DDNSDiscrete Dynamics in Nature and Society1607-887X1026-0226Hindawi Publishing Corporation18976810.1155/2009/189768189768Research ArticleExistence of Positive Solutions for m-Point Boundary Value Problems on Time ScalesZhangYingQiaoShiDongZhangBinggenDepartment of MathematicsShanxi Datong UniversityDatong, Shanxi 037009Chinasxdtdx.edu.cn20092701200920092708200824112008140120092009Copyright © 2009This 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.

We study the one-dimensional p-Laplacian m-point boundary value problem (φp(uΔ(t)))Δ+a(t)f(t,u(t))=0, t[0,1]T, u(0)=0, u(1)=i=1m2aiu(ξi), where T is a time scale, φp(s)=|s|p2s, p>1, some new results are obtained for the existence of at least one, two, and three positive solution/solutions of the above problem by using Krasnoselsklls fixed point theorem, new fixed point theorem due to Avery and Henderson, as well as Leggett-Williams fixed point theorem. This is probably the first time the existence of positive solutions of one-dimensional p-Laplacian m-point boundary value problem on time scales has been studied.

1. Introduction

With the development of p-Laplacian dynamic equations and theory of time scales, a few authors focused their interest on the study of boundary value problems for p-Laplacian dynamic equations on time scales. The readers are referred to the paper .

In 2005, He  considered the following boundary value problems:(φp(uΔ(t)))+a(t)f(u(t))=0,t[0,T]T,u(0)B0uΔ(η)=0,uΔ(T)=0oruΔ(0)=0,u(T)+B1uΔ(η)=0,where T is a time scales, φp(s)=|s|p2s,p>1,η(0,ρ(t))T. The author showed the existence of at least two positive solutions by way of a new double fixed point theorem.

In 2004, Anderson et al.  used the virtue of the fixed point theorem of cone and obtained the existence of at least one solution of the boundary value problem:(g(uΔ(t)))+c(t)f(u)=0,a<t<b,u(a)B0uΔ(γ)=0,uΔ(b)=0.

In 2007, Geng and Zhu  used the Avery-Peterson and another fixed theorem of cone and obtained the existence of three positive solutions of the boundary value problem:(φp(uΔ(t)))+a(t)f(u(t))=0,t[0,T]T,u(0)B0uΔ(η)=0,uΔ(T)=0.Also, in 2007, Sun and Li  discussed the existence of at least one, two or three positive solutions of the following boundary value problem:(φp(uΔ(t)))Δ+h(t)f(uσ(t))=0,t[a,b]T,u(a)B0uΔ(a)=0,uΔ(σ(b))=0.

In this paper, we are concerned with the existence of multiple positive solutions to the m-point boundary value problem for the one dimension p-Laplcaian dynamic equation on time scale T(φp(uΔ(t)))Δ+a(t)f(t,u(t))=0,t[0,1]T,u(0)=0,u(1)=i=1m2aiu(ξi), where T is a time scale, φp(s)=|s|p2s,p>1,0<ξ1<ξ2<<ξm2<1,0ai,i=1,2,,m3,am2>0, and

i=1m2aiξi<1;

fCrd([0,1]T×[0,),[0,));

aCrd([0,1]T,[0,)) and there exists t0(ξm2,1) such that a(t0)>0.

In this paper, we have organized the paper as follows. In Section 2, we give some lemmas which are needed later. In Section 3, we apply the Krassnoselskiifs  fixed point theorem to prove the existence of at least one positive solution to the MBVP(1.5). In Section 4, conditions for the existence of at least two positive solutions to the MBVP (1.5) are discussed by using Avery and Henderson  fixed point theorem. In Section 5, to prove the existence of at least three positive solutions to the MBVP (1.5) are discussed by using Leggett and Williams  fixed point theorem.

For completeness, we introduce the following concepts and properties on time scales.

A time scale T is a nonempty closed subset of R, assume that T has the topology that it inherits from the standards topology on R.

Definition 1.1.

Let T be a time scale, for tT, one defines the forward jump operator σ:TT by σ(t)=inf{sT:s>t}, and the backward jump operator ρ:TT by ρ(t)=sup{sT:s<t}, while the graininess function μ:T[0,) is defined by μ(t)=σ(t)t. If σ(t)>t, one says that is right-scattered, while if ρ(t)<t, one says that tis left-scattered. Also, if t<supT and σ(t)=t, then t is called right-dense, and if t>infT and ρ(t)=t, then t is called left-dense. One also needs below the set Tk as follows: if T has a left-scattered maximum m, then Tk=Tm, otherwise Tk=T. For instance, if supT=, then Tk=T.

