(0)=0,Dtβx(0)=0,x(1)=ax(ξ),
where 0<β≤2, 0<α≤1, 0≤a≤1, and 0<ξ<1. And then, by looking for a more suitable upper and lower solution, Ren and Chen [21] established the existence of positive solutions for four points fractional order boundary value problem:
(5)Dtβ(φp(Dtαx))(t)=f(t,x(t)),t∈(0,1),x(0)=0,x(1)=ax(ξ),Dtαx(0)=0,Dtαx(1)=bDtαx(η),
where Dtα and Dtβ are the standard Riemann-Liouville derivatives, p-Laplacian operator is defined as φp(s)=|s|p-2s, p>1, and the nonlinearity f may be singular at both t=0,1 and x=0.
Inspired by the above work, in this paper, we study the uniqueness of positive solutions for the following fractional order differential system with p-Laplacian operator:
(6)Dtβ(φp1(Dtαx))(t)=λf(t,y(t)),Dtγ(φp2(Dtδy))(t)=ρg(t,x(t)),x(0)=0,x(1)=ax(ξ),Dtαx(0)=0,Dtαx(1)=bDtαx(η),y(0)=0,y(1)=cy(ζ),Dtδy(0)=0,Dtδy(1)=dDtδy(μ),
where Dtα, Dtβ, Dtγ, and Dtδ are the standard Riemann-Liouville derivatives with α,β,γ,δ∈(1,2], a,b,c,d∈[0,1], and ξ,η,ζ,μ∈(0,1), λ and ρ are positive parameters, p-Laplacian operator is defined as φp1(s)=|s|p1-2s, p1>1, (φp1)-1=φq1,1/p1+1/q1=1, and φp2(s)=|s|p2-2s, p2>1, (φp2)-1=φq2, 1/p2+1/q2=1. In the rest of paper, we assume that f,g:[0,1]×ℝ→ℝ are continuous.
Normally, we cannot apply the contraction mapping principle for solving the BVP (1) like ur Rehman and Khan [13] since p-Laplacian operator is nonlinear. In this paper, by using a property of the p-Laplacian operator, we overcome this difficulty and establish the uniqueness of solution for the eigenvalue problem of the fractional differential system (6).
2. Preliminaries and Lemmas
We firstly list the necessary definitions from fractional calculus theory here, which can be found in [10–12].
Definition 1.
Let β>0. The fractional integral operator of a function f:(0,+∞)→ℝ is given by
(7)Iβf(t)=1Γ(β)∫0t(t-s)β-1f(s)ds.
Definition 2.
Let β>0. The Riemann-Liouville fractional derivative of a function f:(0,+∞)→ℝ is given by
(8)Dtβf(t)=1Γ(n-β)(ddt)n∫0t(t-s)n-β-1f(s)ds,
where n=[β]+1, [β] denotes the integer part of the number β, and Γ denotes the gamma function.
where ci∈ℝ(i=1,2,…,n) and n is the smallest integer greater than or equal to β.
The main results of this paper are based on the following property of p-Laplacian operator, which is easy to be proved.
Lemma 3.
(1) If q≥2 and |x|,|y|≤M, then
(11)|φq(x)-φq(y)|≤(q-1)Mq-2|x-y|.
(2) If 1<q<2, xy>0, and |x|,|y|≥m>0, then
(12)|φq(x)-φq(y)|≤(q-1)mq-2|x-y|.
Applying Definitions 1 and 2 and Property 1, we have the following lemma.
Lemma 4.
