Time scale version of the Ważewski retract method
In the paper there are discussed approaches to the Ważewski retract method on time scales. In particular there is presented planar case without a restrictive assumption that the whole boundary of a set of constraints, where we look for solutions, is a set of egress points. One example illustrating t...
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Інститут математики НАН України
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552807195738112 |
|---|---|
| author | Ruszkowski, S. Ruszkowski, S. |
| author_facet | Ruszkowski, S. Ruszkowski, S. |
| author_institution_txt_mv | [
{
"author": "S. Ruszkowski",
"institution": "Faculty of Mathematics and Computer Science, Nicolaus Copernicus University"
}
] |
| author_sort | Ruszkowski, S. |
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| description | In the paper there are discussed approaches to the Ważewski retract method on time scales. In particular there is presented planar case without a restrictive assumption that the whole boundary of a set of constraints, where we look for solutions, is a set of egress points. One example illustrating the main theorem is presented. |
| first_indexed | 2026-08-04T01:05:18Z |
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| fulltext |
Збiрник праць Iн-ту математики НАН України 2013, том 6, N 6, 190–208
Sebastian Ruszkowski
(Faculty of Mathematics and Computer Science, Nicolaus Copernicus
University, ul. Chopina 12/18, Toruń, Poland)
Time scale version of the Ważewski
retract method
sebrus@mat.umk.pl
In the paper there are discussed approaches to the Ważewski retract
method on time scales. In particular there is presented planar case without
a restrictive assumption that the whole boundary of a set of constraints,
where we look for solutions, is a set of egress points. One example illus-
trating the main theorem is presented.
Introduction
In 1947 Tadeusz Ważewski (see [1]) gave a simple but excellent topo-
logical principle, now called the Ważewski retract method, which has
been used by many authors to prove the existence of solutions of a given
differential equation which remain in a prescribed set of constraints. In
particular, the method helps to find bounded solutions in several differen-
tial problems. It generalizes the direct method of Lyapunov and is based
on examining so-called ‘egress’ and ‘strict egress’ points on a boundary
of the set of constraints. It is worth noting that the set does not need to
be an attractor or repellor. It is sufficient to check that the set of egress
points, which is usually assumed to be equal to the set of strict egress
points, is not a retract or, more generally, strong deformation retract of
the whole set. This topological principle became a base and a motivation
for a construction of a very well known and useful topological invariant,
the Conley index (see, e.g., [2] for a comparison of these two topological
tools).
c© Sebastian Ruszkowski, 2013
Sebastian Ruszkowski 191
The Ważewski retract method was generalized and adopted to: dif-
ferential inclusions (see, e.g., [2] or [3] and references therein), difference
equations (e.g. [4, 5]) or, recently, dynamic equations on time scales
([6, 7, 8]). This last area of research has been intensively developed
since 90’s as a unification and generalization of the theory of difference
equations and differential equations, and has found applications in many
mathematical models in biology and physics, where discrete and con-
tinuous dynamics have to be studied simultaneously. Moreover, various
impulsive differential problems can be transformed to dynamic equations
on time scales.
While several results on dynamic equations on time scales are just sim-
ple transformations of continuous or discrete analogs, the ones concerning
qualitative theory are not. The results on the Ważewski topological prin-
ciple for dynamic equations on time scales are still not satisfactory. In
fact, the only cases explored enough are the ones where the set of con-
straints is negatively invariant (see [6, 7]).
When we drop the above simplification, we meet several essential prob-
lems. The main of them is to construct a retraction, which has to be a
continuous map, from an initial section Ωt0 of the tube of constraints
onto the t0-section Et0 of the set of egress points. We need a deep ge-
ometrical study to overcome this problems. The Shöenflies theorem, a
convexity and strict convexity play an important role in proofs of non-
whole boundary case (see [8]).
The paper is organized as follows. In section 2 we recall some infor-
mation on the calculus on time scales that will be useful in the sequel.
Section 3 shows topological ideas contained in [6] and [7]. Section 4
presents results from [8], where the positive or negative invariance as
well as a repulsivity of the set is not assumed anymore. One transparent
example is given to illustrate the results.
2 Preliminaries
2.1 Basics of a calculus on time scales
The interested reader can consult [9, 10] to get a complete introduction
or to find proofs of statements of this section.
A time scale is any closed subset of the set R of real numbers and we
denote it by T.
192 Ważewski retract method
Basic functions describing T are jump operators σ, ρ :T→ T i µ :T→
R, defined as follows:
• σ(t) = inf{s ∈ T : s > t} (forward jump operator)
• ρ(t) = sup{s ∈ T : s < t} (backward jump operator)
• µ(t) = σ(t)− t (graininess function)
where we assume: inf ∅ = sup T and sup ∅ = inf T.
Proposition 2.1 (Induction Principle). Let t0 ∈ T and assume that
{S(t) : t ∈ [t0,∞) ∩ T} is a family of statements satisfying:
• The statemnt S(t0) is true.
• If t ∈ [t0,∞) ∩ T is right-scattered and S(t) is true for all s ∈
[t0, t) ∩ T, then S(σ(t)) is also true.
• If t ∈ [t0,∞) ∩ T is right-dense and S(t) is true, then there is a
neighborhood U of t such that S(s) is true for all s ∈ U ∩(t,∞)∩T.
