# Several Existence Theorems of Nonlinearm-Point BVP for an Increasing Homeomorphism and Homomorphism on Time Scales

Authors: Yanbin Sang; Hua Su; Yafeng Xiao
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

Several existence theorems of positive solutions are established for nonlinearm-point boundary value problem for the following dynamic equations on time scales (ϕ(uΔ))∇+a(t)f(t,u(t))=0, t∈(0,T), ϕ(uΔ(0))=∑i=1m−2aiϕ(uΔ(ξi)), u(T)=∑i=1m−2biu(ξi), where ϕ:R→R is an increasing homeomorphism and homomorphism and ϕ(0)=0. As an application, an example to demonstrate our results is given.


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## Body

## 1. Introduction

In this paper, we study the existence of positive
solutions of the following dynamic equations on time scales:(1.1)(ϕ(uΔ))∇+a(t)f(t,u(t))=0,t∈(0,T),ϕ(uΔ(0))=∑i=1m−2aiϕ(uΔ(ξi)),u(T)=∑i=1m−2biu(ξi), where ϕ:R→R is an increasing homeomorphism and
homomorphism and ϕ(0)=0.A projectionϕ:R→R is called an increasing homeomorphism and
homomorphism if the following conditions are satisfied:(i)
ifx≤y,
then ϕ(x)≤ϕ(y),
for all x,y∈R;(ii)
ϕ is a continuous bijection and its inverse
mapping is also continuous;(iii)
ϕ(xy)=ϕ(x)ϕ(y),
for all x,y∈R.We will assume that the following conditions are
satisfied throughout this paper:(H1)
0<ξ1<⋯<ξm−2<ρ(T),ai,bi∈[0,+∞) satisfy 0<∑i=1m−2ai<1,
and ∑i=1m−2bi<1,T∑i=1m−2bi≥∑i=1m−2biξi;(H2)
a(t)∈Cld((0,T),[0,+∞)) and there exists t0∈(ξm−2,T),
such that a(t0)>0;(H3)
f∈C([0,T]×[0,+∞),[0,+∞)).Recently, there is much attention focused on the
existence of positive solutions for second-order, three-point boundary value
problem on time scales. On the other hand, three-point andm-point boundary value problems with p-Laplacian operators on time scales have also
been studied extensively, for details, see [1–11] and references
therein. But with an increasing homeomorphism and homomorphism, few works were done
as far as we know.A time scaleT is a nonempty closed subset of R. We make the
blanket assumption that 0,T are points in T.
By an interval (0,T), we always mean the intersection of the real interval (0,T) with the given time scale, that is, (0,T)∩T.We would like to mention some results of Anderson et al. [2], He [4, 5], Sun and Li [9], Ma et al. [12],
Wang and Hou [13], Wang and Ge [14], which motivate us to consider our problem.In [2], Anderson et al. considered the
following problem: (1.2)(ϕp(uΔ))∇+a(t)f(u(t))=0,t∈(a,b),u(a)−B0(uΔ(υ))=0,uΔ(b)=0,     where ϕp(u)=|u|p−2u,p>1,υ∈(a,b),f∈Cld([0,+∞),[0,+∞)),a(t)∈Cld((a,b),[0,+∞)),
and Kmx≤B0x≤KMx for some positive constants Km,KM.
They established the existence results of at least one positive solution by
using a fixed point theorem of cone expansion and compression of functional
type.In [4, 5], He considered the existence of positive
solutions of the p-Laplacian dynamic equations on time scales: (1.3)(ϕp(uΔ))∇+a(t)f(u(t))=0,t∈(0,T)  satisfying the
boundary conditions (1.4)u(0)−B0(uΔ(η))=0,uΔ(T)=0  or (1.5)uΔ(0)=0,u(T)−B1(uΔ(η))=0,  where ϕp(u)=|u|p−2u,p>1,η∈(0,ρ(T)),a(t)∈Cld((0,T),[0,+∞)),f∈C([0,+∞),[0,+∞)),
and Ax≤Bix≤Bx(i=0,1) for some positive constants A,B.