Definition 1.2.

Assume f:TR is a function and let tT. Then , one defines fΔ(t) to be the number (provided it exists) with the property that any given ε>0, there is a neighborhood U of t such that|[f(σ(t))f(s)]fΔ(t)[σ(t)s]|ε|σ(t)s|,for all sU. One says that f is delta differentiable (or in short: differentiable) on T provided fΔ(t) exist for all tT.

If T=R, then fΔ(t)=f(t), if T=, then fΔ(t)=Δf(t).

A function f:TR.

If f is continuous , then f is rd-continuous.

The jump operator σ is rd-continuous.

If f is rd-continuous, then so is fσ.

A function F:TR is called an antidervative of f:TR, provided FΔ(t)=f(t) holds for all tTk. One defines the definite integral byabf(t)Δt=F(b)F(a).For all a,bT. If fΔ(t)0, then f is nondecreasing.

2. The Preliminary LemmasLemma 2.1 (see [<xref ref-type="bibr" rid="B8">5</xref>, <xref ref-type="bibr" rid="B9">6</xref>]).

Assume that (H1)–(H3) hold. Then u(t) is a solution of the MBVP (1.5) on [0,1]T if and only ifu(t)=0tφq(0sa(τ)f(τ,u(τ))Δτ)Δsti=1m2ai0ξiφq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi+t01φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi,where φq(s)=|s|q2s,(1/p)+(1/q)=1, and q>1.

Lemma 2.2.

Assume that conditions (H1)–(H3) are satisfied, then the solution of the MBVP (1.5) on [0,1]T satisfiesu(t)0,t[0,1]T.

Lemma 2.3 (see [<xref ref-type="bibr" rid="B8">5</xref>]).

If the conditions (H1)–(H3) are satisfied, thenu(t)γu,t[ξm2,1],where u=supt[0,1]T|u(t)|,γ=min{am2(1ξm2)1am2ξm2,am2ξm2,ξ1}.

Lemma 2.4 (see [<xref ref-type="bibr" rid="B9">6</xref>]).

mint[ξm2,1]Au(t)=min{Au(1),Au(ξm2)}.

Let E denote the Banach space Crd[0,1]T with the norm u=supt[0,1]T|u(t)|.

Define the cone PE, byP={uEu(t)0,t[ξm2,1]minu(t)γu,uisconcave}.

The solutions of MBVP (1.5) are the points of the operator A defined byAu(t)=0tφq(0sa(τ)f(τ,u(τ))Δτ)Δsti=1m2ai0ξiφq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi+t01φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi=u(t).So, APP. It is easy to check that A:PP is completely continuous.

3. Existence of at least One Positive SolutionsTheorem 3.1 (see [<xref ref-type="bibr" rid="B12">8</xref>]).

Let E be a Banach space, and let PE be a cone. Assume Ω1 and Ω2 are open boundary subsets of E with 0Ω1,Ω¯1Ω2, and let A:P(Ω¯2Ω1)P be a completely continuous operator such that either

Auu for uPΩ1,Auu for uPΩ2; or

Auu for uPΩ1,Auu for uPΩ2 hold.

Then A has a fixed point in P(Ω¯2Ω1).

Theorem 3.2.

Assume conditions (H1)–(H3) are satisfied. In addition, suppose there exist numbers 0<r<R< such that f(t,u)φp(m)φp(r), if t[0,σ(1)],0ur, and f(t,u)φp(Mγ)φp(R), if t[ξm2,1],Ru<, whereM=1i=1m2aiξiγξm2ξm21φq(ξm2sa(τ)Δτ)Δs,m=1i=1m2aiξi01φq(0sa(τ)Δτ)Δs.

Then the MBVP (1.5) has at least one positive solution.

Proof.

Define the cone P as in (2.5), define a completely continuous integral operator A:PP byAu(t)=0tφq(0sa(τ)f(τ,u(τ))Δτ)Δsti=1m2ai0ξiφq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi+t01φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi.From (H1)–(H3), Lemmas 2.1 and 2.2, APP. If uP with u=r, then we getAu(t)=0tφq(0sa(τ)f(τ,u(τ))Δτ)Δsti=1m2ai0ξiφq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi+t01φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξit01φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξiφq(φp(m)φp(r))01φq(0sa(τ)Δτ)Δs1i=1m2aiξirm01φq(0sa(τ)Δτ)Δs1i=1m2aiξi=r=u. This implies that Auu. So, if we set Ω1={uCrd([0,1])u<r}, then Auu, for uPΩ1.

