$p$-Regularity Theory. Tangent Cone Description in the Singular Case
We present a new proof of the theorem which is one of the main results of the $p$-regularity theory. This gives us a detailed description of the structure of the zero set of a singular nonlinear mapping. We say that $F : X → Y$ is singular at some point $x_0$, where $X$ and $Y$ are Banach spaces, if...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507968364085248 |
|---|---|
| author | Prusińska, A. Tret’yakov, A. Прусінска, А. Третьяков, А. |
| author_facet | Prusińska, A. Tret’yakov, A. Прусінска, А. Третьяков, А. |
| author_sort | Prusińska, A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:49:28Z |
| description | We present a new proof of the theorem which is one of the main results of the $p$-regularity theory. This gives us a detailed description of the structure of the zero set of a singular nonlinear mapping. We say that $F : X → Y$ is singular at some point $x_0$, where $X$ and $Y$ are Banach spaces, if Im $F′(x_0) ≠ Y$. Otherwise, the mapping $F$ is said to be regular. |
| first_indexed | 2026-03-24T02:17:44Z |
| format | Article |
| fulltext |
UDC 517.5
A. Prusińska (Siedlce Univ. Nat. Sci. and Humanities, Poland),
A. Tret’yakov (Siedlce Univ. Nat. Sci. and Humanities and System Research Inst., Polish Acad. Sci., Warsaw, Poland)
p-REGULARITY THEORY. TANGENT CONE DESCRIPTION
IN SINGULAR CASE
ТЕОРIЯ p-РЕГУЛЯРНОСТI. ОПИС ДОТИЧНОГО КОНУСА
В СИНГУЛЯРНОМУ ВИПАДКУ
We present the new proof of the theorem which is one of the main results of the p-regularity theory. This gives a detailed
description of the structure of the zero set of an singular nonlinear mapping. We say that F : X → Y is singular at some
point x0, where X and Y are Banach spaces if ImF ′(x0) 6= Y. Otherwise, the mapping F is said to be regular.
Наведено нове доведення теореми, що є одним з основних результатiв теорiї p-регулярностi. Дано детальний опис
структури множини нулiв сингулярного лiнiйного вiдображення. Кажуть, що F : X → Y є сингулярним у точцi
x0, де X та Y — банаховi простори, якщо ImF ′(x0) 6= Y. В протилежному випадку вiдображення F називається
регулярним.
A description of the solution set in regular case of the equation
F (x) = 0, (1)
where X, Y are Banach spaces, we may obtain by means of the Lyusternik theorem, which says that
the tangent cone of the solution set is equal to the kernel of the first derivative of F evaluated at
some point x0 ∈ X.
In this paper we consider the case when the regularity condition does not hold, i. e., ImF ′(x0) 6=
6= Y, but the mapping F is p-regular. Let us remind the definition of p-regularity and construction of
p-factor operator [1, 2].
For F : X → Y, p-times Fréchet differentiable mapping, we construct the p-factor operator under
the assumption that Y is decomposed into a direct sum
Y = Y1 ⊕ . . .⊕ Yp, (2)
where Y1 = ImF ′(x0) (the closure of the image of the first derivative of F evaluated at x0), and the
remaining spaces are defined as follows. Let Z1 = Y, Z2 be closed complementary subspace to Y1
(we assume that such closed complement exists), and let PZ2 : Y → Z2 be the projection operator
onto Z2 along Y1. Let Y2 be the closed linear span of the image of the quadratic map PZ2F
(2)(x0)[·]2.
More generally, define inductively,
Yi = span ImPZiF
(i)(x0)[·]i ⊆ Zi, i = 2, . . . , p− 1,
where Zi is a choice of closed complementary subspace for (Y1 ⊕ . . . ⊕ Yi−1) with respect to Y,
i = 2, . . . , p and PZi : Y → Zi is the projection operator onto Zi along (Y1⊕ . . .⊕Yi−1) with respect
to Y, i = 2, . . . , p. Finally, Yp = Zp. The order p is chosen as the minimum number for which (2)
holds. Now, define the following mappings (see [3, 7]):
fi : U → Yi, fi(x) = PYiF (x), i = 1, . . . , p,
c© A. PRUSIŃSKA, A. TRET’YAKOV, 2015
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8 1097
1098 A. PRUSIŃSKA, A. TRET’YAKOV
where U is a neighborhood of x0, PYi : Y → Yi is the projection operator onto Yi along (Y1 ⊕ . . .
