# Positive Solutions for System of First-Order Dynamic Equations

Authors: Da-Bin Wang; Jian-Ping Sun; Xiao-Jun Li
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

We study the existence of positive solutions to the system of nonlinear first-order periodic boundary value problems on time scalesxΔ(t)+P(t)x(σ(t))=F(t,x(σ(t))), t∈[0,T]T, x(0)=x(σ(T)), by using a well-known fixed point theorem in cones. Moreover, we characterize the eigenvalue intervals for xΔ(t)+P(t)x(σ(t))=λH(t)G(x(σ(t))), t∈[0,T]T, x(0)=x(σ(T)).


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## Body

## 1. Introduction

On the one hand, periodic boundary value problems (PBVPs for short) for differential equations and difference equations have received much attention in the literature. See, for example, [1–17] and references therein. On the other hand, recently, the study of dynamic equations on time scales has became a new important branch (see, e.g., [18–22]). Naturally, some authors have focused their attention on the BVPs or PBVPs for dynamic equations on time scales [23–32], in which the works in [25, 27] concerned the singular problems on time scales (concerned the study, theory, and applications of boundary value problems involving singularities of differential equations, please see [33]). In particular, for the first-order PBVP of dynamic equations on time scales
(1.1)xΔ(t)+p(t)x(σ(t))=f(t,x(σ(t))),t∈[0,T]T,x(0)=x(σ(T)),
the works in [24, 30] obtained the existence of at least one solution. The methods involved novel inequalities and the well-known Schaefer fixed point theorem [34].In [31], Sun and Li obtained the some existence and multiplicity criteria of positive solutions to the following first-order PBVP on time scales
(1.2)xΔ(t)+p(t)x(σ(t))=f(x(t)),t∈[0,T]T,x(0)=x(σ(T))
by using Guo-Krasnoselskii fixed point theorem [35], Schauder fixed point theorem [35], and Leggett-Williams fixed point theorem [36].Very recently, Sun and Li [32] considered the following first-order PBVP on time scales
(1.3)xΔ(t)+p(t)x(σ(t))=λf(x(t)),t∈[0,T]T,x(0)=x(σ(T)),
where λ>0. Some existence, multiplicity, and nonexistence criteria of positive solutions were established. The main tool used in [32] is the fixed point index theory [37].However, up to now, there are few works for studying systems of PBVP of dynamic equations on time scales [29]. In [29], Sun and Li considered the following system of nonlinear first-order PBVP on time scales
(1.4)uiΔ(t)+gi(t,u1(σ(t)),u2(σ(t)),…,un(σ(t)))=0,t=[0,T]T,ui(0)=ui(σ(T)),i=1,2,…,n.
By using a fixed point theorem for completely continuous operators [35], they obtained some existence criteria of one positive solution to the system.In this paper, we study the existence of positive solutions for the following system of first-order PBVP on time scale(1.5)xΔ(t)+P(t)x(σ(t))=F(t,x(σ(t))),t∈[0,T]T,x(0)=x(σ(T)),
where T is a time scale, [0,T]T means [0,T]∩T (here T>0 and 0,T∈T), x=(x1,x2,…,xn)𝒯(𝒯 stands for the transpose),P(t)=diag[p1(t),p2(t),…,pn(t)], and F=(f1,f2,…,fn)𝒯. For i∈{1,2,…,n},pi:[0,T]T→(0,∞) is right-dense continuous and fi:[0,T]T×[0,∞)n→[0,∞) is continuous.The main results in this paper are proved by a fixed point theorem (see [37]) for compact maps on conical shells which are different from those used in [24, 29–32]. To do this, we extend the ideas introduced by Lan and Webb in [38] (see also [39]) to the general time scales. This approach was used in [5] for the continuous case and in [6] for the discrete case.As an application, we study the following eigenvalue problem:(1.6)xΔ(t)+P(t)x(σ(t))=λH(t)G(x(σ(t))),t∈[0,T]T,x(0)=x(σ(T)),
where H(t)=diag[h1(t),h2(t),…,hn(t)],G(x)=[g1(x),g2(x),…,gn(x)]𝒯, and λ>0 is a positive parameter. We prove that PBVP (1.6) has at least one positive solution for each λ in an explicit eigenvalue interval. Recently, several eigenvalue characterization for different kinds of boundary value problems have appeared and we refer the readers to [32, 40–42].It is noticed that the results obtained in this paper generalize some results in [30–32] to some degree.In the remainder of this section, we state a fixed point theorem for compact maps on conical shell [37].Now we recall a completely continuous operator which transforms every bounded set into a relatively compact set. IfD is a subset of X, we write DK=D∩K and ∂KD=(∂D)∩K.Theorem 1.1 (see [37]).
LetX be a Banach space with a cone K. Assume that Ω1,Ω2 are open bounded subsets of X with ΩK1≠ϕ,Ω1¯K⊂ΩK2. Let Φ:Ω2¯K→K be a continuous and compact operator such that(i)
∥Φx∥≤∥x∥ for x∈∂KΩ1 (or x∈∂KΩ2);(ii)
there existse∈K∖{0} such that x≠Φx+λe for x∈∂KΩ2 (or x∈∂KΩ1) and λ>0.
ThenΦ has a fixed point in (Ω2¯K∖Ω1¯K).Remark 1.2.
In Theorem1.1, the use of (ii) gives better results than the use of the common assumption ∥Φx∥≥∥x∥ for x∈∂KΩ2 (or x∈∂KΩ1).