He obtained the existence of at least double and triple positive solutions of
the problem (1.3), (1.4), and (1.5) by using a new double fixed point theorem
and triple fixed point theorem, respectively.In recent papers, Ma et al. [12] have obtained
the existence of monotone positive solutions for the following BVP: (1.6)(ϕp(u′))′+a(t)f(t,u(t))=0,t∈(0,1),u′(0)=∑i=1m−2aiu′(ξi),u(1)=∑i=1m−2biu(ξi).     The main tool
is the monotone iterative technique.In [9], Sun and Li studied the following p-Laplacian, m-point BVP on time scales: (1.7)(φp(uΔ))∇+a(t)f(t,u(t))=0,t∈(0,T),u(0)=0,φp(uΔ(T))=∑i=1m−2aiφp(uΔ(ξi)),     where φp(u)=|u|p−2u,p>1,ai≥0 for i=1,…,m−2,0<ξ1<⋯<ξm−2<ρ(T),∑i=1m−2ai<1,a(t)∈Cld((0,T),[0,+∞)),f∈Cld((0,T)×[0,+∞),[0,+∞)).
Some new results are obtained for the existence of at least twin or triple
positive solutions of the problem (1.7) by applying Avery-Henderson
and Leggett-Williams fixed point theorems, respectively.In [15], Sang and xi investigated the existence of
positive solutions of the p-Laplacian dynamic equations on time scales: (1.8)(ϕp(uΔ))∇+a(t)f(t,u(t))=0,t∈(0,T),ϕp(uΔ(0))=∑i=1m−2aiϕp(uΔ(ξi)),u(T)=∑i=1m−2biu(ξi),     where ϕp(s) is p-Laplacian operator, that is, ϕp(s)=|s|p−2s,p>1,ϕp−1=ϕq,1/p+1/q=1,0<ξ1<⋯<ξm−2<ρ(T).
Let (1.9)fγρρ=min{minξm−2≤t≤Tf(t,u)ϕp(ρ):u∈[γρ,ρ]},f0ρ=max{max0≤t≤Tf(t,u)ϕp(ρ):u∈[0,ρ]},γ=∑i=1m−2bi(T−ξi)T−∑i=1m−2biξi,fα=limu→αsupmax0≤t≤Tf(t,u)ϕp(u),fα=limu→αinfmaxξm−2≤t≤Tf(t,u)ϕp(u),(α:=∞or0+),m={11−∑i=1m−2bi∫0Tϕq[∫0sa(τ)∇τ+∑i=1m−2ai∫0ξia(τ)∇τ1−∑i=1m−2ai]Δs}−1,M={T∑i=1m−2bi−∑i=1m−2biξiT(1−∑i=1m−2bi)∫0Tϕq[∫0sa(τ)∇τ+∑i=1m−2ai∫0ξia(τ)∇τ1−∑i=1m−2ai]Δs}−1,  they mainly
obtained the following results.Theorem 1.1.
Assume (H1), (H2),
and (H3) hold, and assume that one of the following
conditions holds:
(H4
)
there existρ1,ρ2∈(0,+∞)withρ1<γρ2 such that(1.10)f0ρ1≤ϕp(m),fγρ2ρ2≥ϕp(Mγ);(H5
)
there existρ1,ρ2∈(0,+∞) with ρ1<ρ2 such that(1.11)f0ρ2≤ϕp(m),fγρ1ρ1≥ϕp(Mγ).
Then, (1.8) have a positive solution.In this paper, we will establish two new theorems of
positive solution of (1.8), our work concentrates on the case when
the nonlinear term does not satisfy the conditions of Theorem 1.1. At the end
of the paper, we will give an example which illustrates that our work is true.


## 2. Preliminaries and Some Lemmas

For convenience, we list the following definitions
which can be found in [16–19].Deffinition.
A time scaleT is a nonempty closed subset of real numbers R.
For t<supT and r>infT,
define the forward jump operator σ and backward jump operator ρ,
respectively, by (2.1)σ(t)=inf{τ∈T∣τ>t}∈T,ρ(r)=sup{τ∈T∣τ<r}∈T,     for all t,r∈T.
If σ(t)>t,t is said to be right scattered; and if ρ(r)<r,r is said to be left scattered; if σ(t)=t,t is said to be right dense; and if ρ(r)=r,r is said to be left dense. If T has a right scattered minimum m,
define Tk=T−{m};
otherwise, set Tk=T.
