# Necessary and Sufficient Conditions for Positive Solutions of Second-Order Nonlinear Dynamic Equations on Time Scales

Authors: Zhi-Qiang Zhu
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

This paper is concerned with the existence of nonoscillatory solutions for
the nonlinear dynamic equation(p(t)(ψ∘x)(xσ)Δ)Δ+q(t)f∘xσ=0 on time scales. By making use of the generalized Riccati transformation technique, we
establish some necessary and sufficient criteria to guarantee the existence. The last examples show that our results can be applied on the differential equations, the difference equations, and the q-difference equations.


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## Body

## 1. Introduction

In the recent decade there have been many literatures to study the oscillatory properties for second-order dynamic equations on time scales; see, for example, [1–11] and the references therein. In particular, the dynamic equation of the form(1.1)(p(t)xΔ)Δ+q(t)f∘xσ=0
has been attracting one’s interesting; see, for example, [3, 5, 8]. Motivated by the papers mentioned as above, in this paper we consider the existence of nonoscillatory solutions for nonlinear dynamic equation(1.2)(p(t)(ψ∘x)(xσ)Δ)Δ+q(t)f∘xσ=0
on a time scale 𝕋, where xσ=x∘σ.Referring to [12, 13], a time scale 𝕋 can be defined as an arbitrary nonempty subset of the set ℝ of real numbers, with the properties that every Cauchy sequence in 𝕋 converges to a point of 𝕋 with the possible exception of Cauchy sequences which converge to a finite infimum or finite supremum of 𝕋. On any time scale 𝕋, the forward and backward jump operators are defined, respectively, by(1.3)σ(t):=inf⁡⁡{s∈T:s>t},ρ(t):=sup⁡⁡{s∈T:s<t},
where inf∅:=sup𝕋 and sup∅:=inf𝕋. A point t∈𝕋 is said to be right-scattered if σ(t)>t, right-dense if σ(t)=t, left-scattered if ρ(t)<t, and left-dense if ρ(t)=t. A derived set from 𝕋 is defined as follows: 𝕋k=𝕋-{m} when 𝕋 has a left-scattered maximum m, otherwise 𝕋k=𝕋.Definition 1.1.
For a functionf:𝕋→ℝ and t∈𝕋k, we define the delta-derivative fΔ(t) of f(t) to be the number (provided it exists) with the property that, for any ɛ>0, there is a neighborhood U of t (i.e., U=(t-δ,t+δ)∩𝕋 for some δ) such that
(1.4)|[f(σ(t))-f(s)]-fΔ(t)[σ(t)-s]|≤ɛ|σ(t)-s|∀s∈U.
We say that f is delta-differentiable (or in short: differentiable) on 𝕋k provided fΔ(t) exists for all t∈𝕋k.For two differentiable functionsf and g at t∈𝕋k with g(t)g(σ(t))≠0, it holds that(1.5)(fg)Δ(t)=fΔ(t)g(t)+f(σ(t))gΔ(t),(1.6)(fg)Δ(t)=fΔ(t)g(σ(t))-f(σ(t))gΔ(t)g(t)g(σ(t)).Definition 1.2.
A functionF:𝕋→ℝ is called an antiderivative of f provided FΔ(t)=f(t) holds for all t∈𝕋k. By the antiderivative, the Cauchy integral of f is defined as ∫abf(s)Δs=F(b)-F(a), and ∫a∞f(s)Δs=limt→∞∫atf(s)Δs.Definition 1.3.
Letf:𝕋→ℝ be a function, where f is called right-dense continuous (rd-continuous) if it is right continuous at right-dense points in 𝕋 and its left-sided limits exist (finite) at left-dense points in 𝕋.To distinguish from the traditional interval inℝ, we define the interval in 𝕋 by(1.7)[a,∞)T:={t∈T:a≤t<∞}.