## 2. Preliminaries

Let(2.1)A={x∣x:[0,T]T→Riscontinuous}.Forhi∈A, we consider the following linear PBVP:
(2.2)xiΔ(t)+pi(t)xi(σ(t))=hi(t),t∈[0,T]T,xi(0)=xi(σ(T)).Lemma 2.1 (see [30]).
Forhi∈A,i=1,2,…,n, the PBVP (2.2) has a unique solution, which can be written by
(2.3)xi(t)=1epi(t,0)[∫0tepi(s,0)hi(s)Δs+1epi(σ(T),0)-1∫0σ(T)epi(s,0)hi(s)Δs],t∈[0,σ(T)]T.Remark 2.2.
By Lemma2.1, for hi∈A,i=1,2,…,n, the PBVP (2.2) has a unique solution:
(2.4)xi(t)=∫0σ(T)Gi(t,s)hi(s)Δs,t∈[0,σ(T)]T,
where
(2.5)Gi(t,s)={epi(s,t)epi(σ(T),0)epi(σ(T),0)-1,0≤s≤t≤σ(T),epi(s,t)epi(σ(T),0)-1,0≤t<s≤σ(T).Lemma 2.3.
LetGi(t,s) be defined as Remark 2.2; then
(2.6)Ai≜1epi(σ(T),0)-1≤Gi(t,s)≤epi(σ(T),0)epi(σ(T),0)-1≜Bi,i=1,2,…,n.Let(2.7)B={x∣x:[0,σ(T)]T→Riscontinuous}
with the norm |x|0=maxt∈[0,σ(T)]T|x(t)|, and X=Bn, for any x=(x1,x2,…,xn)∈X, its norm
(2.8)∥x∥=max{|x1|0,|x2|0,…,|xn|0},
and then X is a Banach space.Let(2.9)K={x=(x1,x2,…,xn)∈X:xi(t)≥0,t∈[0,σ(T)]T,xi(t)≥δi|xi|0,∀i=1,2,…,n},
where δi=Ai/Bi=epi(0,σ(T))∈(0,1). It is not difficult to verify that K is a cone in X.We define an operatorΦ:K→X as follows:
(2.10)(Φx)=(Φ1x,Φ2x,…,Φnx)𝒯,
where
(2.11)(Φix)(t)=∫0σ(T)Gi(t,s)fi(s,x(σ(s)))Δs,t∈[0,σ(T)]T.By Lemma2.1 and Remark 2.2, it is easy to see that fixed points of Φ are the solutions to the PBVP (1.5).Lemma 2.4.
Φ:K→K is completely continuous.Proof.
First, we assert thatΦ:K→X is completely continuous.
The proof is divided into three steps.
Step 1.
Let{xn}n=1∞ be a sequence such that limn→∞xn=x in K. Then for all i∈{1,2,…,n}, we have
(2.12)|(Φixn)(t)-(Φix)(t)|=|∫0σ(T)Gi(t,s)[fi(s,xn(σ(s)))-fi(s,x(σ(s)))]Δs|≤Bi∫0σ(T)|fi(s,xn(σ(s)))-fi(s,x(σ(s)))|Δs.
Sincefi(t,x) is continuous in x, we have |(Φixn)(t)-(Φix)(t)|→0, which leads to |Φixn-Φix|0→0(n→∞), so we get ∥Φxn-Φx∥→0(n→∞). That is, Φ:K→X is continuous.Step 2.
To show thatΦ maps bounded sets into bounded sets in X,let B⊂K be a bounded set. Then, for t∈[0,σ(T)]T and any x∈B, we have
(2.13)|(Φix)(t)|=|∫0σ(T)Gi(t,s)fi(s,x(σ(s)))Δs|≤Bi∫0σ(T)|fi(s,x(σ(s)))|Δs.
In virtue of the continuity of fi(t,x), we can conclude that Φix is bounded uniformly for all i∈{1,2,…,n}, which leads to Φx being bounded uniformly, and so Φ(B) is a bounded set.Step 3.
To show thatΦ maps bounded sets into equicontinuous sets of X,let t1,t2∈[0,σ(T)]T,x∈B, and then for all i∈{1,2,…,n}(2.14)|(Φix)(t1)-(Φix)(t2)|≤∫0σ(T)|Gi(t1,s)-Gi(t2,s)||fi(s,x(σ(s)))|Δs.
The right-hand side tends to uniformly zero as|t1-t2|→0, which imply Φ maps bounded sets into equicontinuous sets of X.
Consequently, Steps1–3 together with the Arzela-Ascoli Theorem show that Φ:K→X is completely continuous.
Next, to show thatΦ maps K into K, let x∈K, by Lemma 2.3; we have
(2.15)(Φix)(t)≥0,i=1,2,…,n,(Φix)(t)≤Bi∫0σ(T)fi(s,x(σ(s)))Δs,
and this implies that
(2.16)|Φix|0=maxt∈[0,σ(T)]T|(Φix)(t)|≤Bi∫0σ(T)fi(s,x(σ(s)))Δs.
On the other hand, from Lemma2.3 we have
(2.17)(Φix)(t)≥Ai∫0σ(T)fi(s,x(σ(s)))Δs.
Therefore,(2.18)(Φix)(t)≥Ai∫0σ(T)fi(s,x(σ(s)))Δs=δiBi∫0σ(T)fi(s,x(σ(s)))Δs≥δi|Φix|0.
That isΦ(K)⊂K.


