An infinite-dimensional Borsuk-Ulam-type generalization of the Leray-Schauder fixed-point theorem and some applications
A generalization of the classical Leray – Schauder fixed point theorem, based on the infinite-dimensional Borsuk – Ulam type antipode construction, is proposed. A new nonstandard proof of the classical Leray – Schauder fixed point theorem and a study of the solution manifold to a nonlinear Hamilto...
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irk-123456789-1645062020-02-10T01:28:04Z An infinite-dimensional Borsuk-Ulam-type generalization of the Leray-Schauder fixed-point theorem and some applications Prykarpatsky, A.K. Статті A generalization of the classical Leray – Schauder fixed point theorem, based on the infinite-dimensional Borsuk – Ulam type antipode construction, is proposed. A new nonstandard proof of the classical Leray – Schauder fixed point theorem and a study of the solution manifold to a nonlinear Hamilton – Jacobi type equation are presented. Запропоновано узагальнення класичної теореми Лерея - Шаудера про нерухому точку, що ґрунтується на нєскінчєнновимірній конструкції антиподiв типу Борсука-Улама. Наведено нестандартне доведення класичної теореми Лерея - Шаудера про нерухому точку та досліджено многовид розв'язків нелінійного рівняння типу Гамільтона-Якобі. 2008 Article An infinite-dimensional Borsuk-Ulam-type generalization of the Leray-Schauder fixed-point theorem and some applications / A.K. Prykarpatsky // Український математичний журнал. — 2008. — Т. 60, № 1. — С.100–106. — Бібліогр.: 28 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164506 517.9 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Prykarpatsky, A.K. An infinite-dimensional Borsuk-Ulam-type generalization of the Leray-Schauder fixed-point theorem and some applications Український математичний журнал |
| description |
A generalization of the classical Leray – Schauder fixed point theorem, based on the infinite-dimensional
Borsuk – Ulam type antipode construction, is proposed. A new nonstandard proof of the classical Leray –
Schauder fixed point theorem and a study of the solution manifold to a nonlinear Hamilton – Jacobi type
equation are presented. |
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Article |
| author |
Prykarpatsky, A.K. |
| author_facet |
Prykarpatsky, A.K. |
| author_sort |
Prykarpatsky, A.K. |
| title |
An infinite-dimensional Borsuk-Ulam-type generalization of the Leray-Schauder fixed-point theorem and some applications |
| title_short |
An infinite-dimensional Borsuk-Ulam-type generalization of the Leray-Schauder fixed-point theorem and some applications |
| title_full |
An infinite-dimensional Borsuk-Ulam-type generalization of the Leray-Schauder fixed-point theorem and some applications |
| title_fullStr |
An infinite-dimensional Borsuk-Ulam-type generalization of the Leray-Schauder fixed-point theorem and some applications |
| title_full_unstemmed |
An infinite-dimensional Borsuk-Ulam-type generalization of the Leray-Schauder fixed-point theorem and some applications |
| title_sort |
infinite-dimensional borsuk-ulam-type generalization of the leray-schauder fixed-point theorem and some applications |
| publisher |
Інститут математики НАН України |
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2008 |
| topic_facet |
Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/164506 |
| citation_txt |
An infinite-dimensional Borsuk-Ulam-type generalization of the Leray-Schauder fixed-point theorem and some applications / A.K. Prykarpatsky // Український математичний журнал. — 2008. — Т. 60, № 1. — С.100–106. — Бібліогр.: 28 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT prykarpatskyak aninfinitedimensionalborsukulamtypegeneralizationofthelerayschauderfixedpointtheoremandsomeapplications AT prykarpatskyak infinitedimensionalborsukulamtypegeneralizationofthelerayschauderfixedpointtheoremandsomeapplications |
| first_indexed |
2025-07-14T17:03:02Z |
| last_indexed |
2025-07-14T17:03:02Z |
| _version_ |
1837642633985392640 |
| fulltext |
UDC 517.9
A. K. Prykarpatsky (Inst. Appl. Problem Mech. and Math. Nat. Acad. Sci. Ukraine, Lviv)
AN INFINITE-DIMENSIONAL BORSUK – ULAM TYPE
GENERALIZATION OF THE LERAY – SCHAUDER FIXED
POINT THEOREM AND SOME APPLICATIONS
НЕСКIНЧЕННОВИМIРНЕ УЗАГАЛЬНЕННЯ ТИПУ
БОРСУКА – УЛАМА ДЛЯ ТЕОРЕМИ ЛЕРЕЯ – ШАУДЕРА
ПРО НЕРУХОМУ ТОЧКУ ТА ДЕЯКI ЗАСТОСУВАННЯ
A generalization of the classical Leray – Schauder fixed point theorem, based on the infinite-dimensional
Borsuk – Ulam type antipode construction, is proposed. A new nonstandard proof of the classical Leray –
Schauder fixed point theorem and a study of the solution manifold to a nonlinear Hamilton – Jacobi type
equation are presented.