Let Crd(𝕋) (or Crd(𝕋,ℝ)) denote the set of all rd-continuous functions defined on 𝕋, and Crd1(𝕋) (or Crd1(𝕋,ℝ)) denote the set of all differentiable functions whose derivative is rd-continuous.Since we are interested in the existence of nonoscillatory solutions of (1.2), we make the blanket assumption that inf𝕋=t0 and sup𝕋=∞. As defined in [1], by a solution of (1.2) we mean a nontrivial real function x(t), with x(t)∈Crd1[t1,∞)𝕋 and p(t)ψ(x(t))(xσ(t))Δ∈Crd1[t1,∞)𝕋 for some t1≥t0, which satisfies (1.2) on [t1,∞)𝕋. A solution of (1.2) is said to be nonoscillatory if it is eventually positive or eventually negative.


## 2. Preliminaries

Let us assume in (1.2) under consideration that(H1)
σ∈Crd1(𝕋,𝕋) with σΔ(t)>0 on 𝕋,(H2)
p,q∈Crd(𝕋,ℝ) and there exists a t1∈𝕋 such that p(t)>0 and q(t)≥0 on [t1,∞)𝕋 and ∫t1∞(Δs/p(s))=∞, ∫t1∞q(s)Δs<∞; also, q is not identically zero on [t1,∞)𝕋,(H3)
ψ∈C1(ℝ,ℝ+) with ψ(x)≤m0 on ℝ for some constant m0>0 and, ψ is nonincreasing on [0,∞) and nondecreasing on (-∞,0],(H4)
f∈C1(ℝ,ℝ) satisfies that xf(x)>0 for x≠0, f′(x)≥m1 on ℝ for some constant m1>0, and f′ is nondecreasing on [0,∞) and nonincreasing on (-∞,0].Suppose that𝕋̃={x(t)∈Crd(𝕋,ℝ):t∈𝕋} is a time scale. By σ̃ and ρ̃ we denote the forward and backward jump operators on 𝕋̃, respectively, and by Δ̃ we denote the derivative on 𝕋̃. Let σ̃2=σ̃∘σ̃. Then, fΔ̃(σ̃(x))/ψ(x) is bounded below on 𝕋̃ by assumptions (H3)-(H4). Indeed, in case σ̃(x)=σ̃2(x), we have f′(σ̃(x))/ψ(x)=fΔ̃(σ̃(x))/ψ(x) and the assertion holds. In case σ̃(x)<σ̃2(x), by the definition of delta-derivative and the mean value theorem, there exits a constant c∈[σ̃(x),σ̃2(x)] such that(2.1)fΔ̃(σ̃(x))=f(σ̃2(x))-f(σ̃(x))σ̃2(x)-σ̃(x)=f′(c),
and then(2.2)fΔ̃(σ̃(x))ψ(x)=f′(c)ψ(x)≥m1m0.Furthermore,fΔ̃(σ̃(x)) is nondecreasing on [0,∞)𝕋̃ by virtue of (H4). Indeed, for any x,y∈[0,∞)𝕋̃ with x≤y, there are four cases to consider. In case σ̃(x)=σ̃2(x) and σ̃(y)=σ̃2(y), fΔ̃(σ̃(x))=f′(σ̃(x))≤f′(σ̃(y))=fΔ̃(σ̃(y)). In case σ̃(x)=σ̃2(x) and σ̃(y)<σ̃2(y), we have fΔ̃(σ̃(x))=f′(σ̃(x)) and(2.3)fΔ̃(σ̃(y))=f(σ̃2(y))-f(σ̃(y))σ̃2(y)-σ̃(y)=f′(cy),
where cy∈[σ̃(y),σ̃2(y)]. Since cy≥σ̃(x), it follows that fΔ̃(σ̃(x))≤fΔ̃(σ̃(y)). The other cases can be shown likewise.As a consequence, we see from assumption(H3) that fΔ(σ̃(x))/ψ(x) is nondecreasing on [0,∞)𝕋̃. Similarly, we can show that fΔ̃(σ̃(x))/ψ(x) is nonincreasing on (-∞,0]𝕋̃ provided σ̃(x)∈(-∞,0]𝕋̃.As thus, we can extract the essences as above and obtain a result as follows.Lemma 2.1.
