DDNSDiscrete Dynamics in Nature and Society1607-887X1026-0226Hindawi Publishing Corporation12328310.1155/2009/123283123283Research ArticleTriple Positive Solutions for Third-Order m-Point Boundary Value Problems on Time ScalesLiuJian1Xu Fuyi2ShaikhetLeonid1School of Statistics and Mathematics ScienceShandong Economics UniversityJinan, Shandong 250014Chinasdie.edu.cn2School of ScienceShandong University of TechnologyZibo, Shandong 255049Chinasdut.edu.cn200929062009200925022009250520092009Copyright © 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 following third-order m-point boundary value problems on time scales (φ(uΔ))+a(t)f(u(t))=0, t[0,T]T, u(0)=i=1m2biu(ξi), uΔ(T)=0, φ(uΔ(0))=i=1m2ciφ(uΔ(ξi)), where φ:RR is an increasing homeomorphism and homomorphism and φ(0)=0, 0<ξ1<<ξm2<ρ(T). We obtain the existence of three positive solutions by using fixed-point theorem in cones. The conclusions in this paper essentially extend and improve the known results.

1. Introduction

The theory of time scales was initiated by Hilger  as a mean of unifying and extending theories from differential and difference equations. The study of time scales has led to several important applications in the study of insect population models, neural networks, heat transfer, and epidemic models; see, for example . Recently, the boundary value problems with p-Laplacian operator have also been discussed extensively in the literature, for example, see .

A time scale T 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.

In , Anderson considered the following third-order nonlinear boundary value problem (BVP): x(t)=f(t,x(t)),t1tt3,x(t1)=x(t2)=0,γx(t3)+δx′′(t3)=0. He used the Krasnoselskii and Leggett-Williams fixed-point theorems to prove the existence of solutions to the nonlinear boundary value problem.

In [9, 10], He considered the existence of positive solutions of the p-Laplacian dynamic equations on time scales (ϕp(uΔ))+a(t)f(u(t))=0,t[0,T]T, satisfying the boundary conditions u(0)-B0(uΔ(η))=0,uΔ(T)=0, or uΔ(0)=0,u(T)-B1(uΔ(η))=0, where η(0,ρ(T)). He obtained the existence of at least double and triple positive solutions of the problems by using a new double fixed point theorem and triple fixed point theorem, respectively.

In , Zhou and Ma firstly studied the existence and iteration of positive solutions for the following third-order generalized right-focal boundary value problem with p-Laplacian operator (ϕp(u′′))(t)=q(t)f(t,u(t)),0t1,u(0)=i=1mαiu(ξi),u(η)=0,u′′(1)=i=1nβiu′′(θi). They established a corresponding iterative scheme for the problem by using the monotone iterative technique.

However, to the best of our knowledge, little work has been done on the existence of positive solutions for the increasing homeomorphism and positive homomorphism operator on time scales. So the goal of the present paper is to improve and generate p-Laplacian operator and establish some criteria for the existence of multiple positive solutions for the following third-order m-point boundary value problems on time scales (φ(uΔ))+a(t)f(u(t))=0,t[0,T]T,u(0)=i=1m-2biu(ξi),uΔ(T)=0,φ(uΔ(0))=i=1m-2ciφ(uΔ(ξi)), where φ:RR is an increasing homeomorphism and homomorphism and φ(0)=0, and bi,ci,a,f satisfy

bi,ci[0,+), 0<ξ1<<ξm-2<ρ(T), 0<i=1m-2bi<1,0<i=1m-2ci<1;

f:[0,+)R+ is continuous, aCld([0,T]T,R+) and there exits t0[0,T)T such that a(t0)>0, where R+=[0,+).

A projection φ:RR is called an increasing homeomorphism and homomorphism, if the following conditions are satisfied:

if xy, then φ(x)φ(y),x,yR;

φ is continuous bijection and its inverse mapping is also continuous;

φ(xy)=φ(x)φ(y),x,yR.

