MINISTRY OF EDUCATION AND TRAINING
VINH UNIVERSITY
LE KHANH HUNG
ON THE EXISTENCE OF FIXED POINT
FOR SOME MAPPING CLASSES
IN SPACES WITH UNIFORM STRUCTURE
AND APPLICATIONS
Speciality: Mathematical Analysis
Code: 62 46 01 02
A SUMMARY OF MATHEMATICS DOCTORAL THESIS
NGHE AN - 2015
Work is completed at Vinh University
Supervisors:
1. Assoc. Prof. Dr. Tran Van An
2. Dr. Kieu Phuong Chi
Reviewer 1:
Reviewer 2:
Reviewer 3:
Thesis will be presented and defended at school - level thesis evaluating Council at
Vinh University
at h date month year
Thesis can be found at:
1. Nguyen Thuc Hao Library and Information Center
2. Vietnam National Library
1
PREFACE
1 Rationale
1.1. The first result on fixed points of mappings was obtained in 1911. At that
time, L. Brouwer proved that: Every continuous mapping from a compact convex
set in a finite-dimensional space into itself has at least one fixed point. In 1922, S.
Banach introduced a class of contractive mappings in metric spaces and proved the

+
→ R
+
is a semi right
upper continuous function and satisfies 0 ≤ ϕ(t) < t for all t ∈ R
+
.
In 2001, B. E. Roades, while improving and extending a result of Y. I. Alber and
2
S. Guerre-Delabriere, gave a contractive condition of the form
(R1) d(T x, T y) ≤ d(x, y) − ϕ

d(x, y)

, for all x, y ∈ X, where ϕ : R
+
→ R
+
is a
continuous, monotone increasing function such that ϕ(t) = 0 if and only if t = 0.
Following the way of reducing contractive conditions, in 2008, P. N. Dutta and B.
S. Choudhury introduced a contractive condition of the form
(DC) ψ

d(T x, T y)

≤ ψ

d(x, y)


+
are continuous functions such that ψ(t) > 0, ϕ(t) > 0 for all t > 0 and ψ(0) =
0 = ϕ(0), moreover, ϕ is a monotone non-decreasing function and ψ is a monotone
increasing function.
In 2012, B. Samet, C. Vetro and P. Vetro introduced the notion of α-ψ-contractive
type mappings in complete metric spaces, with a contractive condition of the form
(SVV) α(x, y)d(T x, T y) ≤ ψ

d(x, y)

, for all x, y ∈ X where ψ : R
+
→ R
+
is
a monotone non-decreasing function satisfying

+∞
n=1
ψ
n
(t) < +∞ for all t > 0 and
α : X × X → R
+
.
1.3. In recent years, many mathematicians have continued the trend of generalizing
contractive conditions for mappings in partially ordered metric spaces. Following this
trend, in 2006, T. G. Bhaskar and V. Lakshmikantham introduced the notion of
coupled fixed points of mappings F : X ×X → X with the mixed monotone property
and obtained some results for the class of those mappings in partially ordered metric


2

,
3
for all x, y, u, v ∈ X with g(x) ≥ g(u), g(y) ≤ g(v) and F (X ×X) ⊂ g(X).
In 2011, V. Berinde and M. Borcut introduced the notion of triple fixed points for
the class of mappings F : X × X × X → X and obtained some triple fixed point
theorems for mappings with mixed monotone property in partially ordered metric
spaces satisfying the contractive condition
(BB) There exists constants j, k, l ∈ [0, 1) such that j + k + l < 1 satisfy
d

F (x, y, z), F (u, v, w)

≤ jd(x, u) + kd(y, v) + ld(z, w), for all x, y, z, u, v, w ∈ X
with x ≥ u, y ≤ v, z ≥ w.
After that, in 2012, H. Aydi and E. Karapinar extended the above result and
obtained some triple fixed point theorems for the class of mapping F : X ×X ×X → X
with mixed monotone property in partially ordered metric spaces and satisfying the
following weak contractive condition
(AK) There exists a function φ such that for all x ≤ u, y ≥ v, z ≤ w we have
d

