# Existence of Positive Solutions form-Point Boundary Value Problems on Time Scales

Authors: Ying Zhang; ShiDong Qiao
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

We study the one-dimensionalp-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=1m−2aiu(ξi), where T is a time scale, φp(s)=|s|p−2s, 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 Krasnosel′skll′s 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.


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## Body

## 1. Introduction

With the development ofp-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 [1–7].In 2005, He [1] considered the following boundary
value problems:(1.1)(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|p−2s,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. [2] used the virtue of the fixed
point theorem of cone and obtained the existence of at least one solution of the boundary value  problem:(1.2)(g(uΔ(t)))∇+c(t)f(u)=0,a<t<b,u(a)−B0uΔ(γ)=0,uΔ(b)=0.In 2007, Geng and Zhu [3] used the Avery-Peterson and another
fixed theorem of cone and obtained the existence of three positive solutions of
the boundary value problem:(1.3)(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 [4]
discussed the existence of at least one, two or three positive solutions of the
following boundary value problem:(1.4)(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 them-point boundary value problem for the one
dimension p-Laplcaian dynamic equation on time scale T(1.5)(φp(uΔ(t)))Δ+a(t)f(t,u(t))=0,t∈[0,1]T,u(0)=0,u(1)=∑i=1m−2aiu(ξi), where T is a time scale, φp(s)=|s|p−2s,p>1,0<ξ1<ξ2<⋯<ξm−2<1,0≤ai,i=1,2,…,m−3,am−2>0, and(H1)
∑i=1m−2aiξi<1;(H2)
f∈Crd([0,1]T×[0,∞),[0,∞));(H3)
a∈Crd([0,1]T,[0,∞)) and there exists t0∈(ξm−2,1) such that a(t0)>0.In this paper, we have organized the paper as follows.
In Section2, we give some lemmas which are needed later. In Section 3, we
apply the Krassnoselskiifs [8] 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 [9] 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 [10] fixed point theorem.For completeness, we introduce the following concepts
and properties on time scales.A time scaleT 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.
LetT be a time scale, for t∈T,
one defines the forward jump operator σ:T→T by σ(t)=inf{s∈T:s>t},
and the backward jump operator ρ:T→T by ρ(t)=sup{s∈T: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=T−m, otherwise Tk=T. For instance, if supT=∞, then Tk=T.Definition 1.2.
Assumef:T→R is a function and let t∈T. 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(1.6)|[f(σ(t))−f(s)]−fΔ(t)[σ(t)−s]|≤ε|σ(t)−s|,for all s∈U. One says that f is delta differentiable (or in short:
differentiable) on T provided fΔ(t) exist for all t∈T.IfT=R, then fΔ(t)=f′(t), if T=ℤ, then fΔ(t)=Δf(t).A functionf:T→R.(i)
Iff is continuous , then f is rd-continuous.(ii)
The jump operatorσ is rd-continuous.(iii)
Iff is rd-continuous, then so is fσ.A functionF:T→R is called an antidervative of f:T→R,
provided FΔ(t)=f(t) holds for all t∈Tk. One defines the definite integral
by(1.7)∫abf(t)Δt=F(b)−F(a).For all a,b∈T.
If fΔ(t)≥0, then f is nondecreasing.


