Variations on some finite-dimensional fixed-point theorems
We give rather elementary topological proofs of some generalizations of fixed-point theorems in $\mathbb{R}^n$ due to Pireddu-Zanolin and Zgliczynski, which are useful in various questions related to ordinary differential equations.
Saved in:
| Date: | 2013 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2013
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/2417 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508301055229952 |
|---|---|
| author | Mawhin, J. Мавхін, Й. |
| author_facet | Mawhin, J. Мавхін, Й. |
| author_sort | Mawhin, J. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:15:01Z |
| description | We give rather elementary topological proofs of some generalizations of fixed-point theorems in $\mathbb{R}^n$ due to Pireddu-Zanolin and Zgliczynski,
which are useful in various questions related to ordinary differential equations. |
| first_indexed | 2026-03-24T02:23:02Z |
| format | Article |
| fulltext |
UDC 517.9
J. Mawhin
(Inst. de Recherche en Mathématique et Physique Université Catholique de Louvain chemin du cyclotron, Belgium)
VARIATIONS ON SOME FINITE-DIMENSIONAL FIXED-POINT THEOREMS
УЗАГАЛЬНЕННЯ ДЕЯКИХ СКIНЧЕННОВИМIРНИХ ТЕОРЕМ
ПРО НЕРУХОМУ ТОЧКУ
We give rather elementary topological proofs of some generalizations of fixed-point theorems in Rn due to Pireddu – Zanolin
and Zgliczyński, which are useful in various questions related to ordinary differential equations.
Наведено елементарнi топологiчнi доведення деяких узагальнень теорем Пiредду – Занолiна та Зглiчинського про
нерухому точку в Rn, якi можуть бути використанi при розглядi рiзних питань, пов’язаних iз звичайними диферен-
цiальними рiвняннями.
1. Introduction. Fixed-point theorems in finite-dimensional spaces have applications, for example,
in mathematical economy, nonlinear difference equations, periodic solutions of ordinary differential
equations using Poincaré’s operator, and in chaos theory.
The present paper finds its inspiration in the interesting papers [11] and [12] of Pireddu – Zanolin,
where applications to topological chaos theory and a large bibliography can be found. Its aim is to
give simple proofs of generalizations of some fixed-point theorems in Rn.
In Section 2, we show that a reduction theorem for Brouwer degree theory developed by the
author [8] (Lemma 1), provides a particular short and natural proof for a result (Theorem 1) containing
as a special case a slight generalization (Theorem 2) of a fixed-point theorem of Pireddu – Zanolin [11]
for mappings of which are expansive on the boundary of a ball in a vector subspace of Rn and
compressive on the boundary of a ball in a direct summand. The precise meaning of ‘expansive’ and
‘compressive’ is given in Section 3. This Theorem 2 is deduced from Theorem 1 in Section 3, and
some special cases are considered and compared to older fixed-point theorems. Possible extensions
to infinite-dimensional normed vector spaces are mentioned.
In Section 4, we describe a generalization of a fixed-point theorem due to Zgliczyński [15] (The-
orem 3), whose original proof is rather complicated and based upon more sophisticated topological
tools. The proof we give here is inspired by the one given in [12], but uses directly Poincaré –
Miranda’s theorem (Lemma 2), instead of an intersection result deduced from this theorem. We also
relate a special case of Theorem 3 with Theorem 2 and its corollaries.
2. A fixed-point theorem for some mappings in a direct sum of vector spaces. For the
reader’s convenience, we first recall a reduction theorem for the Brouwer degree stated and proved
as Theorem 3.1 in [8], and which is essentially a finite-dimensional version of Proposition II.2 in [7].
If U and V are oriented n-dimensional topological vector spaces, Ω ⊂ X an open bounded set with
closure Ω and boundary ∂Ω, and g : Ω ⊂ U → V a continuous mapping such that 0 6∈ g(∂Ω), we
denote the Brouwer degree of g with respect to Ω and 0 by dB[g,Ω, 0] (see e.g. [3] for its definition
and properties). If z is an isolated zero of g, the Brouwer index iB[g, z] of g at z is defined as the
Brouwer degree dB[g,Br(z), 0] for sufficiently r > 0, where Br(z) denotes the open ball in U of
center z and radius r. The symbol ⊂ will always mean ‘non strict inclusion’.