. . .⊕ Yi−1 ⊕ Yi+1 ⊕ . . .⊕ Yp) with respect to Y, i = 1, . . . , p.
If F (i)(x0) = 0, where i = 1, . . . , p− 1, then we say that F is completely degenerate at x0 ∈ X
up to the order p.
Note that
f
(i)
k (x0) = 0, i = 1, . . . , k − 1, k = 1, . . . , p. (3)
Definition 1. The linear operator Λh ∈ L(X,Y1 ⊕ . . .⊕ Yp) is defined for some h ∈ X by
Λh(x) = f ′1(x0)[x] + f ′′2 (x0)[h, x] + . . .+
1
(p− 1)!
f (p)p (x0)[h, . . . , h, x], x ∈ X,
and is called the p-factor operator.
We will also use more exact notation Λh = (Λh,1 + Λh,2 + . . .+ Λh,p) , where
Λh,k =
1
(k − 1)!
f
(k)
k (x0)[h]k−1, k = 1, . . . , p.
It is also convenient to use the following equivalent definition of p-factor operator Λ̃h∈L(X,Y1×. . .
. . .× Yp) for some fixed h ∈ X,
Λ̃h(x) =
(
f ′1(x0)[x], f ′′2 (x0)[h, x], . . . ,
1
(p− 1)!
f (p)p (x0)[h, . . . , h, x]
)
, x ∈ X.
Note that in completely degenerate case the p-factor operator has the form F (p)(x0)[h]p−1.
In other words, we construct a decomposition of „non regular part” of the mapping F on partial
mappings fi in such a way that all of those mappings are completely degenerate up to the order i−1,
where i = 2, . . . , p.
For our further considerations we need the following generalization of the notion of regular
mapping.
Definition 2. We say that the mapping F is p-regular at x0 along h if
Im Λh = Y.
Let us introduce a corresponding nonlinear operator
Ψ[x]p = f ′1(x0)[x] + f ′′2 (x0)[x]2 + . . .+ f (p)p (x0)[x]p
and k-kernel, k = 1, . . . , p of F (k)(x0),
KerkF (k)(x0) =
{
h ∈ X : F (k)(x0)[h]k = 0
}
.
It is easy to see that Ψ[h]p = Λh(h).
Definition 3. We say that the mapping F is p-regular at x0 if it is p-regular along any h from
the set
Hp(x0)
df
=
{
p⋂
i=1
Kerif (i)i (x0)
}
\ {0}.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
p-REGULARITY THEORY. TANGENT CONE DESCRIPTION IN SINGULAR CASE 1099
Let us consider two examples of p-regular mappings.
Example 1. Let us consider the equation (1) with
F (x) =
x21 − x22 + x23 + x32
x21 − x22 + x23 + x2x3 + x33
, x ∈ R3,
F : R3 → R2, x0 = (0, 0)T . It is easy to verify F ′(x0) = (0, 0)T and it is obvious that F ′(x0) is not
surjective. However, the mapping F is 2-regular at x0. Indeed,
F ′′(0) =
2 0 0
0 −2 0
0 0 2
2 0 0
0 −2 1
0 1 2
and
Ker2F ′′(0) = span
1
1
0
,
1
−1
0
, h̄ =
1
1
0
, ¯̄h =
1
−1
0
,
where F ′′(0)h̄ =
(
2 −2 0
2 −2 1
)
, F ′′(0)¯̄h =
(
2 2 0
2 2 −2
)
are not degenerate, which means that
the condition of 2-regularity of F at x0 is valid.
Example 2. Consider the type of (1) equation
∆u− (ε− 10)g(u) = 0
on Ω = [0, π]× [0, π] in R2 with u = 0 on ∂Ω. If we denote F (u, ε) = ∆u− (ε− 10)g(u), then the
mapping F (u, ε) is 2-regular at the point (u0, ε0) = (0, 0) (see [7, p. 403], Example 1).
For a linear surjective operator Λ : X 7→ Y between Banach spaces we denote by Λ−1 its right
inverse. Therefore Λ−1 : Y 7→ 2X and we have
Λ−1(y) = {x ∈ X : Λx = y}.
