AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation51240210.1155/2009/512402512402Research ArticleExistence of Positive Solutions to Singular p-Laplacian General Dirichlet Boundary Value Problems with Sign Changing NonlinearityWeiQiying1SuYou-Hui2LiSubei2YanXing-Xue3EloePaul1College of SciencesChina University of Mining and TechnologyXuzhouJiangsu 221008Chinacumtb.edu.cn2School of Mathematics and PhysicsXuZhou Institute of TechnologyXuzhouJiangsu 221008Chinaxzit.edu.cn3Department of MathematicsHexi UniversityZhangyeGansu 734000Chinahxu.edu.cn20090803200920092712200821022009250220092009Copyright © 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.

By using the well-known Schauder fixed point theorem and upper and lower solution method, we present some existence criteria for positive solution of an m-point singular p-Laplacian dynamic equation on time scales with the sign changing nonlinearity. These results are new even for the corresponding differential (𝕋=) and difference equations (𝕋=), as well as in general time scales setting. As an application, an example is given to illustrate the results.

1. Introduction

Initiated by Hilger in his Ph.D. thesis  in 1988, the theory of time scales has been improved greatly ever since, especially in the unification of the theory of differential equations in the continuous case and the theory of finite difference equations in the discrete case. For the time being, it remains active and attracts many distinguished researchers' attention. The reason is two sided. On the one hand, the calculus on time scales not only can unify differential and difference equations, but also can provide accurate information of phenomena that manifest themselves partly in continuous time and partly in discrete time. On the other hand, it is also widely applied to the research of biology, heat transfer, stock market, wound healing and epidemic models , and so forth. For instance, Hoffacker et al. have used the theory to model how students suffering from the eating disorder bulimia are influenced by their college friends. With the theory on time scales, they can model how the number of sufferers changes during the continuous college term as well as during long breaks . Hence, the dynamic equations on time scales are worth studying theoretically and practically [3, 5, 7].

Here and hereafter, we denote φp(u) is p-Laplacian operator, that is, φp(u)=|u|p-2u for p>1 and (φp)-1=φq, where 1/p+1/q=1. We make the blanket assumption that 0,T are points in 𝕋, by an interval (0,T)𝕋 we always mean (0,T)𝕋. Other types of interval are defined similarly.

Recent research results indicate that considerable work has been made in the existence problems of solutions of boundary value problems on time scales, for details, see  and the references therein. In particular, some of them are considered the existence of positive solutions of p-Laplacian boundary value problems on time scales, see . The main tools used in these papers are the various fixed point theorems in cones. Very recently, when the nonlinear term f is allowed to change sign, Su et al.  proved the existence of positive solutions to p-Laplacian dynamic equations with sign changing nonlinearity on time scales.

Motivated by references , we consider the following m-point singular p-Laplacian boundary value problem on time scales of the form (φp(uΔ(t)))+q(t)f(t,u(t))=0,t(0,T)𝕋,u(0)=0,u(T)=i=1mψi(uΔ(ξi)),m, where f(t,u):(0,T)𝕋×(0,) is continuous and ψi: are continuous, nondecreasing and ψi may be nonlinear, 0ξ1<ξ2<<ξmT. The singularity may occur at u=0,t=0 and t=T, and the nonlinearity is allowed to change sign. In particular, the boundary condition (1.2) includes the Dirichlet boundary condition. We obtain some new existence criteria for positive solutions of the boundary value problem (1.1) and (1.2) by using the upper and lower method. Our results are new even for the corresponding differential (𝕋=) and difference equations (𝕋=), as well as in general time scales setting. As an application, an example is given to illustrate these results. In particular, our results improve and generalize some known results of Agarwal et al. , O'Regan  (p=2) and Lü et al.  when 𝕋=; include the results of Lü et al.  when 𝕋=; extend and include the results of Jiang et al.  in the case of 𝕋=.

For the convenience of statements, now we present some basic definitions and lemmas concerning the calculus on time scales that one needs to read this manuscript, which can be found in [3, 7]. One of other excellent sources on dynamical systems on time scales is from the book in .

Definition 1.1 (see [<xref ref-type="bibr" rid="B6">3</xref>, <xref ref-type="bibr" rid="B5">7</xref>]).

A time scale 𝕋 is a nonempty closed subset of . It follows that the jump operators σ,ρ:𝕋𝕋 defined by σ(t)=inf{τ𝕋:τ>t},ρ(t)=sup{τ𝕋:τ<t} (supplemented by inf:=sup𝕋 and sup:=inf𝕋 ) are well defined. The point t𝕋 is left-dense, left-scattered, right-dense, right-scattered if ρ(t)=t,ρ(t)<t,σ(t)=t,σ(t)>t, respectively. If 𝕋 has a right-scattered minimum m, define 𝕋κ=𝕋-{m}; otherwise, set 𝕋κ=𝕋. If 𝕋 has a left-scattered maximum M, define 𝕋κ=𝕋-{M}; otherwise, set 𝕋κ=𝕋. The forward graininess is μ(t):=σ(t)-t. Similarly, the backward graininess is ν(t):=t-ρ(t).

Definition 1.2 (see [<xref ref-type="bibr" rid="B5">7</xref>]).

We say that a function f:𝕋 is right-increasing at a point t0𝕋{max  𝕋} provided the following conditions hold.

If t0 is right-scattered, then f(σ(t0))>f(t0).

If t0 is right-dense, then there is a neighborhood U of t0 such that f(t)>f(t0) for all tU with t>t0.