Suppose that𝕋̃ is a time scale and fΔ̃(x)/ψ(x) is defined on it. Then it follows that(i)
fΔ̃(σ̃(x))/ψ(x) is positively bounded below and(ii)
fΔ̃(σ̃(x))/ψ(x) is nondecreasing on [0,∞)𝕋̃ and nonincreasing on (-∞,0]𝕋̃.For a givena∈𝕋̃ with a≠0, we introduce a function on 𝕋̃ as follows:(2.4)Γa(x)=∫axψ(ρ̃(u))f(u)Δ̃uforx∈T̃withx≠0.
Then the function Γa(x) possesses the following properties. (i)
Ifa>0, then Γa(x) is strictly increasing for x>0 and Γa-1(y)≥a is strictly increasing for y≥0.(ii)
Ifa<0, then Γa(x) is strictly decreasing for x<0 and Γa-1(y)≤a is strictly decreasing for y≥0.(iii)
Ifa≠0, then F(Γa-1(y)) is nondecreasing for y≥0 (by Lemma 2.1 (ii)). Here F is defined as in (2.6).For the sake of convenience, we let(2.5)R(t)=p(t)ψ(x(t))(xσ(t))Δf(xσ(t)),(2.6)F(x)=fΔ̃(σ̃(x))ψ(x)
whenever they are defined.Note that, ifx(t) is an eventually negative solution of (1.2), then y(t)=-x(t) satisfies that(2.7)(p(t)(ψ̃∘y)(yσ)Δ)Δ+q(t)f̃∘yσ=0,where(2.8)ψ̃(y)=ψ(-y),f̃(y)=-f(-y)
possess the same properties as ψ and f, respectively. Therefore, in what follows we will restrict our attention to the eventually positive solutions of (1.2).


## 3. Main Results

Before entering our main discussions, we remark thatx([t1,∞)𝕋) is a time scale when x∈Crd1[t1,∞)𝕋 and x is monotonic on [t1,∞)𝕋, where t1∈𝕋. In the following discussions, the notations Δ̃,σ̃ and ρ̃ will act upon the time scale x([t1,∞)𝕋), while Δ, σ, and ρ will do on 𝕋. Then σ̃∘x=x∘σ when x is strictly increasing on [t1,∞)𝕋.Lemma 3.1.
Suppose thatt1∈𝕋 and x(t) is a solution of (1.2) with x(t)>0 on [t1,∞)𝕋. Then it holds that
(3.1)R(t)=∫t∞q(s)Δs+∫t∞R(s)R(σ(s))F(x(s))p(s)Δs
for all t∈[t1,∞)𝕋 and
(3.2)R(t)=∫t∞q(s)Δs+∫t∞R(s)R(σ(s))F(Γxσ(t1)-1(∫t1ρ(s)(R(u)Δu/p(u))))p(s)Δs
for all t∈[σ(t1),∞)𝕋.Proof.
Without loss of generality, letp(t)>0 and q(t)≥0 on [t1,∞)𝕋. Then it follows from (1.2) that
(3.3)(p(t)ψ(x(t))(xσ(t))Δ)Δ=-q(t)f(xσ(t))≤0∀t∈[t1,∞)T.
We assert that
(3.4)p(t)ψ(x(t))(xσ(t))Δ>0∀t∈[t1,∞)T.
Otherwise, note from assumption (H2) that q is not identically zero on [t1,∞)𝕋, (3.3) implies that there exits a t2∈[t1,∞)𝕋 and a constant m>0 such that
(3.5)p(t)ψ(x(t))(xσ(t))Δ≤-m∀t∈[t2,∞)T.
Consequently we find that
(3.6)xσ(t)≤xσ(t2)-mm0∫t2tΔsp(s)⟶-∞ast⟶∞,
which contradicts x(t)>0 on [t1,∞)𝕋. Since (3.4) holds, it is clear that
(3.7)(xσ(t))Δ>0,R(t)>0∀t∈[t1,∞)T.