T F (x, y, z), T F (u, v, w)

≤ φ

max


(0) = 0 and 0 < φ
α
(t) < t for all t > 0. Then he introduced the notion of
Φ-contractive mappings, which are mappings T : M → X satisfying
(A) d
α
(T x, T y) ≤ φ
α

d
j(α)
(x, y)

for all x, y ∈ M and for all α ∈ I, where M ⊂ X
and obtained some results on fixed points of the class of those mappings. By intro-
4
ducing the notion of spaces with j-bounded property, V. G. Angelov obtained some
results on the unique existence of a fixed point of the above mapping class.
Following the direction of extending results on fixed points to the class of local
convex spaces, in 2005, B. C. Dhage obtained some fixed point theorems in Banach
algebras by studying solutions of operator equations x = AxBx where A : X → X,
B : S → X are two operators satisfying that A is D-Lipschitz, B is completely
continuous and x = AxBy implies x ∈ S for all y ∈ S, where S is a closed, convex
and bounded subset of the Banach algebra X, such that it satisfies the contractive
condition
(Dh) ||T x − T y|| ≤ φ

||x − y||

for all x, y ∈ X, where φ : R

The thesis is devoted to extend some results on the existence of fixed points in
metric spaces to spaces with uniform structure. We also considered the existence of
solutions of some classes of integral equations with unbounded deviation, which we
can not apply fixed point theorems in metric spaces.
The thesis can be a reference for under graduated students, master students and
Ph.D students in analysis major in general, and the fixed point theory and applications
in particular.
7 Overview and Organization of the research
The content of this thesis is presented in 3 chapters. In addition, the thesis also
consists Protestation, Acknowledgements, Table of Contents, Preface, Conclusions
and Recommendations, List of scientific publications of the Ph.D. student related to
the thesis, and References.
In chapter 1, at first we recall some notions and known results about uniform
spaces which are needed for later contents. Then we introduce the notion of (Ψ, Π)-
contractive mapping, which is an extension of the notion of (ψ, ϕ)-contraction of P. N.
Dutta and B. S. Choudhury in uniform spaces, and obtained a result on the existence
of fixed points of the (Ψ, Π)-contractive mapping in uniform spaces. By introducing
the notion of uniform spaces with j-monotone decreasing property, we get a result
on the existence and uniqueness of a fixed point of (Ψ, Π)-contractive mapping. Con-
tinuously, by extending the notion of α-ψ-contractive mapping in metric spaces to
uniform spaces, we introduce the notion of (β, Ψ
1
)-contractive mappings in uniform
spaces and obtain some fixed point theorems for the class of those mappings. Theo-
rems, which are obtained in uniform spaces, are considered as extensions of theorems
in complete metric spaces. Finally, applying our theorems ab out fixed points of the
class of (β, Ψ
1
)-contractive mapping in uniform spaces, we prove the existence of so-
lutions of a class of nonlinear integral equations with unbounded deviations. Note

and useful results for later parts. Then, we give some fixed point theorems for the class
of (Ψ, Π)-contractive mappings in uniform spaces. In the last part of this chapter, we
extend fixed point theorems for the class of α-ψ-contractive mappings in metric spaces
to uniform spaces. After that, we apply these new results to show a class of integral
equations with unbounded deviations having a unique solution.
1.1 Uniform spaces
In this section, we recall some knowledge about uniform spaces needed for later
presentations.
Let X be a non-empty set, U, V ⊂ X ×X. We denote by
1) U
−1
= {(x, y) ∈ X ×X : (y, x) ∈ U}.
2) U ◦V = {(x, z) : ∃y ∈ X, (x, y) ∈ U, (y, z) ∈ V } and U ◦ U is replaced by U
2
.
3) ∆(X) = {(x, x) : x ∈ X} is said to be a diagonal of X.
4) U[A] = {y ∈ X : ∃x ∈ A such that (x, y) ∈ U}, where A ⊂ X and U[{x}] is
replaced by U [x].
Definition 1.1.1. An uniformity or uniform structure on X is a non-empty family
U consisting of subsets of X ×X which satisfy the following conditions
1) ∆(X) ⊂ U for all U ∈ U.
2) If U ∈ U then U
−1
∈ U.
3) If U ∈ U then there exists V ∈ U such that V
2
⊂ U.
4) If U, V ∈ U then U ∩ V ∈ U.
5) If U ∈ U and U ⊂ V ⊂ X ×X then V ∈ U.
The ordered pair (X, U) is called a uniform space.