## 3. Existence of Positive Solutions for the PBVP (1.5)

In this section, we establish the existence of positive solutions for the PBVP (1.5). First we extend the ideas introduced by Lan and Webb in [38, 39] to the general time scale.Forr>0, we define the open sets
(3.1)Ωr={x∈X∣mint∈[0,σ(T)]Txi(t)<δir∀i=1,2,…,n},Br={x∈X∣∥x∥<r}.Lemma 3.1.
Ωr,Br defined above have the following properties.(a)
ΩKr and BKr are open relative to K.(b)
BKδr⊂ΩKr⊂BKr, here δ=min{δi,i=1,2,…,n}.(c)
x∈∂KΩr if and only if x∈K and mint∈[0,σ(T)]Txj(t)=δjr for some j∈{1,2,…,n} and mint∈[0,σ(T)]Txi(t)≤δir for each i∈{1,2,…,n}.(d)
Ifx∈∂KΩr, then δjr≤xj(t)≤r,t∈[0,σ(T)]T for some j∈{1,2,…,n} and 0≤xi(t)≤r,t∈[0,σ(T)]T for each i∈{1,2,…,n}. Moreover, |xi|0≤r.(e)
For eachρ>r, the following relations hold:
(3.2)ΩKr=(Ωr∩Bρ)K,Ωr¯K=(Ωr∩Bρ¯)K.Proof.
(a) and (c) are obvious. So, we only prove that (b), (d), and (e) hold.
First we assert (b). Letx∈BKδr; then for each i∈{1,2,…,n}, we have |xi|0<δr, so mint∈[0,σ(T)]Txi(t)<δr≤δir, and x∈ΩKr. On the other hand, if x∈ΩKr, then for each i∈{1,2,…,n}, we have mint∈[0,σ(T)]Txi(t)<δir and xi(t)≥δi|xi|0 for t∈[0,σ(T)]T. So |xi|0<r, that is, ΩKr⊂BKr. Hence (b) holds.
Next, we assert (d). Letx∈∂KΩr; so we have from (c) that there exists j∈{1,2,…,n} such that
(3.3)δj|xj|0≤mint∈[0,σ(T)]Txj(t)=δjr.
Thus|xj|0≤r and δjr≤xj(t)≤r,t∈[0,σ(T)]T. Furthermore notice for each i∈{1,2,…,n} that δi|xi|0≤mint∈[0,σ(T)]Txi(t)≤δir, so |xi|0≤r and 0≤xi(t)≤r for t∈[0,σ(T)]T; that is, (d) holds.
Finally we assert (e). From (b), the first equality is obvious. Now we prove the second equality.
Letx∈Ωr¯K; then from (c), we have that
(3.4)δi|xi|0≤mint∈[0,σ(T)]Txi(t)≤δir<δiρ,i=1,2,…,n.
So|xi|0<ρ,i=1,2,…,n, and this implies that x∈(Ωr¯∩Bρ)∩K. Since Ωr and Bρ are open sets, we have Ωr¯∩Bρ⊂Ωr∩Bρ¯. Thus x∈(Ωr∩Bρ¯)K, that is, Ωr¯K⊆(Ωr∩Bρ¯)K. The reverse inclusion is trivial.Remark 3.2.
It is clear that the setsΩr are unbounded sets for each r>0; so we cannot use Theorem 1.1 with Ωr directly. However we will be able to apply Theorem 1.1 with ΩKr since (e) holds.Theorem 3.3.
Suppose the following.(H1)
