# Existence of Nontrivial Solutions and Sign-Changing Solutions for Nonlinear Dynamic Equations on Time Scales

Authors: Huiye Xu
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

By using the topological degree theory and the fixed point index theory,
the existence of nontrivial solutions and sign-changing solutions for a class of boundary value
problem on time scales is obtained. We should point out that the integral operator corresponding
to above boundary value problem is not assumed to be a cone mapping.


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## Body

## 1. Introduction

Let𝕋 be a time scale which has the subspace topology inherited from the standard topology on ℝ. For each interval I of ℝ, we define I𝕋=I∩𝕋.In this paper, we study the existence of nontrivial solutions and sign-changing solutions of the following problem on time scales:(1.1)-[r(t)uΔ(t)]Δ=f(t,u(σ(t))),t∈[a,b]T,αu(a)-βuΔ(a)=0,γu(σ(b))+δuΔ(σ(b))=0,
where α≥0, β≥0, γ≥0, δ≥0, (α2+β2)(γ2+δ2)≠0, r(t)>0, t∈[a,σ(b)]𝕋, and rΔ exists. Some basic definitions on time scales can be found in [1–3].In an ordered Banach space, much interest has developed regarding the computation of the fixed-point index about cone mappings. Based on it, the existence of positive solutions of various dynamic equations has been studied extensively by numerous researchers. The reader is referred to [4–13] for some recent works on second-order boundary value problem on time scales.Dynamic equations on time scales have attracted considerable interest because of their ability to model economic phenomena, see [14–16] and references therein. The existence of mathematics framework to describe plays an important role in the advancement in all the social sciences. Time scales calculus is likely to be applied in many fields of economics, which can combine the standard discrete and continuous models. Historically, two separate approaches have dominated mathematics modeling: differential equation, termed continuous dynamic modeling, and difference equation, termed discrete dynamic modeling. Certain economically important phenomena do not solely possess elements of the continuous or modeling elements of the discrete. For example, in a discrete model, an enterprise obtains some net income in a period of time and decides how many dividends to extend and how many to leave as retained earnings during the same period. Thus, all decisions are assumed to be made at evenly spaced intervals. The time scales to this optimization problem are much more flexible and realistic. For example, an enterprise obtains net income at one point in time, dividend-extending decisions are made at a different point in time, and keeping retained earnings takes place at yet another point in time. Moreover, dividends-extending and retained earning decisions can be modeled to occur with arbitrary, time-varying frequency. It is hard to overestimate the advantages of such an approach over the discrete or continuous models used in economics. In recent years, there is much attention paid to the existence of positive solution for second-order boundary value problems on time scales; for details, see [4–11] and references therein.In [5], Erbe and Peterson considered the existence of positive solutions to the following problem:
(1.2)-xΔΔ(t)=f(t,x(σ(t))),t∈[a,b]T,αx(a)-βxΔ(a)=0,γx(σ(b))+δxΔ(σ(b))=0,
where f:[a,σ(b)]𝕋×ℝ+→ℝ+ is continuous. The following conditions are imposed on f:
(1.3)f0:=lim⁡u→0+f(t,u)u=0,f∞:=lim⁡u→+∞f(t,u)u=+∞,uniformly  ont∈[a,σ(b)]T
or
(1.4)f0:=lim⁡u→0+f(t,u)u=+∞,f∞:=lim⁡u→+∞f(t,u)u=0,uniformly  ont∈[a,σ(b)]T.
The authors obtained the existence of positive solutions for problem (1.2) by means of the cone expansion and compression fixed-point theorem.In [6], Chyan and Henderson were concerned with determining values of λ for which there exist positive solutions of the following nonlinear dynamic equation:
(1.5)xΔΔ(t)+λa(t)f(x(σ(t)))=0,t∈[0,1]T,
satisfying either the conjugate boundary conditions
(1.6)x(0)=x(σ(1))=0
or the right focal boundary conditions
(1.7)x(0)=xΔ(σ(1))=0.
Their analysis still relied on the cone expansion and compression fixed-point theorem.Furthermore, Hong and Yeh [9] generalized the main theorems in [6] to the following problem
(1.8)xΔΔ(t)+λf(t,x(σ(t)))=0,t∈[0,1]T,αu(0)-βxΔ(0)=0,γx(σ(1))+δxΔ(σ(1))=0.
They also applied the above-mentioned fixed-point theorem in a cone to yield positive solutions of (1.8) for λ belonging to a suitable open interval.On the other hand, there is much attention paid to the study of spectral theory of linear problems on time scales, see [12, 13] and references therein. Agarwal et al. [12] obtained a fundamental result on the existence of eigenvalues and the number of the generalized zeros of associated eigenfunctions. This result provides a foundation for the further study of the behavior of eigenvalues of the Sturm-Liouville problems on time scales. In [13], Kong first extended the existence result in [12] to the Sturm-Liouville problems in a more general form and with normalized separated boundary conditions and then explored the dependence of the eigenvalues on the boundary condition. The author showed that the nth eigenvalue λn depends continuously on the boundary condition except at the generalized “Dirichlet” boundary conditions, where certain jump discontinuities may occur. Furthermore, λn as a function of the boundary condition angles is continuously differentiable wherever it is continuous. Formulas for such derivatives were obtained which reveal the monotone properties of λn in terms of the boundary condition angles.We note that Li et al. [10] were concerned with the existence of positive solutions for problem (1.1) under some conditions concerning the first eigenvalue corresponding to the relevant linear operator. Their main results improved and generalized ones in [5–9, 11].We should point out that the nonlinear item that appeared in the above dynamic equations is required to be nonnegative; moreover, the positive solutions can be obtained only by using these tools.However, some existing nonlinear problems cannot be attributed to the cone mappings; thus, the cone mapping theory fails to solve these problems. At the same time, the existence of the sign-changing solutions of the above dynamic equations is being raised by an ever-increasing number of researchers; it is also difficult to deal with sign-changing solution problems by the cone mappings theory. Stimulated by these works, in [17], Sun and Liu used the lattice structure to present some methods of computation of the topological degree for the operator that is quasiadditive on lattice. The operator is not assumed to be a cone mapping. Furthermore, in [18], Liu and Sun established some methods of computation of the fixed-point index and the topological degree for the unilaterally asymptotically linear operators that are not assumed to be cone mappings. Very recently, in [19], Sun and Liu obtained some theorems for fixed-point index about a class of nonlinear operators which are not cone mappings by means of the theory of cone.The main purpose of this paper is to establish some existence theorems of nontrivial solutions and sign-changing solutions of (1.1). Our results generalize and complement ones in [5–11].This paper is organized as follows. Some preliminaries are given in Section2. In Section 3, we presented some results obtained in [19]; then, we apply these abstract results to problem (1.1), the existence of nontrivial solutions and sign-changing solutions for sublinear problem (1.1) is obtained. In Section 4, we are concerned with the existence of sign-changing solutions and nontrivial solutions for superlinear problem (1.1). Our main tool relies on the computation of the topological degree for superlinear operators, which are obtained in [17]. In Section 5, by the use of the abstract results obtained in [18], some existence theorems of sign-changing solutions for unilaterally asymptotically linear problem (1.1) are established.