(α)

, k = 1, 2, . . .
1.2 Fixed points of weak contractive mappings
In the next presentations, (X, P) or X we mean a Hausdorff uniform space whose
uniformity is generated by a saturated family of pseudometrics P = {d
α
(x, y) : α ∈ I},
where I is an index set. Note that, (X, P) is Hausdorff if only if d
α
(x, y) = 0 for all
α ∈ I implies x = y.
Definition 1.2.2. A uniform space (X, P) is said to be j-bounded if for every
α ∈ I and x, y ∈ X there exists q = q(x, y, α) such that d
j
n
(α)
(x, y) ≤ q(x, y, α) <
∞, for all n ∈ N.
Let Ψ = {ψ
α
: α ∈ I} be a family of functions ψ
α
: R
+
→ R
+
which is monotone
non-decreasing and continuous, ψ
α


− ϕ
α

d
j(α)
(x, y)

,
for all x, y ∈ X and for all ψ
α
∈ Ψ, ϕ
α
∈ Π, α ∈ I.
Definition 1.2.5. A uniform space (X, P) is called to have the j-monotone decreasing
property iff d
α
(x, y) ≥ d
j(α)
(x, y) for all x, y ∈ X and all α ∈ I.
Theorem 1.2.6. Let X is a Hausdorff sequentially complete uniform space and
T : X → X. Suppose that
9
1) T is a (Ψ, Π)-contractive map on X.
2) A map j : I → I is surjective and there exists x
0
∈ X such that the sequence
{x
n
} with x

. For every
n = 1, 2, . . . we denote by P
n
: X → R a map is defined by P
n
(x) = x
n
for all
x = {x
n
} ∈ X. Denote I = N
∗
× R
+
. For every (n, r) ∈ I we define a pseudometrics
d
(n,r)
: X ×X → R, which is given by
d
(n,r)
(x, y) = r


P
n
(x) −P
n
(y)



3

(1 −x
1
), 1 −

1 −
2
3.2

(1 −x
2
), . . . , 1 −

1 −
2
3n

(1 −x
n
), . . .

,
for every x = {x
n
} ∈ X.
Applying Theorem 1.2.6, T has a unique fixed point, that is x = {1, 1, . . .}.
Theorem 1.2.9. Let X be a Hausdorff sequentially complete uniform space and
T, S : X → X be mappings satisfying
ψ

n
} with x
2k+1
= T x
2k
, x
2k+2
= Sx
2k+1
, k ≥ 0 satisfies d
α
(x
m+n
, x
m
) ≥
d
j(α)
(x
m+n
, x
m
) for all m, n ≥ 0, α ∈ I.
Then, there exists u ∈ X such that u = Tu = Su.
Moreover, if X has the j-monotone decreasing property, then there exists a unique
point u ∈ X such that u = T u = Su.
10
1.3 Fixed points of (β,Ψ
1
)-contractive type mappings