We define the norm of Λ−1 via the formula
‖Λ−1‖ = sup
‖y‖=1
inf
{
‖x‖ : x ∈ Λ−1(y)
}
.
We say that Λ−1 is bounded if ‖Λ−1‖ <∞.
Lemma 1 [4]. Let X and Y be Banach spaces, and let Λ ∈ L(X,Y ). We set
C(Λ) = sup
y∈Y
(
‖y‖−1 inf{‖x‖ : x ∈ X, Λx = y}
)
.
If Im Λ = Y, then C(Λ) <∞.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
1100 A. PRUSIŃSKA, A. TRET’YAKOV
We shall give a „multivalued” generalization of the contraction mapping principle that is of
independent interest. Let distH(A1, A2) be the Hausdorff distance between sets A1 and A2.
Lemma 2 (contraction multimapping principle) [4]. Let Z be a complete metric space with dis-
tance ρ. Assume that a multimapping
Φ : B(z0, ε) 7→ 2Z ,
on a ball B(z0, ε) = {z : ρ(z, z0) < ε} (ε > 0) where the sets Φ(z) are non-empty and closed for
any z ∈ B(z0, ε). Further, assume that there exists a number θ, 0 < θ < 1, such that
1) distH
(
Φ(z1),Φ(z2)
)
≤ θρ(z1, z2) for any z1, z2 ∈ B(z0, ε),
2) ρ
(
z0,Φ(z0)
)
< (1− θ)ε.
Then, there exists an element z ∈ B(z0, ε) such that
z ∈ Φ(z). (4)
Moreover, among the points z satisfying (4) there exists a point such that
‖z − z0‖ ≤
2
1− θ
ρ
(
z0,Φ(z0)
)
. (5)
Now we introduce an inverse multivalued operator for Λh as follows:
Λ−1h (y) =
{
ξ ∈ X :
(
f ′1(x0)[ξ], f
′′
2 (x0)[h, ξ], . . . ,
1
(p− 1)!
f (p)p (x0)[h, . . . , h, ξ]
)
= (y1, . . . , yp)
}
,
where y = (y1, . . . , yp) and yi ∈ Yi, i = 1, . . . , p.
Definition 4. The mapping F is called strongly p-regular at the point x0 if there exist γ > 0
such that
sup
h∈Hγ
‖Λ−1h ‖ <∞,
where
Hγ
df
=
{
h ∈ X :
∥∥∥f (k)k (x0)[h]k
∥∥∥
Yk
≤ γ, k = 1, . . . , p, ‖h‖ = 1
}
.
The following theorem is a generalization of the Banach open mapping theorem.
Theorem 1. If ‖Λ−1h ‖ <∞, then there exists C(h) ≥ 0 such that
‖Λ−1h y‖ ≤ C(h)
(
‖y1‖+
‖y2‖
‖h‖
+ . . .+ (p− 1)!
‖yp‖
‖h‖p−1
)
,
where h 6= 0, y = (y1, . . . , yp) and yi ∈ Yi, i = 1, . . . , p.
The proof was given in [6].
The next theorem was first given and proved in [3, p. 158]. We give new improved proof of this
theorem.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
p-REGULARITY THEORY. TANGENT CONE DESCRIPTION IN SINGULAR CASE 1101
Theorem 2. Let X and Y be Banach spaces, U a neighborhood of x0 ∈ X and let F : U → Y
be a (p + 1)-times continuously Fréchet differentiable mapping in U. Assume that F is strongly
p-regular at x0. Then there exists a neighborhood U ′ ⊆ U of the point x0, a mapping η → x(η) :
U ′ → X and constants β1 > 0 and β2 > 0, such that for η ∈ U ′ :
F (η + x(η)) = F (x0), (6)
‖x(η)‖X ≤ β1
p∑
i=1
‖fi(η)− fi(x0)‖Yi
‖η − x0‖i−1
(7)
and
‖x(η)‖X ≤ β2
p∑
i=1
‖fi(η)− fi(x0)‖1/iYi
. (8)
Before stating the proof of the theorem, we prove the following lemma.