c© J. MAWHIN, 2013
266 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 2
VARIATIONS ON SOME FINITE-DIMENSIONAL FIXED-POINT THEOREMS 267
Lemma 1. Let X and Z be n-dimensional topological vector spaces (oriented if X 6= Z),
L : X → Z be a linear mapping with kernel N(L) 6= {0}, let Y ⊂ Z be a vector subspace such that
Z = Y ⊕R(L) (with R(L) the range of L), D ⊂ X be an open bounded set, and let r : D → Y ×{0}
be a continuous mapping such that 0 6∈ (L + r)(∂D). Then, for each isomorphism J : N(L) → Y,
and each projector P : X → X such that R(P ) = N(L), one has
dB[L+ r,D, 0] = iB[L+ JP, 0] · dB[J−1r|N(L), D ∩N(L), 0].
Let now p ≥ 0, q ≥ 0 be integers such that p + q = n ≥ 1, so that Rn = Rp ⊕ Rq, let D ⊂ Rn
be a nonempty open bounded set, and let
f : D → Rn = Rp × Rq, (x, y) 7→ (fe(x, y), f c(x, y))
be a continuous mapping.
Theorem 1. Assume that the following conditions hold:
1) ∀(x, y) ∈ ∂D ∀λ ∈ [0, 1) : (λx− fe(x, y), y − λf c(x, y)) 6= (0, 0);
2) p = 0, 0 ∈ D;
3) p ≥ 1, dB[fe(·, 0), D ∩ (Rp × {0}), 0] 6= 0.
Then f has at least one fixed-point in D.
Proof. Assume first that p ≥ 1. Because of assumption 1 with λ = 0, the Brouwer degree
dB[fe(·, 0), D ∩ (Rp × {0}), 0] is well defined. Define
L : Rp ⊕ Rq, (x, y) 7→ (0, y), (1)
and the continuous mapping H : D × [0, 1]→ Rp × Rq by
H(x, y, λ) = (λx− fe(x, y),−λf c(x, y)), (2)
so that the fixed points of f are the zeros of L+H(·, ·, 1).
If f has a fixed point on ∂D, the result is proved. If it is not the case, then assumption 1 holds
for λ ∈ [0, 1]. From the definition of H and the homotopy invariance of Brouwer degree, we get
dB[I − f,D, 0] = dB[L+H(·, ·, 1), D, 0] = dB[L+H(·, ·, 0), D, 0]. (3)
Now, N(L) = Rp × {0} and R(L) = {0} × Rq, so that we can write Rn = N(L)⊕R(L). So
L+H(·, ·, 0) = (0, ·) + (−fu(·, ·), 0)
has the structure requested by Lemma 1. If we define the projector P : Rp × Rq → Rp × Rq by
P (x, y) = (x, 0), then N(L) = R(P ) = Rp×{0}. Thus we can choose the identity on Rp×{0} for
the isomorphism J : N(L)→ N(L) involved in Lemma 1. This implies that
L+ JP = L+ P = I (4)
(I the identity in Rn). Therefore, from Theorem 3.1 of [8], (4) and elementary properties of Brouwer
degree, we obtain
dB[L+H(·, ·, 0), D, 0] = iB[L+ P, 0] · dB[−fe(·, 0), D ∩ (Rp × {0}), 0] =
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 2
268 J. MAWHIN
= (−1)pdB[fe(·, 0), D ∩ (Rp × {0}), 0]. (5)
Using (3) and (5) we obtain
dB[I − f,D, 0] = (−1)pdB[fe(·, 0), D ∩ (Rp × {0}), 0]
and the existence of a fixed point of f in D follows from assumption 3 and the existence property of
Brouwer degree.
If now p = 0, so that f = f c, either f has a fixed point on ∂D or, using assumption 1, for any
λ ∈ [0, 1],
y +H(y, λ) := y − λf(y) 6= 0
for any y ∈ ∂D. By the homotopy invariance of Brouwer degree and assumption 2, we get
dB[I − f,D, 0] = dB[I −H(·, 1), 0] = dB[I −H(·, 0), 0] = dB[I,D, 0] = 1.
The existence of a fixed point of f in D follows from the existence property of Brouwer degree.
Theorem 1 is proved.
Remark 1. If p = 0, Theorem 1 just reduces to a finite-dimensional version of Schaefer’s
version [13] of the Leray – Schauder fixed-point theorem [6] (see also [3]).
Remark 2. It is easily checked that Theorem 2 remains true if Rq is replaced by an infinite-
dimensional normed vector space and f c is assumed to be compact on D. The proof is exactly the
same, except that the use of Lemma 1 has to be replaced by that of Proposition II.12 of [7].