## 2. Preliminaries and Some Lemmas

LetE be an ordered Banach space in which the partial ordering ≤ is induced by a cone P⊂E. P is said to be normal if there exists a positive constant N such that θ≤x≤y implies ∥x∥≤N∥y∥. P is called solid if it contains interior points, that is, intP≠∅. If x≤y and x≠y, we write x<y; if cone P is solid and y-x∈intP, we write x≪y. Operator A is said to be strongly increasing if y<x implies Ay≪Ax. P is called total if E=P-P¯. If P is solid, then P is total. A fixed-point u of operator A is said to be a sign-changing fixed-point if u∉P∪(-P). Let B:E→E be a bounded linear operator. B is said to be positive if B(P)⊂P. For the concepts and the properties about the cones, we refer to [20–23].Forw∈E, let Pw=P+w={x∈E∣x≥w} and Pw=w-P={x∈E∣x≤w}.We callE a lattice under the partial ordering ≤ if sup{x,y} and inf{x,y} exist for arbitrary x, y∈E. For x∈E, let
(2.1)x+=sup⁡{x,θ},x-=sup⁡{-x,θ}.x+ and x- are called positive part and negative part of x, respectively. Taking |x|=x++x-, then |x|∈P, and |x| is called the module of x. One can see [24] for the definition and the properties of the lattice.For convenience, we use the following notations:(2.2)x+=x+,x+=-x-,
and then
(2.3)x+∈P,x-∈(-P),x=x++x-.In the following, we always assume thatE is a Banach space, P is a total cone in E, and the partial ordering ≤ in E is induced by P. We also suppose that E is a lattice in the partial ordering ≤.LetB:E→E be a positive completely continuous linear operator, let r(B) be a spectral radius of B, let B* be the conjugated operator of B, and let P* be the conjugated cone of P. Since P⊂E is a total cone, according to the famous Krein-Rutman theorem (see [25]), we infer that if r(B)≠0, then there exist φ¯∈P∖{θ} and g*∈P*∖{θ}, such that
(2.4)Bφ¯=r(B)φ¯,B*g*=r(B)g*.
Fixed φ¯∈P∖{θ}, g*∈P*∖{θ} such that (2.4) holds. For δ>0, let
(2.5)P(g*,δ)={x∈P,g*(x)≥δ‖x‖}.
Then, P(g*,δ) is also a cone in E.Definition 2.1 (see [23]).
LetB be a positive linear operator. The operator B is said to satisfy H condition if there exist φ¯∈P∖{θ}, g*∈P*∖{θ}, and δ>0 such that (2.4) holds, and B maps P into P(g*,δ).Definition 2.2 (see [23]).
LetD⊂E and A:D→E be a nonlinear operator. A is said to be quasiadditive on lattice if there exists v*∈E such that
(2.6)Ax=Ax++Ax-+v*,∀x∈D,
where x+ and x- are defined by (2.2).Definition 2.3 (see [18]).
LetE be a Banach space with a cone P,  and let A:E→E be a nonlinear operator. We call A a unilaterally asymptotically linear operator along Pw if there exists a bounded linear operator B such that
(2.7)lim⁡x∈Pw,‖x‖→∞‖Ax-Bx‖‖x‖=0.B is said to be the derived operator of A along Pw and will be denoted by APw′.Similarly, we can also define a unilaterally asymptotically linear operator alongPw.For convenience, we list the following definitions on time scales which can be found in [1–3].Definition 2.4.
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
(2.8)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.Definition 2.5.
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
(2.9)|f(σ(t))-f(s)-fΔ(t)(σ(t)-s)|≤ϵ|σ(t)-s|,
for all s∈U.Definition 2.6.
IfGΔ(t)=f(t), then we define the delta integral by
(2.10)∫abf(t)Δt=G(b)-G(a).Lemma 2.7 (see [17]).
Suppose thatB:E→E is a positive bounded linear operator. If the spectral radius  r(B)<1, then  (I-B)-1 exists and is a positive bounded linear operator.Lemma 2.8 (see [5]).
Set(2.11)d=γβr(a)+αδr(σ(b))+αγ∫aσ(b)1r(τ)Δτ≠0.
Let
(2.12)G(t,s)={1dx(t)y(σ(s)),t≤s,1dx(σ(s))y(t),σ(s)≤t,
where
(2.13)x(t)=α∫at1r(τ)Δτ+βr(σ(b)),y(t)=γ∫tσ(b)1r(τ)Δτ+δr(σ(b)).
Then G(t,s) is the Green function of the following linear boundary value problem:
(2.14)-[r(t)uΔ(t)]Δ=0,t∈[a,b]T,αu(a)-βuΔ(a)=0,γu(σ(b))+δuΔ(σ(b))=0.According to Lemma2.8, problem (1.1) can be converted into the equivalent nonlinear integral equation
(2.15)u(t)=∫aσ(b)G(t,s)f(s,u(σ(s)))Δs,t∈[a,σ2(b)]T.
Define the operators
(2.16)Au(t)=∫aσ(b)G(t,s)f(s,u(σ(s)))Δs,t∈[a,σ2(b)]T,(2.17)Ku(t)=∫aσ(b)G(t,s)u(σ(s))Δs,t∈[a,σ2(b)]T.
By in [13, Theorem 2.1], we can know that K has the sequence of eigenvalues:
(2.18)0<λ1<λ2<λ3<⋯<λn<⋯,n∈N0k,
where
(2.19)N0k:={{1,2,…},k=∞,{1,2,…,k},k<∞,
the algebraic multiplicities of every eigenvalue is simple, and the spectral radius r(K)=λ1-1.Let(2.20)E={[r(t)uΔ(t)]Δu:[a,σ2(b)]T⟶R,uΔ(t)iscontinuouson[a,σ2(b)]T,[r(t)uΔ(t)]Δisrightdensecontinuouson[a,b)T}.
Then E is an ordered Banach space with the supremum norm ∥u∥=supt∈[a,σ2(b)]𝕋|u(t)|. Define a cone P⊂E by
(2.21)P={u∈E∣u(t)≥0,t∈[a,σ2(b)]T}.
It is clear that P is a normal solid cone and E becomes a lattice under the natural ordering ≤.Lemma 2.9 (see [10]).
Assume that there existsh(t)∈P∖{θ}, such that
(2.22)G(t,s)≥h(t)G(τ,s),t,τ∈[a,σ(b)]T,s∈[a,b]T.
Suppose that there exists ψ*∈P*∖{θ}, such that ψ*=r-1(K)K*ψ*, ψ*(h(t))≢0. Then operator K satisfies H condition, where K is defined in (2.17).