Lemma 3. Let all the assumptions of Theorem 2 hold. Then for any ε > 0 there exist constants
δ > 0 and R > 0 such that for any h ∈ X, ‖h‖ ≤ δ and for any x1, x2 ∈ X, ‖xi‖ ≤ ‖h‖/R,
i = 1, 2, the following estimation is satisfied:
‖F (x0 + h+ x1)− F (x0 + h+ x2)− Λh(x1 − x2)‖Y =
=
∥∥f1(x0 + h+ x1)− f1(x0 + h+ x2)− f ′1(x0)[h](x1 − x2)
∥∥
Y1
+ . . .
. . .+
∥∥∥∥fp(x0 + h+ x1)− fp(x0 + h+ x2)−
1
(p− 1)!
f (p)p (x0)[h]p−1(x1 − x2)
∥∥∥∥
Yp
≤
≤ ε
p∑
i=1
‖h‖i−1‖x1 − x2‖.
Proof. First we prove that for any k = 1, . . . , p inequality∥∥∥∥fk(x0 + h+ x1)− fk(x0 + h+ x2)−
1
(k − 1)!
f
(k)
k (x0)[h]k−1(x1 − x2)
∥∥∥∥
Yk
≤
≤ sup
θ∈[0,1]
∥∥∥∥fk(x0 + h+ x1 + x2 + θ(x1 − x2))−
1
(k − 1)!
f
(k)
k (x0)[h]k−1
∥∥∥∥
Yk
‖x1 − x2‖X (9)
is satisfied.
By Taylor’s expansion,
fk
(
x0 + h+ x1 + x2 + θ(x1 − x2)
)
=
= f ′k(x0) + . . .+
1
(k − 1)!
f
(k)
k (x0)
[
h+ x2 + θ(x1 − x2)
]k−1
+ ωk(x0, h, x1, x2, θ),
where
∥∥ωk(x0, h, x1, x2, θ)∥∥ = o
(
‖x0 + h+ x1 + x2 + θ‖
)
.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
1102 A. PRUSIŃSKA, A. TRET’YAKOV
Let ε > 0 be sufficiently small number. Define
R = max
{
1,
2
ε
∥∥f ′1(x0)∥∥, . . . , 2
ε
4p−1
(p− 1)!
∥∥f (p)1 (x0)
∥∥}.
By an assumption, xi ≤
‖h‖
R
, i = 1, 2, so
‖x0 + h+ x1 + x2 + θ‖ ≤ 4‖h‖.
The last inequality yields ∥∥ωk(x0, h, x1, x2, θ)∥∥ = o
(
‖h‖k−1
)
.
Then there exists δ > 0 such that for ‖h‖ ≤ δ and xi ≤
‖h‖
R
, i = 1, 2, the inequality (9) holds and
∥∥ωk(x0, h, x1, x2, θ)∥∥ ≤ 2
ε
‖h‖k−1.
By (3), (9) and the last estimation we obtain∥∥∥∥fk(x0 + h+ x1)− fk(x0 + h+ x2)−
1
(k − 1)!
f
(k)
k (x0)[h]k−1(x1 − x2)
∥∥∥∥ ≤
≤ sup
θ∈[0,1]
∥∥∥∥ 1
(k − 1)!
f
(k)
k (x0)[h+ x2θ(x1 − x2)]k−1 −
1
(k − 1)!
f
(k)
k (x0)[h]k−1
∥∥∥∥ ‖x1 − x2‖+
+
2
ε
‖h‖k−1‖x1 − x2‖. (10)
Let us observe that for any k = 1, . . . , p− 1,
f
(k)
k (x0)
[
h+ x2 + θ(x1 − x2)
]k−1
=
k−1∑
i=0
Cik−1f
(k)
k (x0)[h]k−1−i
[
x2 + θ(x1 − x2)
]i
=
= f
(k)
k (x0)[h]k−1 +
k−1∑
i=1
Cik−1f
(k)
k (x0)[h]k−1−i
[
x2 + θ(x1 − x2)
]i
.