Similarly, we say that f is right-decreasing if above in (i), f(σ(t0))<f(t0) and (ii), f(t)<f(t0).

Definition 1.3 (see [<xref ref-type="bibr" rid="B6">3</xref>]).

A function f:𝕋 is called predifferentiable with (region of differential) D provided the following conditions hold:

f is continuous on 𝕋;

D𝕋κ;

𝕋κD is countable and contains no right-scattered elements of 𝕋;

f is differentiable at each tD.

Next, we list some lemmas which will be used in the sequel.

Lemma 1.4 (see [<xref ref-type="bibr" rid="B6">3</xref>, <xref ref-type="bibr" rid="B5">7</xref>]).

Suppose f:𝕋 is a function and let t𝕋κ, then one has the following:

If f is differentiable at t, then f is continuous at t.

If f is continuous at t and t is right-scattered, then f is differentiable at t with fΔ(t)=f(σ(t))-f(t)μ(t)=f(σ(t))-f(t)σ(t)-t.

If f is right-dense, then f is differentiable at t if and only one the limit limst=f(t)-f(s)t-s exists as a finite number. In this case fΔ(t)=limstf(t)-f(s)t-s.

If f is differentiable at t, then f(σ(t))=f(t)+μ(t)fΔ(t)=f(t)+(σ(t)-t)fΔ(t).

Lemma 1.5 (see [<xref ref-type="bibr" rid="B5">7</xref>]).

Suppose f:𝕋 is differentiable at t0𝕋{max𝕋}. If f assumes its local right-minimum at t0, then fΔ(t0)0. If f assumes its local right-maximum at t0, then fΔ(t0)0.

Lemma 1.6 ((Mean Value Theorem) [<xref ref-type="bibr" rid="B5">7</xref>]).

Let f be a continuous function on [a,b] that is differentiable on [a,b). Then there exist ξ*,τ*[a,b) such that fΔ(τ*)f(b)-f(a)b-afΔ(ξ*).

Lemma 1.7 (see [<xref ref-type="bibr" rid="B6">3</xref>]).

Suppose f and g are pre-differential with D. If U is a compact interval with endpoints r,s𝕋, then |f(s)-f(r)|{suptUκD𝕋|fΔ(t)|}|s-r|.

Now, we can obtain the following lemma which is similar to Lemma 1.7. The proofs are similar to the proofs of Lemma 1.7 by a slight modification and we omit the proofs.

Lemma 1.8.

Suppose f(t,u) and g(t,u) are predifferential with D×(0,+). If U is a compact interval with endpoints r,s𝕋, then |f(s,u)-f(r,u)|{suptUκD𝕋|fΔ(t,u)|}|s-r|, here D𝕋κ.

Throughout this paper, it is assumed that

f(t,u):(0,T)𝕋×(0,) is continuous;

qC((0,T)𝕋,(0,)) and qCld[0,T]𝕋;

ψi: are continuous and nondecreasing, here i=1,2,,m.

2. Existence Results

Define the Banach space 𝔹=C[0,T]𝕋 with the norm y=supt[0,T]𝕋|y(t)|.

To demonstrate existence of positive solutions to problem (1.1) and (1.2), we first approximate the singular problem by means of a sequence of nonsingular problems, and by using the lower and upper solution for nonsingular problem together with Schauders fixed point theorem, and then we establish the existence of solutions to each approximating problem. Our results are new even for the corresponding differential (𝕋=) and difference equations (𝕋=), as well as in general time scales setting. If we consider the corresponding differential equation (𝕋=) of problem (1.1) and (1.2) in the method mentioned above, we obtain the same existence results to problem (1.1) and (1.2). In the same way, we consider the corresponding difference equation (𝕋=) of problem (1.1) and (1.2), we obtain the same existence results to problem (1.1) and (1.2). Here, the two same existence results are obtained in different settings by using the essentially same method. Naturally, it is quite necessary to consider the existence results to problem (1.1) and (1.2) in same setting. In this case, we need to solve the problem with the help of calculus on time scales, because it not only can unify differential and difference equations, but also can provide accurate information of phenomena that manifests themselves partly in continuous time and partly in discrete time. For example, we can consider the problem (1.1) and (1.2) on time scales 𝕋={0}{(12)}[12,1][2,3]. However, if t is taken from (2.1), we cannot study the problem (1.1) and (1.2) only in differential case, neither can we study the problem (1.1) and (1.2) only in difference case.

Now we state and prove our main result.

Theorem 2.1.

Let n0{1,2,} be fixed. Assume that (H1)–(H3) hold and the following conditions are satisfied.

For each n{n0,n0+1,}1, there is a constant ρn such that {ρn} is a strictly monotone decreasing sequence with limnρn=0, and q(t)f(t,ρn)0 for t[1/2n+1,T]𝕋;

There exists a function αC[0,T]𝕋CΔ(0,T]𝕋,φp(αΔ)C(0,T)𝕋 with α(0)=0,α(T)-i=1mψi(αΔ(ξi))0,α>0on(0,T]𝕋 and -(φp(αΔ))q(t)f(t,α)for  t(0,T)𝕋;

There exists a function βC[0,T]𝕋CΔ(0,T]𝕋, φp(βΔ)C(0,T)𝕋 with βα,βρn0fort[0,T]𝕋 and β(T)-i=1mψi(βΔ(ξi))>0, with -(φp(βΔ))q(t)f(t,β) for t(0,T)𝕋, and -(φp(βΔ))q(t)f(1/2n0+1,β) for t(0,1/2n0+1)𝕋.