Let us now set Ω2={uCrd([0,1])u<R}.

Then for uP with u<R, by Lemma 2.4 we have u(t)γu,t[ξm2,1]. Therefore, we haveAu(t)Au(ξm2)=0ξm2φq(0sa(τ)f(τ,u(τ))Δτ)Δsξm2i=1m2ai0ξiφq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi+ξm201φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi=ξm201φq(0sa(τ)f(τ,u(τ))Δτ)Δs0ξm2φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi+11i=1m2aiξii=1m2ai(ξi0ξm2φq(0sa(τ)f(τ,u(τ))Δτ)Δsξm20ξiφq(0sa(τ)f(τ,u(τ))Δτ)Δs)ξm201φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi0ξm2φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξiξm2ξm21φq(ξm2sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξiφq(φp(Mγ)φp(R))ξm2ξm21φq(ξm2sa(τ)Δτ)Δs1i=1m2aiξiMγR1i=1m2aiξiξm2ξm21φq(ξm2sa(τ)Δτ)Δs=u.Hence, Auu for uPΩ2. Thus by the Theorem 3.1, A has a fixed point u in P(Ω¯2Ω1). Therefore, the MBVP (1.5) has at least one positive solution.

4. Existence of at least Two Positive Solutions

In this section, we apply the Avery-Henderson fixed point theorem  to prove the existence of at least two positive solutions to the nonlinear MBVP (1.5) .

Theorem 4.1 (see Avery and Henderson [<xref ref-type="bibr" rid="B13">9</xref>]).

Let P be a cone in a real Banach space E. SetP(Φ,ρ3)={uPΦ(u)<ρ3}.

If ν and Φ are increasing, nonnegative continuous functionals on P, let θ be a nonnegative continuous functional on P with θ(0)=0 such that, for some positive constants ρ3 and M>0,Φ(u)θ(u)ν(u) and uMΦ(u), for all uP(Φ,ρ3)¯. Suppose that there exist positive numbers ρ1<ρ2<ρ3 such that θ(λu)=λθ(u) for all 0λ1 and uP(θ,ρ2).

If A:P(Φ,ρ3)¯P is a completely continuous operator satisfying

Φ(Au)>ρ3 for all uP(Φ,ρ3);

θ(Au)<ρ2 for all uP(θ,ρ2);

P(ν,ρ1)ϕ and ν(Au)>ρ1 for all uP(ν,ρ1), then A has at least two fixed points u1 and u2 such that ρ1<ν(u1) with θ(u1)<ρ2 and ρ3<(u2) with Φ(u2)<ρ3.

Let l(0,1)T and 0<ξm2<l<1. Define the increasing, nonnegative and continuous functionals Φ,θ, and ν on P, by Φ(u)=u(ξm2),θ(u)=u(ξm2), and ν(u)=u(l).

From Lemma 2.4, for each uP,Φ(u)=θ(u)ν(u).

In addition, for each uP, Lemma 2.3 implies Φ(u)=u(ξm2)γu.

Thus,u<1γΦ(u),uP.

We also see that θ(0)=0 and θ(λu)=λθ(u) for all 0λ1 and uP(θ,q).

Theorem 4.2.

Assume (H1)–(H3) hold, suppose there exist positive numbers ρ1<ρ2<ρ3, such that the function f satisfies the following conditions:

f(t,u)>φp(mγ)φp(ρ1), for t[ξm2,l] and u[γρ1,ρ1];

f(t,u)<φp(m)φp(ρ2), for t[ξm2,1] and u[0,ρ2];

f(t,u)>φp(Mγ)φp(ρ3), for t[ξm2,l] and u[ρ3,(1/γ)ρ3].

Then the MBVP (1.5) has at least two positive solutions u1 and u2 such that u1(t)>ρ1 with u1(l)<ρ2 and u2(l)>ρ2 with u2(l)<ρ3.

Proof.

We now verify that all of the conditions of Theorem 4.1 are satisfied.

Define the cone P as (2.5), define a completely continuous integral operator A:PP by Au(t)=0tφq(0sa(τ)f(τ,u(τ))Δτ)Δsti=1m2ai0ξiφq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi+t01φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi.