## 2. The Preliminary Lemmas

Lemma 2.1 (see [5, 6]).
Assume that (H1)–(H3) hold. Then u(t) is a solution of the MBVP (1.5) on [0,1]T if and only if(2.1)u(t)=−∫0tφq(∫0sa(τ)f(τ,u(τ))Δτ)Δs−t⋅∑i=1m−2ai∫0ξiφq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi+t⋅∫01φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi,where φq(s)=|s|q−2s,(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 satisfies(2.2)u(t)≥0,t∈[0,1]T.Lemma 2.3 (see [5]).
If the conditions (H1)–(H3) are satisfied, then(2.3)u(t)≥γ∥u∥,t∈[ξm−2,1],where (2.4)∥u∥=supt∈[0,1]T|u(t)|,γ=min{am−2(1−ξm−2)1−am−2ξm−2,am−2ξm−2,ξ1}.Lemma 2.4 (see [6]).
mint∈[ξm−2,1]Au(t)=min{Au(1),Au(ξm−2)}.LetE denote the Banach space Crd[0,1]T with the norm ∥u∥=supt∈[0,1]T|u(t)|.Define the coneP⊂E, by(2.5)P={u∈E∣u(t)≥0,t∈[ξm−2,1]minu(t)≥γ∥u∥,uisconcave}.The solutions of MBVP (1.5) are the points of
the operator A defined by(2.6)Au(t)=−∫0tφq(∫0sa(τ)f(τ,u(τ))Δτ)Δs−t⋅∑i=1m−2ai∫0ξiφq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi+t⋅∫01φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi=u(t).So, AP⊂P. It is easy to check that A:P→P is completely continuous.