3. Fixed-point theorems for expansive-compressive mappings. Let Br be the open ball of
center 0 and radius r > 0 in any Euclidean space with inner product 〈·, ·〉 and corresponding norm
‖ · ‖, and let Br denote its closure. Let p ≥ 0, q ≥ 0 be integers such that p+ q = n ≥ 1, let a > 0,
b > 0 be real numbers, and let f = (fe, f c) : Ba × Bb → Rp × Rq = Rn be a continuous mapping.
The following result is a slight extension of a fixed-point theorem of [11], which is motivated by and
generalizes some results of [1].
Theorem 2. Assume that the following conditions hold:
(i) ∀(x, y) ∈ Rp × Rq : ‖x‖ = a, ‖y‖ ≤ b : ‖fe(x, y)‖ ≥ a;
(ii) ∀(x, y) ∈ Rp × Rq : ‖x‖ ≤ a, ‖y‖ = b : 〈y, f c(x, y)〉 ≤ b2;
(iii) p ≥ 1, dB[fe(·, 0), Ba, 0] 6= 0.
Then f has at least one fixed point in Ba ×Bb.
Proof. Notice that, when p ≥ 1, the Brouwer degree dB[fe(·, 0), Ba, 0] is well defined because,
if ‖x‖ = a, then ‖fe(x, 0)‖ ≥ a 6= 0. Also, assumption 2 of Theorem 1 is trivially satisfied. If f
has a fixed point on ∂(Ba ×Bb), the result is proved. Assume therefore that f has no fixed point on
∂(Ba ×Bb). Notice that
∂(Ba ×Bb) = (∂Ba ×Bb) ∪ (Ba × ∂Bb).
To apply Theorem 1, let λ ∈ [0, 1). If (x, y) ∈ ∂Ba ×Bb, then, using assumption (i),
‖λx‖ = λa < a ≤ ‖fe(x, y)‖,
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 2
VARIATIONS ON SOME FINITE-DIMENSIONAL FIXED-POINT THEOREMS 269
so that (x, y) is not a zero of L+H(·, ·, λ) with L defined in (1) and H defined in (2). If λ ∈ [0, 1)
and (x, y) ∈ Ba × ∂Bb, then, using assumption (ii),
‖y‖2 = b2 > λb2 ≥ λ〈y, f c(x, y)〉,
so that (x, y) is not a zero of L+H(·, ·, λ). Thus, assumption 1 of Theorem 1 is satisfied. Assump-
tion (iii) is just assumption 3 of Theorem 1, and the result follows.
An first consequence of Theorem 2 is the following fixed-point theorem, where assumptions (i)
and (ii) of Theorem 2 have a more symmetric structure, which justifies the name of ‘expansive-
compressive’ mapping given to f.
Corollary 1. Assume that the following conditions hold:
(i) ∀(x, y) ∈ Rp × Rq : ‖x‖ = a, ‖y‖ ≤ b : ‖fe(x, y)‖ ≥ a;
(ii′) ∀(x, y) ∈ Rp × Rq : ‖x‖ ≤ a, ‖y‖ = b : ‖f c(x, y)‖ ≤ b;
(iii) p ≥ 1, dB[fe(·, 0), Ba, 0] 6= 0.
Then f has at least one fixed point in Ba ×Bb.
Proof. It suffices to notice that, for all (x, y) ∈ Rp×Rq such that ‖x‖ ≤ a and ‖y‖ = b, one has
〈y, f c(x, y)〉 ≤ ‖y‖‖f c(x, y)‖ ≤ ‖y‖b = b2,
so that assumption (ii′) implies assumption (ii) of Theorem 2.
Another consequence comes from the other way of ‘symmetrizing’ assumptions (i) and (ii) of
Theorem 2.
Corollary 2. Assume that the following conditions hold:
(i′) ∀(x, y) ∈ Rp × Rq : ‖x‖ = a, ‖y‖ ≤ b : 〈x, fe(x, y)〉 ≥ a2;
(ii) ∀(x, y) ∈ Rp × Rq : ‖x‖ ≤ a, ‖y‖ = b : 〈y, fe(x, y)〉 ≤ b2.
Then f has at least one fixed point in Ba ×Bb.