M and m as in (3.1). To verify that condition (i) of Theorem 4.1 holds, we choose uP(Φ,ρ3), then Φ(u)=ρ3. This implies ρ3u(1/γ)Φ(u). Note that u(1/γ)Φ(u)=(1/γ)ρ3. We have ρ3u(t)(1/γ)ρ3, for t[ξm2,1]T. As a consequence of (B3), f(t,u)>φp(Mγ)φp(ρ3), for t[ξm2,l]T. Since AuP, we have from Lemma 2.2, Φ(Au)=(Au)(ξm2)=0ξm2φq(0sa(τ)f(τ,u(τ))Δτ)Δsξm2i=1m2ai0ξiφq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi+ξm201φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi=ξm201φq(0sa(τ)f(τ,u(τ))Δτ)Δs0ξm2φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi+11i=1m2aiξii=1m2ai(ξi0ξm2φq(0sa(τ)f(τ,u(τ))Δτ)Δsξm20ξiφq(0sa(τ)f(τ,u(τ))Δτ)Δs)ξm201φq(0sa(τ)f(τ,u(τ))Δτ)Δs0ξm2φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξiξm2ξm21φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξiφq(φp(Mγ)φp(ρ3))ξm2ξm21φq(0sa(τ)Δτ)Δs1i=1m2aiξiMγρ3ξm21i=1m2aiξiξm21φq(0sa(τ)Δτ)Δsρ3.Then condition (i) of Theorem 4.1 holds.

Let uP(θ,ρ2). Then θ(u)=ρ2. This implies 0u(t)u(1/γ)ρ2, for t[ξm2,1]. From (B2), we haveθ(Au)=(Au)(ξm2)ξm201φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξiξm201φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξimρ201φq(0sa(τ)Δτ)Δs1i=1m2aiξi=ρ2=u.Hence condition (ii) of Theorem 4.1 holds.

If we first define u(t)=ρ1/2, for t[0,1]T, then ν(u)=ρ1/2<ρ1. So P(ν,ρ1)ϕ.

Now, let uP(ν,ρ1), then ν(u)=u(l)=ρ1. This mean that ρ1/γu(t)uρ1. From (B1) and Lemma 2.4, we getν(Au)=(Au)(l)(Au)(ξm2)=0ξm2φq(0sa(τ)f(τ,u(τ))Δτ)Δsξm2i=1m2ai0ξiφq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi+ξm201φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi=ξm201φq(0sa(τ)f(τ,u(τ))Δτ)Δs0ξm2φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi+11i=1m2aiξii=1m2ai(ξi0ξm2φq(0sa(τ)f(τ,u(τ))Δτ)Δsξm20ξiφq(0sa(τ)f(τ,u(τ))Δτ)Δs)ξm201φq(0sa(τ)f(τ,u(τ))Δτ)Δs0ξm2φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξiξm2ξm21φq(ξm2sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξiφq(φp(mγ)φp(ρ1))ξm2ξm21φq(ξm2sa(τ)Δτ)Δs1i=1m2aiξi=mγρ11i=1m2aiξiξm2ξm21φq(ξm2sa(τ)Δτ)Δsρ1.Then condition (iii) of Theorem 4.1 holds.

Since all conditions of Theorem 4.1 are satisfied, the MBVP (1.5) has at least two positive solutions u1 and u2 such that u1(t)>ρ1 with u1(l)<ρ2 and u2(l)>ρ2 with u2(l)<ρ3.

5. Existence of at least Three Positive Solutions

We will use the Leggett-Williams fixed point theorem  to prove the existence of at least three positive solutions to the nonlinear MBVP (1.5).

Theorem 5.1 (see Leggett and Williams [<xref ref-type="bibr" rid="B14">10</xref>]).

Let P be a cone in the real Banach space E. SetPr={xPx<r},P(Ψ,a,b)={xPaΨ(x),xb}.

Suppose A:P¯rP¯r be a completely continuous operator and be a nonnegative continuous concave functional on P with Ψ(u)u for all uP¯r. If there exists 0<ρ1<ρ2<(1/γ)ρ2ρ3 such that the following condition hold:

{uP(Ψ,ρ2,(1/γ)ρ2)Ψ(u)>ρ2}ϕ and Ψ(Au)>ρ2 for all uP(Ψ,ρ2,(1/γ)ρ2);

Au<ρ1 for uρ1;

Ψ(Au)>ρ2 for uP(Ψ,ρ2,(1/γ)ρ2) with Au>(1/γ)ρ2, then A has at least three fixed points u1,u2 and u3 in P¯r satisfying u1<ρ1,Ψ(u2)>ρ2,ρ1<u3 with Ψ(u2)<ρ2.