Let y∈L1[0,1], 1<α,β≤2, 0<ξ,η<1, and 0≤a,b≤1. The fractional order boundary value problem,
(13)Dtβ(φp1(Dtαx))(t)=h(t),t∈(0,1),x(0)=0,x(1)=ax(ξ),Dtαx(0)=0,Dtαx(1)=bDtαx(η),
has the unique solution
(14)x(t)=∫01K1(t,s)φq1(∫01K2(s,τ)h(τ)dτ)ds,
where
(15)K1(t,s)=k1(t,s)+ak1(ξ,s)tα-11-aξα-1,K2(t,s)=k2(t,s)+b1k2(η,s)tβ-11-b1ηβ-1,k1(t,s)={(t(1-s))α-1-(t-s)α-1Γ(α),0≤s≤t≤1,(t(1-s))α-1Γ(α),0≤t≤s≤1,k2(t,s)={(t(1-s))β-1-(t-s)β-1Γ(β),0≤s≤t≤1,(t(1-s))β-1Γ(β),0≤t≤s≤1,
and b1=bp1-1.
Similar to (14), the fractional order boundary value problem,
(16)Dtγ(φp2(Dtδy))(t)=h(t),t∈(0,1),y(0)=0,y(1)=cy(ζ),Dtγy(0)=0,Dtδy(1)=dDtδy(μ),
has unique solution
(17)y(t)=∫01K3(t,s)φq2(∫01K4(s,τ)h(τ)dτ)ds,
where
(18)K3(t,s)=k3(t,s)+ck3(ζ,s)tδ-11-cζδ-1,K4(t,s)=k4(t,s)+d1k4(μ,s)tγ-11-d1μγ-1,k3(t,s)={(t(1-s))δ-1-(t-s)δ-1Γ(δ),0≤s≤t≤1,(t(1-s))δ-1Γ(δ),0≤t≤s≤1,k4(t,s)={(t(1-s))γ-1-(t-s)γ-1Γ(γ),0≤s≤t≤1,(t(1-s))γ-1Γ(γ),0≤t≤s≤1,
and d1=dp2-1.
Lemma 5.
Let 1<α,β,γ,δ≤2, 0<ξ,ζ,η,μ<1, and 0≤a,b,c,d≤1. The functions Ki(t,s),i=1,2,3,4, are continuous on [0,1]×[0,1] and satisfy
Ki(t,s)≥0,i=1,2,3,4 for t,s∈[0,1];
for t,s∈[0,1],
(19)σ1(s)tβ-1≤K2(t,s)≤σ3(s)tβ-1,σ2(s)tγ-1≤K4(t,s)≤σ4(s)tγ-1,
where
(20)σ1(s)=b1k2(η,s)1-b1ηβ-1,σ3(s)=(1-s)β-1Γ(β)+b1k2(η,s)1-b1ηβ-1,σ2(s)=d1k4(μ,s)1-d1μγ-1,σ4(s)=(1-s)γ-1Γ(γ)+d1k4(μ,s)1-d1μγ-1.
For t,s∈[0,1],
(21)K1(t,s)≤r1(1-s)α-1,K3(t,s)≤r2(1-s)δ-1,
where
(22)r1=1Γ(α)[1+a1-aξα-1],r2=1Γ(δ)[1+c1-cξδ-1].
Proof.
The proof is obvious; we omit the proof.
The basic space used in this paper is E=C([0,1];ℝ)×C([0,1];ℝ), where ℝ is a real number set. Obviously, the space E is a Banach space if it is endowed with the norm as follows:
(23)∥(u,v)∥:=∥u∥+∥v∥,∥u∥=maxt∈[0,1]|u(t)|,∥v∥=maxt∈[0,1]|v(t)|,
for any (u,v)∈E. By Lemma 4, (x,y)∈E is a solution of the fractional order system (1) if and only if (x,y)∈E is a solution of the integral equation
(24)x(t)=λq1∫01K1(t,s)φq1(∫01K2(s,τ)f(s,y(τ))dτ)ds,kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkt∈[0,1],y(t)=ρq2∫01K3(t,s)φq2(∫01K4(s,τ)g(s,x(τ))dτ)ds,kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkt∈[0,1].