If T has a left scattered maximum M,
define Tk=T−{M};
otherwise, set Tk=T.Deffinition.
Forf:T→R and t∈Tk,
the delta derivative of f at the point t is defined to be the number fΔ(t),
(provided it exists), with the property that for each ϵ>0;
there is a neighborhood U of t such that (2.2)|f(σ(t))−f(s)−fΔ(t)(σ(t)−s)|≤ϵ|σ(t)−s|  for all s∈U.
Forf:T→R and t∈Tk,
the nabla derivative of f at t is the number f∇(t),
(provided it exists), with the property that for each ϵ>0;
there is a neighborhood U of t such that (2.3)|f(ρ(t))−f(s)−f∇(t)(ρ(t)−s)|≤ϵ|ρ(t)−s|  for all s∈U.Deffinition.
A functionf is left-dense continuous (i.e., ld continuous) if f is continuous at each left-dense point in T,
and its right-sided limit exists at each right-dense point in T.Deffinition.
IfGΔ(t)=f(t),
then we define the delta integral by (2.4)∫abf(t)Δt=G(b)−G(a).  If F∇(t)=f(t),
then we define the nabla integral by (2.5)∫abf(t)∇t=F(b)−F(a).  To prove the
main results in this paper, we will employ several lemmas. These lemmas are
based on the linear BVP: (2.6)(ϕ(uΔ))∇+h(t)=0,t∈(0,T),ϕ(uΔ(0))=∑i=1m−2aiϕ(uΔ(ξi)),u(T)=∑i=1m−2biu(ξi).
We can prove the following lemmas by the methods of
[15].Lemma 2.5.
Forh∈Cld[0,T],
the BVP (2.6) has the unique solution:(2.7)u(t)=−∫0tϕ−1(∫0sh(τ)∇τ−A)Δs+B,  where
(2.8)A=−∑i=1m−2ai∫0ξih(τ)∇τ1−∑i=1m−2ai,B=∫0Tϕ−1(∫0sh(τ)∇τ−A)Δs−∑i=1m−2bi∫0ξiϕ−1(∫0sh(τ)∇τ−A)Δs1−∑i=1m−2bi.Lemma 2.6.
Assume  (H1)   holds,
Forh∈Cld[0,T] and h≥0,
then the unique solution u of (2.6) satisfies(2.9)u(t)≥0,fort∈[0,T].Lemma 2.7.
Assume  (H1)   holds, ifh∈Cld[0,T] and h≥0,
then the unique solution u of (2.6) satisfies(2.10)inft∈[0,T]u(t)≥γ∥u∥,  where (2.11)γ=∑i=1m−2bi(T−ξi)T−∑i=1m−2biξi,∥u∥=maxt∈[0,T]|u(t)|.Let the norm onCld[0,T] be the maximum norm. Then, the Cld[0,T] is a Banach space. It is easy to see that the
BVP (1.1)  has a solution u=u(t) if and only if u is a fixed point of the operator equation: (2.12)(Au)(t)=−∫0tϕ−1(∫0sa(τ)f(τ,u(τ))∇τ−A˜)Δs+B˜,  where (2.13)A˜=−∑i=1m−2ai∫0ξia(τ)f(τ,u(τ))∇τ1−∑i=1m−2ai,B˜=∫0Tϕ−1(∫0sa(τ)f(τ,u(τ))∇τ−A˜)Δs−∑i=1m−2bi∫0ξiϕ−1(∫0sa(τ)f(τ,u(τ))∇τ−A˜)Δs1−∑i=1m−2bi.  Throughout this paper, we will assume that 0≤μ≤ν≤T. Denote (2.14)K={u∣u∈Cld[0,T],u(t)≥0,inft∈[μ,ν]u(t)≥γ∥u∥},  where γ is the same as in Lemma 2.7. It is obvious
that K is a cone in Cld[0,T].
By Lemma 2.7, A(K)⊂K.
So by applying Arzela-Ascoli theorem on time scales [20], we can obtain that A(K) is relatively compact. In view of Lebesgue's
dominated convergence theorem on time scales [21], it is easy to prove that A is continuous. Hence, A:K→K is completely continuous.Lemma 2.8.