• If t ∈ (t0,∞)∩T is left-dense and S(s) is true for all s ∈ [t0, t)∩T,
then S(t) is true.
Then S(t) is true for all t ∈ [t0,∞) ∩ T.
Definition 2.2. ∆-derivative of a function f : T→ X in a point t, where
X is a linear normed space, is the point f∆(t) ∈ X (if it exists) such that:
∀ε>0 ∃δ>0 ∀s∈B(t,δ)∩T ‖f(σ(t))− f(s)− f∆(t)(σ(t)− s)‖ ≤ ε|σ(t)− s|
Proposition 2.3. If a function f is continuous in t and:
• t = σ(t), then: f∆(t) = lims→t
f(t)−f(s)
t−s
• t 6= σ(t), then: f∆(t) = f(σ(t))−f(t)
µ(t)
• in general, for t ∈ Tκ we have
f∆(t) = lim
s→t T
f(σ(t))− f(s)
σ(t)− s
where Tκ is the set T without the point maxT if this point exists
and is isolated.
Sebastian Ruszkowski 193
2.2 ∆-differential equations
Definition 2.4. By a local solution of a system of equations:{
x∆(t) = f(t, x(t))
x(t0) = x0
(1)
we will mean a continuous function x :T∩ (a, b)→ X such that a < ρ(t0),
b > σ(t0), x(t0) = x0 and for all t ∈ Tκ∩(a, b) equation x∆(t) = f(t, x(t))
is fulfilled.
Definition 2.5. A solution x2 is an extension of a solution x1, if x1
and x2 are local solutions of the same system of equations, Dom(x2) (
Dom(x1) and x2|Dom(x1) = x1 .
If we cannot extend a local solution, then we call it a global solution.
Proposition 2.6. If for all t0 ∈ Tκ and x0 ∈ X there exists a unique
local solution of system (1), then for all t0 ∈ Tκ and x0 ∈ X there exists
a unique global solution of the same equation x : T ∩ (a, b) → X, where
µ(a) = 0 or a = −∞, and b− ρ(b) = 0 or b =∞.
In analogy to standard local processes on R we can define a local
∆-process. We have then a formal definition:
Definition 2.7. A continuous function Π:M → X (where M ⊂ X×T2)
is a local ∆-process if:
P1 ∀x∈X,t∈T∃α<t<β (µ(α) = 0 ∨ α = −∞) ∧ (β−ρ(β) = 0 ∨ β =∞) ∧
∧ {s ∈ T ; (x, t, s) ∈M} = (α, β) ∩ T,
P2 ∀x∈X,t∈T Π(x, t, t) = x,
P3 ∀(x,t,s),(x,t,r)∈M (Π(x, t, s), s, r) ∈M ∧ Π(Π(x, t, s), s, r) = Π(x, t, r).
Definition 2.8. We say that an equation x∆(t) = f(t, x(t)) generates a
local ∆-process Π, if for all x0 ∈ X and t0 ∈ T a function Π(x0, t0, ·) is a
global and unique solution of (1) and Π is a local ∆-process.
Analogously as processes on R, a ∆-process induce homeomorphisms
along trajectories:
There is also possibility of understanding a solution as a function that fulfills the
equation x∆(t) = f(t, x(t)) T-almost everywhere (that concept had been introduced in
[11]). If we accept this definition the next part of this paper needs only nonsignificant
changes.
194 Ważewski retract method
Proposition 2.9. If an equation x∆(t) = f(t, x(t)) generates a local
∆-process Π, and if all solutions of the problem{
x∆(t) = f(t, x(t))
x(t0) ∈ A
exist in time t1, then Π(·, t0, t1)|A is a homeomorphism between A and
its image.
Proof. Π is continuous so Π(·, t0, t1) and Π(·, t1, t0) are continuous on
theirs domains which implies what was to prove.
We will need the preservation of orientation by Π. Below we show a
simple theorem which gives an example of a class of functions implying
that property.
Definition 2.10. A function f : T × X → X is rd-continuous if it is
continuous in all t ∈ T such that µ(t) = 0, that is in so-called right dense
points (this justifies "rd" in the name).
Proposition 2.11. Let f : T×Rn → Rn be rd-continuous and Lipschitz
continuous (with Lipschitz constant L(t)) with respect to the second vari-
able. If for all t ∈ T inequality L(t)µ(t) < 1 is fulfilled, then an equation
x∆(t) = f(t, x(t)) generates a local ∆-process Π and for all t ∈ T we have
that function Π(·, t, σ(t)) preserves an orientation of Rn.
Proof. An equation x∆(t) = f(t, x(t)) has a global and unique solution
(see [9, p.322, 324]) with a continuous dependence on the initial condi-
tions, so this equation generates a local ∆-process Π. Moreover:
Π(x, t, σ(t))−Π(0, t, σ(t)) = x+ µ(t)f(t, x)− (0 + µ(t)f(t, 0)) =
= x+ µ(t)(f(t, x)− f(t, 0))
so, by L(t)µ(t) < 1, for x 6= 0 we have:
〈Π(x, t, σ(t))−Π(0, t, σ(t)), x〉 > 0
which means that vectors Π(x, t, σ(t))−Π(0, t, σ(t)) and x are in the same
halfspace, so Π(·, t, σ(t)) preserves an orientation of Rn.