Запропоновано узагальнення класичної теореми Лерея – Шаудера про нерухому точку, що ґрунту-
ється на нескiнченновимiрнiй конструкцiї антиподiв типу Борсука – Улама. Наведено нестандарт-
не доведення класичної теореми Лерея – Шаудера про нерухому точку та дослiджено многовид
розв’язкiв нелiнiйного рiвняння типу Гамiльтона – Якобi.
1. Introduction. The fixed point theorems are of very importance for many applications
[1 – 3] in modern theories of differential equations and mathematical physics. Especially,
the classical Leray – Schauder theorem and its diverse modifications [1, 4 – 9] in infinite-
dimensional both Banach and Frechet spaces, being nontrivial generalizations of the well
known finite-dimensional Brouwer fixed point theorem, are of special interest [4 – 7, 10,
11] in modern nonlinear mathematical analysis. In particular, there exist many problems
in theories of differential and operator equations [1, 4, 9 – 12], which can be uniformly
formulated as the following equation:
âx = f(x), (1)
where x ∈ E1, â : E1 → E2 is a closed surgective linear operator from Banach space E1
onto Banach space E2, defined on a domain D(â) ⊂ E1 (which can be not dense) and
f : E1 → E2 is a nonlinear continuous mapping, whose domain D(f) = D(a) ∩ Sr(0).
(Here Sr(0) ⊂ E1 is the sphere in E1 of radius r > 0, centered at zero.)
The following problem, important for many applications, is posed.
Problem. Under what conditions on the linear operator â : E1 → E2 and the non-
linear continuous mapping f : E1 → E2 does equation (1) possess a solution x ∈ D(f),
and what is the topological dimension dimN (â, f) of the solution set N (â, f) ⊂ D(f)?
Recall also that the topological dimension of a closed compact set A ⊂ X (X
is a topological space) is defined as the number dim A := inf
{
k ∈ Z+ : there holds
the condition
⋂
j=1,k+2
Uαj = ∅ for any subsets Uαj ∈ {Uαβ
} of all specially chosen
subcoverings {Uαβ
} of any covering {Uα} of the set A
}
.
a) In the case := id and E1 := E2 equation (1) reduces to the standard fixed point
problem f(x) = x, x ∈ Sr(0), studied before [1, 5, 8, 13, 14] by Banach, Leray,
Schauder, Browder and many other mathematicians.
c© A. K. PRYKARPATSKY, 2008
100 ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
AN INFINITE-DIMENSIONAL BORSUK – ULAM TYPE GENERALIZATION ... 101
b) In the odd case when f(−x) = −f(x) for any x ∈ D(f) equation (1) reduces
to an infinite-dimensional generalization of the classical Borsuk – Ulam theorem on the
sphere Sr(0) ⊂ E1, which was recently stated by B. Gelman [11, 15].
Below we will prove a theorem, giving rise to a suitable solution to the Problem
above, and give some its application to studying the solution set to a nonlinear Hamilton –
Jacobi type equation.
2. Main theorem. We will assume further that the following natural conditions are
fulfilled:
i) domain D(f) = D(a) ∩ Sr(0);
ii) the mapping f : E1 → E2 is â-compact that is, it is continuous and for any
bounded set A2 ⊂ E2, any bounded A1 ⊂ D(f) the set f(A1 ∩ â−1(A2)) is relatively
compact in E2 (the empty set ∅ is considered, by definition, compact);
iii) there exists a bounded constant kf > 0, such that
sup
x∈Sr(0)
1
r
‖f(x)‖2 := k−1
f ;
iv) the inequality
k(â) < kf
holds, where, by definition,
k(â) := ‖ã−1‖ = sup
y∈E2
1
‖y‖2
inf
x∈D(â)
{
‖x‖1 : âx = y
}
, (2)
and ã := â|E1/ ker â is an invertible susjective and continuous linear operator from the
factor-space E1/ ker â onto E2.