We define an operator T:E→E by
(25)T(x,y)(t)=(F(x,y),G(x,y)),
where
(26)F(x,y)=λq1∫01K1(t,s)φq1(∫01K2(s,τ)f(s,y(τ))dτ)ds,G(x,y)=ρq2∫01K3(t,s)φq2(∫01K4(s,τ)g(s,x(τ))dτ)ds.
It is easy to see that (x,y) is the solution of the boundary value problem (6) if and only if (x,y) is the fixed point of T. As f,g∈C([0,1]×ℝ,ℝ), we know that T:E→E is a continuous and compact operator.
3. Main Results
Now we here introduce a new concept: the 𝒟-contraction mapping.
Definition 6.
A function ψ:(-∞,+∞)→[0,+∞) is called a nonlinear 𝒟-contraction mapping if it is continuous and nondecreasing and satisfies ψ(r)≤r,r>0.
Theorem 7.
Suppose that p1,p2>2, if there exist nonnegative functions ai(t),i=1,2,3,4, such that
(27)0<∫01δi(t)ai(t)dt<+∞,i=1,2,3,4,
and the following conditions are satisfied:
for any (t,w)∈(0,1)×ℝ,
(28)f(t,w)≥a1(t),g(t,w)≥a2(t),
there exist 𝒟-contraction mappings ψ1, ψ2 as
(29)|f(t,u)-f(t,v)|≤a3(t)ψ1(|u-v|),kkkkklkkka.e.(t,u),(t,v)∈[0,1]×ℝ,|g(t,u)-g(t,v)|≤a4(t)ψ2(|u-v|),kkkkkkkka.e.(t,u),(t,v)∈[0,1]×ℝ.
Then the fractional order differential system (6) has a unique solution provided that
(30)Λ=λq1(q1-1)r1B(α,(β-1)(q1-2)+1)×(∫01δ1(τ)a1(τ)dτ)q1-2∫01δ3(τ)a3(τ)dτ+ρq2(q2-1)r2B(δ,(γ-1)(q2-2)+1)×(∫01δ2(τ)a2(τ)dτ)q2-2∫01δ4(τ)a4(τ)dτ<1.
Proof.
In the case p1,p2>2, we have 1<q1, q2<2. Now we prove that T is a contraction mapping. By (27)-(28) and Lemma 5, we have
(31)∫01K2(s,τ)f(s,y(τ))dτ≥sβ-1∫01δ1(τ)a1(τ)dτ,∫01K4(s,τ)g(s,x(τ))dτ≥sγ-1∫01δ2(τ)a2(τ)dτ.
By (12), (28), and (31), for any (u1,v1),(u2,v2)∈E and for t>0, we have
(32)|φq1(∫01K2(s,τ)f(s,v1(τ))dτ)l-φq1(∫01K2(s,τ)f(s,v2(τ))dτ)|≤(q1-1)(sβ-1∫01δ1(τ)a1(τ)dτ)q1-2×∫01K2(s,τ)|f(τ,v1(τ))-f(τ,v2(τ))|dτ≤(q1-1)(sβ-1∫01δ1(τ)a1(τ)dτ)q1-2×∫01δ3(τ)a3(τ)dτψ1(∥v1-v2∥)≤(q1-1)s(β-1)(q1-2)(∫01δ1(τ)a1(τ)dτ)q1-2×∫01δ3(τ)a3(τ)dτ∥v1-v2∥.
Similarly, we also have
(33)|φq2(∫01K4(s,τ)g(s,u1(τ))dτ)l-φq2(∫01K4(s,τ)g(s,u2(τ))dτ)|≤(q2-1)s(γ-1)(q2-2)(∫01δ2(τ)a2(τ)dτ)q2-2×∫01δ4(τ)a4(τ)dτ∥u1-u2∥.
