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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

famous contraction mapping principle: Each contractive mapping from a complete
metric space (X, d) into itself has a unique fixed point. The birth of the Banach
contraction mapping principle and its application to study the existence of solutions
of differential equations marks a new development of the study of fixed point theory.
After that, many mathematicians have studied to extend the Banach contraction
mapping principle for classes of maps and different spaces. Expanding only contractive
mappings, till 1977, was summarized and compared with 25 typical formats by B.E.
Rhoades.
1.2. The Banach contraction mapping principle associates with the class of con-
tractive mappings T : X → X in complete metric space (X, d) with the contractive
condition
(B) d(T x, T y) ≤ kd(x, y), for all x, y ∈ X where 0 ≤ k < 1.
There have been many mathematicians seeking to extend the Banach contraction
mapping principle for classes of mappings and different spaces. The first extending
was obtained by E. Rakotch by mitigating a contractive condition of the form
(R) d(T x, T y) ≤ θ

d(x, y)

d(x, y), for all x, y ∈ X, where θ : R
+
→ [0, 1) is a
monotone decreasing function.
In 1969, D. W. Boyd and S. W. Wong introduced an extended form of the above
result by considering a contractive condition of the form
(BW) d(T x, T y) ≤ ϕ

d(x, y)

, for all x, y ∈ X, where ϕ : R

+
→ 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)


−ϕ

d(x, y)

, for all x, y ∈ X, where ψ, ϕ : R
+
→
R
+
is a continuous, monotone non-decreasing functions such that ψ(t) = 0 = ϕ(t) if
and only if t = 0.
In 2009, R. K. Bose and M. K. Roychowdhury introduced the notion of new gen-
eralized weak contractive mappings with the following contractive condition in order
to study common fixed points of mappings
(BR) ψ

d(T x, Sy

≤ ψ

d(x, y)

−ϕ

d(x, y)

, for all x, y ∈ X, where ψ, ϕ : R
+
→
R

+
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

spaces satisfying the contractive condition
(BL) There exists k ∈ [0, 1) such that d

F (x, y), F (u, v)

≤
k
2

d(x, u) + d(y, v)

,
for all x, y, u, v ∈ X such that x ≥ u, y ≤ v.
In 2009, by continuing extending coupled fixed point theorems, V. Lakshmikantham
and L. Ciric obtained some results for the class of mappings F : X × X → X with
g-mixed monotone property, where g : X → X from a partially ordered metric space
into itself and F satisfies the following contractive condition
(LC) d

F (x, y), F (u, v)

≤ ϕ

d

g(x), g(u)

+ d

g(y), g(v)


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


d(T x, T u), d(T y, T v), d(T z, T w)


.
1.4. The development of fixed point theory is motivated from its popular ap-
plications, especially in theory of differential and integral equations, where the first
impression is the application of the Banach contraction mapping principle to study
the existence of solutions of differential equations.
In the modern theory of differential and integral equations, proving the existence of
solutions or approximating the solutions are always reduced to applying appropriately
certain fixed point theorems. For boundary problems with bounded domain, fixed
point theorems in metric spaces are enough to do the above work well. However, for
boundary problems with unbounded domain, fixed point theorems in metric spaces are
not enough to do that work. So, in the 70s of last century, along with seeking to extend
to mapping classes, one was looking to extend to classes of wider spaces. One of typical
directions of this expansion is seeking to extend results on fixed points of mappings in
metric spaces to the class of local convex spaces, more broadly, uniform spaces which
has attracted the attention of many mathematical, notably V. G. Angelov.
In 1987, V. G. Angelov considered the family of real functions Φ = {φ
α
: α ∈ I}
such that for each α ∈ I, φ
α
: R
+
→ R
+
is a monotone increasing, continuous,
φ
α