Then the following main theorem [16 – 18] holds.
Theorem 1. Let the dimension dim Ker â ≥ 1 and conditions i) – iv) hold. Then
equation (1) possesses in D(f) ⊂ E1 the nonempty solution setN (â, f), whose topologi-
cal dimension dimN (â, f) ≥dim ker â− 1.
A proof of the theorem is based on the following lemmas.
Lemma 1. For any constant ks > k(â) there exists a continuous odd selection
s : E2 → E1 for the mapping ã−1 : E2 → E1, satisfying the conditions:
1) âs(y) = y for any y ∈ E2;
2) ‖s(y)‖1 ≤ ks‖y‖2, y ∈ E2.
Proof. The lemma can be proved making use of the well known E. Michael theorem
[19] on the selection for a linear surjective and continuous mapping, applied to the
induced mapping ã : E1/ ker â → E2. As the latter is invertible and continuous, there
exists the bounded constant k(â) := ‖ã−1‖ < ∞. The set-valued mapping ã−1 : E2 →
→ E1 is lower semi-continuous with closed convex values. It is clear that ã−1(−y) =
= −ã−1(y) for any y ∈ E2. Consider now, following [11, 15], another set-valaued
mapping ϕ : E2 → E1, such that ϕ(y) = Br(y)(0) for any y ∈ E2, where Br(y)(0) is
the closed ball of radius r(y) = k(â)‖y‖2 +1 in E2. If to define a mapping ϕ : E2 → E1
as ϕ̃(y) := ã−1(y)∩ϕ(y), one can see that ϕ̃(−y) = −ϕ̃(y) for any y ∈ E2. There exists
a theorem proved by E. Michael [19], which says that any below semicontinuous set-
valued mapping ϕ : E2 → E1 of a paracompact space E2 (in particular, of any metrized
or Banach space E2) into a Banach space E1 with closed and convex values possesses
a continuous selection. Moreover, by the theorem on equivariant selections [20] there
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
102 A. K. PRYKARPATSKY
exists an odd selection s : E2 → E1, such that s(y) ∈ ϕ̃(y) for each y ∈ E2, whence
âs(y) = y. This mapping, in general, is nonlinear, if there does not exist the linear
continuous projector from E1 onto ker â ⊂ E1. The selection s : E2 → E1 allows also
a more analytical construction. Really, since the set-valued mapping â−1 : E2 → E1 is
defined on the whole Banach space E2, one can write down that
â−1y = x̄y ⊕Ker â (3)
for any y ∈ E2 and some specified elements x̄y ∈ E1\ ker â, labelled by elements y ∈
∈ E2. If the composition (3) is already specified, we can define a selection s : E2 → E1
as follows:
s(y) :=
1
2
(x̄y − x̄−y)⊕ 1
2
(c̄y − c̄−y), (4)
where the elements c̄y ∈ ker â, y ∈ E2, are chosen arbitrary, but fixed. It is now easy
to check that
s(−y) = −s(y)
and
â s(y) = â
(
1
2
(x̄y − x̄−y)⊕ 1
2
(c̄y − c̄−y)
)
=
=
1
2
âx̄y −
1
2
âx̄−y =
1
2
y − 1
2
(−y) = y
for all y ∈ E2, thereby the mapping (4) satisfies the main conditions i) and ii) above.
