# Monotone Iterative Technique for Partial Dynamic Equations of First Order on Time Scales

Authors: Peiguang Wang; Ping Li
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

This work is concerned with the monotone iterative technique for partial dynamic equations of first order on time scales and for this purpose, the existence, uniqueness, and comparison results are also established.


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## Body

## 1. Introduction

The study of dynamic equations on time scales is an area
of research that has received a lot of attention. Many authors have been
devoted to the qualitative research of ordinary differential equations on time
scales. We can refer to monographs by Bohner and Peterson [1],
Lakshmikantham et al. [2],  and the references cited therein. With the
development of technology, in recent years, some authors have explored the
partial differential equations on time scales. Ahlbrandt and Morian [3] have
introduced definitions of time scale derivatives and integrals for functions of
two variables and obtained the results that include an Euler-Lagrange equation
for double integral variational problems on time scales and a Picone identity
which implies a Sturm-Picone comparison theorem for second-order elliptic
partial differential equations on time scales. Jackson [4] has extended the
existing ideas of the univariate case of the time scales calculus to the
multivariate case.Lakshmikantham et al. [5] investigated the
monotone iterative technique for partial differential equations of first order
in 1984. Nevertheless, up till now, we have not yet seen any research about
monotone iterative technique for partial dynamic equations on time
scales. In this paper, we will investigate the
monotone iterative technique for partial dynamic equations of first order on
time scales. Firstly, in Sections 2 and 3, we will introduce the univariable
calculus and multivariable calculus on time scales that will be the preliminary
work in order to define the partial dynamic equations on time scales. Secondly,
in Sections 4 and 5, we will present existence, uniqueness, and comparison
results and provide the monotone iterative technique for such partial dynamic
equations.


## 2. Univariable Calculus on Time Scales

For the detail of basic notions concerned with time scales, we refer to [1, 2]. To meet the requirements in the next sections, we introduce some notions and lemma
here.A time scale is a nonempty closed subset of the real
numberℝ, and we denote it by the symbol 𝕋. We define the forward and backward jump operators σ,ρ:𝕋→𝕋 by
(2.1)σ(t)=inf{s∈𝕋∣s>t},ρ(t)=sup{s∈𝕋∣s<t}
(supplemented by inf∅=sup𝕋). A point t∈𝕋 is called
right-scattered, right-dense, left-scattered, left-dense if σ(t)>t,σ(t)=t,ρ(t)<t,ρ(t)=t hold,
respectively. The set 𝕋k is defined to
be 𝕋 if 𝕋 does not have a
left-scattered maximum; otherwise it is 𝕋 without this
left-scattered maximum. The graininess μ:𝕋→[0,∞) is defined by μ(t)=σ(t)−t.Iff:𝕋→ℝ is a function
and t∈𝕋k, then 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⊂𝕋 of t such that
(2.2)|f(σ(t))−f(s)−fΔ(t)[σ(t)−s]|≤ϵ|σ(t)−t|∀s∈U.Lemma 2.1 (see [1]).
Letf:ℝ→ℝ be continuously
differentiable and suppose that g:𝕋→ℝ is delta
differentiable. Then f∘g:𝕋→ℝ is delta
differentiable and the formula
(2.3)(f∘g)Δ(t)={∫01f′(g(t)+hμ(t)gΔ(t))dh}gΔ(t)
holds.