Let(2.15)φ(s)=ϕ−1(∫0sa(τ)f(τ,u(τ))∇τ−A˜),  for ξi(i=1,…,m−2),
then (2.16)∫0ξiφ(s)Δs≤ξiT∫0Tφ(s)Δs.Lemma 2.9 ([22]).
LetE be a Banach space, and let K⊂E be a cone. Assume Ω1,Ω2 are open bounded subset of E with 0∈Ω1,Ω1¯⊂Ω2, and let(2.17)F:K∩(Ω2¯∖Ω1)→K  be a completely continuous operator such that
(i)
∥Fu∥≤∥u∥,u∈K∩∂Ω1,and∥Fu∥≥∥u∥,u∈K∩∂Ω2;
or(ii)
∥Fu∥≥∥u∥,u∈K∩∂Ω1, and∥Fu∥≤∥u∥,u∈K∩∂Ω2.
Then, F has a fixed point in K∩(Ω2¯∖Ω1).Now, we introduce the following notations. Let(2.18)A0={11−∑i=1m−2bi∫0Tϕ−1[∫0sa(τ)∇τ+∑i=1m−2ai∫0ξia(τ)∇τ1−∑i=1m−2ai]Δs}−1,B0={T∑i=1m−2bi−∑i=1m−2biξiT(1−∑i=1m−2bi)∫μνϕ−1[∫0sa(τ)∇τ+∑i=1m−2ai∫0ξia(τ)∇τ1−∑i=1m−2ai]Δs}−1.  For l>0,Ωl={u∈K:∥u∥<l},∂Ωl={u∈K:∥u∥=l}, (2.19)α(l)=sup{∥Au∥:u∈∂Ωl},β(l)=inf{∥Au∥:u∈∂Ωl}.  By Lemma 2.6, α and β are well defined.


## 3. Existence Theorems of Positive Solution

Theorem 3.1.
Assume  (H1), (H2), (H3)
hold, and assume that the following conditions
hold:
(A1)
pi∈C([0,+∞),[0,+∞)),i=1,2,and(3.1)liml→0¯p1(l)ϕ(l)<ϕ(A0),liml→∞¯p2(l)ϕ(l)>ϕ(B0γ);(A2)
k1∈L1([0,T],[0,+∞)),k2∈L1([0,T],[0,+∞));(A3)
there exist0<c1≤c2,0≤λ2<1<λ1,
such that(3.2)f(t,l)≤p1(l)+k1(t)[ϕ(l)]λ1,(t,l)∈[0,T]×[0,c1],f(t,l)≥p2(l)−k2(t)[ϕ(l)]λ2,(t,l)∈[μ,ν]×[c2,+∞).Then, the problems (1.1) have at least one positive solution.Theorem 3.2.
Assume  (H1), (H2), (H3)
hold, and assume that the following conditions
hold:
(B1)
pi∈C([0,+∞),[0,+∞)),i=3,4,and(3.3)liml→∞¯p3(l)ϕ(l)<ϕ(A0),liml→0+¯p4(l)ϕ(l)>ϕ(B0γ);(B2)
k3∈L1([0,T],[0,+∞)),k4∈L1([0,T],[0,+∞));(B3)
there exist0<c3≤c4,0≤λ4<1<λ3,
such that(3.4)f(t,l)≤p3(l)+k3(t)[ϕ(l)]λ4,(t,l)∈[0,T]×[c4,+∞),f(t,l)≥p4(l)−k4(t)[ϕ(l)]λ3,(t,l)∈[μ,ν]×[0,c3].Then, the problems (1.1) have at least one positive solution.Proof of Theorem3.1.
Bylim¯l→0(p1(l)/ϕ(l))<ϕ(A0),
then we can get that there exist 0<a¯1≤c1,0<ϵ<ϕ(A0) such that (3.5)p1(l)≤(ϕ(A0)−ϵ)ϕ(l),0≤l≤a¯1.  If 0≤l≤a¯1,u∈∂Ωl,
then 0≤u(t)≤l,0≤t≤T.