+∞

n=1
ψ
n
α
(t) < +∞ for all t > 0.
Denote by β a family of functions β = {β
α
: X ×X → R
+
, α ∈ I}.
Definition 1.3.7. Let (X, P) be a uniform space with P =

d
α
(x, y) : α ∈ I

and
T : X → X be a given mapping. We say that T is an (β, Ψ
1
)-contractive if for every
function β
α
∈ β and ψ
α
∈ Ψ
1
we have
β

i) T is β-admissible.
ii) There exists x
0
∈ X such that for each α ∈ I we have β
α
(x
0
, T x
0
) ≥ 1 and
d
j
n
(α)
(x
0
, T x
0
) < q(α) < +∞ for all n ∈ N
∗
.
Also, assume either
a) T is continuous; or
b) for all α ∈ I, if {x
n
} is a sequence in X such that β
α
(x
n
, x


s, x(s)

ds, (1.27)
where the functions f : R
+
× R → R and G : R
+
× R
+
→ R
+
continuous. The
deviation ∆ : R
+
→ R
+
is a continuous function, in general case, unbounded. Note
that, since deviation ∆ : R
+
→ R
+
is unbounded, we can not apply the known fixed
point theorems in metric space for the above integral equations.
Assumption 1.4.1. A1) There exists a function u : R
2
→ R such that for each
compact subset K ⊂ R
+
, there exist a positive number λ and ψ

u

x
0
(t),

∆(t)
0
G(t, s)f

s, x
0
(s)

ds

≥ 0.
A3) For all t ∈ R
+
, x, y ∈ C(R
+
, R), if u

x(t), y(t)

≥ 0, then
u


∆(t)

∗
, then u(x
n
, x) ≥ 0 for all n ∈ N
∗
.
A5) For each compact subset K ⊂ R
+
, there exists a compact set

K ⊂ R
+
such
that for all n ∈ N
∗
, we have ∆
n
(K) ⊂

K.
Theorem 1.4.3. Suppose that Assumption 1.4 are fulfilled. Then, equation (1.27)
has at least one solution in C

R
+
, R

.
Corollary 1.4.4. Suppose that
1) f : R

12
3) For each compact subset K ⊂ R
+
, there exists a compact set

K ⊂ R
+
such that
for all n ∈ N
∗
, ∆
n
(K) ⊂

K.
Then, the equation (1.27) has a unique solution in C

R
+
, R

.
Example 1.4.5. Consider nonlinear functional integral equation
x(t) =

t
0
G(t, s)f

s, x(s)

1 + x
2
if x < 0
2 + x −
√
1 + x
2
if x ≥ 0,
for every t ∈ R
+
.
Applying Corollary 1.4.4 we get that the equation (1.28) has a unique solution.
Conclusions of Chapter 1
In this chapter, we obtained the following main results
• Give and prove theorems which confirm the existence and unique existence of a
fixed point for the class of (Ψ, Π)-contractive maps in uniform space (Theorem 1.2.6,
1.2.9).
These results are written in the article: Tran Van An, Kieu Phuong Chi and
Le Khanh Hung (2014), Some fixed point theorems in uniform spaces, submitted to
Filomat.
• Give and prove a theorem which confirm the existence and unique existence of a
fixed point for the class of (β, Ψ
1
)-contractive mappings in uniform spaces (Theorem
1.3.11). And apply Theorem 1.3.11 to prove the unique existence of solution of a class
of integral equations with unbounded deviations.
These results are written in the article: Kieu Phuong Chi, Tran Van An, Le
Khanh Hung (2014), Fixed point theorems for (α-Ψ)-contractive type mappings in
uniform spaces and applications, Filomat (to appear).
13

and
if y
1
, y
2
∈ X, y
1
≤ y
2
then F (x, y
1
) ≥ F (x, y
2
).
In this section, we prove some coupled fixed point theorems for generalized con-
tractive mappings in partially ordered uniform spaces.
Let Φ
1
= {φ
α
: R
+
→ R
+
; α ∈ I} be a family of functions with the properties:
i) φ
α
is monotone non-decreasing.
ii) 0 < φ
α

(y, v)
2

,
for all x ≤ u, y ≥ v.
2) For each α ∈ I, there exists φ
α
∈ Φ
1
such that sup{φ
j
n
(α)
(t) : n = 0, 1, . . .} ≤
φ
α
(t) and
φ
α
(t)
t
is non-decreasing.
3) There are x
0
, y
0
∈ X such that x
0
≤ F (x
0

