Adv Syst Sci Appl 2021; 03:113–118
Published online at https://ijassa.ipu.ru.

On Solvability of Equations Defined by Continuous and Smooth
Regular Mappings

Sergey E. Zhukovskiy1*

1V. A. Trapeznikov Institute of Control Sciences of RAS, Moscow, Russia

Abstract: We consider equations defined by continuous mappings acting between finite-
dimensional real vector spaces. It is assumed that the mappings are differentiable in the first
variable. A regularity condition for this type of equations is obtained. It is shown that the
regularity assumption implies the existence of solutions to the considered equations. Systems
of two equations defined by continuous mappings acting between finite-dimensional real vector
spaces are considered. It is assumed that the first mapping is differentiable in the first variable and
the second mapping is differentiable in the second variable. A regularity condition for this type of
systems is obtained. It is shown that the regularity assumption implies the existence of solutions
to the considered system. The proofs of the main results of the paper are based on Brouwer’s fixed
point theorem and global implicit function theorem.

Keywords: nonlinear equations, regularity, covering, fixed point

1. INTRODUCTION

Given positive integers n, k and a continuous mapping f : Rn × Rn → Rk, consider the
following equation

f(x, x) = 0 (1.1)

with unknown x ∈ Rn. Our goal is to obtain sufficient conditions for the existence of a
solution to this equations.

One of the standard approaches to this problem is based on the application of the covering
mappings theory (see, for example, [1, 2]). The corresponding results guarantee that if f
is covering in the first variable and is Lipschitz continuous in the second variable with a
sufficiently small Lipschitz constant, then there exists a solution to equation (1.1). In the
most general settings, these assertions provide sufficient solvability conditions for analogous
equations defined by mappings acting between metric spaces. In this paper, we consider a
specific case of finite-dimensional real linear spaces and differential mappings. We show that
in this specific case, the assumption of Lipschitz continuity is redundant.

The proof of our main result is based on two assertions. One of them is the well-known
Brouwer’s fixed-point theorem (see, for example, Chapter II, §5.7 in [3]). The second is a
global implicit function theorem (see Theorem 2 in [4]). In the second section of this paper,
we recall this implicit function theorem as well as the related concepts and assertions. In the
third section, we present solvability conditions for equation (1.1) and provide a proof of this
result. The last section is devoted to a development of the main result to systems of equations.

∗Corresponding author: s-e-zhuk@yandex.ru



114 S.E. ZHUKOVSKIY

2. PRELIMINARIES

Let us recall the concept of covering constant of a linear operator. Denote byLn×k the space of
linear operators A : Rn → Rk, denote by SLn×k the set of all surjective operators A ∈ Ln×k.
Denote by Bn(r) the closed ball in the space Rn centered at a point x ∈ Rn with a radius
r ≥ 0. Here and below we assume that Rn and Rk are equipped with norms which we denote
by | · |, and the space Ln×k is equipped with the corresponding operator norm.

For a linear operator A ∈ Ln×k, put

covA := sup{α ≥ 0 : Bk(α) ⊂ ABn(1)}.
It is a straightforward task to ensure that covA > 0 if and only if A ∈ SLn×k.

Let us recall the global implicit function theorem from [4]. Given a topological space Σ
and a mapping f : Rn × Σ→ Rk, assume that for every σ ∈ Σ the mapping f(·, σ) : Rn →
Rk is differentiable. For t ≥ 0, put

α(t) := inf

{
cov

∂f

∂x
(x, σ) : x ∈ Bn(t), σ ∈ Σ

}
.

Theorem 2.1:
(see Theorem 2 in [4]) Assume that

(A1) the mapping f(·, ·) is continuous on Rn × Σ, for every σ ∈ Σ the mapping f(·, σ) :

Rn → Rk is differentiable on Rn, the mapping
∂f

∂x
(·, ·) is continuous on Rn × Σ.

If
+∞∫
0

α(t) dt = +∞ or sup
σ∈Σ
|f(0, σ)| <

+∞∫
0

α(t) dt,

then for every ε > 0 there exists a continuous mapping g : Σ→ Rn such that

f(g(σ), σ) = 0 ∀σ ∈ Σ,

|g(σ)|∫
0

α(t) dt ≤ (1 + ε)|f(0, σ)| ∀σ ∈ Σ.

Below we also use the following corollary of Theorem 2.1.
Corollary 2.1:
Let f satisfies the assumption (A1). If there exists r̄ > 0 such that

sup
σ∈Σ
|f(0, σ)| < α(r̄)r̄,

then for every ε > 0 there exists a continuous mapping g : Σ→ Rn such that

f(g(σ), σ) = 0 ∀σ ∈ Σ,

|g(σ)| ≤ (1 + ε)|f(0, σ)|
α(r̄)

∀σ ∈ Σ.

Note that in [4] these assertions were proved under more general assumptions. In
particular, it was assumed that the domain of f in the variable x as well as the target space
are Banach spaces. However, the considered here weak form of implicit function theorem
from [4] is enough for the subsequent constructions.