## 3. Multivariable Calculus on Time Scales

In this section, we generalize existing ideas of the time scales calculus to the
multivariate case.Firstly, consider the product𝕋n=𝕋1×𝕋2×⋯×𝕋n, where 𝕋i is a time scale
for all 1≤i≤n, so 𝕋n is a time scale
too. Then for any x∈𝕋n with x=(x1,x2,…,xn), in which xi∈𝕋i, 1≤i≤n, define the following:
(i)
the forward jump operatorσ:𝕋n→𝕋n by σ(x)=(σ(x1),σ(x2),…,σ(xn)), where σ(xi) represents the
forward jump operator of xi∈𝕋i for all 1≤i≤n. Hereafter, the forward jump operator of xi∈𝕋i will be denoted
by σ(xi)=σi(x);(ii)
the back jump operatorρ:𝕋n→𝕋n by ρ(x)=(ρ(x1),ρ(x2),…,ρ(xn)), where ρ(xi) represents the
backward jump operator of xi∈𝕋i for all 1≤i≤n. The backward jump operator of xi∈𝕋i will be denoted
by ρ(xi)=ρi(x);(iii)
the graininess functionμ:𝕋n→ℝn by μ(x)=(μ(x1),μ(x2),…,μ(xn)), where μ(xi) represents the
graininess function of xi∈𝕋i on the time
scale 𝕋i for all 1≤i≤n. Again, from this point on the graininess function of
the time scale 𝕋i for xi∈𝕋i will be denoted
by μ(xi)=μi(x);(iv)
𝕋kn=𝕋1k×𝕋2k⋯×𝕋nk.Secondly, we introduce the definitions of time-scale
derivatives for the functionu(t,x), where t∈𝕋 and x∈𝕋n. The forward jump operator σ:𝕋→𝕋 and τ:𝕋n→𝕋n are defined by σ(t)=inf{s1∈𝕋∣s1>t} and τ(x)=inf{s2∈𝕋n∣s2>x}. The
graininess h:𝕋→ℝ is defined as h(t)=σ(t)−t. The graininess k:𝕋n→𝕋n is defined as k(x)=τ(x)−x. We will use the notation uσ(t,x)=u(σ(t),x),uτ(t,x)=u(t,τ(x)) and uστ(t,x)=u(σ(t),τ(x)).Because we will need notation for partial derivatives
with respect to time-scale variablest and x, we employ lexigraphic ordering for consistency. Let uΓ(t,x) denote the
time-scale partial derivative with respect to t, in order to
distinguish for the sign of Δ-derivative in
Section 2, let uΔ2(t,x) denote the
time-scale partial derivative with respect to x.The following definitions of these partial derivatives
are now given.Definition 3.1.
Letu be a
real-valued function on 𝕋×𝕋n. At (t,x)∈𝕋k×𝕋kn, we say that u has a “Γ partial
derivative” uΓ(t,x), with respect to t, if for each ε>0, there exists a neighborhood 𝒰t  (open in the
relative topology of 𝕋 of t) such that
(3.1)|u(σ(t),x)−u(s1,x)−uΓ(t,x)[σ(t)−s1]|≤ε|σ(t)−s1|
for all s1∈𝒰t. Similarly, we say that u has a “Δ partial
derivative” denoted by uΔ2(t,x)∈ℝ, with respect to x, if for each ε>0 there exists a
neighborhood 𝒰x of x such that
(3.2)|u(t,τ(x))−u(t,s2)−uΔ2(t,x)[τ(x)−s2]|≤ε|τ(x)−s2|
for all s2∈𝒰x.
Finally, we define the partial derivative ofu(t,x) at x with respect to
the variable xi, where t∈𝕋 and x=(x1,x2,…,xn)∈𝕋n. Having defined the multivariable calculus as
earlier, we set
(3.3)uσi(t,x)=u(t,x1,x2,…,xi−1,σi(x),xi+1,…,xn),uis3(t,x)=u(t,x1,x2,…,xi−1,s3,xi+1,…,xn).Definition 3.2.
Letu:𝕋×𝕋n→ℝ be a
real-valued function and let x=(x1,x2,…,xn)∈𝕋kn. Then define uΔ2i(t,x) to be the
number (provided it exists) with the property that given any ε>0, there exists a neighborhood  𝕌 of xi, with 𝕌=(xi−δ,xi+δ)∩𝕋i for δ>0 such that
(3.4)|[uσi(t,x)−uis3(t,x)]−uΔ2i(t,x)[σi(x)−s3]|≤ε|σi(x)−s3|
for all s3∈𝕌. uΔ2i(t,x) is called the
partial delta derivative of u(t,x) at x with respect to
variable xi.