(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

+
→ R
+
is a non-
decreasing continuous function, φ(0) = 0.
1.5. Recently, together with the appearance of classes of new contractive mappings,
and new types of fixed points in metric spaces, the study trend on the fixed point
theory has advanced steps of strong development. With above reasons, in order to
extend results in the fixed point theory for classes of spaces with uniform structure,
we chose the topic ‘‘On the existence of fixed points for some mapping classes
in spaces with uniform structure and applications” for our doctoral thesis.
2 Objective of the research
The purpose of this thesis is to extend results on the existence of fixed points
in metric spaces to some classes of mappings in spaces with uniform structure and
apply to prove the existence of solutions of some classes of integral equations with
unbounded deviation.
3 Subject of the research
Study objects of this thesis are uniform spaces, generalized contractive map-
pings in uniform spaces, fixed points, coupled fixed points, triple fixed points of some
mapping classes in spaces with uniform structure, some classes of integral equations.
4 Scope of the research
The thesis is concerned with study fixed point theorems in uniform spaces and
apply to the problem of the solution existence of integral equations with unbounded
deviational function.
5
5 Methodology of the research
We use the theoretical study method of functional analysis, the method of the
differential and integral equation theory and the fixed point theory in process of study-
ing the topic.
6 Contribution of the thesis

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

that, when we consider a class of integral equations with unbounded deviations, we
can not apply known fixed point theorems in metric spaces. Main results of Chapter
6
1 is Theorem 1.2.6, Theorem 1.2.9, Theorem 1.3.11 and Theorem 1.4.3.
In Chapter 2, we consider extension problems in partially ordered uniform spaces.
Firstly, in section 2.1, we obtain results on couple fixed points for a mapping class
in partially ordered uniform spaces when we extend (LC)-contractive condition of V.
Lakshmikantham and L. Ciric for mappings in uniform spaces. In section 2.2, by
extending the contractive condition (AK) of H. Aydi and E.Karapinar for mappings
in uniform spaces, we get results on triple fixed points of a class in partially ordered
uniform spaces. In section 2.3, by introducing notions of upper (lower) couple, upper
(lower) triple solution, and applying results in section 2.1, 2.2, we prove the unique
existence of solution of some classes of non-linear integral equations with unbounded
deviations. Main results of Chapter 2 are Theorem 2.1.5, Corollary 2.1.6, Theorem
2.2.5, Corollary 2.2.6, Theorem 2.3.3 and Theorem 2.3.6.
In Chapter 3, at first we present systematically some basic notions about locally
convex algebras needed for later sections. After that, in section 3.2, by extending the
notion of D-Lipschitz maps for mappings in locally convex algebras and by basing on
known results in Banach algebras, and uniform spaces, we prove a fixed point theorem
in locally convex algebras which is an extension of an obtained result by B. C. Dhage.
Finally, in section 3.3, applying obtained theorems, we prove the existence of solution
of a class of integral equations in locally convex algebras with unbounded deviations.
Main results of Chapter 3 are Theorem 3.2.5, Theorem 3.3.2.
In this thesis, we also introduce many examples in order to illustrate our results
and the meaning of given extension theorems.
7
CHAPTER 1
UNIFORM SPACES
AND FIXED POINT THEOREMS
In this chapter, firstly we present some basic knowledge about uniform spaces

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.

In this section, we also present the concept of topology generated by uniform struc-
ture, uniform space with uniform structure generated by a family of pseudometrics,
8
Cauchy sequence, sequentially complete uniform space and the relationship between
them.
Remark 1.1.8. 1) Let X be a uniform space. Then, uniform topology on X is
generated by the family of uniform continuous pseudometrics on X
2) If E is locally convex space with a saturated family of seminorms {p
α
}
α∈I
,
then we can define a family of associate pseudometrics ρ
α
(x, y) = p
α
(x − y) for every
x, y ∈ E. The uniform topology generated the family of associate pseudometrics
coincides with the original topology of the space E. Therefore, as a corollary of our
results, we obtain fixed point theorems in the locally convex space.
3) Let j : I → I be an arbitrary mapping of the index I into itself. The iterations
of j can be defined inductively
j
0
(α) = α, j
k
(α) = j

j
k−1

(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

(t) = 0 if only if t = 0, for all α ∈ I.
Denote Π = {ϕ
α
: α ∈ I} be a family of functions ϕ
α
: R
+
→ R
+
, α ∈ I such that
ϕ
α
is continuous, ϕ
α
(t) = 0 if only if t = 0.
Definition 1.2.4. Let X be a uniform space. A map T : X → X is called a
(Ψ, Π)-contractive on X if
ψ
α

d
α
(T x, T y)