By condition (A3),
we have
(3.6)f(t,u(t))≤p1(u(t))+k1(t)[ϕ(u(t))]λ1≤(ϕ(A0)−ϵ)ϕ(u(t))+k1(t)[ϕ(u(t))]λ1≤(ϕ(A0)−ϵ)ϕ(∥u∥)+k1(t)[ϕ(∥u∥)]λ1=(ϕ(A0)−ϵ)ϕ(l)+k1(t)[ϕ(l)]λ1.  Let M(l,τ)=(ϕ(A0)−ϵ)ϕ(l)+k1(τ)[ϕ(l)]λ1 so that
(3.7)∫0sa(τ)f(τ,u(τ))∇τ−A˜=∫0sa(τ)f(τ,u(τ))∇τ+∑i=1m−2ai∫0ξia(τ)f(τ,u(τ))∇τ1−∑i=1m−2ai≤∫0sa(τ)M(l,τ)∇τ+∑i=1m−2ai∫0ξia(τ)M(l,τ)∇τ1−∑i=1m−2ai.  Therefore, (3.8)∥Au∥≤B˜=11−∑i=1m−2bi(∫0Tφ(s)Δs−∑i=1m−2bi∫0ξiφ(s)Δs)≤11−∑i=1m−2bi∫0Tφ(s)Δs≤11−∑i=1m−2bi∫0Tϕ−1[∫0sa(τ)M(l,τ)∇τ+∑i=1m−2ai∫0ξia(τ)M(l,τ)∇τ1−∑i=1m−2ai]Δs.  It follows that (3.9)α(l)l≤11−∑i=1m−2bi∫0Tϕ−1[∫0sa(τ)M(l,τ)ϕ(l)∇τ+∑i=1m−2ai∫0ξia(τ)(M(l,τ)/ϕ(l))∇τ1−∑i=1m−2ai]Δs.  Noticing λ1>1,
we have (3.10)liml→0+¯α(l)l≤11−∑i=1m−2bi∫0Tϕ−1[∫0sa(τ)(ϕ(A0)−ϵ)∇τ+∑i=1m−2ai∫0ξia(τ)(ϕ(A0)−ϵ)∇τ1−∑i=1m−2ai]Δs=ϕ−1[ϕ(A0)−ϵ]1−∑i=1m−2bi∫0Tϕ−1[∫0sa(τ)∇τ+∑i=1m−2ai∫0ξia(τ)∇τ1−∑i=1m−2ai]Δs≤ϕ−1[ϕ(A0)]A0−1=A0⋅A0−1=1.  Therefore,
there exist 0<a1<a¯1 such that α(a1)<a1.
It implies that ∥Au∥<∥u∥,u∈∂Ωa1.
On the other hand, bylim̲l→∞(p2(l)/ϕ(l))>ϕ(B0/γ),
we can get that there exist ϵ′>0,
and c2≤d<+∞ such that (3.11)p2(l)≥[ϕ(B0γ)+ϵ′]ϕ(l),d≤l≤+∞,ford≤γl<+∞,u∈∂Ωl,  then d≤γl≤u(t)≤l,μ≤T≤ν.
Letm(l,τ)=[ϕ(B0/γ)+ϵ′]ϕ(γl)−k2(τ)[ϕ(l)]λ2.
By Lemma 2.8 and condition (A3),
applying Lemma 2.8, it follows that (3.12)∥Au∥=max0≤t≤T(Au)(t)≥−∫0Tφ(s)Δs+11−∑i=1m−2bi(∫0Tφ(s)Δs−∑i=1m−2bi∫0ξiφ(s)Δs)=∑i=1m−2bi1−∑i=1m−2bi∫0Tφ(s)Δs−∑i=1m−2bi∫0ξiφ(s)Δs1−∑i=1m−2bi≥∑i=1m−2bi1−∑i=1m−2bi∫0Tφ(s)Δs−∑i=1m−2biξiT(1−∑i=1m−2bi)∫0Tφ(s)Δs=T∑i=1m−2bi−∑i=1m−2biξiT(1−∑i=1m−2bi)∫0Tφ(s)Δs≥T∑i=1m−2bi−∑i=1m−2biξiT(1−∑i=1m−2bi)∫μνϕ−1[∫0sa(τ)m(l,τ)∇τ+∑i=1m−2ai∫0ξia(τ)m(l,τ)∇τ1−∑i=1m−2ai]Δs.  It follows that (3.13)β(l)l≥T∑i=1m−2bi−∑i=1m−2biξiT(1−∑i=1m−2bi)∫μνϕ−1[∫0sa(τ)m(l,τ)ϕ(l)∇τ+∑i=1m−2ai∫0ξia(τ)(m(l,τ)/ϕ(l))∇τ1−∑i=1m−2ai]Δs.  Noticing 0≤λ2<1,
we get
(3.14)liml→∞¯β(l)l≥T∑i=1m−2bi−∑i=1m−2biξiT(1−∑i=1m−2bi)×∫μνϕ−1[∫0sa(τ)ϕ(B0/γ)ϕ(γl)ϕ(l)∇τ+∑i=1m−2ai∫0ξia(τ)((ϕ(B0/γ)ϕ(γl))/ϕ(l))∇τ1−∑i=1m−2ai]Δs=ϕ−1ϕ(B0)B0−1=B0B0−1=1.