2. Preliminaries and Lemmas

For convenience, we list the following definitions which can be found in .

Definition 2.1.

A time scale T 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 σ(t)=inf{τTτ>t}T,ρ(r)=sup{τTτ<r}T, for all t,rT. 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.

Definition 2.2.

For f:TR and tTk, 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 |f(σ(t))-f(s)-fΔ(t)(σ(t)-s)|ϵ|σ(t)-s|, for all sU.

For f:TR and tTk, the nabla derivative of f at t, denoted by f(t) (provided it exists) with the property that for each ϵ>0, there is a neighborhood U of t such that |f(ρ(t))-f(s)-f(t)(ρ(t)-s)|ϵ|ρ(t)-s|, for all sU.

Definition 2.3.

A function f 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.

Definition 2.4.

If ϕΔ(t)=f(t), then we define the delta integral by abf(t)Δt=ϕ(b)-ϕ(a). If F(t)=f(t), then we define the nabla integral by abf(t)t=F(b)-F(a).

Definition 2.5.

Let E be a real Banach space over R. A nonempty closed set PE is said to be a cone provided that

uP, a0 implies auP;

u,-uP implies u=0.

Definition 2.6.

Given a cone P in a real Banach space E, a functional ψ:PP is said to be increasing on P, provided ψ(x)ψ(y), for all x,yP with xy.

Definition 2.7.

Given a cone P in a real Banach space E, we define for each a>0 the set Pa={xPx<a}.

Definition 2.8.

A map α is called nonnegative continuous concave functional on a cone P of a real Banach space X if α:P[0,+) is continuous and α(λx+(1-λ)y)λα(x)+(1-λ)α(y) for all x,yP and λ[0,1]. Similarly we say that the map β is called nonnegative continuous concave functional on a cone P of a real Banach space X if β:P[0,+) is continuous and β(λx+(1-λ)y)λβ(x)+(1-λ)β(y) for all x,yP and λ[0,1].

Let γ,θ be nonnegative continuous convex functionals on P, let α be a nonnegative continuous concave functional on P, and let ψ be a nonnegative continuous functional on P. For nonnegative real numbers a,b,k, and c we define the following convex set: P(γ,c)={xP:γ(x)<c},P(α,b;γ,c)={xP:bα(x),γ(x)c},R(ψ,a;γ,c)={xP:aψ(x),γ(x)c},P(α,b;θ,k;γ,c)={xP:bα(x),θ(x)k,γ(x)c}.

Theorem 2.9 ([<xref ref-type="bibr" rid="B21">17</xref>]).

Let P be a cone in a real Banach space X. Let γ and θ be nonnegative continuous convex functionals on P, let α be a nonnegative continuous concave functional on P, and let ψ be a nonnegative continuous functional on P satisfying ψ(λx)λψ(x) for 0λ1, such that for some positive numbers M and c, α(x)ψ(x) and xMγ(x) for all xP(γ,c)¯. Suppose that Φ:P(γ,c)¯P(γ,c)¯ is a completely continuous operator and there exist nonnegative numbers a,b, and k with 0<a<b such that

{xP(α,b;θ,k;γ,c):α(x)>b} and α(Φ(x))>b for xP(α,b;θ,k;γ,c);

α(Φ(x))>b for xP(α,b;γ,c) with θ(Φ(x))>k;

0¯R(ψ,a;γ,c) and ψ(Φ(x))<a for xR(ψ,a;γ,c) with ψ(x)=a.

Then Φ has at least three fixed points x1,x2,x3P(γ,c)¯ satisfying γ(xi)c,i=1,2,3,b<α(x1),α(x2)<b,a<ψ(x2),ψ(x3)<a.

Theorem 2.10 ([<xref ref-type="bibr" rid="B20">18</xref>]).