## 4. Existence, Uniqueness, and Comparison Result

In this section, we consider the initial value problem (IVP) for partial dynamic
equation of first order on time scales(4.1)uΓ+f(t,x)uΔ2=g(t,x,u),u(0,x)=ϕ(x),
where f∈C(Ω,ℝn), g∈C(Ω×ℝ,ℝ), Ω={(t,x):0≤t≤T,a≤x≤b,t∈𝕋k,a,b,x∈𝕋kn}, fuΔ2=∑i=1nfi(t,x)uΔ2i and ϕ∈C1([a,b],ℝ).We define the set𝔻={u∈C[Ω1,ℝ],uΓ,uΔ2,anduΔ2iarecontinuouson𝕋k×𝕋kn}, Ω1={(t,x),0≤t≤T,a≤x≤σ(b),t∈𝕋k,a,b,andx∈𝕋kn}.Now, we begin to prove the following comparison result.Theorem 4.1.
Assume that(A0)
αΓ+f(t,x)αΔ2≤g(t,x,α),α(0,x)≤ϕ(x), and   βΓ+f(t,x)βΔ2≥g(t,x,β),β(0,x)≥ϕ(x) for (t,x)∈Ω1, where α,β∈𝔻.(A1)
f(t,x) is
quasimonotone nonincreasing in x for each i and f(t,a)≤0, f(t,b)≥0.(A2)
g(t,x,u1)−g(t,x,u2)≤L(u1−u2) whenever u1≥u2 for some L≥0.
Thenα(t,x)≤β(t,x) on Ω1.Proof.
Let us prove the theorem for strict inequalities firstly. For example,
we suppose thatαΓ+f(t,x)αΔ2<g(t,x,α) and α(0,x)<ϕ(x) on Ω1, and prove that α(t,x)<β(t,x) on Ω1. If this conclusion is not true, then consider the set
(4.2)ℤ={(t,x)∈Ω1∣α(t,x)≥β(t,x)}.
Let ℤt be the
projection of ℤ on the t-axis and let t0=infℤt. Clearly, t0>0 and there
exists an x0∈[a,σ(b)] such that
(4.3)α(t0,x0)=β(t0,x0),α(σ(t0),x0)≥β(σ(t0),x0),α(t0,x)≤β(t0,x)fora≤x≤σ(b).
It then follows that αΓ(t0,x0)≥βΓ(t0,x0) and if a≤x≤σ(b), we also have αΔ2i(t0,x0)=βΔ2i(t0,x0),i=1,2,…,n. In this case,
we get the following contradiction:
(4.4)g(t0,x0,α(t0,x0))>αΓ+f(t0,x0)αΔ2≥βΓ+f(t0,x0)βΔ2≥g(t0,x0,β(t0,x0)).
If, on the other hand, for some j, x0,j=bj and x0,i<bi,i≠j, and a<x0, then we have αΔ2i(t0,x0)=βΔ2i(t0,x0),i≠j, and α(t0,σ(x0))≥β(t0,σ(x0)), then we have αΔ2j(t0,x0)≥βΔ2j(t0,x0). Hence using the assumption
(A1), we obtain fj(t0,x0)≥fj(t0,b)≥0 and fj(t0,x0)αΔ2j(t0,x0)≥fj(t0,x0)βΔ2j(t0,x0). Consequently, we get the inequality
(4.5)g(t0,x0,α(t0,x0))>αΓ+f(t0,x0)αΔ2≥βΓ+f(t0,x0)βΔ2≥g(t0,x0,β(t0,x0)),
which is a contradiction as before. This proves that α(t,x)<β(t,x) on Ω1. If one of the inequalities in (A0) is not strict, and on one