0
, x
0
)

< 2p(α) < ∞
for all α ∈ I, n ∈ N.
Also, assume either
a) F is continuous; or,
b) X has the property
i) If a non-decreasing sequence {x
n
} in X converges to x then x
n
≤ x for all n.
ii) If a non-increasing sequence {y
n
} in X converges to y then y
n
≥ y for all n.
Then, F has a coupled fixed point.
Moreover, if X is j-bounded and for every (x, y), (z, t) ∈ X ×X there exists (u, v) ∈
X ×X which is comparable to them, then F has a unique coupled fixed point.
Corollary 2.1.6. In addition to hypotheses of Theorem 2.1.5, if x
0
and y
0
are compa-
rable then F has a unique fixed point, that is, there exists x ∈ X such that F (x, x) = x.
2.2 Triple fixed points in partially ordered uniform spaces

2
⇒ F (x, y
1
, z) ≥ F (x, y
2
, z)
and
z
1
, z
2
∈ X, z
1
≤ z
2
⇒ F (x, y, z
1
) ≤ F (x, y, z
2
).
Definition 2.2.2. Let F : X
3
→ X. An element (x, y, z) is called a triple fixed point
of F if
F (x, y, z) = x, F (y, x, y ) = y and F (z, y, x) = z.
In this section, with Φ
1
is the function family defined in Section 2.2, we prove some
tripled fixed point theorems for generalized contractive mappings in uniform spaces.
We also give some examples to show that our results are effective.

α
∈ Φ
1
such that
d
α

T F (x, y, z), T F (u, v, w)

≤ φ
α

max

d
j(α)
(T x, T u), d
j(α)
(T y, T v), d
j(α)
(T z, T w)


,
for all x ≤ u, y ≥ v and z ≤ w.
2) For each α ∈ I, there exists φ
α
∈ Φ
1
such that sup{φ

, x
0
, y
0
), z
0
≤
F (z
0
, y
0
, x
0
) and
max

d
j
n
(α)

T x
0
, T F (x
0
, y
0
, z
0
)

)


< p(α) < ∞,
for every α ∈ I, n ∈ N.
Also, assume either
a) F is continuous; or
b) X has the property
i) If a non-decreasing sequence {x
n
} in X converges to x then x
n
≤ x for all n.
ii) If a non-increasing sequence {y
n
} in X converges to y then y
n
≥ y for all n.
Then, F has a triple fixed point.
Moreover, if X is j-bounded and for every (x, y, z), (u, v, w) ∈ X
3
there exists
(a, b, c) ∈ X
3
which is comparable to them, then F has a unique triple fixed point.
Corollary 2.2.6. In addition to hypotheses of Theorem 2.2.5, if x
0
≤ y
0
and z

+
× R
+
, R

, f, g ∈ C

R
+
× R, R

and the unknown functions
x ∈ C

R
+
, R). The deviation ∆ : R
+
→ R
+
is a continuous function, in general case,
17
unbounded. Note that, since deviation ∆ is unbounded, we can not apply the known
coupled fixed point theorems in metric space for the above integral equations.
We shall adopt the following assumptions.
Assumption 2.3.1. B1) K
1
(t, s) ≥ 0 and K
2
(t, s) ≤ 0 for all t, s ≥ 0.

(t, s) − K
2
(t, s)

ds ≤
1
2
.
B3) For each compact subset K ⊂ R
+
, there exists a compact set

K ⊂ R
+
such
that ∆
n
(K) ⊂

K, for all n ≥ 0.
B4) For each compact subset K ⊂ R
+
, there exists φ
K
∈ Φ
1
such that
φ
K
(t)


f(s, α(s)) + g(s, β(s))

ds
+

∆(t)
0
K
2
(t, s)

f(s, β(s)) + g(s, α(s))

ds
and
β(t) ≥ h(t) +

∆(t)
0
K
1
(t, s)

f(s, β(s)) + g(s, α(s))

ds
+

∆(t)