## 3. Topological Degree for Sublinear Problem (1.1)

ForR>0, define TR={x∈E∣∥x∥<R}.In [17], Sun and Liu obtained the following abstract results.Lemma 3.1 (see [17]).
LetE be a Banach space, and let  P be a normal solid cone in  E,  w0∈E, and let  A:Pw0→Pw0 be a completely continuous operator. Suppose that there exists a positive bounded linear operator  B:E→E with  r(B)<1 and  u0∈P such that
(3.1)Ax≤Bx+u0,∀x∈Pw0.
Then there exists R0>0, such that the fixed-point index i(A,TR∩Pw0,Pw0)=1 for all R>R0.Lemma 3.2 (see [17]).
LetE be a Banach space, and let  P be a normal solid cone in  E,  w0∈E, and let  A:Pw0→Pw0 be a completely continuous operator. Suppose that there  exists  a positive bounded linear operator  B and a positive completely continuous operator  B1,  u0∈P,  r∈ℝ,  r>0, such that
(3.2)Ax≤Bx+u0,∀x∈P,Ax≥B1x,∀x∈∂Tr∩P.
If r(B)<1 and r(B1)≥1, then A has at least one nonzero positive fixed-point in P.Lemma 3.3 (see [17]).
LetE be a Banach space, let  P be a normal solid cone in  E,  w0∈int(-P), and let  A:Pw0→Pw0 be a completely continuous operator. Suppose that (i)
there  exists  a positive bounded linear operatorB with  r(B)<1 and  u0∈P such that
(3.3)Ax≤Bx+u0,∀x∈Pw0,(ii)
Aθ=θ, the Fréchet derivative  Aθ′ of  A at  θ exists, and  1 is not an eigenvalue of  Aθ′.
Then(1)
if the sum of the algebraic multiplicities for all eigenvalues ofAθ′ in (0,1) is an odd number, A has at least one nonzero fixed-point;(2)
ifr(Aθ′)>1 and A(P)⊂P, A(-P)⊂(-P), A has at least two nonzero fixed-points, one of which is positive, the other is negative;(3)
ifr(Aθ′)>1, the sum of the algebraic multiplicities for all eigenvalues of Aθ′ in (0,1) is an even number, and
(3.4)A(P∖{θ})⊂int⁡(P),A((-P)∖{θ})⊂int⁡(-P),
then A has at least three nonzero fixed-points, one of which is positive another is negative, and the third fixed-point is sign-changing.By means of the above abstract results, we have the following theorems.Theorem 3.4.
Suppose that there existM>0 and  η>0 such that
(3.5)f(t,u)≥-M,∀t∈[a,σ(b)]T,u≥0,(3.6)lim⁡|u|→∞sup⁡f(t,u)u≤λ1-η,uniformly ont∈[a,σ(b)]T.
Then problem (1.1)  has at least one solution.Proof.
It is easy to check thatA:E→E is a completely continuous operator. Choose 0<ɛ<η. Denote c=λ1-η+ɛ<λ1. According to (3.6), we can know that there exists R¯>0 such that
(3.7)f(t,u)u≤c,t∈[a,σ(b)]T,|u|≥R¯.
This implies that there exists constant number M0>0 such that
(3.8)f(t,u)≥cu-M0,t∈[a,σ(b)]T,u≤0,(3.9)f(t,u)≤cu+M0,t∈[a,σ(b)]T,u≥0.
Denote B=cK, where K is defined as (2.17). It is obvious that B:E→E is a positive bounded linear operator, and r(B)=cr(K)<λ1r(K)=1. By Lemma 2.7, we have that (I-B)-1 exists and is a positive bounded linear operator.
Setb>0, and let
(3.10)b(t)≡b,l(t)≡M0+M,w0(t)=-[(I-B)-1(Kl+b)](t),t∈[a,σ(b)]T.
It is obvious that w0∈(-P), and
(3.11)w0(t)-(Bw0)(t)=-(Kl)(t)-b(t),t∈[a,σ(b)]T.
For any u0≤0, it follows from (3.5) and (3.8) that
(3.12)f(t,u)≥cu0-M-M0,u≥u0,
which together with w0≤0 implies that
(3.13)f(t,u(t))≥cw0(t)-l(t),u(t)≥w0(t).
Note that K(P)⊂P. Combining (3.11) with (3.13) and (2.16), we have
(3.14)Au(t)=∫aσ(b)G(t,s)f(s,u(σ(s)))Δs≥∫aσ(b)G(t,s)[cw0(σ(s))-l(σ(s))]Δs≥∫aσ(b)G(t,s)[cw0(σ(s))-l(σ(s))]Δs-b(t)=w0(t),∀u(t)≥w0(t),t∈[a,σ2(b)]T.
This implies that A(Pw0)⊂Pw0.
Let(3.15)ϑ=sup⁡{|w0(t)|:t∈[a,σ(b)]T},κ=sup⁡{|f(t,u)|:t∈[a,σ(b)]T,-ϑ≤u≤0}.
It is easy to know from (3.9) that
(3.16)f(t,u)≤cu+M0+κ+cϑ,t∈[a,σ(b)]T,u≥-ϑ.
Thus,
(3.17)f(t,u(t))≤cu(t)+M0+κ+cϑ,∀u(t)≥w0(t),t∈[a,σ(b)]T.
Let e0(t)≡M0+κ+cϑ, u0(t)=(Ke0)(t), t∈[a,σ(b)]𝕋. It is clear that u0∈P. It follows from (3.17) that
(3.18)Au(t)≤Bu(t)+u0(t),∀u∈Pw0.
Thus, all conditions in Lemma 3.1 are satisfied; the conclusion then follows from Lemma 3.1.Theorem 3.5.
Suppose that(3.19)f(t,u)≥0,∀t∈[a,σ(b)]T,u≥0,
and there exists η>0 such that
(3.20)lim⁡u→∞sup⁡f(t,u)u≤λ1-η,lim⁡u→0inf⁡f(t,u)u≥λ1+η,uniformlyont∈[a,σ(b)]T.
Then problem (1.1)  has at least one positive solution.Proof.
It is clear thatA:P→P is a completely continuous operator. Choose 0<ε<η.
Denote(3.21)c=λ1-η+ε,c1=λ1+η-ε.
then c<λ1,c1>λ1. By (3.20), we obtain that there exists D>d>0 such that