On the other hand, we see from (3.3) that R(t) is nonincreasing on [t1,∞)𝕋, which, associated with (3.7), means that limt→∞R(t) exists as a finite number.
Next we will show that(3.8)∫t1∞R(s)R(σ(s))F(x(s))p(s)Δs<∞.
Note that xσ is strictly increasing due to assumption (H1) and (3.7); by (1.2), (1.5) and (1.6) and the chain rule [13, Theorem 1.93] we have
(3.9)RΔ(t)=[p(t)ψ(x(t))(xσ(t))Δ]Δf(xσ2(t))f(xσ(t))f(xσ2(t))-[p(σ(t))ψ(xσ(t))(xσ2(t))Δ]fΔ̃(xσ(t))(xσ(t))Δf(xσ(t))f(xσ2(t))=-q(t)-p(σ(t))ψ(xσ(t))(xσ2(t))Δf(xσ2(t))⋅fΔ̃(xσ(t))[p(ψ∘x)(xσ)Δ](t)p(t)ψ(x(t))f(xσ(t))=-q(t)-R(σ(t)R(t))F(x(t))p(t).
Taking Δ-integral on (3.9) from t1 to t, we obtain that
(3.10)R(t)-R(t1)=-∫t1tq(s)Δs-∫t1tR(s)R(σ(s))F(x(s))p(s)Δs.
Now that the limit ofR(t) exists as t→∞, by assumption (H2) and (3.10) we see that (3.8) holds. Note that F(x(t)) is positive bounded below (see Lemma 2.1(i)), (3.8) infers that limt→∞R(t)=0. To sum up, it is easy to see that (3.10) yields (3.1).
Next we prove (3.2). Again note that xσ is strictly increasing on [t1,∞)𝕋, by the substitution theorem [13, Theorem 1.98], it follows that
(3.11)∫t1tR(t)Δsp(t)=∫t1tψ(x(s))(xσ(s))ΔΔsf(xσ(s))=∫xσ(t1)xσ(t)ψ(ρ̃(u))Δ̃uf(u),
which, together with the definition of Γa, induces
(3.12)x(t)=Γxσ(t1)-1(∫t1ρ(t)R(s)Δsp(s))∀t∈[σ(t1),∞)T,
where we have used ρ(σ(t))=t (see assumption (H1)). Now let us substitute x(t) in (3.12) into (3.1) and then we obtain (3.2). The proof is complete.Theorem 3.2.
Suppose thatt1∈𝕋. Then x(t) is a solution of (1.2) with x(t)>0 on [t1,∞)𝕋 (or, on [σ(t1),∞)𝕋) if and only if there exists a constant α>0 and a function φ∈Crd([t1,∞)𝕋,ℝ+) such that
(3.13)φ(t)≥∫t∞q(s)Δs+∫t∞φ(s)φ(σ(s))p(s)F(Γα-1(∫t1ρ(s)φ(u)Δup(u)))Δs
for all t∈[σ(t1),∞)𝕋.Proof.
Suppose thatx(t) is a solution of (1.2) with x(t)>0 on [t1,∞)𝕋. Then, by Lemma 3.1, R defined as in (2.5) satisfies R∈Crd([t1,∞)𝕋,ℝ+) and (3.13) holds, where α=xσ(t1).
Conversely, suppose that there exists a constantα>0 and a function φ∈Crd(𝕋,ℝ+) such that (3.13) holds. Let y0(t)≡0 and define a sequence of functions {yn(t)}n=0∞ on [σ(t1),∞)𝕋 as follows:
(3.14)yn+1(t)=∫t∞q(s)Δs+∫t∞yn(s)yn(σ(s))F(Γα-1(∫t1ρ(s)(yn(u)Δu/p(u))))p(s)Δs.
It is clear that 0≤y1(t)≤φ(t) on [σ(t1),∞)𝕋. Suppose that yn-1(t)≤yn(t)≤φ(t) on [σ(t1),∞)𝕋. Then, by the monotone of F(Γa-1(y)) for y≥0, we learn that
(3.15)yn+1(t)≥∫t∞q(s)Δs+∫t∞yn-1(s)yn-1(σ(s))F(Γα-1(∫t1ρ(s)(yn-1(u)Δu/p(u))))p(s)Δs=yn(t)
as well as
(3.16)yn+1≤φ(t).