Theorem 5.2.

Assume (H1)–(H3) hold . Suppose that there exist constants 0<ρ1<ρ2<(1/γ)ρ2ρ3 such that

f(t,u)φp(m)φp(ρ3), for t[ξm2,l] and u[0,ρ3];

f(t,u)>φp(Mγ)φp(ρ2), for t[ξm2,l] and u[ρ2,(1/γ)ρ2];

f(t,u)<φp(m)φp(ρ1), for t[ξm2,1] and u[0,ρ1].

Then the MBVP (1.5) has at least three positive solutions u1,u2, and u3 such that u1(ξ)<ρ1,u2(l)>ρ2,u3(ξ)>ρ1 with u3(l)<ρ2.

Proof.

The conditions of Theorem 5.1 will be shown to be satisfied. Define the nonnegative continuous concave functional Ψ:P[0,) to be Ψ(u)=u(ξm2), the cone P as in (2.5), M and m as in (3.1). We have Ψ(u)u for all uP. If uP¯ρ3, then uρ3, and from assumption (C1) , then we have(Au)(t)=0tφq(0sa(τ)f(τ,u(τ))Δτ)Δsti=1m2ai0ξiφq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi+t01φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξit01φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξiφq(φp(m)φp(ρ3))01φq(0sa(τ)Δτ)Δs1i=1m2aiξimρ301φq(0sa(τ)Δτ)Δs1i=1m2aiξi=ρ3.

This implies that Auρ3. Thus, we have A:P¯ρ3P¯ρ3. Since (1/γ)ρ2P(Ψ,ρ2,(1/γ)ρ2) and Ψ((1/γ)ρ2)=(1/γ)ρ2>ρ2,{uP(Ψ,ρ2,(1/γ)ρ2)|Ψ(u)>ρ2}ϕ. For uP(Ψ,ρ2,(1/γ)ρ2) we have ρ2u(ξm2)u(1/γ)ρ2. Using assumption (C2), f(t,u)>φp(Mγ)φp(ρ2), we obtainΨ(Au)=(Au)(ξm2)=0ξm2φq(0sa(τ)f(τ,u(τ))Δτ)Δsξm2i=1m2ai0ξiφq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi+ξm201φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi=ξm201φq(0sa(τ)f(τ,u(τ))Δτ)Δs0ξm2φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξi+11i=1m2aiξii=1m2ai(ξi0ξm2φq(0sa(τ)f(τ,u(τ))Δτ)Δsξm20ξiφq(0sa(τ)f(τ,u(τ))Δτ)Δs)ξm201φq(0sa(τ)f(τ,u(τ))Δτ)Δs0ξm2φq(0sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξiξm2ξm21φq(ξm2sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξiφq(φp(Mγ)φp(ρ2))ξm2ξm21φq(ξm2sa(τ)Δτ)Δs1i=1m2aiξiMγρ21i=1m2aiξiξm2ξm21φq(ξm2sa(τ)Δτ)Δsρ2.Hence, condition (i) of Theorem 5.1 holds.

If uρ1, from assumption (C3), we obtain(Au)(t)φq(φp(m)φp(ρ1))01φq(0sa(τ)Δτ)Δs1i=1m2aiξimρ101φq(0sa(τ)Δτ)Δs1i=1m2aiξi=ρ1.This implies that Auρ1.

Consequently, condition (ii) of Theorem 5.1 holds.

We suppose that uP(Ψ,ρ2,ρ3), with Au>(1/γ)ρ2. Then we getΨ(Au)=(Au)(ξm2)ξm2ξm21φq(ξm2sa(τ)f(τ,u(τ))Δτ)Δs1i=1m2aiξiφq(φp(Mγ)φp(ρ2))ξm2ξm21φq(ξm2sa(τ)Δτ)Δs1i=1m2aiξiMγρ21i=1m2aiξiξm2ξm21φq(ξm2sa(τ)Δτ)Δsρ2.Hence, condition (iii) of Theorem 5.1 holds.

Because all of the hypotheses of the Leggett-Williams fixed point theorem are satisfied, the nonlinear MBVP (1.5) has at least three positive solutions u1,u2, and u3 such that u1(ξ)<ρ1,u2(l)>ρ2, and u3(ξ)>ρ1 with u3(l)<ρ2.

Acknowledgment

This work is supported by the Research and Development Foundation of College of Shanxi Province (no. 200811043).

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