To state the continuity of the mapping (4), we will consider below expression (2) for the
norm ‖ã−1‖ = k(â) of the inverse mapping ã−1 : E2 → E1. We can easily write down
the following inequality:∥∥s(y)
∥∥
1
=
∥∥∥∥1
2
(x̄y − x̄−y)⊕ 1
2
(c̄y − c̄−y)
∥∥∥∥
1
=
=
1
2
‖(x̄y ⊕ c̄y)− (x̄−y ⊕ c̄−y)‖1 ≤
≤ 1
2
(‖(x̄y ⊕ c̄y)‖1 + ‖(x̄−y ⊕ c̄−y)‖1) ≤
≤ 1
2
ks ‖y‖2 +
1
2
ks ‖y‖2 = ks ‖y‖2 ,
giving rise to the continuity of mapping (4), where we have assumed that there exists
such a constant ks > 0, that ∥∥(x̄y ⊕ c̄y)
∥∥
1
≤ ks ‖y‖2 ,
for all y ∈ E2. This constant ks > k(â) strongly depends on the choice of elements
c̄y ∈ ker â, y ∈ E2, what one can observe from definition (2). Really, owing to the
definition of infimum, for any ε > 0 and all y ∈ E2 there exist elements x̄
(ε)
y ⊕ c̄
(ε)
y ∈ E1,
such that
k(â) ≤
∥∥∥x̄
(ε)
y ⊕ c̄
(ε)
y
∥∥∥
1
‖y‖2
< k(â) + ε := ks. (5)
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
AN INFINITE-DIMENSIONAL BORSUK – ULAM TYPE GENERALIZATION ... 103
Now making now use of formula (4), we can construct a selection sε : E2 → E1 as
follows:
sε(y) :=
1
2
(
x̄(ε)
y − x̄
(ε)
−y
)
⊕ 1
2
(
c̄(ε)
y − c̄
(ε)
−y
)
,
satisfying, owing to inequalities (5), the searched for conditions i) and ii):
âsε(y) = y, ‖sε(y)‖1 ≤ ks ‖y‖2
for all y ∈ E2 and ks := k(â) + ε, ε > 0. Moreover, the mapping sε : E2 → E1 is, by
construction, continuous [15, 19, 20] and odd that finishes the proof.
Lemma 2. Let a mapping fr : E1 → E2 be defined as
fr(x) :=
‖x‖1
r
f
(
rx
‖x‖1
)
, if x 6= 0;
0, if x = 0.
Then the equation
t(t2 + ε2)−1fr(ts(y) + t2c̄) = y, (6)
where c̄ ∈ ker â, is solvable for any ε 6= 0 with respect to (t, y) ∈ [−1, 1]× S1(0), such
that ‖y‖2 + t2 = 1. Moreover, the corresponding solution (tε, yε) satisfies the limiting
condition: lim infε→0 |tε| = α0 ∈ (0, 1).
Proof. Proof is based on a Borsuk – Ulam type theorem of [11, 15] and some standard
functional-analytic resasonings.
As a consequence of Lemmas 1 and 2 one deduces the proof of the main Theorem 1.
In particular, the solution setN (â, f) depends on the kernel ker â, and whose topological
dimension dimN (â, f) ≥dim ker â− 1, following from the form of equation (6).
3. Applications. 3.1. The classical Leray – Schauder fixed point theorem. The
following classical Leray – Schauder fixed point theorem holds.
Theorem 2. Let a compact mapping f̄ : B → B in a Banach space B be such
that there exists a cloesed convex and bounded set M ⊂ M, for which f̄(M) ⊆ M.
Then there exists a fixed point x̄ ∈ M, such that f̄(x̄) = x̄.
Proof. A proof of the theorem can be obtained from the main Theorem 1. Really,
put, by definition, E1 := B ⊕ R and E2 := B. For any point x ∈ B one can define the
set-valued projection mapping (metric projection)
B 3 x → Pf̄ (x) ⊂ Mf̄ ⊂ B, (7)
where Mf̄ := conv f̄(M) ⊆ M and
inf
y∈Mf̄
‖x− y‖ :=
∥∥x− Pf̄ (x)
∥∥. (8)
The constructed mapping (7) is well-defined [1, 21, 22] and below semi-continuous,
owing to the compactness, closedness and convexity of the set Mf̄ ⊂ B. Take now
the unite sphere S1(0) ⊂ E1, a compact surjective linear operator b̂ : B → B, whose
dim ker b̂ ≥ 1, a continuous selection P̄f̄ : B → Mf̄ for the set-valued mapping (7),
existing owing to the aboave mentioned E. Michael theorem [19], and construct a
mapping f : S1(0) → E2, where, by definition, for any (x, τ) ∈ S1(0) and λ ∈ R
f(x, τ) := f̄(P̄f̄ (x))− P̄f̄ (x) + λb̂x. (9)
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
104 A. K. PRYKARPATSKY
If to define now a related with (8) mapping â : E1 → E2 as
â(x, τ) := λb̂x
for any (x, τ) ∈ E1, the fixed point problem for the mapping f̄ : B → B becomes
equivalent to the following equation:
â (x, τ) = f(x, τ) ⇐⇒ f̄(P̄f̄ (x)) = P̄f̄ (x).
The following simple lemma holds.