So by the mathematical induction we obtain that
(3.17)0≤yn(t)≤yn+1(t)≤φ(t)∀[σ(t1),∞)T,
which means that there exists a function y such that
(3.18)lim⁡n→∞yn(t)=y(t)≤φ(t)∀t∈[σ(t1),∞)T.
Now by Lebesgue's domination convergence theorem on time scales [14, Chapter 5], we may deduce from (3.14) that(3.19)y(t)=∫t∞q(s)Δs+∫t∞y(s)y(σ(s))F(Γα-1(∫t1ρ(s)(y(u)Δu/p(u))))p(s)Δs.
Let
(3.20)x(t)=Γα-1(∫t1ρ(t)y(s)Δsp(s)).
Then x(t)≥α on [σ(t1),∞)𝕋 and xσ(t) is strictly increasing on [t1,∞)𝕋. Moreover, x([σ(t1),∞)𝕋) is another time scale with σ̃∘x=x∘σ. Therefore, by the theorem on derivative of the inverse [13, Theorem 1.97] we learn that
(3.21)(xσ(t))Δ=f(xσ(t))y(t)ψ(x(t))p(t)fort∈[t1,∞)T.
Now from (3.20) we have
(3.22)Γα(xσ(t))=∫t1ty(s)Δsp(s)fort∈[t1,∞)T,
which implies that
(3.23)p(t)ψ(x(t))(xσ(t))Δ=y(t)f(xσ(t)),
and this results in
(3.24)(p(t)ψ(x(t))(xσ(t))Δ)Δ=yσ(t)fΔ̃(xσ(t))(xσ(t))Δ+f(xσ(t))yΔ(t)=yσ(t)fΔ̃(xσ(t))(xσ(t))Δ+f(xσ(t))[-q(t)-y(t)y(σ(t))F(x(t))p(t)]=yσ(t)fΔ̃(xσ(t))(xσ(t))Δ-q(t)f(xσ(t))-[p(t)ψ(x(t))(xσ(t))Δ]yσ(t)F(x(t))p(t)=-q(t)f(xσ(t)),
where we have imposed formulas (3.19) and (3.20) for the second equal sign and (3.23) for the third, respectively.
Now we see from (3.24) that x(t) defined by (3.20) is a positive solution of (1.2). The proof is complete.In the remainder of this section, we define formally a sequence of functions{Qn(t)}n=0∞ as follows. Let t1∈𝕋, α>0, and(3.25)Q0(t)=∫t∞q(s)Δs,(3.26)Q1(t)=∫t∞Q0(s)Q0(σ(s))F(Γα-1(∫t1ρ(s)Q0(u)Δu/p(u)))p(s)Δs
as well as(3.27)Qn+1(t)=∫t∞[Q0(s)+Qn(s)][Q0(σ(s))+Qn(σ(s))]F(Γα-1(∫t1ρ(s)((Q0(u)+Qn(u))/p(u)Δu)))p(s)Δs.
Then Q1(t)≤Q2(t). By induction, it follows that(3.28)0<Qn(t)≤Qn+1(t)forn=1,2,3,….Ifx(t) is a solution of (1.2) with x(t)>0 on [t1,∞)𝕋 for some t1∈𝕋, then (3.1) holds by Lemma 3.1 and hence(3.29)Q0(t)≤R(t)∀t∈[σ(t1),∞)T,
which, together with (3.26), results in(3.30)Q1(t)≤∫t∞R(s)R(σ(s))F(Γa-1(∫t1ρ(s)(R(u)Δu/p(u))))p(s)Δs∀t∈[σ(t1),∞)T.
Let us now define R0(t) by(3.31)R0(t)=∫t∞R(s)R(σ(s))F(Γa-1(∫t1ρ(s)(R(u)Δu/p(u))))p(s)Δs,t∈[σ(t1),∞)T.