Then the boundary value problem (1.1) and (1.2) has a positive solution uC[0,T]𝕋CΔ(0,T]𝕋,φp(uΔ)C(0,T)𝕋 with uα for t[0,T]𝕋.

Proof.

It follows from the condition (A1) that 1/2n+1(0,T]𝕋 for each n1. That is, (0,T]𝕋 is not empty. Without loss of generality, fix n1. If ξ1>0, then we can suppose that mint[ξ1,T]𝕋α(t)ρn, let tn(0,ξ1)𝕋 be such that α(tn)=ρn,αρnfort[0,tn]𝕋. If ξ1=0, then we can suppose that mint[ξ2,T]𝕋α(t)ρn, let tn(0,ξ2)𝕋 be such that (2.2) holds. Define αn(t)={ρnift[0,tn]𝕋,αift[tn,T]𝕋,hereα(tn)=ρn. We denote en=[1/2n+1,T]𝕋,ωn(t)=max{1/2n+1,t}fort[0,T]𝕋 and fn(t,x)=max{f(t,x),f(ωn(t),x)}. Define a sequence hn0(t,x)=fn0(t,x) and hn(t,x)=min{fn0(t,x),,fn(t,x)},n=n0+1,n0+2,. Then f(t,x)hn+1(t,x)hn(t,x)hn0(t,x)for(t,x)(0,T)𝕋×(0,),hn(t,x)=f(t,x)for(t,x)en×(0,).

Consider the p-Laplacian boundary value problem (φp(uΔ(t)))+q(t)hn0*(t,u(t))=0,t(0,T)𝕋,u(0)=ρn0,u(T)-i=1mψi*(uΔ(ξi))=ρn0, where hn0*(t,u(t))={hn0(t,αn0(t))+r(αn0(t)-u(t)),u(t)αn0(t),hn0(t,u(t)),αn0(t)u(t)β,hn0(t,β(t))+r(β(t)-u(t)),u(t)β,ψi*(zi)={ψi(αΔ(ξi)),ziαn0Δ(ξi)=αΔ(ξi),ψi(zi),αn0Δ(ξi)ziβΔ(ξi),ψi(βΔ(ξi)),ziβΔ(ξi),i=1,,m, and r:[-1,1] is the radial retraction function defined by r(u)={u,|u|1,u|u|,|u|>1.

Suppose C0[0,T]𝕋={uC[0,T]𝕋:u(0)=0}Cρn0Δ[0,T]𝕋={uCΔ[0,T]𝕋:u(0)=ρn0}. We define the mappings Lp;F:Cρn0Δ[0,T]𝕋C0[0,T]𝕋× be such that Lpu(t)=(φp(uΔ(t))-φp(uΔ(0)),u(T)),Fu(t)=(-0tq(x)hn0*(x,u(x))x,i=1mψi*(uΔ(ξi))+ρn0). By using the Arzela-Ascoli theorem on time scales , we can show that F is continuous and compact. By using the (2.7), (2.8), (2.13) and (2.14), we obtain (φp(uΔ(t))-φp(uΔ(0)),u(T))=(-0tq(x)hn0*(x,u(x))x,i=1mψi*(uΔ(ξi))+ρn0), that is Lpu(t)=Fu(t). If Lpv=(u,γ)foruC0[0,T]𝕋,γ=ρn0+0Tφq(u(x)-u(T))Δx, then v(t)=ρn0+0tφq(u(x)-u(T))Δx, hence Lp-1 exists and is continuous. So u(t)=Lp-1Fu(t). It is clear that solving the boundary value problem (2.7) and (2.8) is equivalent to finding a fixed point of u=Lp-1FuNu, where N=Lp-1F:Cρn0Δ[0,T]𝕋Cρn0Δ[0,T]𝕋 is compact. Schauder's fixed point theorem guarantees that the boundary value problem (2.7) and (2.8) has a solution un0(t)CΔ[0,T]𝕋 with φp(un0Δ(t))C(0,T)𝕋.

We first show that αn0(t)un0(t)fort[0,T]𝕋. If (2.19) is not true, the function un0(t)-αn0(t) has a negative minimum for some τ(0,T]𝕋. We consider two cases, namely, τ(0,T)𝕋 and τ=T.

Case 1.

Assume that τ(0,T)𝕋, then we claim (φp(un0Δ))(τ)(φp(αn0Δ))(τ). Since un0(t)-αn0(t) has a negative minimum for some τ(0,T)𝕋, in view of Definition 1.2, Lemmas 1.4 and 1.5, we have un0Δ(τ)-αn0Δ(τ)0 and there exists a δ with τ-δ[0,τ)𝕋 such that un0Δ(t)-αn0Δ(t)0fort[τ-δ,τ)𝕋. Thus φp(un0Δ(t))-φp(αn0Δ(t))φp(un0Δ(τ))-φp(αn0Δ(τ))fort[τ-δ,τ)𝕋, which leads to φp(un0Δ(t))-φp(un0Δ(τ))t-τφp(αn0Δ(t))-φp(αn0Δ(τ))t-τfort[τ-δ,τ)𝕋.

If τ is left-dense, in view of Lemma 1.4(φp(un0Δ))(τ)=limt[τ-δ,τ)τφp(un0Δ(t))-φp(un0Δ(τ))t-τlimt[τ-δ,τ)τφp(αn0Δ(t))-φp(αn0Δ(τ))t-τ=(φp(αn0Δ))(τ).