Therefore,
there existsb with b>a1>0 such that β(b)>b.
It implies that ∥Au∥>∥u∥ for u∈∂Ωb.
By Lemma2.9, we assert that the operator A has one fixed point u*∈K such that a1≤∥u*∥≤b.
Therefore, u* is positive solution of the problems (1.1).Proof of Theorem3.2.
Bylim¯l→∞(p3(l)/ϕ(l))<ϕ(A0),
then we can get that there exist d≥λ4,0<ϵ<ϕ(A0) such that (3.15)p3(l)≤(ϕ(A0)−ϵ)ϕ(l),l≥d.  For l≥d,u∈∂Ωl,
then 0≤u(t)≤l,0≤t≤T.
By condition (B3),
we have
(3.16)f(t,u(t))≤p3(u(t))+k3(t)[ϕ(u(t))]λ4≤(ϕ(A0)−ϵ)ϕ(u(t))+k3(t)[ϕ(u(t))]λ4≤(ϕ(A0)−ϵ)ϕ(∥u∥)+k3(t)[ϕ(∥u∥)]λ4=(ϕ(A0)−ϵ)ϕ(l)+k3(t)[ϕ(l)]λ4.  Let M1(l,τ)=(ϕ(A0)−ϵ)ϕ(l)+k3(τ)[ϕ(l)]λ4 so that (3.17)∫0sa(τ)f(τ,u(τ))∇τ−A˜≤∫0sa(τ)M1(l,τ)∇τ+∑i=1m−2ai∫0ξia(τ)M1(l,τ)∇τ1−∑i=1m−2ai.  Denote (3.18)m1=max{f(τ,u(τ)):τ∈[0,s]⋃[0,ξi],u(τ)≤d}  satisfying (3.19)m1+ϕ−1[ϕ(A0)−ϵ]≤A0.  Therefore, (3.20)∥Au∥≤m1A0−1+11−∑i=1m−2bi∫0Tϕ−1[∫0sa(τ)M1(l,τ)∇τ+∑i=1m−2ai∫0ξia(τ)M1(l,τ)∇τ1−∑i=1m−2ai]Δs.  It follows that (3.21)α(l)l≤m1A0−1+11−∑i=1m−2bi∫0Tϕ−1[∫0sa(τ)M1(l,τ)ϕ(l)∇τ+∑i=1m−2ai∫0ξia(τ)(M1(l,τ)/ϕ(l))∇τ1−∑i=1m−2ai]Δs.  Noticing 0≤λ4<1,
we have (3.22)liml→∞¯α(l)l≤m1A0−1+11−∑i=1m−2bi∫0Tϕ−1[∫0sa(τ)(ϕ(A0)−ϵ)∇τ+∑i=1m−2ai∫0ξia(τ)(ϕ(A0)−ϵ)∇τ1−∑i=1m−2ai]Δs=m1A0−1+ϕ−1[ϕ(A0)−ϵ]1−∑i=1m−2bi∫0Tϕ−1[∫0sa(τ)∇τ+∑i=1m−2ai∫0ξia(τ)∇τ1−∑i=1m−2ai]Δs=m1A0−1+ϕ−1[ϕ(A0)−ϵ]A0−1≤A0⋅A0−1=1.  Therefore,
there exists a1≥d such that α(a1)<a1.
It implies that ∥Au∥<∥u∥,u∈∂Ωa1.