Then, in view of (3.2) and (3.30), we have(3.32)Q0(t)+Q1(t)≤Q0(t)+R0(t)=R(t)∀t∈[σ(t1),∞)T.
By the mathematical induction again we educe(3.33)Q0(t)+Qn(t)≤R(t)∀t∈[σ(t1),∞)T,n=1,2,3,….Now we learn from (3.28) and (3.33) that {Qn(t)}0∞ is well defined and converges to some function Q(t) when x(t) is a solution of (1.2) with x(t)>0 on [t1,∞)𝕋.Conversely, suppose that{Qn(t)}n=0∞ is well defined and(3.34)lim⁡n→∞Qn(t)=Q(t)<∞on⁡[σ(t1),∞)T.
Then, by (3.28), we have Qn(t)≤Q(t) for all n≥1. Hence, by Lebesgue's domination convergence theorem on time scales, we may obtain from (3.27) that(3.35)Q(t)=∫t∞[Q0(s)+Q(s)][Q0(σ(s))+Q(σ(s))]F(Γα-1(∫t1ρ(s)((Q0(u)+Q(u))/p(u))Δu))p(s)Δs.Letφ(t)=Q0(t)+Q(t). Then, from (3.35) we see that (3.13) holds. Furthermore, by means of Theorem 3.2, (1.2) has a solution x(t)>0 on [σ(t1),∞)𝕋.To sum up, we obtain our last result as follows.Corollary 3.3.
Suppose thatt1∈𝕋. Then x(t) is a solution of (1.2) with x(t)>0 on [t1,∞)𝕋 (or, on [σ(t1),∞)𝕋) if and only if there exists a constant α>0 such that the sequence of functions {Qn(t)}n=0∞ defined as in (3.25)–(3.27) is well defined and
(3.36)lim⁡n→∞Qn(t)=Q(t)<∞on⁡[σ(t1),∞)T.Example 3.4.
Let𝕋=[0,∞). Suppose in (1.2) that p(t)=t+1, q(t)=1/(4t2), ψ(x)=1/2(1+|x|), and f(x)=x. Let φ(t)=1/(2t). It is easy to verify that
(3.37)Γ1-1(∫1sφ(s)dsp(s))=s,F(Γ1-1(∫1sφ(u)dup(u)))=12(1+|s|)fors≥1
as well as
(3.38)∫t∞q(s)ds+∫t∞φ2(s)p(s)F(Γ1-1(∫1sφ(u)dup(u)))ds=38t≤φ(t),t≥1.
By Theorem 3.2, (1.2) has a solution x(t)>0 on [1,∞).Example 3.5.
Leth>0 and q0>1, and let ℕ0 be a set of nonnegative integers. Suppose in (1.2) that ψ(x)=1 and f(x)=x.
In case𝕋=hℕ0, let(3.39)p(t)=1,q(t)=1(2t+2h)(2t+h).
Then, for a given φ(t)=1/(2t+h) and any given α>0, we have
(3.40)∫t∞q(s)Δs≤∫t∞Δtσ(2s+h)(2s+h)=12(2t+h),∫t∞φ(s)φ(σ(s))p(s)F(Γα-1(∫0ρ(s)φ(u)Δup(u)))Δs=∫t∞Δs(2s+h)(2σ(s)+h)≤12(2t+h)
and hence (3.13) holds. By means of Theorem 3.2, (1.2) has a solution x(t)>0 on 𝕋=hℕ0.
In case𝕋={q0k:k∈ℕ0}, let(3.41)p(t)=2,φ(t)=1t,q(t)=12σ(t)t.
Then, for any given α>0, we have
(3.42)∫t∞q(s)Δs+∫t∞φ(s)φ(σ(s))p(s)F(Γα-1(∫1ρ(s)φ(u)Δup(u)))Δs=φ(t),
and hence Theorem 3.2 implies that (1.2) has a solution x(t)>0 on 𝕋={q0k:k∈ℕ0}.



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