Proof. If condition (i′) holds, then for all (x, y) ∈ Rp ×Rq such that ‖x‖ = a and ‖y‖ ≤ b, one
has
a‖fe(x, y)‖ = ‖x‖‖fe(x, y)‖ ≥ 〈x, fe(x, y)〉 ≥ a2,
and hence assumption (i′) implies assumption (i) of Theorem 2. Furthermore, if we define the homo-
topy
F : Ba × [0, 1]→ Rp, (x, λ) 7→ (1− λ)x+ λfe(x, 0),
we have, for all x such that ‖x‖ = a and all λ ∈ [0, 1], using again assumption (i′),
(1− λ)‖x‖2 + λ〈x, fe(x, 0)〉 ≥ (1− λ)a2 + λa2 = a2 > 0
so that F(·, λ) has no zero on ∂Ba when λ ∈ [0, 1]. Hence, the homotopy invariance of Brouwer
degree implies that
dB[fe(·, 0), Ba, 0] = dB[F(·, 1), Ba, 0] = dB[F(·, 0), Ba, 0] = dB[I,Ba, 0] = 1,
and assumption (iii) of Theorem 2 is satisfied.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 2
270 J. MAWHIN
Remark 3. For p = 0, Theorem 2 reduces to a fixed point generally attributed to Krasnosel’skii,
and which, in the finite-dimensional case, follows directly from the proof of Brouwer fixed-point
theorem given in 1910 by Hadamard [5, p. 472]. For p = 0, Corollary 1 reduces to Rothe’s fixed-
point theorem [3] in a finite-dimensional space, already mentioned in 1922 by Birkhoff – Kellogg
([2, p. 100], footnote). For q = 0 and n = 2, Theorem 2 was already obtained by Dolcher [4] in
1948. In this case, condition (i) is not sufficient to get a fixed point, as shown by the simple example
of the one-dimensional mapping f : [−1, 1] → R, x 7→ x + 1, which has no fixed point and is such
that
|f(−1)| = 1, |f(1)| = 3 > 1,
so that assumption (i) holds. On the other hand,
dB[I + 2, (−1, 1), 0] = 0.
Remark 4. It follows from Remark 2 that Theorem 2 and its corollaries remain valid if one
replaces Rq by an infinite-dimensional normed vector space under the assumption that f c is compact
on Ba ×Bb.
4. Poincaré – Miranda’s and fixed-point theorems. Let I = [−1, 1], n ≥ 1 an integer, and for
j ∈ {1, . . . , n} and ε = ±1, denote by [xj = ε] the set {x ∈ In : xj = ε}, i.e., the j-th opposite
faces of In. For the reader’s convenience, let us recall Poincaré – Miranda’s theorem [10, 14], a
n-dimensional generalization of Bolzano’s theorem, for which an elementary proof is given in [9].
Lemma 2. Let g = In → Rn be a continuous mapping such that there exists a finite sequence
{ε1, ε2, . . . , εn} in {−1,+1} with the property that, for each j ∈ {1, . . . , n}, one has
εjgj(x) ≤ 0 ∀x ∈ [xj = −1] and εjgj(x) ≥ 0 ∀x ∈ [xj = 1].
Then g has at least one zero in In.
The following result generalizes a fixed-point theorem due to Zgliczyński [15], with a proof
inspired by [12], but using directly Lemma 2 instead of some of its topological consequences.
Theorem 3. Let f : In → Rn be a continuous map, and supppose there exists a finite sequence
of indexes 1 ≤ i1 < i2 < . . . < ik ≤ n, such that the following conditions hold:
(a) for every j ∈ {i1, i2, . . . , ik},
(a1) [−1, 1] ⊂
[
max[xj=−1] fj ,min[xj=+1] fj
]
or
(a2) [−1, 1] ⊂
[
max[xj=1] fj ,min[xj=−1] fj
]
;
(b) for every j ∈ {1, . . . , n} \ {i1, i2, . . . , ik},
min
[xj=−1]
fj ≥ −1 and max
[xj=1]
fj ≤ 1
Then f has at least a fixed point in In.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 2
VARIATIONS ON SOME FINITE-DIMENSIONAL FIXED-POINT THEOREMS 271
Proof. Let g : In → Rn be the continuous map defined by g = I − f. It suffices to prove that g
has a zero in In. From assumption (a), we see that, for j ∈ {i1, i2, . . . , ik}, we have either
max
[xj=−1]
fj ≤ −1 ≤ 1 ≤ min
[xj=1]
fj , (6)
or
max
[xj=1]
fj ≤ −1 ≤ 1 ≤ min
[xj=−1]
fj . (7)
Now, (6) is equivalent to
fj(x) ≤ −1 ∀x ∈ [xj = −1], fj(x) ≥ 1 ∀x ∈ [xj = 1],
and hence to
gj(x) = xj − fj(x) = −1− fj(x) ≥ 0 ∀x ∈ [xj = −1],
gj(x) = xj − fj(x) = 1− fj(x) ≤ 0 ∀x ∈ [xj = 1].