(3.22)f(t,u)≤cu,t∈[a,σ(b)]T,u≥D,(3.23)f(t,u)≥c1u,t∈[a,σ(b)]T,0≤u≤d.
According to (3.22), we have
(3.24)f(t,u)≤cu+M¯,t∈[a,σ(b)]T,u≥0,
where
(3.25)M¯=sup⁡t∈[a,σ(b)]T,|u|≤D|f(t,u)|.
Denote B=cK, B1=c1K, where K is still defined as (2.17); then, B,B1:E→E are both completely continuous positive linear operators, and
(3.26)r(B)=cr(K)<λ1r(K)=1,r(B1)=c1r(K)>λ1r(K)=1.
Let v0=K(M¯). Then, v0∈P. It follows from (2.16), (3.19), (3.23), and (3.24) that
(3.27)Au(t)≤∫aσ(b)G(t,s)[cu(σ(s))+M¯]Δs=Bu(t)+v0(t),∀u∈P,t∈[a,σ(b)]T,Au(t)≥∫aσ(b)G(t,s)m1u(σ(s))Δs=B1u(t),∀u∈P,t∈[a,σ(b)]T,‖u‖≤d.
By Lemma 3.2, problem (1.1) has at least one positive solution.Remark 3.6.
Condition (3.20) implies that
(3.28)lim⁡u→+∞sup⁡f(t,u)u<λ1,lim⁡u→0+inf⁡f(t,u)u>λ1,uniformlyont∈[a,σ(b)]T.
Hence, Theorem 3.5 becomes Theorem 2.1 in [10]. Therefore, our main result generalizes Theorem 2.1 in [10].Theorem 3.7.
Assume that  (3.5)  and  (3.6)  hold. In addition, suppose that
(3.29)f(t,0)≡0,∀t∈[a,σ(b)]T,(3.30)lim⁡u→0f(t,u)u=λ,uniformlyont∈[a,σ(b)]T.
Then(i)
ifλn<λ<λn+1, where  n is an odd number, then problem  (1.1)  has at least one nontrivial solution;(ii)
ifλ>λ1 and
(3.31)f(t,u)u≥0,∀t∈[a,σ(b)]T,u∈R,
then problem (1.1)  has at least two nontrivial solutions, one of which is positive and the other is negative.Proof.
It suffices to verify that all the conditions of Lemma3.3 are satisfied.
Letw0 be defined as (3.10). Since Kl+b∈intP, we have w0∈int(-P). It follows from (3.5), (3.6), and the proof of Theorem 3.4 that A:Pw0→Pw0 is completely continuous and condition (i) of Lemma 3.3 holds. Furthermore, (3.29) implies that Aθ=θ; (3.30) implies that
(3.32)Aθ′u(t)=∫aσ(b)G(t,s)fu′(s,0)u(σ(s))Δs=λ∫aσ(b)G(t,s)u(σ(s))Δs,t∈[a,σ2(b)]T.
This shows that Aθ′=λK. Let ι denote the sum of the algebraic multiplicities for all eigenvalues of Aθ′ in (0,1). Since λn<λ<λn+1, we can obtain that 1 is not an eigenvalue of Aθ′ and ι=n is an odd number. Conclusion (1) of Lemma 3.3 holds.
Equqtion (3.31) implies that A(P)⊂P,A(-P)⊂(-P). By λ>λ1, we can know that 1 is not an eigenvalue of Aθ′, and r(Aθ′)=λr(K)>λ1r(K)=1. This implies that the conclusion (2) of Lemma 3.3 holds.Theorem 3.8.
Assume that  (3.5), (3.6), (3.29), and  (3.30)  hold. In addition, suppose that
(3.33)λn<λ<λn+1,wherenisanevennumber,f(t,u)u>0,∀t∈[a,σ(b)]T,u≠0,u∈R.
Then the problem (1.1)  has at least three nontrivial solutions, one of which is positive, another is negative, and the third solution is sign changing.Proof.
For convenience, let us introduce another ordered Banach space.
Letting,e1 be the first normalized eigenfunction of K corresponding to its first eigenvalue λ1, then ∥e1∥=1. It follows from [10, Lemma 1.4] that e1>0. Let
(3.34)E¯={x∈E∣thereexistsν>0,-νe1(t)≤x(t)≤νe1(t)}.
According to [20], we can obtain that E¯ is an ordering Banach space, P¯=E¯∩P is a normal solid cone, and K:E→E¯ is a linear completely continuous operator satisfying K(P∖{θ})⊂intP¯, where
(3.35)int⁡P¯={x∈P¯∣thereexistς1>0,ς2>0,ς1e1(t)≤x(t)≤ς2e1(t)}.
Lettingl(t) and B be as in Theorem 3.4, then l∈P∖{θ}. Let w0=-(I-B)-1(Kl). Since K(P)⊂P¯, we have Kl∈intP¯. Furthermore, we obtain w0∈int(-P¯). Noting that K(E)⊂E¯, we have A(E)⊂E¯.
Similarly to the proof of Theorem3.4, we have
(3.36)A(Pw0)⊂Pw0.
Furthermore, we can know that
(3.37)A(P¯w0)⊂P¯w0.
Since K:E→E¯ is completely continuous, we have that A:P¯w0→P¯w0 is completly continuous. According to Theorem 3.7, it is easy to show that conditions (i) and (ii) of Lemma 3.3 are satisfied, and Aθ′=λK.
Sinceλn<λ<λn+1, we have that 1 is not an eigenvalue of Aθ′ and λ>λ1. It follows that
(3.38)r(Aθ′)=λr(K)>λ1r(K)=1.
Moreover, we can know that the sum of the algebraic multiplicities for all eigenvalues of Aθ′ in (0,1) is n; this is an even number.
In the following, we check the condition (3.4).
By (3.33), we have
(3.39)f(t,u)>0,∀u>0,f(t,u)<0,∀u<0,∀t∈[a,σ(b)]T.
Notice that
(3.40)K(P∖{θ})⊂int⁡P¯.
It follows that
(3.41)A(P¯∖{θ})⊂int⁡P¯,A((-P¯)∖{θ})⊂int⁡(-P¯).
Therefore, condition (3.4) holds. The conclusion follows from Lemma 3.3 (3).