Sebastian Ruszkowski 195
3 Ważewski method for the whole boundary
egress set
There are shown two approaches to basic Ważewski Theorem in this
chapter, which means: the case of set Ω, for which all trajectories starting
from the boundary immediately leaves that set. In this section we assume,
that X = Rn.
3.1 Approach 1.
If local ∆-process Π generated by ∆-equation is not well defined in
(x, t0, t1), it means, that solution starting in (x, t0) reaches to boundary
of Rn - infinity (one point compactification of Rn). This observation leads
to convenient notation for points (x, t0, t1) outside of the domain of local
∆-process Π: Π(x, t0, t1) :=∞.
We will use a function of positively closest point of change of interval
charakter of T.
Definition 3.1. Essential forward jump operator is a function essσ :T→
T ∪ supT with formula:
essσ(t) := inf{s ∈ T ; s > t ∧ (µ(s) > 0 ∨ µ(t) > 0)}.
Let Ω̃ be closed subset of R × Rn, such that for each r ∈ R the set
Ω̃r := {x ∈ Rn ; (r, x) ∈ Ω̃} is nonempty and bounded, ∂(Ω̃r) is not a re-
trakt of Ω̃r and {r} × ∂(Ω̃r) is a retrakt of ∂(Ω̃). We will use curtailment
of Ω̃ to the time scale T:
Ω :=
⋃
t∈T
{t} × Ω̃t.
Theorem 3.2. For above set Ω and equation x∆(t) = f(t, x(t)), which
generates local ∆-process Π, if for all t ∈ T and for all s ∈ (t, essσ(t)]
we have Π(cl(Ωc)t, t, s) ⊂ (Ωs)
c() then for each t0 ∈ T there exists point
x0 ∈ Ωt0 , such that the solution starting from (t0, x0) remain in Ω for
every t ∈ T bigger than t0.
Similar theorem is proved in [7], but they are focused on simple time scales (with
values of grainies function equal 0 or bigger than ε), and then using inverse systems
and analitic means, they obtain general case.
This condition means that starting from the outside of set Ω there is no trajektory
such that enters set Ω up to time of essential forward jump of starting time, which
means that the whole boundary of Ω is a set of egress points in a specyfic sens.
196 Ważewski retract method
Proof. We will prove by induction principle for time scales (Proposition
2.1), that Π(Ωt, t, s) ⊂ Ωs for s, t ∈ T where s 6 t.
Obviously Π(Ωt, t, t) = Ωt.
We have essσ(t) = σ(t) for right-scattered points, therefor:
Π(cl(Ωc)t, t, σ(t)) ∈ (Ωσ(t))
c, so Π(Ωσ(t), σ(t), t) ⊂ Ωt.
For points t in compact interval in time scale, that are not right bound-
ery of that interval we have that essσ(t) is right boundery of that interval.
In particular we have essσ(t) − t > ε > 0, so for s ∈ (t, t + ε) we have
Π(Ωt, t, s) ⊂ Ωs.
For other right-dense points t we know that ther exists sequence (tn) ⊂
T diminishing to t such that µ(tn) > 0 for all n, for which we have
Π(Ωσ(tn), σ(tn), tn) ⊂ Ωtn , so by continuity of Π we find ε > 0 such that
for s ∈ (t, t+ ε) ∩ T we have Π(Ωt, t, s) ⊂ Ωs.
For left-dense points we obtain needed property also by continuity.
By induction we have Π(Ωt, t, s) ⊂ Ωs for s, t ∈ T where s 6 t.
Let us fix t0 ∈ T. We can choose sequence (tn)n=0..∞ ⊂ T increasing
to supT and define (Ωn)n=1..∞ by:
Ωn := Π(Ωtn , tn, t0).
We know, that:
Ωn+1 = Π(Π(Ωtn+1 , tn+1, tn), tn, t0) ⊂ Ωn,
therefore (Ωn)n=1..∞ is descending family of compact sets. Intersec-
tion of descending family of nonemty compact sets is nonempty and we
choose x0 in that intersection. We now know, that trajektory starting
in (t0, x0) is in Ω up to any time tn, and tn → supT, so this is the
searched trajektory.
3.2 Approach 2.
Let bi, ci : T → R (for i = 1..n) be ∆-differentiable functions where
bi < ci. We define Ω using that functions:
Ω := {(t, x) ∈ T× Rn ; bi(t) 6 xi 6 ci(t) for all i}
and we will use notoation:
Sebastian Ruszkowski 197
∂TΩ := {(t, x) ∈ T× Rn ; bi(t) 6 xi 6 ci(t) for all i
wherein at least one inequality is equality}.
All points p ∈ ∂TΩ can be presented in one of the following ways:
p = (t, x1, ..., xi−1, bi(t), xi+1, ..., xn) ∈ Ωib
or
p = (t, x1, ..., xi−1, ci(t), xi+1, ..., xn) ∈ Ωic.
Theorem 3.3. Let bi, ci :T→ R be ∆-differentiable and f :T×Rn → Rn
generates local ∆-process Π. If for all (t, x) ∈ Ωib we have f(t, x) < b∆i (t)
and for all (t, x) ∈ Ωic we have f(t, x) > c∆i (t) then for each t0 ∈ T there
exists x0 ∈ Ωt0 , such that solution starting in (t0, x0) remains in Ω for
every t ∈ T bigger than t0.