On the other hand, bylim̲l→0+(p4(l)/ϕ(l))>ϕ(B0/γ),
we can get that there exist ϵ′>0,
and 0<e<c3 such that (3.23)p4(l)≥[ϕ(B0γ)+ϵ′]ϕ(l),for0≤l≤e,u∈∂Ωl,  then γl≤u(t)≤l,μ≤T≤ν.
Letm1(l,τ)=[ϕ(B0/γ)+ϵ′]ϕ(γl)−k4(τ)[ϕ(l)]λ3.
By Lemma 2.8 and condition (B3),
applying Lemma 2.8, it follows that (3.24)∥Au∥≥T∑i=1m−2bi−∑i=1m−2biξiT(1−∑i=1m−2bi)∫μνϕ−1[∫0sa(τ)m1(l,τ)∇τ+∑i=1m−2ai∫0ξia(τ)m1(l,τ)∇τ1−∑i=1m−2ai]Δs.  It follows that (3.25)β(l)l≥T∑i=1m−2bi−∑i=1m−2biξiT(1−∑i=1m−2bi)∫μνϕ−1[∫0sa(τ)m1(l,τ)ϕ(l)∇τ+∑i=1m−2ai∫0ξia(τ)(m1(l,τ)/ϕ(l))∇τ1−∑i=1m−2ai]Δs.  Noticing λ3>1,
we can get (3.26)liml→0+¯β(l)l≥1.  Therefore,
there exists
b with 0<b<a1 such that β(b)>b.
It implies that ∥Au∥>∥u∥ for u∈∂Ωb.
By Lemma2.9, we assert that the operator A has one fixed point u*∈K such that b≤∥u*∥≤a1.
Therefore, u* is positive solution of the problems (1.1).


## 4. Example

In this section, we present a simple example to
explain our results.Letf(t,0)≡0,T=R,T=1.
Consider the following BVP: (4.1)(ϕ(uΔ))∇+f(t,u(t))=0,t∈(0,T),ϕ(uΔ(0))=14ϕ(uΔ(13)),u(T)=12u(13),     where (4.2)ϕ(u)={−u2,u≤0,u2,u>0,f(t,u)={45u3+522min{1t(1−t),4u}u5,(t,u)∈[0,1]×[0,2],400,(t,u)∈[0,1]×[2,5],175u3−52min{1t(1−t),2u5}u3,(t,u)∈[25,12]×[5,+∞).  It is easy to
check that f:[0,1]×[0,+∞)→[0,+∞) is continuous. In this case, a(t)≡1,m=3,a1=1/4,b1=1/2,ξ1=1/3,
it follows from a direct calculation that (4.3)A0=[11−b1∫0Tϕ−1(s+a1ξ11−a1)Δs]−1=[11−(1/2)∫01(s+1/4⋅1/31−(1/4))1/2Δs]−1=3⋅274(1010−1),γ=b1(T−ξ1)T−b1ξ1=1/2(1−(1/3))1−(1/2⋅1/3)=25,B0=[Tb1−b1ξ1T(1−b1)∫2/51/2ϕ−1(s+a1ξ11−a1)Δs]−1=[1/2−(1/2⋅1/3)1−(1/2)∫2/51/2(s+1/4⋅1/31−(1/4))1/2Δs]−1=[4933×2522−69×41158100]−1.  Choose c1=2,c2=5,b=5,λ1=5/4,λ2=3/4,p1(u)=45u3,p2(u)=(17/5)u3,k1(t)=(52/2)(1/t(1−t)),k2(t)=(5/2)(1/t(1−t)),
it is easy to check that (4.4)f(t,u)≤p1(u)+k1(t)(u2)5/4,(t,u)∈[0,1]×[0,2],f(t,u)≥p2(u)−k2(t)(u2)3/4,(t,u)∈[25,12]×[5,+∞),limu→0¯p1(u)ϕ(u)=limu→0¯45u3u2=0<A02=[3⋅274(1010−1)]2,limu→∞¯p2(u)ϕ(u)=limu→∞¯(17/5)u3u2=+∞>ϕ(B0γ).  It follows that f satisfies the conditions (A1)–(A3) of Theorem 3.1, then problems (1.1)
have at least one positive solution.



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*Source: 252876-2008-05-26.xml*