Lemma 3. The mapping (9) is continuoes, â-compact and satisfying for some
nonzero value λ ∈ R the condition kf > k(â).
Thereby, based on main Theorem 1 there exists a point (xτ , τ) ∈ S1(0) ⊂ E1, such
that
f̄(P̄f̄ (xτ )) = P̄f̄ (xτ ) ⇐⇒ f̄(x̄) = x̄,
where x = P̄f̄ (xτ ) ∈ Mf̄ , prooving the theorem.
Remark 1. There exists [16 – 18] another nonstandard proof of the classical Leray –
Schauder fixed point theorem, based on the measure theory and a Krein – Milman type
theorem about a representation of convex compact sets by means of their extreme points.
3.2. A Hamilton – Jacobi type nonlinear equation in Rn. There is considered the
Cauchy problem to the following nonlinear Hamilton – Jacobi type equation in Rn :
∂u
∂t
+
1
2
(
|ux|2 + βu|x|2
)
= 0, (10)
where x ∈ Rn, t ∈ R+, β ∈ R is a constant parameter and
u|t=+0 = v
for v : Rn → R being a given mapping. The corresponding classical and generalized
solutions to equation (10), when v ∈ BSC(Rn) is a below semi-continuous function,
can be represented [2, 23 – 27] for t ∈ R+ as
u(x, t) = inf
y∈Rn
{
v(y)− 1
2
〈y, α̇〉|τ=0 −
β
16
(
|x|4 − |y|4
)
+
1
2
〈x, α̇〉|τ=t
}
,
where we denoted “ · ” :=
d
dτ
, “ · ·” :=
d2
dτ2
and α : Rn×R+→ Rn is the vector-valued
solution to the following set of nonlinear ordinary differential equations:
−α̈ = β
(
uα +
1
2
|α|2α̇
)
, (11)
u̇ =
1
2
(
|α̇|2 − βu|α|2
)
under the boundary conditions
α|τ=+0 = y, α|τ=t = x, (12)
u|τ=+0 = v(y)
for any x, y ∈ Rn and t ∈ R+. The problems like (11) are of very importance in the
mathematical theory of nonlinear oscillations [3] and were before extensively studied in
[2, 3, 28] by A. M. Samoilenko and his co-workers.
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
AN INFINITE-DIMENSIONAL BORSUK – ULAM TYPE GENERALIZATION ... 105
To show that problem (11) and (12) is solvable, we rewrite it in the following
canonical form:
â(α, u) = fβ(α, u), (13)
where (α, u) ∈ H(0, t; Rn) ⊕ H(0, t; R) := E1, D(â) = H2(0, t; Rn) ⊕ H1(0, t; R),
E2 := H(0, t; Rn)⊕H(0, t; R) and
â(α, u) := (−α̈, u̇),
fβ(α, u) :=
(
β
(
uα +
1
2
|α|2α̇
)
,
1
2
(
|α̇|2 − βu|α|2
))
.
(14)
The corresponding solution set N (â, fβ) ∈ D(â) to problem (13) can be studied making
use of the main Theorem 1. Namely, the following theorem holds.
Theorem 3. Let a parameter β ∈ R be chosen in such a way that kfβ
> k(â),
where
k−1
fβ
:= sup
‖(α,u)‖1=r
1
r
‖fβ(α, u)‖2,
k(â) := ‖ã−1‖ = sup
‖w‖2=1
inf
(α,u)∈D(â)
{
‖(α, u)‖1 : (−α̈, u̇) = w
}
,
for some r > 0. Then there exists a nonempty solution set N (â, fβ) ∈D(â) to equati-
on (14), whose topological dimension dimN (â, fβ)≥2.
Thereby, the Cauchy problem for problem (11) and (12) is solvable and the space
of the corresponding solutions is not trivial (in general, it is nonunique!). Based now
on Theorem 3 the searched for solvability of the Cauchy problem to equation (10) is
completely stated.
4. Acknowledgements. The author is much appreciated to Prof. Charles Chidume
(ICTP, Trieste, Italy) jointly with whom there were obtained some of results presented.
Many cordial thanks belong to Prof. A. M. Samoilenko (Institute of Mathematics at
the National Academy of Sciences, Kyiv, Ukraine) and Lech Górniewicz (J. Schauder
Center at Toruń University, Toruń, Poland) for discussions and very useful comments
during preparation of this work.
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Received 08.08.07
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