For eachi=1,2,…,n, there exist a constant α>0 and a continuous function ψi:[0,T]T→(0,∞) such that
(3.5)fi(t,x)≥δiαψi(t),∀t∈[0,T]T,0≤xl≤α(l∈{1,2,…,n}∖{i}),δiα≤xi≤α,mint∈[0,σ(T)]T∫0σ(T)Gi(t,s)ψi(s)Δs≥1.(H2)
For eachi=1,2,…,n, there exist a constant β>0 and a continuous function χi:[0,T]T→(0,∞) such that
(3.6)fi(t,x)≤βχi(t),∀t∈[0,T]T,0≤xi≤β,maxt∈[0,σ(T)]T∫0σ(T)Gi(t,s)χi(s)Δs≤1.
Then, the following results hold:(a)
ifβ<δα, then the PBVP (1.5) has at least one positive solution x satisfying
(3.7)β≤∥x∥≤α;(b)
ifβ>α, then the PBVP (1.5) has at least one positive solution x satisfying
(3.8)δα≤∥x∥≤β.Proof.
Now we assert that the conditions of Theorem1.1 are satisfied.
First, we assert that∥Φx∥≤∥x∥ for x∈∂KBβ.
For anyx∈∂KBβ, we have |xi|0≤β for each i∈{1,2,…,n}. Fix i∈{1,2,…,n}. Then from (H2) we obtain, for each t∈[0,σ(T)]T,(3.9)(Φix)(t)=∫0σ(T)Gi(t,s)fi(s,x(σ(s)))Δs≤β∫0σ(T)Gi(t,s)χi(s)Δs≤βmaxt∈[0,σ(T)]T∫0σ(T)Gi(t,s)χi(s)Δs≤β.
Hence,|Φix|0≤∥x∥ for each i∈{1,2,…,n}. This implies ∥Φx∥≤∥x∥ for x∈∂KBβ.
Next, we assert that there existse∈K∖{0} such that x≠Φx+λe, for all x∈∂KΩα and all λ>0.
Lete(t)≡1; so e∈K∖{0}. Suppose that there exist x∈∂KΩα and λ>0 such that x=Φx+λe. Since x∈∂KΩα, then from Lemma 3.1(d) there exists j∈{1,2,…,n} with δjα≤xj(t)≤α,t∈[0,σ(T)]T, and 0≤xi(t)≤α for t∈[0,σ(T)]T and i∈{1,2,…,n}∖{j}.
Hence, from (H1) we have
(3.10)xj(t)=(Φjx)(t)+λ=∫0σ(T)Gj(t,s)fj(s,x(σ(s)))Δs+λ≥αδj∫0σ(T)Gj(t,s)ψj(s)Δs+λ≥αδjmint∈[0,σ(T)]T∫0σ(T)Gj(t,s)ψj(s)Δs+λ≥αδj+λ.
Thus,mint∈[0,σ(T)]Txj(t)≥αδj+λ>αδj, contradicting the statement of Lemma 3.1(c). That is, there exists e∈K∖{0} such that x≠Φx+λe, for all x∈∂KΩα and all λ>0.
Ifβ<δα, then from Lemma 3.1 we have that Bβ¯K⊂BKδα⊂ΩKα, and therefore it follows from Theorem 1.1 that Φ has at least one fixed point x∈ΩKα¯∖BKβ. Hence ∥x∥≥β and mint∈[0,σ(T)]Txi(t)≤δiα for each i∈{1,2,…,n}. On the other hand, δi|xi|0≤mint∈[0,σ(T)]Txi(t)≤δiα and therefore |xi|0≤α for each i∈{1,2,…,n}. This implies that ∥x∥≤α.
Ifβ>α, then we have that Ωα¯K⊂BKβ, and therefore Theorem 1.1 guarantees the existence of at least one fixed point x∈Bβ¯K∖ΩKα of Φ. So, we obtain δα≤∥x∥≤β.