## 3. Existence of at least One Positive Solutions

Theorem 3.1 (see [8]).
LetE be a Banach space, and let P⊂E be a cone. Assume Ω1 and Ω2 are open boundary subsets of E with 0∈Ω1,Ω¯1⊂Ω2, and let  A:P∩(Ω¯2∖Ω1)→P be a completely continuous operator such that
either
(i)
∥Au∥≤∥u∥ for u∈P∩∂Ω1,∥Au∥≥∥u∥ for u∈P∩∂Ω2; or(ii)
∥Au∥≥∥u∥ for u∈P∩∂Ω1,∥Au∥≤∥u∥ for u∈P∩∂Ω2 hold.
Then A has a fixed point in P∩(Ω¯2∖Ω1).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)],0≤u≤r, and f(t,u)≥φp(Mγ)φp(R), if t∈[ξm−2,1],R≤u<∞, where(3.1)M=1−∑i=1m−2aiξiγξm−2∫ξm−21φq(∫ξm−2sa(τ)Δτ)Δs,m=1−∑i=1m−2aiξi∫01φq(∫0sa(τ)Δτ)Δs.Then the MBVP (1.5) has at least one positive
solution.Proof.
Define
the coneP as in (2.5), define a completely continuous
integral operator A:P→P by(3.2)Au(t)=−∫0tφq(∫0sa(τ)f(τ,u(τ))Δτ)Δs−t⋅∑i=1m−2ai∫0ξiφq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi+t⋅∫01φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi.From (H1)–(H3), Lemmas 2.1 and 2.2, AP⊂P.
If u∈P with ∥u∥=r, then we get(3.3)Au(t)=−∫0tφq(∫0sa(τ)f(τ,u(τ))Δτ)Δs−t⋅∑i=1m−2ai∫0ξiφq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi+t⋅∫01φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi≤t⋅∫01φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi≤φq(φp(m)φp(r))⋅∫01φq(∫0sa(τ)Δτ)Δs1−∑i=1m−2aiξi≤rm⋅∫01φq(∫0sa(τ)Δτ)Δs1−∑i=1m−2aiξi=r=∥u∥. This implies that ∥Au∥≤∥u∥. So, if we set  Ω1={u∈Crd([0,1])∣∥u∥<r}, then ∥Au∥≤∥u∥, for u∈P∩∂Ω1.
Let us now setΩ2={u∈Crd([0,1])∣∥u∥<R}.
Then foru∈P with ∥u∥<R, by Lemma 2.4 we have u(t)≥γ∥u∥,t∈[ξm−2,1]. Therefore, we have(3.4)∥Au(t)∥≥Au(ξm−2)=−∫0ξm−2φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs−ξm−2∑i=1m−2ai∫0ξiφq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi+ξm−2⋅∫01φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi=ξm−2∫01φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs−∫0ξm−2φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi+11−∑i=1m−2aiξi∑i=1m−2ai(ξi∫0ξm−2φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs−ξm−2∫0ξiφq(∫0sa(τ)f(τ,u(τ))Δτ)Δs)≥ξm−2∫01φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi−∫0ξm−2φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi≥ξm−2∫ξm−21φq(∫ξm−2sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi≥φq(φp(Mγ)φp(R))ξm−2∫ξm−21φq(∫ξm−2sa(τ)Δτ)Δs1−∑i=1m−2aiξi≥MγR1−∑i=1m−2aiξi⋅ξm−2∫ξm−21φq(∫ξm−2sa(τ)Δτ)Δs=∥u∥.Hence, ∥Au∥≥∥u∥ for u∈P∩∂Ω2. Thus by the Theorem 3.1, A has a fixed point u in P∩(Ω¯2∖Ω1). 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 [9] to prove the existence of at least two positive solutions to
the nonlinear MBVP (1.5) .Theorem 4.1 (see Avery and Henderson [9]).
LetP be a cone in a real Banach space E. Set(4.1)P(Φ,ρ3)={u∈P∣Φ(u)<ρ3}.Ifν 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 ∥u∥≤MΦ(u), for all u∈P(Φ,ρ3)¯.
Suppose that there exist positive numbers ρ1<ρ2<ρ3 such that θ(λu)=λθ(u) for all 0≤λ≤1 and u∈∂P(θ,ρ2).IfA:P(Φ,ρ3)¯→P is a completely continuous operator satisfying(i)
Φ(Au)>ρ3 for all u∈∂P(Φ,ρ3);(ii)
θ(Au)<ρ2 for all u∈∂P(θ,ρ2);(iii)
P(ν,ρ1)≠ϕ and ν(Au)>ρ1 for all u∈∂P(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.Letl∈(0,1)T and 0<ξm−2<l<1. Define the increasing, nonnegative and