So it follows from (14), (17), and (31)-(32) that
(34)|F(u1,v1)(t)-F(u2,v2)(t)|=|λq1∫01K1(t,s)kkkkkkkkk×[φq1(∫01K2(s,τ)f(s,v1(τ))dτ)kkkkkkkklklk-φq1(∫01K2(s,τ)f(s,v2(τ))dτ)]ds|≤λq1r1∫01(1-s)α-1kkkkkkkklk×|φq1(∫01K2(s,τ)f(s,v1(τ))dτ)kkkkkkkklklk-φq1(∫01K2(s,τ)f(s,v2(τ))dτ)|ds≤λq1r1(q1-1)∫01(1-s)α-1s(β-1)(q1-2)ds×(∫01δ1(τ)a1(τ)dτ)q1-2×∫01δ3(τ)a3(τ)dτ∥v1-v2∥≤λq1(q1-1)r1B(α,(β-1)(q1-2)+1)×(∫01δ1(τ)a1(τ)dτ)q1-2×∫01δ3(τ)a3(τ)dτ∥v1-v2∥,|G(u1,v1)(t)-G(u2,v2)(t)|=|ρq2∫01K3(t,s)kkkkkkkkk×[φq2(∫01K4(s,τ)g(s,u1(τ))dτ)kkkkkkkkkkk-φq2(∫01K4(s,τ)g(s,u2(τ))dτ)]ds|≤ρq2(q2-1)r2B(δ,(γ-1)(q2-2)+1)×(∫01δ2(τ)a2(τ)dτ)q2-2×∫01δ4(τ)a4(τ)dτ∥u1-u2∥.
Hence
(35)|T(u1,v1)-T(u2,v2)|=|(F(u1,v1)-F(u2,v2),G(u1,v1)-G(u2,v2))|≤∥F(u1,v1)-F(u2,v2)∥+∥G(u1,v1)-G(u2,v2)∥≤λq1(q1-1)r1B(α,(β-1)(q1-2)+1)×(∫01δ1(τ)a1(τ)dτ)q1-2×∫01δ3(τ)a3(τ)dτ∥v1-v2∥+ρq2(q2-1)r2B(δ,(γ-1)(q2-2)+1)×(∫01δ2(τ)a2(τ)dτ)q2-2×∫01δ4(τ)a4(τ)dτ∥u1-u2∥≤Λ(∥v1-v2∥+∥u1-u2∥)=Λ∥(u1,v1)-(u2,v2)∥,
where
(36)Λ=λq1(q1-1)r1B(α,(β-1)(q1-2)+1)×(∫01δ1(τ)a1(τ)dτ)q1-2∫01δ3(τ)a3(τ)dτ+ρq2(q2-1)r2B(δ,(γ-1)(q2-2)+1)×(∫01δ2(τ)a2(τ)dτ)q2-2∫01δ4(τ)a4(τ)dτ.
Noticing that 0<Λ<1, we obtain that F:C[0,1]→C[0,1] is a contraction mapping. By means of the Banach contraction mapping principle, we get that T has a unique fixed point in E which implies that the fractional order differential system (6) has a unique solution.
Theorem 8.
Suppose that 1<p1, p2≤2, if there exist nonnegative functions bi(t), i=1,2,3,4, such that
(37)0<∫01δi(t)bi(t)dt<+∞,i=1,2,3,4,
and the following conditions are satisfied:
for any (t,w)∈(0,1)×ℝ,
(38)|f(t,w)|≤b3(t),g(t,w)≤b4(t),
there exist 𝒟-contraction mappings ϕ1, ϕ2 as
(39)|f(t,u)-f(t,v)|≤b1(t)ϕ1(|u-v|),kkkkkkkka.e.(t,u),(t,v)∈[0,1]×ℝ,|g(t,u)-g(t,v)|≤b2(t)ϕ2(|u-v|),kkkkkkkka.e.(t,u),(t,v)∈[0,1]×ℝ.
Then the fractional order differential system (6) has a unique solution provided that
(40)Λ~=λq1(q1-1)r1B(α,(β-1)(q1-2)+1)×(∫01δ3(τ)b3(τ)dτ)q1-2∫01δ3(τ)b1(τ)dτ+ρq2(q2-1)r2B(δ,(γ-1)(q2-2)+1)×(∫01δ4(τ)b4(τ)dτ)q2-2∫01δ4(τ)b2(τ)dτ<1.