If τ is left-scattered, by Lemma 1.4 and (2.22) we obtain (φp(un0Δ))(τ)=φp(un0Δ(τ))-φp(un0Δ(ρ(τ)))τ-ρ(τ)φp(αn0Δ(τ))-φp(αn0Δ(ρ(τ)))τ-ρ(τ)=(φp(αn0Δ))(τ). Hence, (2.20) is established.

However, by (2.3), (2.9) and un0(τ)<αn0(τ), we obtain (φp(un0Δ(τ)))-(φp(αn0Δ(τ)))=-[q(τ)hn0(τ,αn0(τ))+q(τ)r(αn0(τ)-un0(τ))+(φp(αn0Δ(τ)))]={-[q(τ)hn0(τ,α(τ))+q(τ)r(α(τ)-un0(τ))+(φp(αΔ(τ)))],τ[tn0,T)𝕋,-[q(τ)hn0(τ,ρn0)+q(τ)r(ρn0-un0(τ))],τ(0,tn0)𝕋.

Assume that τ[1/2n0+1,T]𝕋, then gn0(τ,x)=f(τ,x) for x(0,), by (A1) and (A2), we have (φp(un0Δ(τ)))-(φp(αn0Δ(τ)))={-[q(τ)f(τ,α(τ))+q(τ)r(α(τ)-un0(τ))+(φp(αΔ(τ)))],τ[tn0,T)𝕋,-[q(τ)f(τ,ρn0)+q(τ)r(ρn0-un0(τ))],τ(0,tn0)𝕋,<0, which implies a contraction.

Assume that τ(0,1/2n0+1)𝕋, then hn0(τ,x)=max{f(1/2n0+1,x),f(τ,x)}, in view of (A1), (A2) and q(τ)>0, we have (φp(un0Δ(τ)))-(φp(αn0Δ(τ))){-[q(τ)f(τ,α(τ))+q(τ)r(α(τ)-un0(τ))+(φp(αΔ(τ)))],τ[tn0,T)𝕋,-[q(τ)f(12n0+1,ρn0)+q(τ)r(ρn0-un0(τ))],τ(0,tn0)𝕋,<0. which implies a contraction.

Case 2.

Assume that τ=T. That is, αn0(T)-un0(T)>0, by (2.3), (2.8) and (2.10) together with α(T)i=1mψi(αΔ(ξi)), we have the following three subcases.

(a) If un0Δ(ξi)αΔ(ξi)fori=1,2,,m, then 0<αn0(T)-un0(T)=α(T)-i=1mψi*(un0Δ(ξi))-ρn0<i=1mψi(αΔ(ξi))-i=1mψi*(un0Δ(ξi))=i=1mψi(αΔ(ξi))-i=1mψi(αΔ(ξi))=0, this is a contradiction.

(b) If αΔ(ξi)<un0Δ(ξi)fori=1,2,,m. Assume that un0Δ(ξi)βΔ(ξi)fori=1,2,,m, then i=1mψi*(un0Δ(ξi))=i=1mψi(un0Δ(ξi)).

Assume that βΔ(ξi)<un0Δ(ξi)fori=1,2,,m, then i=1mψi*(un0Δ(ξi))=i=1mψi(βΔ(ξi)).

Assume that there exist sequences {il1}={1,2,,l1} and {ik1}={1,2,,k1} such that βΔ(ξil1)<un0Δ(ξil) and un0Δ(ξik1)βΔ(ξik1), here l1+k1=m, then i=1mψi*(un0Δ(ξi))=ik1=1k1ψik1(un0Δ(ξik1))+il1=1l1ψil1(βΔ(ξil1)). Hence, by (2.29), (2.30) and (2.31) together with the monotonicity of ψi, we have 0<αn0(T)-un0(T)=α(T)-i=1mψi*(un0Δ(ξi))-ρn0i=1mψi(αΔ(ξi))-i=1mψi*(un0Δ(ξi))-ρn0<0, this is a contradiction.

(c) If there exist sequences {il}={1,2,,l} and {ik}={1,2,,k} such that αΔ(ξil)<un0Δ(ξil) and un0Δ(ξik)αΔ(ξik), here l+k=m. Essentially the same reasoning as before we have 0<αn0(T)-un0(T)=α(T)-i=1mψi*(un0Δ(ξi))-ρn0<0, this is a contradiction.

Thus, Cases 12 imply (2.19) is established. In particular, since α(t)αn0(t) for t[0,T]𝕋, we obtain α(t)αn0(t)un0(t)fort[0,T]𝕋.

Essentially the same reasoning as the proof of inequality (2.19) we obtain un0(t)βfort[0,T]𝕋.

Hence α(t)αn0(t)un0(t)β(t)fort[0,T]𝕋.

Now, we discuss the boundary value problem (φp(uΔ(t)))+q(t)hn0+1*(t,u(t))=0,t(0,T)𝕋,u(0)=ρn0+1,u(T)-i=1mψi*(uΔ(ξi))=ρn0+1, where hn0+1*(t,u(t))={hn0+1(t,αn0+1(t))+r(αn0+1(t)-u(t)),u(t)αn0+1(t),hn0+1(t,u(t)),αn0+1(t)u(t)un0(t),hn0+1(t,un0(t))+r(un0(t)-u(t)),u(t)un0(t),ψi*(zi)={ψi(αΔ(ξi)),ziαn0+1Δ(ξi)=αΔ(ξi),ψi(zi),αn0+1Δ(ξi)ziun0Δ(ξi),ψi(un0Δ(ξi)),ziun0Δ(ξi),i=1,,m. Schauder's fixed point theorem guarantees that the boundary value problem (2.34) has a solution un0+1(t)CΔ[0,T]𝕋 with φp(un0+1Δ(t))C(0,T)𝕋.