Proof. (ad absurdum)
Let us notice, that for (t, x) ∈ Ωib, where µ(t) > 0, we have
Π(x, t, σ(t)) = x +
∫ σ(t)
t
f(τ, x)∆τ < x +
∫ σ(t)
t
b∆i (τ)∆τ = bi(σ(t)), and
similarly for µ(t) = 0 we have Π(x, t, t + ε) = x +
∫ t+ε
t
f(τ, x)∆τ <
x +
∫ t+ε
t
b∆i (τ)∆τ = bi(t + ε), which means that trajectories starting
in Ωib immiedietly leaves set Ω. By analogy, trajectories starting in Ωic
also immiedietly leaves set Ω.
Let us fix t0 ∈ T. We extend linearly Ω to continuous tube on
[t0, supT] ∩ R:
Ω∗ := Ω ∪ {(t, x) ∈ ([t0, supT] ∩ R \ T)× Rn ;
bi(ta) + (bi(tb)− bi(ta))
t− ta
tb − ta
6 xi 6 ci(ta) + (ci(tb)− ci(ta))
t− ta
tb − ta
for all i}
where ta, tb ∈ T are such that ta < t < tb and (ta, tb) ∩ T = ∅.
Naturally, for above ta and tb we know that set Ω∗[ta,tb] is convex.
Note that r :∂Ω∗ → {t0} × ∂Ωt0
r(t, x) := (t0, (bi(t0) +
ci(t0)− bi(t0)
ci(t)− bi(t)
(xi − bi(t)))i=1..n)
This is Theorem from [6], with assumptions given in the the language of ∆-
processes.
198 Ważewski retract method
is a retraction.
Now we will find continuous function from the set Ωt0 to the boundary
of Ω∗.
Negation of the thesis means that for all x ∈ Ωt0 we have finite time
of exit
te(x) := sup{t ∈ T ; ∀s∈T∩[t0,t](t,Π(x, t0, t)) ∈ Ω} < supT.
If µ(te(x)) = 0, then (te(x),Π(x, t0, te(x))) is in boundary of Ω∗.
If µ(te(x)) 6= 0, then (te(x),Π(x, t0, te(x))) ∈ Ω and
(σ(te(x)),Π(, t0, σ(te(x)))) /∈ Ω and by convexity of Ω∗[te(x),σ(te(x))] we ob-
tain unique intersection of Ω∗[te(x),σ(te(x))] with interval connecting points
(te(x),Π(x, t0, te(x))) and (σ(te(x)),Π(x, t0, σ(te(x)))). Denote this point
by (t∗e(x), x∗e).
Therefore, we can define function p :Ωt0 → ∂Ω∗:
p(x) :=
{
(te(x),Π(x, t0, te(x))), gdy µ(te(x)) = 0
(t∗e(x), x∗e), gdy µ(te(x)) > 0.
By continuities of Π and tube Ω∗ we have continuous dependence
(t∗e(x), x∗e) and (te(x),Π(x, t0, te(x))) in respect to x. It is enough to show
continuous dependence beetwen (t∗e(x), x∗e) and (te(x),Π(x, t0, te(x))).
For point x0 such that te(x0) is left-dense and right-scatered and
(te(x0),Π(x0, t0, te(x0))) ∈ Ω we have p(x) → p(x0) for x → x0 with
the time of exit te(x) < te(x0), and p(x) → p(x0) for x → x0 with the
time of exit te(x) = te(x0). By continuity of Π we can choose small
enough neighborhood of x0, which do not contains another points, so p
is continuous in x0.
By analogy, for left-scattered and rigth-dense points te(x0) we obtain
continuity of p in such points. Therefore p is continuous.
Note that function R :{t0} × Ωt0 → {t0} × ∂Ωt0
R(t0, x) := r(p(x))
is a composition of continuous functions, so it is a retraction, which is in
contradiction with the construction of set Ω.
4 Ważewski method for non-whole boundary
egress set
In this section there are presented results from [8].
Sebastian Ruszkowski 199
4.1 Notation
Let B(x, r) denote an open ball centered in x ∈ R2 and with a radius
r, D(x, r) = clB(x, r), S(x, r) = ∂B(x, r) and S1 := S(0, 1).
Proposition 4.1 (Shöenflies theorem). Any homeomorphism h : S1 →
h(S1) ⊂ R2 can be extended to a homeomorphism h̃ : R2 → R2.
In particular, for any homeomorphism h : S1 → h(S1) ⊂ R2 there
exists a homeomorphism ĥ : D(0, 1)→ ĥ(D(0, 1)) ⊂ R2 such that the set
ĥ(S1) is a boundary of ĥ(D(0, 1)) and the equality h(x) = ĥ(x) holds for
all x ∈ S1.
Let A ⊂ T× R2. Then we define:
At := {x ∈ R2 ; (t, x) ∈ A}.
Let Θ : T× S1 → R2 be a continuous function such that:
• Θt : S1 → Θt(S
1) ⊂ R2, where Θt(x) = Θ(t, x), is a homeomor-
phism,
• Θ(t, s) = Θ(σ(t), s).
For all t ∈ T let Ωt be a closure of a bounded open set surrounded by
the curve Θ(t, S1) and
Ω :=
⋃
t∈T
{t} × Ωt.