Let A be a bounded closed convex subset of a Banach space E. Assume that A1,A2 are disjoint closed convex subsets of A and U1,U2 are nonempty open subsets of A with U1A1 and U2A2. Suppose that Φ:AA is completely continuous and the following conditions hold:

Φ(A1)A1,Φ(A2)A2;

Φ has no fixed points in (A1U1)(A2U2).

Then Φ has at least three points x1,x2,x3 such that x1U1,x2U2, and x3A(A1A2).

Lemma 2.11.

If condition (H1) holds, then for hCld[0,T]T, the boundary value problem (BVP) uΔ+h(t)=0,t(0,T),u(0)=i=1m-2biu(ξi),uΔ(T)=0 has the unique solution u(t)=0t(T-s)h(s)s+i=1m-2bi0ξi(T-s)h(s)s1-i=1m-2bi.

Proof.

By caculating, we can easily get (2.12). So we omit it.

Lemma 2.12.

If condition (H1) holds, then for hCld[0,T]T, the boundary value problem (BVP) (φ(uΔ))+h(t)=0,t(0,T),u(0)=i=1m-2biu(ξi),uΔ(T)=0,φ(uΔ(0))=i=1m-2ciφ(uΔ(ξi)) has the unique solution u(t)=0t(T-s)φ-1(0sh(r)r+C)s+i=1m-2bi0ξi(T-s)φ-1(0sh(r)r+C)s1-i=1m-2bi, where C=i=1m-2ci0ξih(r)r/(1-i=1m-2ci), φ-1(s) is the inverse function to φ(s).

Proof.

Integrating both sides of equation in (2.13) on [0,t], we have φ(uΔ(t))=φ(uΔ(0))-0th(r)r. So, φ(uΔ(ξi))=φ(uΔ(0))-0ξih(r)r. By boundary value condition φ(uΔ(0))=i=1m-2ciφ(uΔ(ξi)), we have φ(uΔ(0))=-i=1m-2ci0ξih(r)r1-i=1m-2ci. By (2.15) and (2.17) we know uΔ(t)=-φ-1(i=1m-2ci0ξih(r)r1-i=1m-2ci+0th(r)r). This together with Lemma 2.11 implies that u(t)=0t(T-s)φ-1(0sh(r)r+C)s+i=1m-2bi0ξi(T-s)φ-1(0sh(r)r+C)s1-i=1m-2bi, where C=i=1m-2ci0ξih(r)r/(1-i=1m-2ci). The proof is complete.

Lemma 2.13.

Let condition (H1) hold. If hCld[0,T]T and h(t)0, then the unique solution u(t) of (2.13) satisfies u(t)0,t[0,T]T.

Proof.

By uΔ(t)=-φ-1((i=1m-2ci0ξih(r)r/(1-i=1m-2ci))+0th(r)r)0, we can know that the graph of u(t) is concave down on (0,T)T and uΔ(t) is nonincreasing on [0,T]T. This together with the assumption that the boundary condition is uΔ(T)=0 implies that uΔ(t)0 for t[0,T]T. This implies that u=u(T),mint[0,T]Tu(t)=u(0). So we only prove u(0)0. By condition (H1) we have u(0)=i=1m-2bi0ξi(T-s)φ-1(0sh(r)r+C)s1-i=1m-2bi0. The proof is completed.

3. Triple Positive Solutions

In this section, some existence results of positive solutions to BVP (1.6) are established by imposing some conditions on f and defining a suitable Banach space and a cone.

Let E=Cld[0,T]T be endowed with the ordering xy if x(t)y(t) for all t[0,T]T, and u=maxt[0,T]T|u(t)| is defined as usual by maximum norm. Clearly, it follows that (E,u) is a Banach space.