Essentially the same reasoning as the proof of inequality (2.33), we have α(t)αn0+1(t)un0+1(t)un0(t)fort[0,T]𝕋.

If there exists uk(t) for some k{n0+1,n0+2,} satisfying αk(t)uk(t)uk-1(t) for t[0,T]𝕋. Then we investigate the boundary value problem (φp(uΔ(t)))+q(t)hk+1*(t,u(t))=0,t(0,T)𝕋,u(0)=ρk+1,u(T)-i=1mψi*(uΔ(ξi))=ρk+1, where hk+1*(t,u(t))={hk+1(t,αk+1(t))+r(αk+1(t)-u(t)),u(t)αk+1(t),hk+1(t,u(t)),αk+1(t)u(t)uk(t),hk+1(t,uk(t))+r(uk(t)-u(t)),u(t)uk(t),ψi*(zi)={ψi(αΔ(ξi)),ziαk+1Δ(ξi)=αΔ(ξi),ψi(zi),αk+1Δ(ξi)ziukΔ(ξi),ψiΔ(uk(ξi)),ziuk(ξi),i=1,,m. It follows from Schauder's fixed point theorem that the boundary value problem (2.37) has a solution uk+1(t)CΔ[0,T]𝕋 with φp(uk+1Δ(t))C(0,T)𝕋.

By using the similar arguments as above, we have α(t)αk+1(t)uk+1(t)uk(t)fort[0,T]𝕋. Hence, for each n{n0,n0+1,}, the mathematical induction implies that α(t)αn(t)un(t)un-1(t)un0(t)β(t)fort[0,T]𝕋. Denote Rn0=sup{|f(t,y)|:t[12n0+1,T]𝕋,α(t)yun0(t)}. It follows from Lemma 1.6 that there exist τ1*,τ2*[1/2n0+1,T)𝕋 satisfy unΔ(τ1*)un(T)-un(1/2n0+1)T-1/2n0+1unΔ(τ2*). From (2.42), we have φp(unΔ(t))=φp(unΔ(τ1*))-τ1*tq(s)f(s,u(s))sφp(un(T)-un(1/2n0+1)T-1/2n0+1)+Rn01/2n0+1Tq(s)sfort[12n0+1,T]𝕋,φp(unΔ(t))=φp(unΔ(τ2*))-τ2*tq(s)f(s,u(s))sφp(un(T)-un(1/2n0+1)T-1/2n0+1)-Rn01/2n0+1Tq(s)sfort[12n0+1,T]𝕋.

So there exists a positive number K0 such that |unΔ(t)|K0. By Lemma 1.8, we have {un(t)}n=n0+1isabounded,equicontinuousfamilyont[12n0+1,T]𝕋. The Arzela-Ascoli theorem on time scales  guarantees the existence of a subsequence n0 of integers and a function zn0(t)C[1/2n0+1,T]𝕋 with un(t) converging uniformly to zn0(t) on [1/2n0+1,T]𝕋 as n through n0. Similarly {un(t)}n=n0+1isabounded,equicontinuousfamilyont[12n0+2,T]𝕋. Thus there is a subsequence n0+1 of n0 and a function zn0+1(t)C[1/2n0+2,T]𝕋 with un(t) converging uniformly to zn0+1(t) on [1/2n0+2,T]𝕋 as n through n0+1. Since n0+1n0, we have zn0+1(t)=zn0(t) on [1/2n0+1,T]𝕋. Proceed inductively to obtain subsequence of integers n0n0+1n and functions zn(t)C[1/2n+1,T]𝕋 with un(t)converginguniformlytozn(t)on[1/2n+1,T]𝕋 as nthroughn and zn(t)=zn-1(t)on[1/2n,T]𝕋.

Now, we define a function u:[0,T][0,) with u(t)=zn(t) on [1/2n+1,T]𝕋 and u(0)=0. Notice, u(t) is well defined and α(t)u(t)un0(t)β for t(0,T)𝕋. Nextly fix t(0,T)𝕋 and let l{n0,n0+1,} be such that t(1/2l+1,T)𝕋, let l*={nl:nl}, we have ψi*(un(ξi))=ψi(un(ξi)),hn*(t,un(t))=hn(t,un(t))=f(t,un(t))fornl*. Hence, for nl*, we have unwhich is the positive solution of the following boundary value problem (φp(unΔ(t)))+q(t)f(t,un(t))=0,t(12n,T)𝕋,un(0)=ρn,un(T)-i=1mψi(unΔ(ξi))=ρn. Let n through l*, we have u(t)that satisfies (φp(uΔ(t)))+q(t)f(t,u(t))=0,t(0,T)𝕋,u(0)=0,u(T)-i=1mψi(uΔ(ξi))=0. It remains to show that u(t) is continuous at 0. Now by limnun(0)=0, there exists n1{n0,n0+1,} with un1(0)<ε/2. Since un1(t)C[0,T]𝕋 there exists δn1(0,T)𝕋 with un1(t)<ε/2 for t[0,δn1)𝕋. By the monotonicity of {un(t)}n0 for each t[0,T]𝕋, we have α(t)un(t)un1(t)<ε/2fort[0,δn1)𝕋andnn1, which means α(t)u(t)<ε/2 for t[0,δn1)𝕋. So u(t) is continuous at 0.

If we replace t[1/2n+1,T]𝕋 with t[1/2n+1,T-1/2n+1]𝕋, the singularity occurs at u=0,t=0 and t=T.

If we replace t[1/2n+1,T]𝕋 with t[0,T-1/2n+1]𝕋, the singularity occurs at u=0 and t=T.