## 4. Topological Degree for Superlinear Problem (1.1)

In [17], Sun and Liu discussed computation for the topological degree about superlinear operators which are not cone mappings. The main tools are the partial ordering relation and the lattice structure. Their main results are the following theorems.Lemma 4.1 (see [17]).
LetE be a Banach space, and let  A:E→E be a completely continuous operator satisfying  A=BF, where  F is quasiadditive on lattice and  B is a positive bounded linear operator satisfying  H condition. Moreover, suppose that  P is a solid cone in  E and (i)
there exista1>r-1(B) and y1∈P, such that
(4.1)Fx≥a1x-y1,∀x∈P,(ii)
there exist0<a2<r-1(B) and y2∈P, such that
(4.2)Fx≥a2x-y2,∀x∈(-P).
In addition, there exist a bounded open set Ω, which contains θ, and a positive bounded linear operator B0 with r(B0)≤1, such that
(4.3)|Ax|≤B0|x|,∀x∈∂Ω.
Then A has at least one nonzero fixed-point.Lemma 4.2 (see [17]).
LetE be a Banach space, and let  A:E→E be a completely continuous operator satisfying  A=BF, where  F is quasiadditive on lattice and  B is a positive bounded linear operator satisfying  H condition. Moreover, suppose that  P is a solid cone in  E and conditions  (i)  and  (ii)  of  Theorem  4.1 are satisfied. In addition, assume that  Aθ=θ, the Fréchet derivative  Aθ′ of  A at  θ exists,  and  1 is not an eigenvalue of  Aθ′. Then  A has at least one nonzero fixed-point.
In [23], Sun obtained the following further result concerning sign-changing fixed-point.Lemma 4.3 (see [23]).
Let the conditions of  Theorem  4.2be satisfied. In addition, suppose that the sum of the algebraic multiplicities for all eigenvalues ofAθ′ in the interval  (0,1) is an even number, and
(4.4)A(P∖{θ})⊂int⁡P,A((-P)∖{θ})⊂-int⁡P.
Then A has at least three nonzero fixed-points, one of which is positive, another is negative, the third one is a sign-changing fixed-point.In the following, we will apply the above three abstract theorems to problem (1.1); our main results are as follows.Theorem 4.4.
Assume thatf:[a,σ(b)]𝕋×ℝ→ℝ is continuous. Suppose that there exists  η>0 such that
(4.5)lim⁡u→-∞sup⁡f(t,u)u≤λ1-η,uniformlyont∈[a,σ(b)]T,(4.6)lim⁡u→+∞inf⁡f(t,u)u≥λ1+η,uniformlyont∈[a,σ(b)]T,(4.7)lim⁡u→0sup⁡|f(t,u)u|≤λ1-η,uniformlyont∈[a,σ(b)]T.
Then problem (1.1)  has at least one nontrivial solution.Proof.
It is clear thatA:E→E is completely continuous. According to (4.6) and (4.5), we have that there exists R1>0, such that
(4.8)f(t,u)≥(λ1+12η)u,t∈[a,σ(b)]T,u≥R1,f(t,u)≥(λ1-12η)u,t∈[a,σ(b)]T,u≤-R1.
Hence, there exists M¯0>0 such that
(4.9)f(t,u)≥(λ1+12η)u-M¯0,t∈[a,σ(b)]T,u≥0,f(t,u)≥(λ1-12η)u-M¯0,t∈[a,σ(b)]T,u≤0.
According to the proof of Theorem 3.1 in [10], we have
(4.10)G(t,s)≥x(t)y(t)max⁡t∈[a,σ2(b)]Tx(t)max⁡t∈[a,σ2(b)]Ty(t)G(τ,s),t,τ∈[a,σ2(b)]T,s∈[a,σ(b)]T.
By Lemma 2.9, we can know that linear operator K satisfies the H condition.
LetFu(t)=f(t,u(t)) for u∈E. It is easy to check that F satisfies (2.6) and A=KF. It follows from (4.9) that (4.1) and (4.2) hold, where a1=λ1+(1/2)η>λ1=r-1(K) and a2=λ1-(1/2)η<λ1=r-1(K).
By (4.7), we obtain that there exists r0>0 such that
(4.11)|f(t,u)|≤(λ1-12η)|u|,t∈[a,σ(b)]T,|u|≤r0.
Therefore,