continuous functionals Φ,θ, and ν on P, by Φ(u)=u(ξm−2),θ(u)=u(ξm−2), and ν(u)=u(l).From Lemma2.4, for each u∈P,Φ(u)=θ(u)≤ν(u).In addition, for eachu∈P, Lemma 2.3 implies Φ(u)=u(ξm−2)≥γ∥u∥.Thus,(4.2)∥u∥<1γΦ(u),∀u∈P.We also see thatθ(0)=0 and θ(λu)=λθ(u) for all 0≤λ≤1 and u∈∂P(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:
(B1)
f(t,u)>φp(mγ)φp(ρ1), for t∈[ξm−2,l] and u∈[γρ1,ρ1];(B2)
f(t,u)<φp(m)φp(ρ2), for t∈[ξm−2,1] and u∈[0,ρ2];(B3)
f(t,u)>φp(Mγ)φp(ρ3), for t∈[ξm−2,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 Theorem4.1 are satisfied.
Define the coneP as (2.5), define a completely continuous
integral operator A:P→P by (4.3)Au(t)=−∫0tφq(∫0sa(τ)f(τ,u(τ))Δτ)Δs−t⋅∑i=1m−2ai∫0ξiφq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi+t⋅∫01φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi.
M and m as in (3.1). To verify that condition
(i) of Theorem 4.1 holds, we choose u∈∂P(Φ,ρ3),
then Φ(u)=ρ3. This implies ρ3≤∥u∥≤(1/γ)Φ(u). Note that ∥u∥≤(1/γ)Φ(u)=(1/γ)ρ3. We have ρ3≤u(t)≤(1/γ)ρ3, for t∈[ξm−2,1]T. As a consequence of (B3), f(t,u)>φp(Mγ)φp(ρ3), for t∈[ξm−2,l]T. Since Au∈P, we have from Lemma 2.2, (4.4)Φ(Au)=(Au)(ξm−2)=−∫0ξm−2φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs−ξm−2⋅∑i=1m−2ai∫0ξiφq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi+ξm−2⋅∫01φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi=ξm−2∫01φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs−∫0ξm−2φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi+11−∑i=1m−2aiξi∑i=1m−2ai(ξi∫0ξm−2φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs−ξm−2∫0ξiφq(∫0sa(τ)f(τ,u(τ))Δτ)Δs)≥ξm−2∫01φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs−∫0ξm−2φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi≥ξm−2∫ξm−21φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi≥φq(φp(Mγ)φp(ρ3))ξm−2∫ξm−21φq(∫0sa(τ)Δτ)Δs1−∑i=1m−2aiξi≥Mγρ3ξm−21−∑i=1m−2aiξi∫ξm−21φq(∫0sa(τ)Δτ)Δs≥ρ3.Then condition (i) of Theorem
4.1 holds.
Letu∈∂P(θ,ρ2).
Then θ(u)=ρ2. This implies 0≤u(t)≤∥u∥≤(1/γ)ρ2, for t∈[ξm−2,1]. From (B2), we have(4.5)θ(Au)=(Au)(ξm−2)≤ξm−2⋅∫01φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi≤ξm−2⋅∫01φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi≤mρ2⋅∫01φq(∫0sa(τ)Δτ)Δs1−∑i=1m−2aiξi=ρ2=∥u∥.Hence condition (ii) of Theorem
4.1 holds.
If we first defineu(t)=ρ1/2, for t∈[0,1]T, then ν(u)=ρ1/2<ρ1. So P(ν,ρ1)≠ϕ.
Now, letu∈∂P(ν,ρ1), then ν(u)=u(l)=ρ1. This mean that ρ1/γ≤u(t)≤∥u∥≤ρ1. From (B1) and Lemma 2.4, we get(4.6)ν(Au)=(Au)(l)≥(Au)(ξm−2)=−∫0ξm−2φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs−ξm−2⋅∑i=1m−2ai∫0ξiφq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi+ξm−2⋅∫01φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi=ξm−2∫01φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs−∫0ξm−2φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi+11−∑i=1m−2aiξi∑i=1m−2ai(ξi∫0ξm−2φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs−ξm−2∫0ξiφq(∫0sa(τ)f(τ,u(τ))Δτ)Δs)≥ξm−2∫01φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs−∫0ξm−2φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi≥ξm−2∫ξm−21φq(∫ξm−2sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi≥φq(φp(mγ)φp(ρ1))ξm−2∫ξm−21φq(∫ξm−2sa(τ)Δτ)Δs1−∑i=1m−2aiξi=mγρ11−∑i=1m−2aiξi⋅ξm−2∫ξm−21φq(∫ξm−2sa(τ)Δτ)Δs≥ρ1.Then condition (iii) of Theorem