If we replace t[1/2n+1,T]𝕋 with t[0,T]𝕋, the singularity occurs at u=0.

So it is easily obtain the analogue of Theorem 2.1 in this section. See the following remark.

Remark 2.2.

If (A3) is appropriately adjusted, we can replace t[1/2n+1,T]𝕋 in (A1) by t[12n+1,T-12n+1]𝕋,t[0,T-12n+1]𝕋, or t[0,T]𝕋. For example, if (2.49) occurs, (A3) is replaced by

(A3) There exists a function βC[0,T]𝕋CΔ(0,T]𝕋,φp(βΔ)C(0,T)𝕋 such that βαandβρn0 for t[0,T]𝕋,β(T)-i=1mψi(βΔ(ξi))>0,-(φp(βΔ))q(t)f(t,β)fort(0,T)𝕋,-(φp(βΔ))q(t)f(1/2n0+1,β) for t(0,1/2n0+1)𝕋 and -(φp(βΔ))q(t)f(T-1/2n0+1,β)fort(T-1/2n0+1,T)𝕋.

Assume that (H1)–(H3), (A1) and (A2) hold, and in addition suppose the following conditions are satisfied:

(A4) -(φp(αΔ))<q(t)f(t,u) for (t,u)(0,T]𝕋×{uC[0,T]𝕋CΔ(0,T]𝕋:0<uα};

(A5) There exists a function βC[0,T]𝕋CΔ(0,T]𝕋,φp(βΔ)C(0,T)𝕋 such that βρn0 for t[0,T]𝕋,β(T)-i=1mψi(βΔ(ξi))>0,-(φp(βΔ))q(t)f(t,β) for t(0,T)𝕋 and -(φp(βΔ))q(t)f(1/2n0+1,β) for t(0,1/2n0+1)𝕋;

(A6) β(T)α(T).

Then the result in Theorem 2.1 is also true. This follows immediately from Theorem 2.1 if we show (A3) holds. That is to say, if we show βα for t[0,T]𝕋, then the result holds. Assume it is not true, in view of (A6) we obtain β-α has a negative minimum for some τ6(0,T)𝕋, so (β-α)Δ(τ6)0 and essentially the same reasoning as the proof of inequality (2.20), we have (φp(αΔ))(τ6)(φp(βΔ))(τ6). However, by (A4), (A5) and α(τ6)>β(τ6)>0, we obtain -(φp(αΔ))(τ6)<q(τ6)f(τ6,β(τ6)). Hence (φp(αΔ))(τ6)-(φp(βΔ))(τ6)(φp(αΔ))(τ6)+q(τ6)f(τ6,β(τ6))>0, which implies a contradiction.

Corollary 2.3.

Let n0{1,2,} be fixed, suppose (H1)–(H3), (A1), (A2) and (A4)–(A6) hold, then the boundary value problem (1.1) and (1.2) has a solution uC[0,T]𝕋CΔ(0,T]𝕋,φp(uΔ)C(0,T)𝕋 with uα for t[0,T]𝕋.

3. Construction of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M380"><mml:mrow><mml:mi mathvariant="bold-italic">α</mml:mi></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M381"><mml:mrow><mml:mi mathvariant="bold-italic">β</mml:mi></mml:mrow></mml:math></inline-formula>

In this section, we consider how to construct a lower solution α and an upper solution β in certain circumstances. In this section, we assume that i=1mψi(xi)0forxi.

Lemma 3.1.

Assume that there exists a nonincreasing positive sequence {εn} with limnεn=0, then there exist a function λ(t)CΔ[0,T]𝕋 satisfying

φp(λΔ(t))C[0,T]𝕋,λ(t)>0 for t(0,T]𝕋 and maxt[0,T]𝕋|(φp(λΔ(t)))|>0;

λ(0)=0,λ(T)-i=1mψi(λΔ(ξi))<0 and 0<λ(t)εn for t(0,T]𝕋.

Proof.

Let en=[1/2n+1,T]𝕋(nn0). Assume that r:[0,T]𝕋[0,) be such that r(0)=0,r(t)=εnp-1/(2T)p+1 for tenen-1,nn0 and r(t)=εn0p-1/(2T)p+1 for t[1/2n0,T]𝕋. Let u(t)=0tr(s)Δs,v(t)=[0tu(s)s]1/(p-1),w(t)=0tv(s)Δs. Suppose τ7enen-1for  nn0,τ8(0,T]𝕋 satisfy τ7<τ8 and 2τ8-Tτ7. It is easy to show that u,v,w:[0,τ7]𝕋[0,) are continuous and increasing. Denote a(t)=[c0(τ8-t)+c1t]1/(p-1)fort[τ7,T]𝕋, here c0=-τ7τ8u(τ7)+1τ8(v(τ7))p-1,c1=τ8-τ7τ8u(τ7)+1τ8(v(τ7))p-1. Hence, a(t)>0 for t[τ7,T]𝕋 and is nondecreasing. Define b(t)=τ7ta(s)Δs+w(τ7)fort[τ7,τ8]𝕋,P(t)={b(t),t[τ7,τ8]𝕋,b(2τ8-t),t[τ8,T]𝕋,λ(t)={w(t),t[0,τ7]𝕋,P(t),t[τ7,T]𝕋. We can easily prove w(τ7)=P(τ7),wΔ(τ7)=PΔ(τ7),(φp(wΔ))(τ7)=(φp(PΔ))(τ7) and wCΔ[0,τ7]𝕋,PCΔ[τ7,T]𝕋,φp(wΔ)C[0,τ7]𝕋,φp(PΔ)C[τ7,T]𝕋. Thus, we have λCΔ[0,T]𝕋 and φp(λΔ)C[0,T]𝕋 with max0tT|(φp(λΔ))(t)|>0. Now since w(t)>0 for t(0,τ7]𝕋 and P(t)>0 for t[τ7,T]𝕋, we have λ(t)>0 for t(0,T]𝕋. On the other hand, u(τ7)=0τ7r(s)Δsτ7εnp-1(2T)p+1<εnp-1(2T)p,v(τ7)=[0τ7u(s)s]1/(p-1)<(τ7εnp-1(2T)p)1/(p-1)<εn2p/(p-1)T,w(τ7)<τ7×εn2p/(p-1)T<εn2, by the monotonicity of P(t) on [τ7,τ8]𝕋,[τ8,T]𝕋, respectively, we have λ(τ8)=maxt[τ7,T]𝕋λ(t)=τ7τ8a(s)Δs+w(τ7)(τ8-τ7)maxt[τ7,τ8]𝕋[c0(τ8-t)+c1t]1/(p-1)+w(τ7)(τ8-τ7)[(τ8-τ7)u(τ7)+(v(τ7))p-1]1/(p-1)+w(τ7)<T[Tεnp-1(2T)p+εnp-12(2T)p-1]1/(p-1)+εn2<εn2+εn2=εn. Consequently, 0<λ(t)εn,t(0,T]𝕋.