For such construction we will say that Ω is Θ-bounded. In particular Ω
can be a constant tube Ω = T×Ω0, where Ω0 is homeomorphic to D(0, 1).
We consider the following parts of the set Ω:
∂TΩ := Θ(T× S1) =
⋃
t∈T
{t} × ∂(Ωt)
∂TΩ+ :=
⋃
t∈T
{t}×cl{x ∈ R2 ; (t, x) ∈ ∂TΩ∧∃r>0∀y∈B(x,r)∩Ωt∀λ∈(0,1) λx+
+(1− λ)y ∈ intΩ}
It is a well kown property of planes, which is presented for example in [12, pp.
68,72].
200 Ważewski retract method
In other words, (∂TΩ+)t is a closure of the set of points in ∂(Ωt) that
have strictly convex neighborhoods in Ωt.
For any maps f :T× R2 → R2 and g :R2 → R2 we define where it
makes sense:
ft(x) := f(t, x),
Φg(·, ·), a local flow generated by the equation y′ = g(y)
wg(x), a duration of a solution in a local flow Φg started in x
We focus our attention at the following subsets of Ω:
• Set of egress points:
E :={(t, x) ∈ ∂TΩ ; y′ = ft(y) generates a local flow Φft
and Φft(x, (0, s]) 6⊂ Ωt for any s ∈ (0, wft(x))}
• Set of escape points:
Es := {(t, x) ∈ Ω ; µ(t) 6= 0 and x+ µ(t)f(t, x) 6∈ intΩσ(t)}
4.2 Theorems
Now we will prove the main theorem of the paper.
Theorem 4.2. Let f : T× R2 → R2 be a map such that:
(H0) equation x∆(t) = f(t, x(t)) generates a local ∆-process Π,
(H1) for all t ∈ T a function Π(·, t, σ(t)) preserves an orientation of R2,
(H2) Ω is Θ-bounded (see section 4.1),
(H3) there exists a closed set W ( S1 such that W is not a retract of
D(0, 1) and Θ(T×W ) = E,
(H4) if µ(t) 6= 0, then
(H4a) Π(Et, t, σ(t)) ∩ Ωσ(t) = ∅,
(H4b) Π(intΩt, t, σ(t)) ∩ ∂TΩσ(t) ⊂ Eσ(t).
Then, for all t0 ∈ T, there exists a point x0 ∈ Ωt0 such that the solution
starting in (t0, x0) remains in Ω for all t ∈ T bigger than t0.
Sebastian Ruszkowski 201
Proof. (ad absurdum)
Let us fix for a while point t ∈ T such that µ(t) 6= 0.
By assumption (H4a), we know that Et ⊂ Est.
By definition of Es we also know that Π(Eσ(t), σ(t), t)∩Ωt ⊂ ∂(Est).
By assumptions (H2) and (H3) we know that Ωt = Ωσ(t) and Et =
Eσ(t). This allows us to define the following continuous tube:
Ẽ = {(s, x) ∈ R× R2 ; (sup{t ∈ T ; t < s}, x) ∈ E}.
In other words we fill the interstices caused by a time scale.
We will construct a continuous function wt :Est → [t, σ(t)]× Et ⊂ Ẽ
such that:
w1 ∀x∈Et wt(x) = (t, x)
w2 ∀x∈Π(Eσ(t),σ(t),t)∩Ωt wt(x) = (σ(t),Π(x, t, σ(t)))
which we will use to construct a continuous function from Ωt0 to Ẽ
and by that, a retraction from D(0, 1) to W .
Let I be the set of indices of connected components Eit of Π(Et, t, σ(t))
By assumption (H4a) we know that each set Eit is contained in a corre-
sponding connected component γi of Π(∂TΩt, t, σ(t)) \ ∂TΩt. The curve
γi cuts from cl(Ωct) a closed bounded connected set denoted by Esit.
Figure 1
By assumption (H3) we know that Et 6= ∂TΩt so a boundary of Esit is
a closed curve and a sum of four curves: θ1
i , Eit , θ2
i and ∂TΩ ∩Esit (each
of them homeomorphic to line segments), so it is homeomorphic to S1
(see Figure 1). By assumption (H1) the sets ∂TΩt and Π(∂TΩt, t, σ(t))
202 Ważewski retract method
have the same orientation, and therefore Eit and ∂TΩ ∩ Esit have an op-
posite orientation on the boundary of Esit, so we can parameterize that
boundary to obtain a homeomorphism hi :∂Ti → ∂(Esit) such that:
• Ti is a trapezoid with vertices (0, 0), (1, 0), (ai, 1), (bi, 1) where
[ai, bi] ⊂ [0, 1], ,
• hi([0, 1], 0) = Eit ,
• hi([ai, bi], 1) = ∂TΩt ∩ Esit,
• ∀x∈[ai,bi]hi(x, 0) = Π(hi(x, 1), t, σ(t)).