4.1 holds.
Since all conditions of Theorem4.1 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
[10] to prove the existence of at least three positive solutions to the nonlinear MBVP (1.5).Theorem 5.1 (see Leggett and Williams [10]).
LetP be a cone in the real Banach space E. Set(5.1)Pr={x∈P∣∥x∥<r},P(Ψ,a,b)={x∈P∣a≤Ψ(x),∥x∥≤b}.SupposeA:P¯r→P¯r be a completely continuous operator and be a
nonnegative continuous concave functional on P with Ψ(u)≤∥u∥ for all u∈P¯r. If there exists 0<ρ1<ρ2<(1/γ)ρ2≤ρ3 such that the following condition hold:(i)
{u∈P(Ψ,ρ2,(1/γ)ρ2)∣Ψ(u)>ρ2}≠ϕ and Ψ(Au)>ρ2 for all u∈P(Ψ,ρ2,(1/γ)ρ2);(ii)
∥Au∥<ρ1 for ∥u∥≤ρ1;(iii)
Ψ(Au)>ρ2 for u∈P(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)ρ2≤ρ3 such that
(C1)
f(t,u)≤φp(m)φp(ρ3), for t∈[ξm−2,l] and u∈[0,ρ3];(C2)
f(t,u)>φp(Mγ)φp(ρ2), for t∈[ξm−2,l] and u∈[ρ2,(1/γ)ρ2];(C3)
f(t,u)<φp(m)φp(ρ1), for t∈[ξm−2,1] 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 Theorem5.1 will be shown to be satisfied. Define the nonnegative
continuous concave functional Ψ:P→[0,∞) to be Ψ(u)=u(ξm−2), the cone P as in (2.5), M and m as in (3.1). We have Ψ(u)≤∥u∥ for all u∈P. If u∈P¯ρ3,
then ∥u∥≤ρ3, and from assumption (C1) , then we have(5.2)(Au)(t)=−∫0tφq(∫0sa(τ)f(τ,u(τ))Δτ)Δs−t⋅∑i=1m−2ai∫0ξiφq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi+t⋅∫01φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi≤t⋅∫01φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi≤φq(φp(m)φp(ρ3))⋅∫01φq(∫0sa(τ)Δτ)Δs1−∑i=1m−2aiξi≤mρ3⋅∫01φq(∫0sa(τ)Δτ)Δs1−∑i=1m−2aiξi=ρ3.
This implies that∥Au∥≤ρ3. Thus, we have A:P¯ρ3→P¯ρ3.
Since (1/γ)ρ2∈P(Ψ,ρ2,(1/γ)ρ2) and Ψ((1/γ)ρ2)=(1/γ)ρ2>ρ2,{u∈P(Ψ,ρ2,(1/γ)ρ2)|Ψ(u)>ρ2}≠ϕ. For u∈P(Ψ,ρ2,(1/γ)ρ2) we have ρ2≤u(ξm−2)≤∥u∥≤(1/γ)ρ2. Using assumption (C2), f(t,u)>φp(Mγ)φp(ρ2), we obtain(5.3)Ψ(Au)=(Au)(ξm−2)=−∫0ξm−2φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs−ξm−2⋅∑i=1m−2ai∫0ξiφq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi+ξm−2⋅∫01φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi=ξm−2∫01φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs−∫0ξm−2φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi+11−∑i=1m−2aiξi∑i=1m−2ai(ξi∫0ξm−2φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs−ξm−2∫0ξiφq(∫0sa(τ)f(τ,u(τ))Δτ)Δs)≥ξm−2∫01φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs−∫0ξm−2φq(∫0sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi≥ξm−2∫ξm−21φq(∫ξm−2sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi≥φq(φp(Mγ)φp(ρ2))ξm−2∫ξm−21φq(∫ξm−2sa(τ)Δτ)Δs1−∑i=1m−2aiξi≥Mγρ21−∑i=1m−2aiξi⋅ξm−2∫ξm−21φq(∫ξm−2sa(τ)Δτ)Δs≥ρ2.Hence, condition (i) of Theorem
5.1 holds.
If∥u∥≤ρ1, from assumption (C3), we obtain(5.4)(Au)(t)≤φq(φp(m)φp(ρ1))⋅∫01φq(∫0sa(τ)Δτ)Δs1−∑i=1m−2aiξi≤mρ1⋅∫01φq(∫0sa(τ)Δτ)Δs1−∑i=1m−2aiξi=ρ1.This implies that ∥Au∥≤ρ1.
Consequently, condition (ii) of Theorem5.1 holds.
We suppose thatu∈P(Ψ,ρ2,ρ3), with ∥Au∥>(1/γ)ρ2. Then we get(5.5)Ψ(Au)=(Au)(ξm−2)≥ξm−2∫ξm−21φq(∫ξm−2sa(τ)f(τ,u(τ))Δτ)Δs1−∑i=1m−2aiξi≥φq(φp(Mγ)φp(ρ2))ξm−2∫ξm−21φq(∫ξm−2sa(τ)Δτ)Δs1−∑i=1m−2aiξi≥Mγρ21−∑i=1m−2aiξi⋅ξm−2∫ξm−21φq(∫ξm−2sa(τ)Δτ)Δs≥ρ2.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.



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*Source: 189768-2009-03-08.xml*