We may estimate the second term in the last equation. Since∥∥x2 + θ(x1 − x2)
∥∥ ≤ 3‖h‖
R
,
then ∥∥∥∥∥
k−1∑
i=1
Cik−1f
(k)
k (x0)[h]k−1−i[x2 + θ(x1 − x2)]i
∥∥∥∥∥ ≤
≤
∥∥∥f (k)k (x0)
∥∥∥ k−1∑
i=1
Cik−1‖h‖k−1−i
3i‖h‖i
Ri
≤
∥∥∥f (k)k (x0)
∥∥∥ ‖h‖k−1 4k−1
R
.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
p-REGULARITY THEORY. TANGENT CONE DESCRIPTION IN SINGULAR CASE 1103
Thus by definition of R,∥∥∥∥∥
k−1∑
i=1
Cik−1f
(k)
k (x0)[h]k−1−i
[
x2 + θ(x1 − x2)
]i∥∥∥∥∥ ≤ (k − 1)!
ε
2
‖h‖k−1.
The last inequalities and (10) yield for k = 1, . . . , p∥∥∥∥fk(x0 + h+ x1)− fk(x0 + h+ x2)−
1
(k − 1)!
f
(k)
k (x0)[h]k−1(x1 − x2)
∥∥∥∥ ≤
≤
(
ε
2
‖h‖k−1 +
ε
2
‖h‖k−1
)
‖x1 − x2‖ = ε‖h‖k−1‖x1 − x2‖.
Hence for k = 1, . . . , p∥∥∥∥fk(x0 + h+ x1)− fk(x0 + h+ x2)−
1
(k − 1)!
f
(k)
k (x0)[h]k−1(x1 − x2)
∥∥∥∥ ≤
≤ ε‖h‖k−1‖x1 − x2‖. (11)
Adding up the inequalities in (11) for k = 1, . . . , p finishes proof of the lemma.
Proof of Theorem 2. Let us consider U as a sufficiently small neighborhood of x0 and divide
our consideration into 2 cases.
Case 1. We consider η ∈ U such that hη =
η − x0
‖η − x0‖
6∈ Hγ , that is there exists k ≤ p such that
∥∥∥f (k)k (x0)[hη]
k
∥∥∥ > γ
or ∥∥∥f (k)k (x0)[η − x0]k
∥∥∥ > γ‖η − x0‖k.
Taking x(η) = η − x0 we get
‖fk(η)− fk(x0)‖ =
∥∥∥∥ 1
k!
f
(k)
k (x0)[η − x0]k + ωk(η)
∥∥∥∥ ≥ γ
2k!
‖η − x0‖k,
where ‖ωk(η)‖ = o
(
‖η − x0‖k
)
. Hence,
‖x(η)‖ = ‖η − x0‖ ≤
2k!
γ
‖fk(η)− fk(x0)‖
‖η − x0‖k−1
≤ 2k!
γ
p∑
i=1
‖fi(η)− fi(x0)‖
‖η − x0‖i−1
,
which finishes the proof of the first case.
Case 2. We consider all η ∈ U such that hη =
η − x0
‖η − x0‖
∈ Hγ , and hence for i = 1, . . . , p
∥∥∥f (i)i (x0)[η − x0]k
∥∥∥ ≤ γ‖η − x0‖i,
where γ is a sufficiently small number that does not depend on η. We also use notation h = η − x0.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
1104 A. PRUSIŃSKA, A. TRET’YAKOV
Let suph∈Hγ ‖Λ
−1
h ‖ = C < ∞. Note that hη = η − x0 and hη =
h
‖h‖
∈ Hγ , the statement of
Theorem 1 holds with C(h) = C for all h = η − x0.
Let ε > 0 be a sufficiently small number such that p!Cε < 1. Then Lemma 3 and inequalities (11)
imply that there exist numbers δ andR, such thatB2δ(0) ⊂ (U−x0), 0 < δ ≤ r, and for k = 1, . . . , p,
and h ∈ Bδ(0) x1, x2 ∈ B‖h/R‖(0),
C(k − 1)!
‖h‖k−1
∥∥∥∥fk(x0 + h+ x1)− fk(x0 + h+ x2)−
1
(k − 1)!
f
(k)
k (x0)[h]k−1(x1 − x2)
∥∥∥∥ ≤
≤ C(k − 1)!ε‖x1 − x2‖. (12)
Adding up the inequalities in (12) for k = 1, . . . , p we obtain
p∑
k=1
C(k − 1)!
‖h‖k−1
∥∥∥∥fk(x0 + h+ x1)− fk(x0 + h+ x2)−
1
(k − 1)!
f
(k)
k (x0)[h]k−1(x1 − x2)
∥∥∥∥ ≤
≤ α‖x1 − x2‖ (13)
with α = p!Cε. Note that 0 < α < 1.