hand, if μ(t)=0, we set α(t,x)=α(t,x)−εe2Lt and note that α<α. Then using (A0) and (A2), we have
(4.6)αΓ+f(t,x)αΔ2=αΓ+f(t,x)αΔ2−2Lεe2Lt≤g(t,x,α)−2Lεe2Lt≤g(t,x,α)+Lεe2Lt−2Lεe2Lt<g(t,x,α),
and α(0,x)<ϕ(x) on Ω1. On the other hand, if μ(t)≠0, we set α(t,x)=α(t,x)−εeLt and note that α<α. Then using (A0), (A2), and Lemma 2.1, we have
(4.7)αΓ+f(t,x)αΔ2=αΓ+f(t,x)αΔ2−ε(eLtμ(eLμ−1))≤g(t,x,α)−εeLtμ(eLμ−1)≤g(t,x,α)+LεeLt−εeLtμ(eLμ−1)=g(t,x,α)+Lμ+1−eLμμεeLt<g(t,x,α),
and α(0,x)<ϕ(x) on Ω1. Thus the foregoing arguments imply that α(t,x)<β(t,x) on Ω1. Taking limit as ε→0, we then get α(t,x)≤β(t,x) on Ω1, and the proof is complete.Theorem 4.2.
Assume that (A1) and (A2) hold. Suppose further that (A3) for each (t0,x0)∈Ω, there exists a unique solution x(t0,x0) of
(4.8)xΔ=f(t,x),x(t0)=x0,
on 0≤t≤T, x(t,t0,x0) is  continuously differentiable with respect to (t0,x0). Assume (A4) for each x0∈[a,b] and y0∈ℝ, there exists a unique solution y(t,0,y0;x0) of
(4.9)yΔ=g(t,x(t,0,x0),y),y(0)=y0,
on 0≤t≤T, where x(t,0,x0) is the unique
solution of (4.8) and y(t,0,y0;x0) is continuously
differentiable with respect to (y0,x0).
Then there exists a unique solutionu(t,x)=y(t,0,ϕ(x(0,t,x));x(0,t,x)) for the problem
(4.1) on Ω1.Proof.
By (A3) and (A4), x(t,t0,x0),y(t,0,y0;x0) are unique
solutions of (4.8) and (4.9), respectively, on 0≤t≤T. Choose y0=ϕ(x0) and note that
if x=x(t,0,x0), then because of uniqueness, x0=x(0,t,x). Hence, y(t,0,ϕ(x(0,t,x));x(0,t,x)) is a unique
solution of (4.9). Then we have
(4.10)yΔ=yΓ+yΔ2xΔ=yΓ+f(t,x)yΔ2=g(t,x,y),y(0,x)=ϕ(x),
that is, u(t,x)=y(t,0,ϕ(x(0,t,x));x(0,t,x)) satisfies (4.1)
and consequently, u(t,x) is a solution
of the problem (4.1).
To show uniqueness of solution of (4.1), suppose that u1(t,x),u2(t,x) are two
solutions of (4.1) on Ω1. Then setting α=u1, β=u2 and applying
Theorem 4.1, we get u1(t,x)≤u2(t,x) on Ω1. Similarly we can prove that u1(t,x)≥u2(t,x) on Ω1. Hence the proof is complete.