Without loss of generality, i=1mψi(λΔ(ξi))εn>λ(T). We have λ(T)-i=1mψi(λΔ(ξi))<0.

Now we discuss how to construct a lower solution α(t) in (A2) and (A4).

(A7) For each n{1,2,}, there exist a constant k0 and a strictly monotone decreasing sequence {ρn} with limnρn=0, and q(t)f(t,u)k0 for (t,u)[1/2n+1,T]𝕋×{uC[0,T]𝕋CΔ(0,T]𝕋:0<u    ρn};

(A8) There exists a function βC[0,T]𝕋CΔ(0,T]𝕋,φp(βΔ)C(0,T)𝕋 such that β0 for t[0,T]𝕋,β(T)-i=1mψi(βΔ(ξi))>0, -(φp(βΔ))q(t)f(t,β)fort(0,T)𝕋 and -(φp(βΔ))q(t)f(1/2n0+1,β) for t(0,1/2n0+1)𝕋.

Theorem 3.2.

Let n0{3,4,} be fixed. If (H1)–(H3), (3.1) and (A7)-(A8) hold, then boundary value problem (1.1) and (1.2) has a solution uC[0,T]𝕋CΔ(0,T]𝕋 with φp(uΔ)C(0,T)𝕋 and u(t)>0 for t(0,T]𝕋.

Proof.

By Corollary 2.3, we need only show that conditions (A1), (A2), (A4)–(A6) are satisfied. Without loss of generality, suppose β(t)>ρn0fort[0,T]𝕋,β(T)-i=1mψi(βΔ(ξi))>ρn0, by (A7), (A8) and (3.8), we obtain that (A1) and ( A5) hold.

From Lemma 3.1 there exists a function λ(t)CΔ[0,T]𝕋 satisfying

φp(λΔ(t))C[0,T]𝕋,λ(t)>0 for t(0,T]𝕋 and R1=maxt[0,T]𝕋|(φp(λΔ(t)))|>0.

λ(0)=0,λ(T)-i=1mψi(λΔ(ξi))<0 and 0<λ(t)ρn for t(0,T]𝕋.

Assume m=min{1,(k0/2R1)1/(p-1),ρn0/|λ|}. Let α(t)=mλ(t) for t[0,T]𝕋. Then α(t)C[0,T]𝕋CΔ(0,T]𝕋,φp(αΔ(t))C(0,T)𝕋,α(0)=0 with 0<α(t)λ(t)ρn for t(1/2n+1,T]𝕋. Without loss of generality, we have α(T)-i=1mψi(αΔ(ξi))<0. For arbitrary (t,u)(0,T]𝕋×{uC[0,T]𝕋CΔ(0,T]𝕋:0<uα(t)}, there exists n{n0,n0+1,} such that (t,u)[1/2n+1,T]𝕋×{uC[0,T]𝕋CΔ(0,T]𝕋:0<uα(t)}. We have q(t)f(t,u)+(φp(αΔ(t)))k0+(φp(mλΔ(t)))=k0+mp-1(φp(λΔ(t)))k0-mp-1|(φp(λΔ(t)))|k0-k02R1|(φp(λΔ(t)))|k0-k02R1maxt[0,T]|(φp(λΔ(t)))|=k02>0. Thus (A4) holds and (A2) is also true if u(t)=α(t). Also since α(T)supt[0,T]𝕋|α(t)|=m  supt[0,T]𝕋|λ|ρn0, we have β(T)ρn0α(T), then (A6) is fulfilled. By Corollary 2.3, the boundary value problem (1.1) and (1.2) has a solution u(t)C[0,T]𝕋CΔ(0,T]𝕋,φp(uΔ(t))C(0,T)𝕋 with u(t)0 for t(0,T]𝕋.

We can replace t[1/2n+1,T]𝕋 with t[0,T-1/2n+1]𝕋 or t[1/2n+1,T-1/2n+1]𝕋. So it is easily obtain (see Remark 2.2) the analogue of Theorem 3.2 in this section.

Looking at Theorem 3.2, it is difficulty for us to discuss examples in constructing β(t) in (A8). The following theorem removes (A8) and replaces it with an easy verified condition.