We define a cone by P={u:uE,u(t)isconcave,nondecreasing,andnonnegativeon[0,T]T,uΔ(T)=0}. Let η=max{tT:tT2}, and fix lT such that 0<η<l<T, and define the nonnegative continuous convex functionals γ and θ, the nonnegative continuous concave functional α, and the nonnegative continuous functional ψ on the cone P by γ(u)=θ(u)=maxt[0,l]Tu(t)=u(l),α(u)=mint[η,T]Tu(t)=u(η),ψ(u)=maxt[0,η]T=u(η). For notational convenience, denote Ĉ=i=1m-2ci0ξia(r)r1-i=1m-2ci,mη=0η(T-s)φ-1(0sa(r)r+Ĉ)s,m=i=1m-2bi0ξi(T-s)φ-1(0sa(r)r+Ĉ)s1-i=1m-2bi,Ml=0l(T-s)φ-1(0sa(r)r+Ĉ)s+i=1m-2bi0ξi(T-s)φ-1(0sa(r)r+Ĉ)s1-i=1m-2bi,Mη=0η(T-s)φ-1(0sa(r)r+Ĉ)s+i=1m-2bi0ξi(T-s)φ-1(0sa(r)r+Ĉ)s1-i=1m-2bi,M=0T(T-s)φ-1(0sa(r)r+Ĉ)s+i=1m-2bi0ξi(T-s)φ-1(0sa(r)r+Ĉ)s1-i=1m-2bi.

Lemma 3.1 ([<xref ref-type="bibr" rid="B10">9</xref>]).

If uP, then

u(t)(t/T)||u|| for all t[0,T]T;

u(s)/su(t)/t for t,s[0,T]T with st.

Define an operator Φ:PE by (Φu)(t)=0t(T-s)φ-1(0sa(r)f(u(r))r+C̃)s+i=1m-2bi0ξi(T-s)φ-1(0sa(r)f(u(r))r+C̃)s1-i=1m-2bi, where C̃=i=1m-2ci0ξia(r)f(u(r))r/(1-i=1m-2ci). Then, u is a solution of boundary value problem (1.6) if and only if u is a fixed point of operator Φ. Obviously, for uP one has (Φu)(t)0 for t[0,T]T. In addition, (Φu)Δ(t)0 for t[0,T]T and (Φu)Δ(T)=0, this implies ΦPP. With standard argument one may show that Φ:PP is completely continuous.

Theorem 3.2.

Suppose conditions (H1) and (H2) hold, and there exist positive numbers a<(η/T)b<b<(l/T)c,Mlb<mc such that

f(u)φ(c/Ml),u[0,Tc/l];

f(u)>φ(b/mη),u[b,T2b/l2];

f(u)<φ(a/Mη),u[0,Ta/η].

Then, the BVP (1.6) has at least three positive solutions u1,u2,u3P(γ,c)¯ satisfying γ(ui)c,i=1,2,3,b<α(u1),α(u2)<b,a<ψ(u2),ψ(u3)<a.

Proof.

Based on Lemma 3.1, it is clear that for uP and λ[0,1], there are α(u)=ψ(u),ψ(λu)=λψ(u) and u(T/l)u(l)=(T/l)γ(u). Furthermore, ψ(0)=0<a and therefore 0¯R(ψ,a;γ,c).

Take uP(γ,c)¯, then 0uu(T/l)γ(u)(T/l)c. By means of (B1) one derives γ(Φu)=Φu(l)=0l(T-s)φ-1(0sa(r)f(u(r))r+C̃)s+i=1m-2bi0ξi(T-s)φ-1(0sa(r)f(u(r))r+C̃)s1-i=1m-2bicMl(0l(T-s)φ-1(0sa(r)r+Ĉ)s+i=1m-2bi0ξi(T-s)φ-1(0sa(r)r+Ĉ)s1-i=1m-2bi)=c. Thus Φ:P(γ,c)¯P(γ,c)¯.

Set uTb/l and k=Tb/l, it follows that α(u)=u(η)=Tbl>b,θ(u)=u(l)=Tbl,γ(u)=Tbl<c, which means {uP(α,b;θ,Tb/l;γ,c):α(u)>b}.