(8)
Similarly, (7) is equivalent to
fj(x) ≤ −1 ∀x ∈ [xj = 1], fj(x) ≥ 1 ∀x ∈ [xj = −1],
gj(x) = xj − fj(x) = 1− fj(x) ≥ 2 ∀x ∈ [xj = 1], (9)
gj(x) = xj − fj(x) = −1− fj(x) ≤ −2 ∀x ∈ [xj = −1].
Now, for j ∈ {1, . . . , n} \ {i1, i2, . . . , ik}, it follows from assumption (b) that
fj(x) ≥ −1 ∀x ∈ [xj = −1] and fj(x) ≤ 1 ∀x ∈ [xj = 1].
Hence,
gj(x) = xj − fj(x) = −1− fj(x) ≤ 0 ∀x ∈ [xj = −1],
gj(x) = xj − fj(x) = 1− fj(x) ≥ 0 ∀x ∈ [xj = 1].
(10)
Conditions (8), (9) and (10) show that, for each j ∈ {1, . . . , n}, gj takes opposite signs on the
opposite faces [xj = −1] and [xj = 1] of In. It follows from Lemma 2 that g has a zero in In.
Remark 5. Assumption (b) is obviously satisfied if[
min
[xj=−1]
fj , max
[xj=1]
fj
]
⊂ [−1, 1] or
[
max
[xj=1]
fj , min
[xj=−1]
fj
]
⊂ [−1, 1].
In this case, one sees that the interval constructed on the values of fj on the opposite jth-faces of In
covers [−1, 1] if j ∈ {i1, i2, . . . , ik}, and is contained in [−1, 1] if j ∈ {1, . . . , n} \ {i1, i2, . . . , ik}.
Remark 6. Theorem 3 is distinct from, but related to Theorem 2 and Corollary 2. Indeed, if in
Theorem 3 we take n = 2, k = 1 and i1 = 1, then the assumptions (a) and (b) become
(a′) f1(−1, y) ≤ −1 ∀y ∈ [−1, 1], f1(1, y) ≥ 1 ∀y ∈ [−1, 1],
or
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 2
272 J. MAWHIN
f1(1, y) ≤ −1 ∀y ∈ [−1, 1], f1(−1, y) ≥ 1 ∀y ∈ [−1, 1];
(b′) f2(x,−1) ≥ −1 ∀x ∈ [−1, 1], f2(x, 1) ≤ 1 ∀x ∈ [−1, 1].
The first condition in assumption (a′) can be written
(a′′) xf1(x, y) ≥ 1 ∀(x, y) : |x| = 1, |y| ≤ 1,
and assumption (b′) can be written equivalenly
(b′′) yf2(x, y) ≤ 1 ∀(x, y) : |x| ≤ 1, |y| = 1.
In this case, its statement is a special case of Corollary 2. Now, the second condition in assumption
(a′) can be written
(a′′′) xf1(x, y) ≤ −1 ∀(x, y) : |x| = 1, |y| ≤ 1,
which is not covered by Corollary 2 but implies that
|f1(x, y)| ≥ 1 ∀(x, y) : |x| = 1, |y| ≤ 1,
and that
dB[f1(·, 0), (−1, 1), 0] = −1.
So, the existence of a fixed point of f = (f1, f2) under the second condition in assumption (a′) and
assumption (b′) follows also from Theorem 2.
Remark 7. Lemma 2 and Theorem 3, stated and proved for In for simplicity and to make more
easy the comparison with Theorem 2 and its Corollaries. They hold equally, with trivial adaptations,
when In is replaced by [a1, b1]× . . .× [an, bn].
1. Andres J., Gaudenzi M., Zanolin F. A transformation theorem for periodic solutions of nondissipative systems //
Rend. Semin. mat. Univ. politecn. Torino. – 1990. – 48. – P. 171 – 186.
2. Birkhoff G. D., Kellogg O. D. Invariant points in function spaces // Trans. Amer. Math. Soc. – 1922. – 23. – P. 96 – 115.