## 4. Eigenvalue Interval of PBVP (1.6)

In this section, we characterize the eigenvalue intervals of system (1.6) by employing Theorem 3.3.First we establish one existence result for the following system:(4.1)xΔ(t)+P(t)x(σ(t))=H(t)G(x(σ(t))),t∈[0,T]T,x(0)=x(σ(T)),
where H(t)=diag[h1(t),h2(t),…,hn(t)],G(x)=[g1(x),g2(x),…gn(x)]𝒯.For eachi=1,2,…,n, we assume the following.(H3)
gi:[0,∞)n→[0,∞) is continuous with gi(x)>0 for ∥x∥>0.(H4)
hi:[0,T]T→[0,∞) is continuous and ∫0σ(T)Gi(t,s)hi(s)Δs>0.Theorem 4.1.
Suppose that conditions (H3) and (H4) hold. Then the PBVP (4.1) has at least one positive solution x with x not identically vanishing on [0,σ(T)]T if one of the following conditions holds:(H5)
0≤g0i<Ci-1 and Di-1<g∞i≤∞, i=1,2,…,n;(H6)
0≤g∞i<Ci-1 and Di-1<g0i≤∞, i=1,2,…,n;where g0i=limx→0+(gi(x)/∥x∥),g∞i=limx→∞(gi(x)/∥x∥),i=1,2,…,n, and
(4.2)Ci=maxt∈[0,σ(T)]T∫0σ(T)Gi(t,s)hi(s)Δs,Di=mint∈[0,σ(T)]T∫0σ(T)Gi(t,s)hi(s)Δs.Proof.
To see this, we will apply Theorem3.3 with fi(t,x)=hi(t)gi(x),i=1,2,…,n. Suppose that (H5) holds; then there exists β>0 such that gi(x)≤Ci-1β for 0<∥x∥≤β.
Chooseχi(t)=Ci-1hi(t) for i=1,2,…,n. Fix i∈{1,2,…,n}. Then fi(t,x)=hi(t)gi(x)≤Ci-1βhi(t)=βχi(t) if t∈[0,σ(T)]T and 0<xi≤β and
(4.3)∫0σ(T)Gi(t,s)χi(s)Δs=Ci-1∫0σ(T)Gi(t,s)hi(s)Δs≤Ci-1maxt∈[0,σ(T)]T∫0σ(T)Gi(t,s)hi(s)Δs=1.
Thus hypothesis (H2) holds.
From the second part of (H5), there exists α>0 such that δiα>β and gi(x)≥Di-1δiα for xi≥δiα,i=1,2,…,n.
Chooseψi(t)=Di-1hi(t); then
(4.4)fi(t,x)=hi(t)gi(x)≥Di-1δiαhi(t)=δiαψi(t),ift∈[0,σ(T)]T,xi≥δiα
(so in particular for δiα≤xi≤α) and
(4.5)∫0σ(T)Gi(t,s)ψi(s)Δs=Di-1∫0σ(T)Gi(t,s)hi(s)Δs≥Di-1mint∈[0,σ(T)]T∫0σ(T)Gi(t,s)hi(s)Δs=1.
This implies that hypothesis (H1) holds. The result now follows from Theorem 3.3.
The case when (H6) holds is similar. So we omit here.Remark 4.2.