By the Shöenflies theorem we can extend hi to a homeomorphism
ĥi : Ti → Esit. If Π(Est, t, σ(t)) \
⋃
i∈I Es
i
t 6= ∅, then with other con-
Est
4 denotes ΩtEst
1
Est
2
Est
Est
3
Est
denotes image
of Ωt by Π(·, t, σ(t))
denotes Et
I = {1,2,3}
J = {4}
Figure 2
nected components (indexed with elements of some set J) we make sim-
ilar sets Esjt , each of them bounded by a part of ∂(Ωt) and a part of
Π(∂(Ωt), t, σ(t)), so bounded by a curve homeomorphic to S1 (see Figure
2). Then we have a homeomorphism hj :∂Tj → ∂(Esjt ) such that:
• Tj is a triangle with vertices (1/2, 1/2), (0, 1), (1, 1),
• hj([0, 1], 1) = ∂TΩt ∩ Esjt .
and again by the Shöenflies theorem we can extend hj to a homeomor-
phism ĥj :Tj → Esjt .
With that construction θ1
i and θ2
i are the images of the side edges of Ti.
In that construction point hj(1/2, 1/2) is free to choose.
Sebastian Ruszkowski 203
Now we know that Π(Est, t, σ(t)) ⊂
⋃
i∈I Es
i
t∪
⋃
j∈J Es
j
t which is the
sum of disjoint sets, therefore we can define wt :Est → [t, σ(t)]×Et ⊂ Ẽ,
wt(x) :=
(t+ µ(t)p2(·),Π(ĥi(p1(·), 0), σ(t), t))(ĥi
−1
(Π(x, t, σ(t)))),
Π(x, t, σ(t)) ∈ Esit (i ∈ I)
(t+ µ(t)p2(·),Π(ĥi(p1(·), 1), σ(t), t))(ĥi
−1
(Π(x, t, σ(t)))),
Π(x, t, σ(t)) ∈ Esjt (j ∈ J)
(σ(t),Π(x, t, σ(t)))
Π(x, t, σ(t)) ∈ Eσ(t)
where p1 and p2 are projections respectively onto the first and second
variables. By construction it is a continuous function.
Notice that for x ∈ Et there exists a unique i ∈ I such that
Π(x, t, σ(t)) ∈ Eit , and consequently y := ĥ−1
i (Π(x, t, σ(t))) ∈ [0, 1] ×
{0}. Since p1(y) = y and p2(y) = 0, we obtain that wt(x) =
(t,Π(ĥi(y), σ(t), t)) = (t, x). Hence property w1 is satisfied.
Moreover, if x ∈ Π(Eσ(t), σ(t), t) ∩ Ωt, then Π(x, t, σ(t)) ∈ Eσ(t), so
also property w2 is fulfilled.
Falsity of thesis means that for every x ∈ Ωt0 we have:
te(x) := sup{t ∈ T ; ∀s∈T∪[t0,t](t,Π(x, t0, t)) ∈ Ω} < supT.
If µ(te(x)) = 0, then (te(x),Π(x, t0, te(x))) is already in E ⊂ Ẽ.
If µ(te(x)) 6= 0, then (te(x),Π(x, t0, te(x))) ∈ Ω and
(σ(te(x)),Π(x, t0, σ(te(x))) 6∈ Ω. So we can use the function wte(x) to
it.
Therefore, we can define r :Ωt0 → Ẽ,
r(x) :=
{
(te(x),Π(x, t0, te(x))), if µ(te(x)) = 0,
wte(x)(Π(x, t0, te(x))), if µ(te(x)) 6= 0.
Take any point x such that µ(te(x)) = 0.
Then for each ε > 0 there exists τ ∈ T such that 0 < τ − t0 < ε and
Π(x, t0, τ) 6∈ Ωτ so, by the continuity of Π, for each ε > 0 there exist
δ > 0 and τ ∈ T such that 0 < τ − t0 < ε and Π(B(x, δ), t0, τ) ∩ Ωτ = ∅.
Therefore
∀ε>0∃δ>0∀y∈B(x,δ) te(y) < te(x) + ε.
Similarly we show that for each point x such that te(x)−ρ(te(x)) = 0
we have
∀ε>0∃δ>0∀y∈B(x,δ) te(y) > te(x)− ε.
204 Ważewski retract method
Using a continuity of Π we get a continuity of r in all x such that te(x)
is a dense point of T. Furthermore, properties w1 and w2 guarantee a
continuity of r in every point x such that µ(te(x)) 6= 0 or te(x)−ρ(te(x)) 6=
0.
Hence r is continuous for all points in Ωt0 .
By the Shöenflies theorem we can extend Θt to Θ̂t :D(0, 1) → Ωt for
every t ∈ T. Using this we define a map R :D(0, 1)→W :
R(y) := Θ−1
te(Θ̂t0 (y))
◦ p2 ◦ r(Θ̂t0(y)).
For all y ∈ W we have R(y) = Θ−1
t0 p2r(Θt0(y)) =
Θ−1
t0 (Π(Θt0(y), t0, t0)) = y and, by the continuity of r, we get that
R is a retraction, what contradicts assumption (H3).
The geometric assumption (H4’) in the next theorem corresponds to
the assumption (H4) and may occure to be easier to check.
Theorem 4.3. Assume that:
(H0) equation x∆(t) = f(t, x(t)) generates a local ∆-process Π,
(H1) for all t ∈ T a function Π(·, t, σ(t)) preserves an orientation of R2,
(H2) Ω is Θ-bounded,
(H3’) there exists a closed set W ( S1 such that W is not a retract of
D(0, 1) and Θ(T×W ) = ∂TΩ+ = E,
(H4’) if µ(t) 6= 0, then Ωt ⊂ x + TΩt(x) for all x ∈ int∂TΩtEt, where
TΩt(x) is a Bouligand tangent cone of the set Ωt ∈ R2 in a point x
(TK(x) = {v ∈ X ; lim infh→0+
d(x+hv,K)
h = 0}).