## 5. Monotone Iterative Technique

we are now in a position to describe the monotone iterative technique which yields
monotone sequences on time scales. We prove the following result specifically.Theorem 5.1.
Assume that (A0), (A1), and (A3) hold with α≤β on Ω1. Suppose further (A5) for some M>0, g(t,x,u1)−g(t,x,u2)≥−M(u1−u2) whenever α≤u2≤u1≤β on Ω1. Assume (A6) for each x0∈[a,b] and y0∈ℝ, there exists a unique solution y(t,0,y0;x0) of
(5.1)yΔ=g(t,x(t,0,x0),y;η(t,x(t,0,x0)))−M(y−η),y(0)=y0,
whenever α≤η≤β.
Then there exist monotone sequences{αn(t,x)},{βn(t,x)}, and the functions ρ(t,x), r(t,x) such that if u is any solution
of (4.1),  then
(5.2)α≤α1≤⋯≤αn≤ρ≤u≤r≤βn≤⋯≤β1≤βonΩ1.Proof.
Consider the linear IVP(5.3)uΓ+f(t,x)uΔ2=G(t,x,u;η),u(0,x)=ϕ(x),
where G(t,x,u;η)=g(t,x,η)−M(u−η) and η∈C(Ω1,ℝ) is such that α≤η≤β on Ω1.
By (A5), we have
(5.4)g(t,x,α)−G(t,x,α;η)=g(t,x,α)−g(t,x,η)+M(u−η)≤M(η−α)+M(α−η)=0,
so it follows that αΓ+f(t,x)αΔ2≤G(t,x,α;η),α(0,x)≤ϕ(x)onΩ1. Similarly, we obtain that βΓ+f(t,x)βΔ2≤G(t,x,β;η),β(0,x)≤ϕ(x)onΩ1. Hence (A0) holds. Also, if u1≥u2,
(5.5)G(t,x,u1;η)−G(t,x,u2;η)=−M(u1−u2)≤L(u1−u2),
and therefore (A2) is satisfied for G. Furthermore, by (A6), (A4) is satisfied relative to
(5.6)yΔ=G(t,x(t,0,x0),y;η(t,x(t,0,x0))),y(0)=y0.
As a result, by
Theorem 4.2, there exists a unique solution u(t,x) of (5.3) on Ω1 for every η∈C(Ω1,ℝ) such that α≤η≤β on Ω1.
Defining a mappingA by Aη=u, where u is the unique
solution of (5.3) corresponding to η. Concerning this mapping A, we will show that (i) α≤Aα, β≥Aβ, and (ii) A is monotone on the sector [α,β], namely, if α≤η1≤η2≤β, then Aη1≤Aη2.
Letη=α and let Aα=α1, where α1 is the unique
solution of (5.3). Then we have αΓ+f(t,x)αΔ2≤G(t,x,α;α),α(0,x)≤ϕ(x), and α1Γ+f(t,x)α1Δ2=G(t,x,α;α),α1(0,x)=ϕ(x) on Ω1. By Theorem 4.1, we see that α≤α1=Aα. Similarly, we can show that β≥Aβ.
To prove (ii), letη1,η2∈C[Ω1,ℝ] be such that α≤η1≤η2≤β and let Aη1=u1, Aη2=u2, where u1, u2 are the unique
solutions of (5.3) corresponding to η=η1, η=η2, respectively. Then, u1Γ+f(t,x)u1Δ2=G(t,x,u1;η1)≤G(t,x,u1;η2) and u2Γ+f(t,x)u2Δ2=G(t,x,u2;η2). Also, u1(0,x)=u2(0,x). Hence by Theorem 4.1, we have u1≤u2 on Ω1. This proves Aη1≤Aη2.
Consider the sequence{αn(t,x)} and note that αn(t,x)=yn[t,0,ϕ(x(0,t,x));x(0,t,x)] on Ω1, where yn=yn(t,0,y0;x0) is the unique
solution of (5.6) such that y0=ϕ(x0). Thus αn(t,x(t,0,x0))=yn(t,0,ϕ(x0);x0) and α≤yn≤β. Since {yn} is monotone
sequence, it is easy to conclude that yn(t,0,ϕ(x0);x0) converges
uniformly and monotonically as n→∞. Suppose that limn→∞yn(t,0,ϕ(x0);x0)=y(t,0,ϕ(x0);x0) on 0≤t≤T. Then it is clear that yΔ(t,0,ϕ(x0);x0)=g(t,x(t,0,x0),y(t,0,ϕ(x0));x0), y(0)=ϕ(x0). Consequently, we can now define ρ(t,x)=y(t,0,ϕ(x(0,t,x));x(0,t,x)) on Ω1. Similar arguments hold relative to the sequence {βn(t,x)} and one defines r(t,x)=y(t,0,ϕ(x(0,t,x));x(0,t,x)) on Ω1.
Finally, we show thatα≤ρ≤u≤r≤β on Ω1, where u is any solution
of (4.1) such that α≤u≤β on Ω1. For this, it is enough to show that αn≤u≤βn on Ω1 and this we do
by induction. Suppose that αk≤u≤βk for some k on Ω1. Then we have
(5.7)αk+1Γ+f(t,x)αk+1Δ2=G(t,x,αk+1;αk),αk+1(0,x)=ϕ(x),uΓ+f(t,x)uΔ2=g(t,x,u)≥G(t,x,u;αk),u(0,x)=ϕ(x).
By Theorem 4.1, we have αk+1≤u on Ω1. We can show similarly that u≤βk+1. Hence it follows that αk≤u≤βk on Ω1 for all n, which prove the claim.Remark 5.2.
We note that if (A4) holds, then ρ and r are actually
solutions of (4.1).



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