For uP(α,b;θ,Tb/l;γ,c), we have bu(t)T2b/l2 for t[η,T]T. By condition (B2) we have α(Φu)=Φu(η)=0η(T-s)φ-1(0sa(r)f(u(r))r+C̃)s+i=1m-2bi0ξi(T-s)φ-1(0sa(r)f(u(r))r+C̃)s1-i=1m-2bibmη0η(T-s)φ-1(0sa(r)r+Ĉ)s=b. So, (i) of Theorem 2.9 is fulfilled.

If uP(α,b;γ,c) and θ(Φu)>c, then due to (2) of Lemma 3.1α(Φu)=Φu(η)ηl(Φu)(l)=ηlθ(Φu)>ηcl>Tηbl2>b. Therefore, (ii) of Theorem 2.9 is fulfilled.

Take uR(ψ,a;γ,c) and ψ(u)=a, then 0uu(T/η)u(η)=(T/η)ψ(u)=Ta/η, it then follows from (B3) that ψ(Φu)=Φu(η)=0η(T-s)φ-1(0sa(r)f(u(r))r+C̃)s+i=1m-2bi0ξi(T-s)φ-1(0sa(r)f(u(r))r+C̃)s1-i=1m-2biaMη(0η(T-s)φ-1(0sa(r)r+Ĉ)s+i=1m-2bi0ξi(T-s)φ-1(0sa(r)r+Ĉ)s1-i=1m-2bi)=a. As a result, all the conditions of Theorem 2.9 are verified. This completes the proof.

Theorem 3.3.

Suppose that conditions (H1) and (H2) hold. Let 0<a<b<c,  Mb<mc and assume that the following conditions are satisfied:

f(u)<φ(a/M), u[0,a];

there exists a number d>c such that f(u)<φ(d/M), u[0,d];

φ(b/m)<f(u)<φ(c/M), u[b,c].

Then, the BVP (1.6) has at least three positive solutions u1,u2, and u3 such that b<u1(t)<c,u2<a,u3>a. Where for real number b, ϕb:[0,T]T[0,+) is continuous, ϕb(t)=b, for t[0,T]T.

Proof.

We first show that A(P¯a)PaP¯a if condition (C1) holds. If uP¯a, then 0uua, which implies f(u)<φ(a/M). We have Φu=(Φu)(T)0T(T-s)φ-1(0sa(r)f(u(r))r+C̃)s+i=1m-2bi0ξi(T-s)φ-1(0sa(r)f(u(r))r+C̃)s1-i=1m-2bi<aM(0T(T-s)φ-1(0sa(r)r+Ĉ)s+i=1m-2bi0ξi(T-s)φ-1(0sa(r)r+Ĉ)s1-i=1m-2bi)=a. This implies that Φ(P¯a)PaP¯a.

Next, condition (C2) indicates that there exists d>c such that Φ(P¯d)P¯d. Now we let A=P¯d,A1=[φb,φc],U1=int(A1),A2=P¯a,U2=Pa, where int(A1) is the interior of A1. Then we have Φ(A)A,Φ(A2)A2. Moreover, Φ(P¯a)PaP¯a means Φ(A2)U2A2. Thus Φ has no fixed point in (A2U2).