(4.12)|Au|≤(λ1-12η)K|u|,∀u∈E,‖u‖=r0.
Thus, (4.3) holds. An application of Lemma 4.1 shows that problem (1.1) has at least one nontrivial solution.Remark 4.5.
If we assume thatf:[a,σ(b)]𝕋×ℝ+→ℝ+, condition (4.5), (4.6) will be removed. Therefore, Theorem 4.4 generalizes and extends Theorem 3.4 in [10].Theorem 4.6.
Assume thatf:[a,σ(b)]𝕋×ℝ→ℝ is continuous and  (4.5) holds. In addition, suppose that
(4.13)f(t,0)≡0,∀t∈[a,σ(b)]T,lim⁡u→0f(t,u)u=λ,uniformlyont∈[a,σ(b)]T,
and λ∉{λ1,λ2,…,λn,…}. Then problem (1.1)  has at least one nontrivial solution.Proof.
By Theorem4.4, we only need to verify the condition “moreover” of Lemma 4.2. According to (4.13), we can know that Aθ=θ and Aθ′=λK. Since K has the sequence of eigenvalues {λn}n=1∞, it is easy to see that {λn/λ}n=1∞ is the sequence of all eigenvalues of Aθ′. Therefore, 1 is not an eigenvalue of Aθ′. The conclusion of Theorem 4.6 follows from Lemma 4.2.Theorem 4.7.
Assume thatf:[a,σ(b)]𝕋×ℝ→ℝ and  (4.5), (4.6), and (4.13) hold. If
(4.14)λn<λ<λn+1,wherenisanevennumber,f(t,u)u>0,∀t∈[a,σ(b)]T,u≠0,u∈R.
Then the problem (1.1) has at least three nontrivial solutions, one of which is positive, another is negative, and the third solution is sign changing.Proof.
From the proof of Theorem4.6, it is easy to show that all conditions of Lemma 4.2 are satisfied and Aθ′ has the sequence of eigenvalues {λn/λ}n=1∞.
Sinceλn<λ<λn+1 and n is an even number, we can know that the sum of the algebraic multiplicities for all eigenvalues of Aθ′ in (0,1) is an even number. Condition (4.14) implies that
(4.15)A((-P)∖{θ})⊂-int⁡P,A(P∖{θ})⊂int⁡P.
An application of Lemma 4.3 shows that problem (1.1) has at least three nontrivial solutions, one of which is positive, another is negative, and the third solution is sign changing.
Now let us end this section by the following two examples.Example 4.8.
Let𝕋=ℤ. Considering the following BVP:
(4.16)Δ2u(t-1)-0.381966u+4u2+u4=0,t∈[1,4],u(0)=u(5)=0,
where [1,4] is the discrete interval {1,2,3,4}, Δu(t)=u(t+1)-u(t), Δ2u(t)=Δ(Δu(t)).
By [26, Remark 2.3 and Lemma 2.9], we can know that λk=4sin2(kπ/10), k=1,2,3,4. In addition, the algebraic multiplicity of each eigenvalue λk is equal to 1.
Settingδ=0.00000001, it is easy to check that
(4.17)lim⁡sup⁡u→0|-0.381966u+4u2+u4u|=0.381966=0.38196601-0.00000001=λ1-δ,lim⁡sup⁡u→-∞-0.381966u+4u2+u4u=lim⁡sup⁡u→-∞(-0.381966+4u+u3)=-∞,lim⁡sup⁡u→+∞(-0.381966+4u+u3)=+∞.
Therefore, all conditions in Theorem 4.4 are satisfied. By Theorem 4.4, BVP (4.16) has at least one nontrivial solution.Example 4.9.
Let𝕋=ℤ. Consider the following BVP:
(4.18)Δ2u(t-1)+u(0.39+0.007arctan⁡u)=0,t∈[1,4],u(0)=u(5)=0.
In virtue of Example4.8, we can know that λ1=0.38196601, λ2=1.381966011, λ3=3.553057584, and λ4=6.316546812.
Settingδ=0.00000001, by direct calculation, we have
(4.19)lim⁡sup⁡u→-∞u(0.39+0.007arctan⁡u)u=0.39-0.007π2=0.39-0.010995574≤0.38196601-0.00000001=λ1-δ,lim⁡sup⁡u→+∞u(0.39+0.007arctan⁡u)u=0.39+0.007×π2=0.400995574≥0.38196601+0.00000001=λ1-δ,lim⁡u→0(0.39+0.007arctan⁡u)=0.39∉{0.38196601,1.381966011,3.553057584,6.316546812}.
Therefore, all conditions in Theorem 4.6 are satisfied. By Theorem 4.6, BVP (4.18) has at least one nontrivial solution.