≤ ψ
α

d
j(α)
(x, y)


− 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

n
= T x
n−1
, n = 1, 2, . . . satisfying d
α
(x
m
, x
m+n
) ≥ d
j(α)
(x
m
, x
m+n
) for all
m, n ≥ 0, all α ∈ I.
Then, T has at least one fixed point. X.
Moreover, if X has j-monotone decreasing property, then T has a unique fixed
point.
Example 1.2.7. Let X = R
∞
=

x = {x
n
} : x
n
∈ R, n = 1, 2, . . .


. 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)



, for every (x, y) ∈ X.
Then, the collection of pseudometrics {d
(n,r)
: (n, r) ∈ I} generated a uniformity on
X.
Now for every (n, r) ∈ I we consider the functions, which is given by ψ
(n,r)
(t) =
2(n −1)
2n −1
t, for all t ≥ 0, and put Ψ = Φ = {ψ
(n,r)
: (n, r) ∈ I}. Denote by j : I → I
a map is defined by j(n, r) =

n, r

1 −
1
2n


, for all (n, r) ∈ I and define a mapping
T : X → X which is defined by
T x =

1 −

1 −
2

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

α

d
α
(T x, Sy)

≤ ψ
α

d
j(α)
(x, y)

− ϕ
α

d
j(α)
(x, y)

,
for all x, y ∈ X, where ψ
α
∈ Ψ, ϕ
α
∈ Π for all α ∈ I.
Suppose j : I → I be a surjective map and for some ti x
0
∈ X such that the
sequence {x

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

Denote Ψ
1
= {ψ
α
: α ∈ I} be a family of functions with the properties
(i) ψ
α
: R
+
→ R
+
is monotone non-decreasing and ψ
α
(0) = 0.
(ii) for each α ∈ I, there exists ψ
α
∈ Ψ
1
such that
sup

ψ
j
n
(α)
(t) : n = 0, 1, . . .

≤ ψ
α
(t) and

+∞

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

α
(x, y).d
α
(T x, T y) ≤ ψ
α

d
j(α)
(x, y)

,
for all x, y ∈ X.
Definition 1.3.8. Let T : X → X. We say that T is a β-admissible if for all x, y ∈ X
and α ∈ I, β
α
(x, y) ≥ 1 implies β
α
(T x, T y) ≥ 1.
Theorem 1.3.11. Let X be a set and P =

d
α
(x, y) : α ∈ I

be a family of
pseudometrics on X such that (X, P) is a Hausdorff sequentially complete uniform
space. Let T : X → X be an (β, Ψ
1
)-contractive mapping satisfying the following
conditions

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

n+1
) ≥ 1 for all n
and x
n
→ x ∈ X as n → +∞, then β
α
(x
n
, x) ≥ 1 for all n ∈ N
∗
.
Then, T has a fixed point.
Moreover, if X is j-bounded and for every x, y ∈ X, there exists z ∈ X such that
β
α
(x, z) ≥ 1 and β
α
(y, z) ≥ 1 for all α ∈ I, then T has a unique fixed point.
We also give some examples to illustrate for our results.
11
1.4 Applications to nonlinear integral equations
In this section, we wish to investigate the existence of a unique solution to nonlinear
integral equations, as an application to the fixed point theorems proved in the Section
1.3.
Let us consider the following integral equations
x(t) =