Proof.
In the case 1<p1, p2≤2, we get q1,q2≥2; here we still prove that T is a contraction mapping if the conditions of theorem are satisfied. By (37)-(38) and Lemma 5, for any (x,y)∈E, we have
(41)∫01K2(s,τ)f(s,y(τ))dτ≤sβ-1∫01δ3(τ)b3(τ)dτ,∫01K4(s,τ)g(s,x(τ))dτ≤sγ-1∫01δ4(τ)b4(τ)dτ.
By (11), (39), and (41), for any (u1,v1),(u2,v2)∈E and for t>0, we have
(42)|φq1(∫01K2(s,τ)f(s,v1(τ))dτ)l-φq1(∫01K2(s,τ)f(s,v2(τ))dτ)|≤(q1-1)(sβ-1∫01δ3(τ)b3(τ)dτ)q1-2×∫01K2(s,τ)|f(τ,v1(τ))-f(τ,v2(τ))|dτ≤(q1-1)(sβ-1∫01δ3(τ)b3(τ)dτ)q1-2×∫01δ3(τ)b1(τ)dτϕ1(∥v1-v2∥)≤(q1-1)s(β-1)(q1-2)(∫01δ3(τ)b3(τ)dτ)q1-2×∫01δ3(τ)b1(τ)dτ∥v1-v2∥.
Similarly, we also have
(43)|φq2(∫01K4(s,τ)g(s,u1(τ))dτ)l-φq2(∫01K4(s,τ)g(s,u2(τ))dτ)|≤(q2-1)s(γ-1)(q2-2)×(∫01δ4(τ)b4(τ)dτ)q2-2×∫01δ4(τ)b2(τ)dτ∥u1-u2∥.
So it follows from (14), (17), and (42)-(43) that
(44)|F(u1,v1)(t)-F(u2,v2)(t)|=|λq1∫01K1(t,s)kkkkk×[φq1(∫01K2(s,τ)f(s,v1(τ))dτ)kkkkkklk-φq1(∫01K2(s,τ)f(s,v2(τ))dτ)]ds|≤λq1r1∫01(1-s)α-1kkkkk×|φq1(∫01K2(s,τ)f(s,v1(τ))dτ)kkkkkkkl-φq1(∫01K2(s,τ)f(s,v2(τ))dτ)|ds≤λq1r1(q1-1)∫01(1-s)α-1s(β-1)(q1-2)ds×(∫01δ3(τ)b3(τ)dτ)q1-2×∫01δ3(τ)b1(τ)dτ∥v1-v2∥≤λq1(q1-1)r1B(α,(β-1)(q1-2)+1)×(∫01δ3(τ)b3(τ)dτ)q1-2×∫01δ3(τ)b1(τ)dτ∥v1-v2∥,|G(u1,v1)(t)-G(u2,v2)(t)|=|ρq2∫01K3(t,s)kkkkkkk×[φq2(∫01K4(s,τ)g(s,u1(τ))dτ)kkkkkkkkkl-φq2(∫01K4(s,τ)g(s,u2(τ))dτ)]ds|≤ρq2(q2-1)r2B(δ,(γ-1)(q2-2)+1)×(∫01δ4(τ)b4(τ)dτ)q2-2×∫01δ4(τ)b2(τ)dτ∥u1-u2∥.