Then, for all t0 ∈ T, there exists a point x0 ∈ Ωt0 such that a solution
starting in (t0, x0) remains in Ω for all t ∈ T bigger than t0.
Proof. To use Theorem 4.2 it is sufficient to show that, if µ(t) 6= 0, then
Π(Et, t, σ(t)) ∩ Ωσ(t) = ∅ and Π(intΩt, t, σ(t)) ∩ ∂TΩσ(t) ⊂ Eσ(t).
Let us fix t ∈ T such that µ(t) 6= 0.
We have that Π(x, t, σ(t)) = x+ µ(t)ft(x) and, for all x ∈ Et, vectors
ft(x) are directed outside the set Ωt so a local strict convexity of points
in Et (assumption (H3’)) guarantees that each connected component Et,i
Sebastian Ruszkowski 205
of Et has no common points with Π(Et,i, t, σ(t)). Moreover, assumption
(H4’) ensures that the set Π(Et,i, t, σ(t)) is outside of the rest of Ωσ(t),
so the first part is fulfilled.
A connected component It,i of ∂TΩt \ Et has only points without
strictly convex neighborhoods in Ωt. All of that points are not egress
points in a local flow, so ft(x) are directed inside the set Ωt. If there
were y ∈ It,i ∩Π(It,i, σ(t), t) and [y,Π(y, t, σ(t))] 6⊂ ∂TΩt, then It,i would
be a part of a spiral shaped curve which end is a beginning of a part of
Et, which would contradict assumption (H4’). Therefore there exists a
small enough neighborhood Ot,i of It,i such that an image of Ot,i ∩ intΩt
has no common points with It,i.
∂TΩt is homeomorphic to Π(∂TΩt, t, σ(t)) so, if Et,i borders on It,j ,
then their images have to border as well. Therefore an image of It,j cuts
out subset Ωjt of Ωt that contains It,j . Moreover the image of ∂TΩt does
not have selfintersections so, in particular, an image of It,j is the only
part of the image of Ωt that can have common points with Ωjt . It means
that
Π(intΩt, t, σ(t)) ∩ (∂TΩσ(t) \ Eσ(t)) = ∅
what was needed to prove.
Example 4.4. Let T =
⋃
n∈N [2n, 2n + 1] and f(t, (x, y)) =(
e−t(1−|y| sin(tπ))−2x
3 , 2y+sin x
5
)
. We are interested in existence of trajec-
tory convergent to (0, 0).
We will want to use Theorem 4.2 taking Ω :=
⋃
n∈N
⋃
t∈[2n,2n+1] {t}×
{(x1, x2) ∈ R2 ; −e(n−t)/3 6 xi 6 e(n−t)/3}.
Firstly, for t = 2n+ 1 and x1, x2, y1, y2 ∈ R we have
‖f(t, (x1, y1))− f(t, (x2, y2))‖ =
= ‖(2(x2 − x1)/3, (2(y1 − y2) + sin(x1)− sin(x2))/5)‖ ≤
≤ ‖(2/3, 3/5)‖ ‖(x1 − x2, y1 − y2)‖ < ‖(x1, y1)− (x2, y2)‖,
therefor we have L(t)µ(t) < 1, so assumptions of Proposition 2.11 are
met.
The set Ω is selected so that Ωt = Ωσ(t) and is Θ-bounded, where
Θ(t, (x, y)) = e(n−t)/3
sup{|x|,|y|} (x, y), for t ∈ [2n, 2n+ 1].
We will find E.
206 Ważewski retract method
For all t ∈ [2n, 2n + 1] and −e(n−t)/3 ≤ x ≤ e(n−t)/3 = |y| we have
(t, (x, y)) ∈ ∂TΩ and | sin(x)| < |y|, therefore these are the egress points.
Whereas for each t ∈ [2n, 2n + 1] i −e(n−t)/3 < y < e(n−t)/3 = |x| we
have (t, (x, y)) ∈ ∂TΩ and |e−t(1 − |y| sin(tπ))| 6 e−t < e(n−t)/3 = |x|,
therefore vectors f(t, (x, y)) areare directed to the center of set Ωt and∣∣∣ e−t(1−|y| sin(tπ))−2x
3
∣∣∣ > | 13x| = | ddte
(n−t)/3|, thus these points are entry
points.
For t = 2n+1 i (x, y) ∈ Et we have Π((x, y), t, σ(t)) = (x+ e−t−2x
3 , y+
2y+sin(x)
5 ) /∈ Ωσ(t), so the assumption (H4a) is met.
For t = 2n + 1 i (x, y) ∈ Ωt we have similarly: Π((x, y), t, σ(t)) =
(x + e−t/3−2x
3 , y + 2y+sin(x)
5 ), therefore the first coordinate is inside the
segment [ e
−t/3−en/3−t/3
3 , e
−t/3+en/3−t/3
3 ], which means that there are no
common points with ∂TΩt \ Et, so assumption (H4b) is met too.
All assumptions are satisfied, therefore there exists trajectory remain-
ing in the set Ω, which is convergent to (0, 0) (from the selection of the
set Ω).