Copyright © 2021 ASSA. Adv Syst Sci Appl (2021)



SOLVABILITY OF EQUATIONS DEFINED BY CONTINUOUS AND SMOOTH MAPPINGS 115

3. SOLVABILITY CONDITION FOR EQUATIONS

Let us turn back to equation (1.1). Assume that for every x2 ∈ Rn the mapping f(·, x2) is
differentiable. For t > 0 put

a(t, r) := inf

{
cov

∂f

∂x
(x1, x2) : x1 ∈ Bn(t), x2 ∈ Bn(r)

}
,

b(r) := sup
x2∈Bn(r)

|f(0, x2)|.

Theorem 3.1:
Assume that

(A) the mapping f(·, ·) is continuous on Rn × Rn, for every x2 ∈ Rn the mapping f(·, x2) :

Rn → Rk is differentiable on Rn, the mapping
∂f

∂x
(·, ·) is continuous on Rn × Rn.

If there exists r̄ > 0 such that

b(r̄) <

r̄∫
0

a(t, r̄) dt, (3.2)

then there exists a point x̄ ∈ Bn(r̄) such that such that

f(x̄, x̄) = 0.

Proof
Apply Theorem 2.1 to the mapping f with Σ = Bn(r̄). We have

α(t) = a(t, r) ∀ r > 0.

Therefore, assumption (3.2) implies that

sup
x2∈Bn(r̄)

|f(0, x2)| = b(r̄) <

r∫
0

a(t, r̄) dt =

+∞∫
0

α(t) dt.

Take an arbitrary ε > 0 such that

(1 + ε)b(r̄) <

r̄∫
0

a(t, r̄) dt.

It follows from Theorem 2.1 that there exists a continuous mapping g : Bn(r̄)→ Rn such
that

f(g(x2), x2) = 0,

|g(x2)|∫
0

a(t, r̄) dt ≤ (1 + ε)|f(0, x2)| ∀x2 ∈ Bn(r̄). (3.3)

Obviously the function a(·, r̄) is decreasing. Thus, the inequality in (3.3) and the
assumption (3.2) imply that |g(x2)| ≤ r for every x2 ∈ Bn(r̄). So,

g(x2) ∈ Bn(r̄) ∀x2 ∈ Bn(r̄).

Copyright © 2021 ASSA. Adv Syst Sci Appl (2021)



116 S.E. ZHUKOVSKIY

Therefore, by virtue of continuity of g Brouwer’s fixed-point theorem implies that there exists
a point x̄ ∈ Bn(r) such that

x̄ = g(x̄).

We have
f(x̄, x̄) = f(g(x̄), x̄) = 0.

So, the point x̄ is the desired one.

Let us derive a stronger but simpler solvability condition for the equation (1.1).

Corollary 3.1:
Let the assumption (A) hold. If there exist ᾱ > 0 and β̄ ≥ 0 such that

β̄ < ᾱ ≤ cov
∂f

∂x
(x1, x2) ∀x1 ∈ Rn, ∀x2 ∈ Rn,

|f(0, x2)| ≤ |f(0, 0)|+ β̄|x2| ∀x2 ∈ Rn,

then there exists a point x̄ ∈ Rn such that

f(x̄, x̄) = 0, |x̄| ≤ |f(0, 0)|
ᾱ− β̄

. (3.4)

Proof
We have

a(t, r) ≥ ᾱ, b(r) ≤ |f(0, 0)|+ β̄r ∀ t > 0, ∀ r ≥ 0.

Take

rj :=
|f(0, 0)|
ᾱ− β̄

+
1

j
, j = 1, 2, ... .

By construction we have

b(rj) ≤ |f(0, 0)|+ β̄
|f(0, 0)|
ᾱ− β̄

+
β̄

j
< ᾱ

(
|f(0, 0)|
ᾱ− β̄

+
1

j

)
=

rj∫
0

a(t, rj) dt.

Therefore, Theorem 3.1 implies that there exists a point x̄j ∈ Bn(rj) such that f(x̄j, x̄j) = 0
for every j = 1, 2, ... . By virtue of the compactness of Bn(r1) there exists a subsequence
{x̄ji} of the sequence {x̄j} which converges to a point x̄. Obviously, the point x̄ satisfies the
inequality in (3.4). Passing to the limit in the equalities f(x̄ji , x̄ji) = 0 as i to∞ we obtain
that f(x̄, x̄) = 0.

4. SOLVABILITY CONDITION FOR SYSTEMS OF EQUATIONS

Consider now the following system {
f1(x1, x2) = 0,
f2(x1, x2) = 0. (4.5)

Here f1, f2 : Rn × Rn → Rk are given mappings. Let us derive solvability conditions for the
system (4.5) analogous to those in Section 3.