Hence
(45)|T(u1,v1)-T(u2,v2)|=|(F(u1,v1)-F(u2,v2),G(u1,v1)-G(u2,v2))|≤∥F(u1,v1)-F(u2,v2)∥+∥G(u1,v1)-G(u2,v2)∥≤λq1(q1-1)r1B(α,(β-1)(q1-2)+1)×(∫01δ3(τ)b3(τ)dτ)q1-2×∫01δ3(τ)b1(τ)dτ∥v1-v2∥+ρq2(q2-1)r2B(δ,(γ-1)(q2-2)+1)×(∫01δ4(τ)b4(τ)dτ)q2-2×∫01δ4(τ)b2(τ)dτ∥u1-u2∥≤Λ~(∥v1-v2∥+∥u1-u2∥)=Λ~∥(u1,v1)-(u2,v2)∥,
where
(46)Λ~=λq1(q1-1)r1B(α,(β-1)(q1-2)+1)×(∫01δ3(τ)b3(τ)dτ)q1-2∫01δ3(τ)b1(τ)dτ+ρq2(q2-1)r2B(δ,(γ-1)(q2-2)+1)×(∫01δ4(τ)b4(τ)dτ)q2-2∫01δ4(τ)b2(τ)dτ.
Noticing that 0<Λ~<1, we obtain that F:C[0,1]→C[0,1] is a contraction mapping. By means of the Banach contraction mapping principle, we get that T has a unique fixed point in E which implies that the fractional order differential system (6) has a unique solution.
It follows from Theorems 7 and 8 that the following corollaries for mixed cases hold.
Corollary 9.
Suppose that p1>2 and 1<p2≤2 if there exist nonnegative functions ai(t),i=1,2,3,4, such that
(47)0<∫01δi(t)ai(t)dt<+∞,i=1,2,3,4,
and the following conditions are satisfied:
for any (t,w)∈(0,1)×ℝ,
(48)f(t,w)≥a1(t),|g(t,w)|≤a2(t),
there exist 𝒟-contraction mappings ψ1, ψ2 as
(49)|f(t,u)-f(t,v)|≤a3(t)ψ1(|u-v|),kkkkklkkka.e.(t,u),(t,v)∈[0,1]×ℝ,|g(t,u)-g(t,v)|≤a4(t)ψ2(|u-v|),kkkkklkkka.e.(t,u),(t,v)∈[0,1]×ℝ.
Then the fractional order differential system (6) has a unique solution provided that
(50)Λ~1=λq1(q1-1)r1B(α,(β-1)(q1-2)+1)×(∫01δ1(τ)a1(τ)dτ)q1-2∫01δ3(τ)a3(τ)dτ+ρq2(q2-1)r2B(δ,(γ-1)(q2-2)+1)×(∫01δ4(τ)a2(τ)dτ)q2-2×∫01δ4(τ)a4(τ)dτ<1.
Corollary 10.
Suppose that p2>2 and 1<p1≤2 if there exist nonnegative functions ai(t), i=1,2,3,4, such that
(51)0<∫01δi(t)ai(t)dt<+∞,i=1,2,3,4,
and the following conditions are satisfied:
for any (t,w)∈(0,1)×ℝ,
(52)|f(t,w)|≤a1(t),g(t,w)≥a2(t),
there exist 𝒟-contraction mappings ψ1, ψ2 as
(53)|f(t,u)-f(t,v)|≤a3(t)ψ1(|u-v|),kkkkklkkka.e.(t,u),(t,v)∈[0,1]×ℝ,|g(t,u)-g(t,v)|≤a4(t)ψ2(|u-v|),kkkkklkkka.e.(t,u),(t,v)∈[0,1]×ℝ.
Then the fractional order differential system (6) has a unique solution provided that
(54)Λ~2=λq1(q1-1)r1B(α,(β-1)(q1-2)+1)×(∫01δ3(τ)a1(τ)dτ)q1-2∫01δ3(τ)a3(τ)dτ+ρq2(q2-1)r2B(δ,(γ-1)(q2-2)+1)×(∫01δ2(τ)a2(τ)dτ)q2-2×∫01δ4(τ)a4(τ)dτ<1.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors were supported financially by Ministry of Education, State Administration of Foreign Experts “111 Project of Innovation and Intelligence Introducing Planning” (B08039).
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