4.3 Remarks
At first we notice that holes homeomorphic to balls in Ωt are avail-
able in Theorem 4.2. Indeed, for n holes we can consider: Θ :
T ×
⊕n
j=0 S
1 → R2 such that Θ({t} ×
⊕n
k=0 S
1) is homeomorphic to
S(0, 1) ∪
⋃n
k=1 S((0, (k − 1)/n), 1/3n)).
In the second remark we observe that properties of a local ∆-process
Π are essential, not of f itself, so in all approaches we can change our
understanding of a solution of x∆(t) = f(t, x(t)) and treat it as a function
that fulfills the equation T-almost everywhere (in a Sobolev space on a
time scale). Moreover, we can change assumptions to a T-almost every-
where form. It is important when we look for possible generalizations to
differential inclusions or multivalued ∆-processes.
A proof technique presented in Section 4 cannot be repeated in higher
dimensions because the Shöenflies theorem does not raise up to them.
The following open problem appears:
That concept had been introduced in [11].
Sebastian Ruszkowski 207
Open problem:
Is it possible to use in higher dimensional spaces the geometric idea
presented in the proof of Theorem 4.2 under some additional restrictions
to Θ or Π?
Nevertheless, this geometric idea opens new perspectives in the
Ważewski retract method on time scales and allows us to study more
classes of systems (for example hyperbolic systems).
References
[1] T. Ważewski, Sur un principe topologique de l’examen de l’allure
asymptotique des intégrales des équations différentielles ordinaires,
Ann. Soc. Polon. Math. 20 (1947), 279-313.
[2] R. Srzednicki, Ważewski method and Conley index, Handbook of dif-
ferential equations, 591-684, Elsevier/North-Holland, Amsterdam,
2004.
[3] G. Gabor and M. Quincampoix, On existence of solutions to differ-
ential inclusions remaining in a prescribed closed subset of a finite-
dimensional space, J. Differential Equations 185 (2002), No. 2, 483-
512.
[4] C.V. Coffman, Asymptotic behavior of solution of ordinary difference
equations, Trans. Amer. Math. Soc., 110 (1964), 22-51.
[5] J. Diblik, Anti-Lyapunov method for systems of discrete equations,
Nonlinear Anal. 57 (2004), no. 7-8, 1043-1057.
[6] J. Diblik, M.Ru̇žičková, Z. Šmarda, Ważewski’s metod for systems
of dynamic equations on time scales, Nonlinear Analysis 71 (2009),
e1124-e1131.
[7] L. Adamec, A theorem of Ważewski and dynamic equations on time
scales, J. Difference Equ. Appl. 13 (2007), no. 1, 63-78.
[8] G. Gabor, S. Ruszkowski Ważewski theorem on time scales with a
set of egress points that is not the whole boundary, Nonlinear Anal.,
Vol. 75 no. 18, 6541-6549.
208 Ważewski retract method
[9] M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An
Introduction with Applications, Birkhäuser, Boston, 2001.
[10] M. Bohner, A. Peterson, Advances in dynamic equations on time
scales, Birkhäuser, Boston, 2003.
[11] A. Cabada, D.R. Vivero, Expression of the Lebesgue ∆-integral on
time scales as a usual Lebesgue integral; application to the calculus
of ∆-antiderivatives, Math. Comput. Modelling 43 (2006), no. 1-2,
194-207.
[12] E.E. Moise, Geometric Topology in Dimensions 2 and 3, Springer-
Verlag, New York, Berlin, Heidelberg, 1977
|
| id | oai:trim.imath.kiev.ua:article-304 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:05:18Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/bd/4b9a7b8a248f8c4841ee2630058955bd.pdf |
| spelling | oai:trim.imath.kiev.ua:article-3042018-02-10T20:56:26Z Time scale version of the Ważewski retract method Часова версія ретрактного методу Важевського Ruszkowski, S. Ruszkowski, S. In the paper there are discussed approaches to the Ważewski retract method on time scales. In particular there is presented planar case without a restrictive assumption that the whole boundary of a set of constraints, where we look for solutions, is a set of egress points. One example illustrating the main theorem is presented. В роботі обговорюються підходи до ректрактного методу Важевського за часом. Зокрема, представлений плоский випадок без припущень, що вся межа множини обмежень, де ми шукаємо розв'язок, є множиною точок виходу. Також наведений один з прикладів, що ілюструє основну теорему. Інститут математики НАН України 2013-06-26 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/304 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 10 No. 6 (2013): Brasilian-Polish Topology Workshop; 190-208 Сборник Трудов Института математики НАН Украины; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 190-208 Збірник Праць Інституту математики НАН України; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 190-208 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/304/286 Авторське право (c) 2013 Праці Інституту математики НАН України |
| spellingShingle | Ruszkowski, S. Ruszkowski, S. Time scale version of the Ważewski retract method |
| title | Time scale version of the Ważewski retract method |
| title_alt | Часова версія ретрактного методу Важевського |
| title_full | Time scale version of the Ważewski retract method |
| title_fullStr | Time scale version of the Ważewski retract method |
| title_full_unstemmed | Time scale version of the Ważewski retract method |
| title_short | Time scale version of the Ważewski retract method |
| title_sort | time scale version of the ważewski retract method |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/304 |
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