+
, R

and suppose that Assumption 2.3 is fulfilled. Then
the existence of a coupled lower and upper solution for (2.49) provides the existence
of a unique solution of (2.49) in C

R
+
, R

.
The next, we wish to investigate the existence of a unique solution to a class of
nonlinear integral equations, as an application of the tripled fixed point theorems
18
proved in the previous section. Let us consider the following integral equations
x(t) = k(t) +

∆(t)
0

K
1
(t, s) + K
2
(t, s) + K
3
(t, s)

×

, f, g, h ∈ C

R
+
×R, R

and an unknown function
x ∈ C

R
+
, R), the deviation ∆ : R
+
→ R
+
is a continuous function,.
We assume that the functions K
1
, K
2
, K
3
, f, g, h fulfill the following conditions.
Assumption 2.3.4. C1) K
1
(t, s) ≥ 0, K
2
(t, s) ≤ 0 and K
3
(t, s) ≥ 0 for all t, s ≥ 0.

K
1
(t, s) − K
2
(t, s) + K
3
(t, s)

ds ≤
1
3
.
C3) For each compact subset K ⊂ R
+
, there exists a compact set

K ⊂ R
+
such
that for all n ∈ N, ∆
n
(K) ⊂

K.
C4) For each compact subset K ⊂ R
+
, there exists φ
K
∈ Φ
1


is called
a tripled lower and upper solution of the integral equation (2.50) if for every t ∈ R
+
we have α(t) ≤ β(t), γ(t) ≤ β(t) and
α(t) ≤ k(t)+

∆(t)
0
K
1
(t, s)

f

s, α(s)

+ g

s, β(s)

+ h

s, γ(s)


ds
+

∆(t)


+ g

s, β(s)

+ h

s, α(s)


ds,
β(t) ≥ k(t)+

∆(t)
0
K
1
(t, s)

f

s, β(s)

+ g

s, α(s)

+ h

s, β(s)

3
(t, s)

f

s, β(s)

+ g

s, α(s)

+ h

s, β(s)


ds,
19
γ(t) ≤ k(t)+

∆(t)
0
K
1
(t, s)

f

s, γ(s)



ds
+

∆(t)
0
K
3
(
t, s
)

f

s, α
(
s
)

+
g

s, β
(
s
)

+
h


R
+
, R

and suppose that Assumption 2.3 is
fulfilled. Then the existence of a tripled lower and upper solution for (2.50) provides
the existence of a unique solution of (2.50) in C

R
+
, R

.
Conclusions of Chapter 2
In this chapter, we obtained the following main results
• Give and prove results which confirm the existence and unique existence of cou-
pled fixed points for a class of contractive mappings in partially ordered uniform spaces
(Theorem 2.1.5, Corollary 2.1.6).
• Give and prove results which confirm the existence and unique existence of triple
fixed points for a class of contractive mappings in partially ordered uniform spaces
(Theorem 2.2.5, Corollary 2.2.6).
• Apply Theorem 2.1.5 to prove the unique existence of solution of a class of
integral equations with unbounded deviation. Apply Theorem 2.2.5 to prove the
unique existence of solution of a class of integral equations with unbounded deviations.
These results were published in the articles
• Tran Van An, Kieu Phuong Chi and Le Khanh Hung (2014), Coupled fixed point
theorems in uniform spaces and application, Journal of Nonlinear Convex Analysis,
Vol. 15, No. 5, 953-966.
• Le Khanh Hung (2014), Triple fixed points in ordered uniform spaces, Bulletin of
Mathematical Analysis and Applications, Vol. 6, Issue 2, 1-22.