Let us define r(h) = ‖h‖/R and a neighborhood V ⊂ Bδ(0) such that
p∑
k=1
C(k − 1)!
‖h‖k−1
‖fk(x0 + h)− fk(x0)‖ < (1− α)r(h) (14)
for all h ∈ V. Such a neighborhood exists by the definition of the set Hγ and property (3).
Now let us fix an element h ∈ V and consider multivalued mapping
Φh : Br(h)(0)→ 2X , Φh(x) = x− Λ−1h
(
F (x0 + h+ x)− F (x0)
)
.
Because of the choice of the number r(h) and the neighborhood V for all x ∈ Br(h)(0) and h ∈ V
we have x0 + h+ x ∈ U.
For any y ∈ Y the set Λ−1h y is a linear manifold parallel to KerΛh. Thus the set Φh(x) is closed
for any x ∈ Br(h)(0).
Next we verify that all assumptions of Lemma 2 are satisfied for Φh(x) with Br(h)(0) and ω0 = 0.
First, it will be shown that assumption 1 of the mentioned lemma holds for all x1, x2 ∈ Br(h)(0).
We have
distH(Φh(x1),Φh(x2)) = inf
{
‖z1 − z2‖ : zi ∈ Φh(xi), i = 1, 2
}
=
= inf
{
‖z1 − z2‖ : Λhzi = Λhxi − F (x0 + h+ xi) + F (x0), i = 1, 2
}
=
= inf
{
‖z‖ : f ′1(x0)z = f ′1(x0)(x1 − x2)− f1(x0 + h+ x1) + f1(x0 + h+ x2), . . .
. . . ,
1
(p− 1)!
f (p)p (x0)[h]p−1z =
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
p-REGULARITY THEORY. TANGENT CONE DESCRIPTION IN SINGULAR CASE 1105
=
1
(p− 1)!
f (p)p (x0)[h]p−1(x1 − x2)− fp(x0 + h+ h1) + fp(x0 + h+ x1)
}
. (15)
By Theorem 1 and (13) we get
distH(Φh(x1),Φh(x2)) =
=
∥∥∥∥∥∥∥∥∥∥
Λ−1h
f ′1(x0)(x1 − x2)− f1(x0 + h+ x1) + f1(x0 + h+ x2)
...
1
(p− 1)!
f
(p)
p (x0)[h]p−1(x1 − x2)− fp(x0 + h+ h1) + fp(x0 + h+ x1)
∥∥∥∥∥∥∥∥∥∥
≤
≤
p∑
k=1
C(k − 1)!
‖h‖k−1
∥∥∥∥fk(x0 + h+ x1)− fk(x0 + h+ x2)−
1
(k − 1)!
f
(k)
k (x0)[h]k−1(x1 − x2)
∥∥∥∥ ≤
≤ α‖x1 − x2‖
for all x1, x2 ∈ Br(h)(0).
Hence the assumption 1 of the lemma holds.
Next we verify that assumption 2 is also satisfied. Using the approach similar to above we get
by (3), (14), and (15),
distH(0,Φh(0)) = inf
{
‖z‖ : f ′1(x0)z = f1(x0 + h)− f1(x0), . . .
. . . ,
1
(p− 1)!
f (p)p (x0)[h]k−1[h]p−1z = fp(x0 + h)− fp(x0)
}
≤
≤
p∑
k=1
C(k − 1)!
‖h‖k−1
‖fk(x0 + h)− fk(x0)‖ < (1− α)r(h).
Then for any h ∈ V the mapping Φh(x) satisfies assumptions of the Lemma 2. Hence there exists
x(h) ∈ Φh(x(h)) which means that
0 ∈ Λ−1h
(
F (x0 + h+ x(h))− F (x0)
)
.
Thus F (x0 + h+ x(h)) = F (x0) for all h ∈ V.
In addition, by (5),
‖x(h)‖ ≤ 2
1− α
distH(0,Φh(0)) ≤ 2
1− α
p∑
k=1
C(k − 1)!
‖h‖k−1
‖fk(x0 + h)− fk(x0)‖ ≤
≤ β1
p∑
k=1
‖fk(x0 + h)− fk(x0)‖
‖h‖k−1
,
for all h ∈ V, and β1 =
Cp!