3. Deimling K. Nonlinear functional analysis. – Berlin: Springer, 1985.
4. Dolcher M. Due teoremi sull’esistenza di punti uniti nelle transformazioni piane continue // Rend. Semin. mat. Univ.
Padova. – 1948. – 17. – P. 97 – 101.
5. Hadamard J. Sur quelques applications de l’indice de Kronecker // Introduction à la théorie des fonctions d’une
variable / J. Tannery. – 2e éd. – Paris: Hermann, 1910. – 2.
6. Leray J., Schauder J. Topologie et équations fonctionnelles // Ann. sci. Ecole norm. supér. – 1934. – 51. – P. 45 – 78.
7. Mawhin J. Topological degree methods in nonlinear boundary value problems // CBMS Region. Conf. – Providence
RI: Amer. Math. Soc., 1979. – 40.
8. Mawhin J. Reduction and continuation theorems for Brouwer degree and applications to nonlinear difference
equations // Opusc. Math. – 2008. – 28. – P. 541 – 560.
9. Mawhin J. Variations on Poincaré – Miranda’s theorem // Adv. Nonlinear Stud. – 2013. – 13 (to appear).
10. Miranda C. Un’ osservazione su un teorema di Brouwer // Bol. Unione. mat. ital. – 1940–1941. – 3, № 2. – P. 5 – 7.
11. Pireddu M., Zanolin F. Fixed points for dissipative-repulsive systems and topological dynamics of mappings defined
on N -dimensional cells // Adv. Nonlinear Stud. – 2005. – 5. – P. 411 – 440.
12. Pireddu M., Zanolin F. Cutting surfaces and applications to periodic points and chaotic-like dynamics // Top. Meth.
Nonlinear Anal. – 2007. – 30. – P. 279 – 319.
13. Schaefer H. H. Über die Methode der a-priori Schranken // Math. Ann. – 1955. – 129. – S. 415 – 416.
14. Poincaré H. Sur certaines solutions particulières du problème des trois corps // C. R. Acad. Sci. Paris. – 1883. – 97. –
P. 251 – 252.
15. Zgliczyński P. On periodic points for systems of weakly coupled 1-dim maps // Nonlinear Anal. – 2001. – 46. –
P. 1039 – 1062.
Received 10.01.13
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 2
|
| id | umjimathkievua-article-2417 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:23:02Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/bb/03e5169926c34d67141bb24ab3d5ffbb.pdf |
| spelling | umjimathkievua-article-24172020-03-18T19:15:01Z Variations on some finite-dimensional fixed-point theorems Узагальнення деяких скiнченновимiрних теорем про нерухому точку Mawhin, J. Мавхін, Й. We give rather elementary topological proofs of some generalizations of fixed-point theorems in $\mathbb{R}^n$ due to Pireddu-Zanolin and Zgliczynski, which are useful in various questions related to ordinary differential equations. Наведено елементарнi топологiчнi доведення деяких узагальнень теорем Пiредду – Занолiна та Зглiчинського про нерухому точку в $\mathbb{R}^n$, якi можуть бути використанi при розглядi рiзних питань, пов’язаних iз звичайними диференцiальними рiвняннями. Institute of Mathematics, NAS of Ukraine 2013-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2417 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 2 (2013); 266-272 Український математичний журнал; Том 65 № 2 (2013); 266-272 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2417/1601 https://umj.imath.kiev.ua/index.php/umj/article/view/2417/1602 Copyright (c) 2013 Mawhin J. |
| spellingShingle | Mawhin, J. Мавхін, Й. Variations on some finite-dimensional fixed-point theorems |
| title | Variations on some finite-dimensional fixed-point theorems |
| title_alt | Узагальнення деяких скiнченновимiрних теорем про нерухому точку |
| title_full | Variations on some finite-dimensional fixed-point theorems |
| title_fullStr | Variations on some finite-dimensional fixed-point theorems |
| title_full_unstemmed | Variations on some finite-dimensional fixed-point theorems |
| title_short | Variations on some finite-dimensional fixed-point theorems |
| title_sort | variations on some finite-dimensional fixed-point theorems |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2417 |
| work_keys_str_mv | AT mawhinj variationsonsomefinitedimensionalfixedpointtheorems AT mavhínj variationsonsomefinitedimensionalfixedpointtheorems AT mawhinj uzagalʹnennâdeâkihskinčennovimirnihteoremproneruhomutočku AT mavhínj uzagalʹnennâdeâkihskinčennovimirnihteoremproneruhomutočku |