+
→ R
+
, α ∈ I} be a class of increasing and continuous
functions satisfying 0 < φ
α
(t) < t for all t > 0 and φ
α
(0) = 0; Ψ = {ψ
α
: R
+
→
R
+
, α ∈ I} be a class of increasing and continuous functions and ψ
α
(0) = 0.
Let X be a locally convex algebra with a saturated family of seminorms {p
α
}
α∈I
.
Definition 3.2.4. The mapping T : X → X is D-Lipschitz with the family of
functions {ψ
α
}
α∈I
if
p

X, B : S → X be two operators such that
1) A is D-Lipschitzian with the family of functions {ψ
α
}
α∈I
.
2) B is completely continuous and for every y ∈ S, x = AxBy implies x ∈ S.
3) p
j(α)
(x −y) ≤ p
α
(x −y) for every x, y ∈ S and α ∈ I.
4) For every x ∈ X and for every α ∈ I, there exists q(α, x) such that
p
j
k
(α)
(x) ≤ q(α, x) < ∞
for all k = 0, 1, 2, . . ., in particular p
j
k
(α)
(x) ≤ q(α) < ∞ for every x ∈ S and for all
k = 0, 1, 2, . . .
5) For each α ∈ I, then M
α
ψ
α
(t) < t for all t > 0 and there exists φ
α

dorff sequentially complete. Let S be a closed, convex and bounded subset of X and
A : X → X, B : S → X be two operators such that
1) A is Lipschitzian with the family of Lipschitz constants {k
α
}
α∈I
.
2) B is completely continuous and for every y ∈ S, x = AxBy implies x ∈ S.
22
3) p
j(α)
(x −y) ≤ p
α
(x −y) for every x, y ∈ S and α ∈ I.
4) For every x ∈ X and for every α ∈ I, there exists q(α, x) such that
p
j
k
(α)
(x) ≤ q(α, x) < ∞
for all k = 0, 1, 2, . . ., in particular p
j
k
(α)
(x) ≤ q(α) < ∞ for every x ∈ S and for all
k = 0, 1, 2, . . .
5) For each α ∈ I, then M
α
k
α

t,

∆
1
(t)
0
x(s)ds, . . . ,

∆
m
(t)
0
x(s)ds, x

τ
1
(t)

, . . . , x

τ
n
(t)


×

q(t) +

t

By a solution of the (3.3), we mean a continuous function x : R
+
→ R that satisfies
(3.3) on R
+
.
Let X = C(R
+
, R) be the lo cally convex algebra (in fact Frechet algebra) of all
continuous real valued functions on R
+
with family of seminorms
p
[0,n]
(x) = max

|x(t)| : t ∈ [0, n]

, n ∈ N
∗
.
We shall adopt the following assumptions.
Assumption 3.3.1. D1) The functions ∆
i
(t) : R
+
→ R
+
, i = 1, 2, . . . , m; τ
l

, . . . , u
m
, v
1
, . . . , v
n
) −F (t, u
1
, . . . , u
m
, v
1
, . . . , v
n
)


≤ Ω

t, |u
1
− u
1
|, . . . , |u
m
− u
m
|, |v
1
− v

y
is non-decreasing in y.
D3) q : R
+
→ R is uniformly continuous on R
+
, q
∞
= sup
t∈R
+
|q(t)| < 1 and

+∞
0


f

s, x(s)



ds < 1 − q
∞
for every x ∈ C(R
+
, R) with |x(t)| ≤ 1 for all t.
Theorem 3.3.2 Under assumptions D1), D2) and D3), then equation (3.3) has at
least one solution x = x(t) which belongs to the space C(R


ds

, (3.9)
where τ(t) is continuous function on R
+
and τ(t) ≤ t for all t ∈ R
+
.
Applying Theorem 3.3, we proved that the equation (3.9) has a solution in C(R
+
, R).
Conclusions of Chapter 3
In this chapter, we obtained the following main results
• Give and prove a fixed point theorem in locally convex algebras, basing on ideals
of known results in Banach algebras and uniform spaces (Theorem 3.2.5).
• Apply Theorem 3.2.5 to prove the existence of solution of a class of integral
equations with unbounded deviations.
These results were published in the article
Le Khanh Hung (2015), Fixed point theorems in locally convex algebras and ap-
plications to nonlinear integral equations, Fixed point theory and applications, DOI
10.1186/s13663-015-0310-9.