To show Φ(A1)A1 and Φ has no fixed point in (A1U1), set uA1, following the definition of φb, we can know bu(t)c, for t[0,T]T. Condition (C3) then gives rise to φ(b/m)<f(u)<φ(c/M), which in turn produces (Φu)(t)(Φu)(0)=i=1m-2bi0ξi(T-s)φ-1(0sa(r)f(u(r))r+C̃)s1-i=1m-2bi>bmi=1m-2bi0ξi(T-s)φ-1(0sa(r)r+Ĉ)s1-i=1m-2bi=b,(Φu)(t)(Φu)(T)20T(T-s)φ-1(0sa(r)f(u(r))r+C̃)s+i=1m-2bi0ξi(T-s)φ-1(0sa(r)f(u(r))r+C̃)s1-i=1m-2bi<cM(0T(T-s)φ-1(0sa(r)r+Ĉ)s+i=1m-2bi0ξi(T-s)φ-1(0sa(r)r+Ĉ)s1-i=1m-2bi)=c. Combining the above two inequalities one achieves ϕb(t)=b<(Φu)(t)<c=ϕc(t), for t[0,T]T. That is, ΦuU1. So Φ(A1)U1A1 and Φ has no fixed point in (A1U1). Therefore, all conditions of Theorem 2.10 are fulfilled, and the BVP (1.6) has at least three positive solutions u1,u2, and u3 such that b<u1(t)<c,u2<a,u3>a.

4. Some Examples

In the section, we present some simple examples to explain our results. We only study the case T=R,(0,T)T=(0,1).

Example 4.1.

Consider the following third-order three-point boundary value problem: (φ(u′′))+a(t)f(u)=0,0<t<1,u(0)=13u(12),u(1)=0,φ(u′′(0))=14φ(u′′(12)), where φ(x)=x,a(t)1,b1=1/3,c1=1/4,ξ1=1/2.

We choose η=1/2, by computing we can know mη=11/24,Ml=351/256,Mη=33/48. Let a=100,b=245,c=770,l=7/8, then a<ηb<b<lc. Obviously, Mlb<mc. We define a nonlinearity f as follows: f(u)={140,u[0,200],140+41045(u-200),u[200,245],550,u[245,320],550+5560(u-320),u[320,+). Then, by the definition of f, we have

f(u)φ(c/Ml)557.2,u[0,880];

f(u)>φ(b/mη)534.5,u[245,320];

f(u)<φ(a/Mη)145.4,u[0,200].

By Theorem 3.2, BVP (4.1) has at least three positive solutions.

Example 4.2.

Consider the following third-order three-point boundary value problem: (φ(u′′))+a(t)f(u)=0,0<t<1,u(0)=13u(12),u(1)=0,φ(u′′(0))=14φ(u′′(12)), where φ(x)=x,a(t)1,b1=1/3,c1=1/4,ξ1=1/2,η=1/2.

By computing, we can know m=11/48,M=83/48. Let a=7,b=12,c=336,l=7/8, then a<b<c. Obviously, Mb<mc. We define a nonlinearity f as follows: f(u)={3,u[0,7],3+9725(u-7)2,u[7,12],100,u[12,336],100+11001764(u-336),u[336,+). Then, by the definition of f, we have

f(u)<φ(a/M)4.2,u[0,7];

and there exists d=2100>c such that f(u)φ(d/M)1214.4,u[0,2100];

φ(b/m)52.4<f(u)<φ(c/M)194.3,u[12,336].

By Theorem 3.3, BVP (4.3) has at least three positive solutions.

Remark 4.3.

Consider following nonlinear m-point boundary value problem: (φ(uΔ))+a(t)f(u(t))=0,0<t<T,u(0)=i=1m-2biu(ξi),uΔ(T)=0,φ(uΔ(0))=i=1m-2ciφ(uΔ(ξi)), where φ(u)={u3,u0,u2,u>0,f and a satisfy the conditions (H1) and (H2). It is clear that φ:RR is an increasing homeomorphism and homomorphism and φ(0)=0. Because p-Laplacian operators are odd, they do not apply to our example. Hence we generalize boundary value problem with p-Laplacian operator, and the results [811, 1315] do not apply to the example.

Remark 4.4.

In a similar way, we can get the corresponding results for the following boundary value problem: (φ(uΔ))+a(t)f(u(t))=0,t(0,T)T,u(T)=i=1m-2aiu(ξi),uΔ(0)=0,φ(uΔ(0))=i=1m-2ciφ(uΔ(ξi)).

Acknowledgment

The project was supported by the National Natural Science Foundation of China (10471075).

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