## 5. Topological Degree for Unilaterally Asymptotically Linear Problem (1.1)

In [18], Liu and Sun presented some methods of computing the topological degree for unilaterally asymptotically linear operators by using the lattice structure. Their main results are as follows.Lemma 5.1 (see [18]).
LetP be a normal cone in  E,  and let  A:E→E be completely continuous and quasiadditive on lattice. Suppose that there exist  u*,u1∈P and a positive bounded linear operator  L0:E→E with  r(L0)<1, such that
(5.1)Ax≥-u*,∀x∈P,Ax≥L0x-u1,∀x∈(-P).
In addition, suppose that AP′ exists, Aθ=θ, the Fréchet derivative Aθ′ of A at θ exists, and 1 is not an eigenvalue of Aθ′. Then A has at least one nontrivial fixed-point, provide that one of the following is satisfied:(i)
r(AP′)>1 and 1 is not an eigenvalue of AP′ corresponding to a positive eigenvector;(ii)
r(AP′)<1 and the sum of the algebraic multiplicities for all eigenvalues of Aθ′, lying in the interval (0,1), is an odd number.Lemma 5.2 (see [18]).
Suppose that(i)
A is strongly increasing on P and -P;(ii)
bothAP′ and A(-P)′ exist with r(AP′)>1 and r(A(-P)′)>1; 1 is not an eigenvalue of AP′ or A(-P)′ corresponding to a positive eigenvector;(iii)
Aθ=θ; the Fréchet derivative Aθ′ of A at θ is strongly positive, and r(Aθ′)<1;(iv)
the Fréchet derivativeA∞′ of A at ∞ exists; 1 is not an eigenvalue of A∞′; the sum of the algebraic multiplicities for all eigenvalues of A∞′, lying in the interval (0,1) is an even number.
ThenA has at least three nontrivial fixed-points, containing one sign-changing fixed-point.Through this section, the following hypotheses are needed:(E1)
limu→+∞(f(t,u)/u)=p uniformly on t∈[a,σ(b)]𝕋;(E2)
limu→-∞(f(t,u)/u)=q uniformly on t∈[a,σ(b)]𝕋;(E3)
f(t,0)≡0, limu→0(f(t,u)/u)=λ uniformly on t∈[a,σ(b)]𝕋.Our main results are as follows.Theorem 5.3.
Suppose thatf satisfies  (E1)–(E3). Then problem  (1.1)  has at least one nontrivial solution provided one of the following conditions is satisfied:(i)
p>λ1, 0<q<λ1,p, λ≠λk, k=1,2,…;(ii)
0<p<λ1, q>λ1, q, λ≠λk, k=1,2,…;(iii)
0<p<λ1, 0<q<λ1, λn<λ<λn+1, n is an odd number.Proof.
It follows from Definition2.3 that
(5.2)AP′u(t)=pKu(t),∀t∈[a,σ2(b)]T.A(-P)′u(t)=qKu(t),∀t∈[a,σ2(b)]T.
Furthermore, we have
(5.3)Aθ′u(t)=∫aσ(b)G(t,s)fu′(s,0)u(σ(s))Δs=λ∫aσ(b)G(t,s)u(σ(s))Δs=λKu(t),∀t∈[a,σ2(b)]T.
Assume that condition (i) is satisfied. By p>0, we can obtain that there exists M¯1>0 such that
(5.4)f(t,u)≥-M¯1,∀t∈[a,σ(b)]T,u≥0.
So, we have
(5.5)Au(t)≥-KM¯1,∀u≥0.
Since q<λ1, we can obtain that q+η¯<λ1 for some η¯>0. (E2) implies that there exists μ0>0 such that
(5.6)f(t,u)≥(q+η¯)u-μ0,∀t∈[a,σ(b)]T,u≤0.
Consequently,
(5.7)Au(t)=∫aσ(b)G(t,s)f(t,u(σ(s)))Δs≥∫aσ(b)G(t,s)[(q+η¯)u(σ(s))-μ0]Δs=(q+η¯)Ku-Kμ0,∀u∈(-P).
Let D0=(q+η¯)K, then
(5.8)r(D0)=(q+η¯)r(K)<λ1r(K)=1.
Equation (5.5) and (5.7) imply that (5.1) holds.
Since the sequences of all eigenvalues ofAθ′, AP′, and A(-P)′ are {λn/λ}n=1∞, {λn/p}n=1∞, and {λn/q}n=1∞, respectively. It is easy to see from p,λ≠λk, k=1,2,… that 1 is not an eigenvalue of Aθ′ or AP′. Lemma 5.1(i) assures that A has at least one nontrivial fixed-point, and hence problem (1.1) has at least one nontrivial solution. Similarly, we can prove that the conclusion holds in the case that condition (ii) or (iii) is satisfied.Theorem 5.4.