Theorem 3.3.

Let n0{1,2,} be fixed. If (H1)–(H3), (A1) and (A2) hold, in addition suppose that the following conditions are satisfied: M1>0,M2>max{supt[0,T]α(t),ρn0},hereM1,M2areconstants,q(t)f(t,M1t+M2)0fort(0,T)𝕋,q(t)f(12n0+1,M1t+M2)0fort(0,12n0+1)𝕋,M1T+M2-i=1mψi(M1)>0. Then boundary value problem (1.1) and (1.2) has a solution uC[0,T]𝕋CΔ(0,T]𝕋 with φp(uΔ)C(0,T)𝕋 and u>0 for t(0,T]𝕋.

Proof.

Denote β(t)=M1t+M2 for t[0,T]𝕋, then β(t)C[0,T]𝕋CΔ(0,T]𝕋,φp(βΔ(t))C(0,T)𝕋 with β(t)α(t) and β(t)ρn0 for t[0,T]𝕋,β(T)-i=1mψi(βΔ(ξi))>0, with -(φp(βΔ(t)))q(t)f(t,β)fort(0,T)𝕋,-(φp(β(t)Δ(t)))q(t)f(12n0+1,β(t))fort(0,12n0+1)𝕋, then (A3) holds. By Theorem 2.1 the result holds.

From Theorems 3.2 and 3.3 we have the following theorem.

Theorem 3.4.

Let n0{1,2,} be fixed. If (H1)–(H3), (3.1) and (A7) hold, in addition suppose there exist constants M1,M2>0 such that (3.11) and (3.5) are true. Then the problem (1.1) and (1.2) has a solution uC[0,T]𝕋CΔ(0,T]𝕋 with φp(uΔ)C(0,T)𝕋 and u>0 for t(0,T]𝕋.

Proof.

Without loss of generality suppose ρn0<M2, by (A7) we have (A1) which holds and M2>ρn0>ρn0+1>,limnρn=0. By the similar way as the proof of the Theorem 3.2, there exists a function αC[0,T]𝕋CΔ(0,T]𝕋,φp(αΔ)C(0,T)𝕋 with α(0)=0,α(T)-i=1mψi(αΔ(ξi))<0,α(t)>0 for t(0,T]𝕋, such that -(φp(αΔ(t)))q(t)f(t,α(t)) for t(0,T)𝕋 and α(t)supt[0,T]𝕋|α|ρn0. This together with (3.14) we have M2>max{supt[0,T]α(t),ρn0}. Thus all the conditions of the Theorem 3.3 are fulfilled.

4. An Example

In this section, we present an example to illustrate our results. Let 𝕋={0}{(12)}[12,1]. Consider the following boundary value problem -(|uΔ(t)|2uΔ(t))=q(t)f(t,u(t))fort(0,1)𝕋,u(0)=0,u(1)-15uΔ(18)-110uΔ(14)-15uΔ(34)-110uΔ(1)=0. It is obvious that T=1,p=4,m=4, ψ1(x)=ψ3(x)=(1/5)x,ψ2(x)=ψ4(x)=(1/10)x. Denote q(t)=t8+5andf(t,u(t))=t/u7(t)+u7(t)-λ2, here λ22 is constant. Let n0{1,2,},ρn=(1/2n+1(λ2+a1))1/7 and k0=a1>0isaconstant. We have ρn01. Note that (H1)–(H3) and (3.1) hold. For n{1,2,},t[1/2n+1,1]𝕋 and 0<uρn, we have q(t)f(t,u)(t8+5)(12n+1ρn7-λ2)(t8+5)(λ2+a1-λ2)>a1, which implies (A7) is satisfied.

Now we show that (A8) holds with β=t1/7.

Notice that if t(1/2,1]𝕋, then βΔ(t)=β(t)=(1/7)t-6/7,|βΔ(t)|2βΔ(t)=1343t-18/7,(|βΔ(t)|2βΔ(t))=-182401t-25/70.

If t=1/2, then βΔ(t)=(1/7)t-6/7 and |βΔ(t)|2βΔ(t)=(1/343)t-18/7, (|βΔ(t)|2βΔ(t))=2343(12)-18/7-2343(14)-18/7-0.17140.

If t=1/2n(n=2,3,), then σ(t)=2t,ρ(t)=t/2, μ(t)=t,ν(t)=t/2, we have βΔ(t)=1t[(2t)1/7-t1/7],|βΔ(t)|2βΔ(t)=1t3[(2t)1/7-t1/7]3, by induction, one gets (|βΔ(t)|2βΔ(t))=24n+1[(12n-1)1/7-(12n)1/7]3-24n+4[(12n)1/7-(12n+1)1/7]30. Thus, for t(0,1)𝕋, we have (|βΔ(t)|2βΔ(t))+q(t)f(t,β)(t8+5)(tt+(t1/7)7-λ2)(t8+5)(2-λ2)0,(|βΔ(t)|2βΔ(t))+q(t)f(12n0+1,β)(t8+5)(12n0+1(t1/7)+(t1/7)7-λ2)(t8+5)(2-λ2)0fort(0,12n0+1)𝕋. Now β(1)-ψ1(βΔ(18))-ψ2(β(14))-ψ3(βΔ(34))-ψ4(β(1))>0. Hence, all conditions of the Theorem 3.2 are satisfied. As a result, the problem (4.2) has a positive solution.

Acknowledgments

This paper is supported by XZIT under Grant XKY2008311 and DEGP under Grant 0709-03.

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