By the proof of Theorem4.1, we emphasize that Theorem 3.3 is very easy to apply; roughly speaking, it only requires an integral representation of the considered equation and some bounds for the kernel of the equivalent integral equation. So, in this way, the corresponding existence result, that is, [31, Theorem  4.1], is improved.Theorem 4.3.
Suppose that conditions (H3) and (H4) hold. Then the PBVP (1.6) has at least one positive solution for each
(4.6)λ∈(1Dmini=1,2,…,n{g∞i},1Cmaxi=1,2,…,n{g0i})
if 1/Dmini=1,2,…,n{g∞i}<1/Cmaxi=1,2,…,n{g0i}. The same result remains valid for each
(4.7)λ∈(1Dmini=1,2,…,n{g0i},1Cmaxi=1,2,…,n{g∞i})
if 1/Dmini=1,2,…,n{g0i}<1/Cmaxi=1,2,…,n{g∞i}, where
(4.8)C=max{Ci,i=1,2,…,n},D=min{Di,i=1,2,…,n},
and one writes 1/gαi=0 if gαi=∞ and 1/gαi=∞ if gαi=0, here α=0,∞.Proof.
We consider the case (4.6). The case (4.7) is similar.
If (4.6) holds, then
(4.9)λg0i≤λmaxi=1,2,…,n{g0i}<1C≤1Ci,i=1,2,…,n,λg∞i≥λmini=1,2,…,n{g∞i}>1D≥1Di,i=1,2,…,n.
Thus Theorem4.1 applies directly.Remark 4.4.
By Theorem4.3, the corresponding existence results in [32] are improved.


## 5. Example

For convenience, the example is given here whenn=1.Example 5.1.
LetT=[0,1]∪[2,3]. We consider the following problem:
(5.1)xΔ(t)+p(t)x(σ(t))=λh(t)g(x(σ(t))),t∈[0,3]T,x(0)=x(3),
where p(t)≡1,T=3,h(t)≡1, and g(x)=x2; it is easy to see that h and g satisfy the conditions (H3) and (H4).
Then we get(5.2)g0=limx→0+g(x)x=0,g∞=limx→∞g(x)x=∞.
By∫0σ(T)G(t,s)h(s)Δs≡1, we have
(5.3)C=maxt∈[0,σ(T)]T∫0σ(T)G(t,s)h(s)Δs=mint∈[0,σ(T)]T∫0σ(T)G(t,s)h(s)Δs=D=1.
Then we get g0=0<1=C- and D-=1<g∞; that is, condition (H5) is satisfied. So, by Theorem 4.1 the problem (5.1) has at least one positive solution when λ=1. Furthermore, for all λ∈(0,∞), the problem (5.1) has at least one positive solution from Theorem 4.3.



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