Lettingf satisfy  (E1)–(E3) and  p=q=:χ. In addition, suppose that (i)
f(t,u) is strictly increasing on u for fixed t∈[a,σ(b)]𝕋;(ii)
λn<χ<λn+1 and n is an even number;(iii)
0<λ<λ1.
Then problem  (1.1)  has at least three nontrivial solutions, containing a sign-changing solution.Proof.
Similarly to the proof of Theorem3.8, we still need to use another ordered Banach space E¯.
From the proof of Theorem5.3 and Theorem 3.8, we know that Aθ′=λK and AP′=A(-P)′=χK. P¯⊂P gives that AP¯′=A(-P¯)′=χK, where P¯ is as in Theorem 3.8. It is easy to see that A∞′=χK. Evidently, e1∈intP¯. It follows from the condition (i) and K(P∖{θ})⊂intP¯ that A is strongly increasing.
Condition (ii) implies that 1 is not an eigenvalue ofAP′ or A(-P)′, and r(AP′)=rA(-P)′=χr(K)>λ1r(K)=1. By the same method, we can show that Aθ′ is strongly increasing. Moreover, by (iii), we have r(Aθ′)=λr(K)<λ1r(K)=1. At last, from the proof of Theorem 5.3, we obtain that the sequence of all eigenvalues of A∞′ is {λn/χ}n=1∞. According to (ii), we can know that the sum of the algebraic multiplicities for all eigenvalues of A∞′ in (0,1) is an even number. Hence, all the conditions of Lemma 5.2 are satisfied. The conclusion follows from Lemma 5.2.


## 6. Conclusion

In this paper, some existence results of sign-changing solutions for dynamic equations on time scales are established. As far as we know, there were few papers that studied this topic. Therefore, our results of this paper are new. It is natural for us to apply the method of this paper to more general form of dynamic equations. However, we note that the study of sign-changing solutions for time scales boundary value problems relies on the spectrum structure of linear dynamic equations on time scales. The reader is referred to [27, 28] for further works.In most papers on time scales boundary value problem, in order to obtain the existence of positive solutions by using fixed-point theorems on a cone, the nonlinear term, which appears in the right-hand side of the equation, is required to be nonnegative. For our main results, the nonlinear term may be a sign-changing function, thus, the integral operator is not necessary to be a cone mapping. We should point out that Sun and Zhang [29] have studied nontrivial solutions of the singular superlinear Sturm-Liouville problems of ordinary differential equations; the ideas of this paper come from [17–19, 29].In [30], Luo focused on the application of spectral theory of linear dynamic problems to the existence of positive solutions for nonlinear weighted eigenvalue problem on time scales. The author also discussed the existence of nodal solutions and the global structure of the solution for nonlinear eigenvalue problem on time scales by using the eigenvalues of the corresponding linear problem and global bifurcation theory. Up to now, no results on the spectrum structure of linear dynamic equations have been established for multipoint boundary value problems. This is an open problem.



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