1− α
.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
1106 A. PRUSIŃSKA, A. TRET’YAKOV
Taking x(η) = x(h) and U ′ = x0 + V we obtain that (6) and (7) hold for all η ∈ U ′.
Note that
fk(x0 + h)− fk(x0)
‖h‖k−1
≤ C1‖fk(x0 + h)− fk(x0)‖1/k
for all k = 1, . . . , p and h ∈ V, with some C1 ≥ 0. Then inequality (7) implies (8) with β2 = C1pβ1
which ends the proof of the theorem.
1. Brezhneva O. A., Tret’yakov A. A. Implicit function theorems for nonregular mappings in Banach spaces. Exit from
singularity // Banach Spaces and Appl Anal. – 2007. – P. 1 – 18.
2. Brezhneva O. A., Tret’yakov A. A., Marsden J. E. Higher-order implicit function theorem and degenerate nonlinear
boundary-value problems // Communs Pure and Appl. Anal. – 2003. – 2. – P. 425 – 445.
3. Izmailov A. F, Tret’yakov A. A. Factor-analysis of nonlinear mappings. – Moscow: Nauka, 1994 (in Russian).
4. Ioffe A. D., Tihomirov V. M. Theory of extremal problems // Stud. Math. and Appl. – 1979.
5. Kantorovitch L. V., Akilov G. P. Functional analysis. — Oxford: Pergamon Press, 1982.
6. Prusińska A., Tret’yakov A. A. Generalization of the solutions existence theorem for degenerate mappings // Sci. Bull.
Chelm. Sec. Math. and Comput. Sci. – 2006. – 1. – P. 107 – 115.
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non-zero p-kernel // Set-Valued Anal. – 2011. – 19. – P. 399 – 416.
Received 08.01.13,
after revision — 30.06.15
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
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| id | umjimathkievua-article-2048 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:17:44Z |
| publishDate | 2015 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/9a/7f7b13512485dbc1292fa870d364d69a.pdf |
| spelling | umjimathkievua-article-20482019-12-05T09:49:28Z $p$-Regularity Theory. Tangent Cone Description in the Singular Case Теорія $p$-регулярност! опис дотичного конуса в сингулярному випадку Prusińska, A. Tret’yakov, A. Прусінска, А. Третьяков, А. We present a new proof of the theorem which is one of the main results of the $p$-regularity theory. This gives us a detailed description of the structure of the zero set of a singular nonlinear mapping. We say that $F : X → Y$ is singular at some point $x_0$, where $X$ and $Y$ are Banach spaces, if Im $F′(x_0) ≠ Y$. Otherwise, the mapping $F$ is said to be regular. Наведено нове доведення теореми, що є одним з основних результата теорії $p$-регулярності. Дано детальний опис структури множини нулів сингулярного лінійного відображення. Кажуть, що $F : X → Y$ є сингулярним у точці $x_0$, де $X$ та $Y$ — банахові простори, якщо Im $F′(x_0) ≠ Y$. В протилежному випадку відображення $F$ називається регулярним. Institute of Mathematics, NAS of Ukraine 2015-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2048 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 8 (2015); 1097-1106 Український математичний журнал; Том 67 № 8 (2015); 1097-1106 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2048/1113 https://umj.imath.kiev.ua/index.php/umj/article/view/2048/1114 Copyright (c) 2015 Prusińska A.; Tret’yakov A. |
| spellingShingle | Prusińska, A. Tret’yakov, A. Прусінска, А. Третьяков, А. $p$-Regularity Theory. Tangent Cone Description in the Singular Case |
| title | $p$-Regularity Theory. Tangent Cone Description in the Singular Case |
| title_alt | Теорія $p$-регулярност! опис дотичного конуса в сингулярному випадку |
| title_full | $p$-Regularity Theory. Tangent Cone Description in the Singular Case |
| title_fullStr | $p$-Regularity Theory. Tangent Cone Description in the Singular Case |
| title_full_unstemmed | $p$-Regularity Theory. Tangent Cone Description in the Singular Case |
| title_short | $p$-Regularity Theory. Tangent Cone Description in the Singular Case |
| title_sort | $p$-regularity theory. tangent cone description in the singular case |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2048 |
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