Copyright © 2021 ASSA. Adv Syst Sci Appl (2021)



SOLVABILITY OF EQUATIONS DEFINED BY CONTINUOUS AND SMOOTH MAPPINGS 117

Assume that f1 is differentiable in x1 and f2 is differentiable in x2. Given numbers r̄1 > 0
and r̄2 > 0, denote

a1 := inf

{
cov

∂f1

∂x1

(x1, x2) : x1 ∈ Bn(r̄1), x2 ∈ Bn(r̄2)

}
,

b1 := sup
x2∈Bn(r̄2)

|f1(0, x2)|,

a2 := inf

{
cov

∂f1

∂x1

(x1, x2) : x1 ∈ Bn(r̄1), x2 ∈ Bn(r̄2)

}
,

b2 := sup
x1∈Bn(r̄1)

|f2(x1, 0)|.

Theorem 4.1:
Assume that mappings f1(·, ·) and f2(·, ·) are continuous on Rn × Rn, for every x1, x2 ∈ Rn

the mappings f1(·, x2), f2(x1, ·) : Rn → Rk are differentiable on Rn, the mappings
∂f1

∂x1

(·, ·)

and
∂f2

∂x2

(·, ·) are continuous on Rn × Rn.

If
b1 < a1r̄1, b2 < a2r̄2, (4.6)

then there exists a solution (x̄1, x̄2) ∈ Bn(r̄1)×Bn(r̄2) to the system (4.5), i.e.{
f1(x̄1, x̄2) = 0,
f2(x̄1, x̄2) = 0.

Proof
Take ε > 0 such that

(1 + ε)b1

a1

≤ r̄1,
(1 + ε)b2

a2

≤ r̄2.

The existence of such number ε follows from the assumption (4.6).
Since b1 < a1r̄1, applying Corollary 2.1 to f = f1 and Σ = Bn(r̄2) we obtain that there

exists a continuous mapping g1 : Bn(r̄2)→ Rn such that

f1(g1(x2), x2) = 0 ∀x2 ∈ Bn(r̄2),

|g1(x2)| ≤ (1 + ε)|f1(0, x2)|
a1

≤ (1 + ε)b1

a1

≤ r̄1 ∀x2 ∈ Bn(r̄2).

Since b2 < a2r̄2, applying Corollary 2.1 to f = f2 and Σ = Bn(r̄1) we obtain that there exists
a continuous mapping g2 : Bn(r̄1)→ Rn such that

f2(g2(x1), x1) = 0 ∀x1 ∈ Bn(r̄1),

|g2(x1)| ≤ (1 + ε)|f2(x1, 0)|
a1

≤ (1 + ε)b2

a2

≤ r̄2 ∀x1 ∈ Bn(r̄1).

Consider the mapping

g : Bn(r̄1)→ Bn(r̄1), g(x1) = g1(g2(x1)), x1 ∈ Bn(r̄1).

This mapping is well-defined, since the above relations imply g2(x1) ∈ Bn(r̄1) and
g1(g2(x1)) ∈ Bn(r̄1) for all x1 ∈ Bn(r̄1). Moreover, g is continuous since it is a composition

Copyright © 2021 ASSA. Adv Syst Sci Appl (2021)



118 S.E. ZHUKOVSKIY

of continuous mappings g1 and g2. Therefore, it follows from Brouwer’s fixed point theorem
that there exists a point x̄1 ∈ Bn(r̄1) such that x̄1 = g(x̄1). Take x̄2 := g2(x̄1). Let us show
that (x̄1, x̄2) is a desired point.

Obviously x̄2 ∈ Bn(r̄2). Moreover,

f1(x̄1, x̄2) = f1(g(x̄1), x̄2) = f1(g1(g2(x̄1)), g2(x̄1)) = 0,

f2(x̄1, x̄2) = f2(x̄1, g2(x̄1)) = 0.

So, (x̄1, x̄2) is a desired point.

ACKNOWLEDGEMENTS

The research is supported by the grant of the President of Russian Federation (Project No
MD-2658.2021.1.1) and by the RFBR grant (Project No 19-01-00080). The results in Section
4 were obtained under the financial support of the Russian Science Foundation (Project No
20-11-20131).

REFERENCES

1. Arutyunov, A.V., Avakov, E.R. & Zhukovskiy, S.E. (2009) Covering mappings and
their applications to differential equations unsolved for the derivative, Differ. Equations,
45(5), 627–649.

2. Arutyunov, A., de Oliveira, V.A., Pereira, F.L., Zhukovskiy, E. & Zhukovskiy, S. (2015)
On the solvability of implicit differential inclusions, Appl. Anal., 94(1), 129–143.

3. Granas, A. & Dugundji, J. (2003) Fixed Point Theory, N.Y., USA: Springer.
4. Arutyunov, A.V. & Zhukovskiy, S.E. (2021) On Global Solvability of Nonlinear

Equations with Parameters, Doklady Mathematics, 103(1), 57–60.

Copyright © 2021 ASSA. Adv Syst Sci Appl (2021)


	Introduction
	Preliminaries
	Solvability condition for equations
	Solvability condition for systems of equations