∆(t)
0
G(t, s)f


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

K
∈ Ψ
1
such that for
all t ∈ R
+
, for all a, b ∈ R with u(a, b) ≥ 0, we have


f(t, a) −f (t, b)


≤ λψ
K

|a −b|

and λ sup
t∈K

∆(t)
0
G(t, s)ds ≤ 1.
A2) There exists x
0
∈ C(R
+
, R) such that for all t ∈ R
+
, we have

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)

0
G(t, s)f

s, x(s)

ds,

∆(t)
0
G(t, s)f

s, y(s)

ds

≥ 0.
A4) If {x
n
} is a sequence in C(R
+
, R) such that x
n
→ x ∈ C(R
+
, R) and u(x
n
, x
n+1
) ≥
0 for all n ∈ N

∗
, 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

+
× R → R
+
is continuous and non-decreasing according to the second
variable.
2) For each compact subset K ⊂ R
+
there exist the positive number λ and ψ
K
∈ Ψ
1
such that for all t ∈ R
+
, for all a, b ∈ R with a ≤ b, we have


f(t, a) −f (t, b)


≤ λψ
K

|a −b|

and λ sup
t∈K

∆(t)
0
G(t, s)ds ≤ 1.

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)


ds, (1.28)
where G : R
+
× R
+
→ R
+
is given by
G(t, s) =

3
4
e
s−t
if t ≥ s ≥ 0
3
4
e
t−s
if s ≥ t ≥ 0
and f : R
+
× R → R
+
is defined by
f(t, x) =

x +
√

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

CHAPTER 2
FIXED POINTS OF SOME MAPPING CLASSES
IN PARTIALLY ORDERED UNIFORM SPACES
AND APPLICATIONS
In this chapter, we prove some fixed point theorems for generalized contractive
mappings in uniform spaces and apply them to study the existence problem of solutions
for a class of nonlinear integral equations with unbounded deviations. We also give
some examples to show that our results are effective.
2.1 Coupled fixed points in partially ordered uniform
spaces
In 2006, T. G. Bhaskar and V. Lakshmikantham introduced the notion of coupled
fixed points of mappings F : X ×X → X with mixed monotone property and obtained
some results for the class of those mappings in partially ordered metric spaces.
Definition 2.1.1. Let (X, ≤) be a partially ordered set and F : X × X → X.
The mapping F is said to have the mixed monotone property if F is monotone non-
decreasing in its first argument and is monotone non-increasing in its second argument,
that is, for any x, y ∈ X
if x
1
, x
2
∈ X, x
1
≤ x
2
then F (x
1
, y) ≤ F (x
2
, y)

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

(t) < t for all t > 0 and φ
α
(0) = 0.
14
Let (X, ≤) be a partially ordered set. Then, we consider the partial order on X ×X
that defined by
for (x, y), (u, v) ∈ X ×X, (x, y) ≤ (u, v) if only if x ≤ u, y ≥ v.
Theorem 2.1.5. Let (X, ≤) be a partially ordered set and P = {d
α
(x, y) : α ∈ I} be
a family of pseudometrics on X such that (X, P) is a Hausdorff sequentially complete
uniform space. Let F : X ×X → X be a mapping having the mixed monotone property
on X. Suppose that
1) For every α ∈ I, there exists φ
α
∈ Φ
1
such that
d
α

F (x, y), F (u, v)

≤ φ
α

d
j(α)
(x, u) + d
j(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

, y
0
), y
0
≥ F (y
0
, x
0
) and
d
j
n
(α)

x
0
, F (x
0
, y
0
)

+ d
j
n
(α)

y
0
, F (y

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

In 2011, V. Berinde and M. Borcut introduced the notion of triple fixed points
for a class of mapping F : X × X × X → X and obtained some triple fixed point
theorems for mappings with mixed monotone property in partially ordered metric
spaces. After that, in 2012, H. Aydi and E. Karapinar extended the above result and
15
obtained some triple fixed point theorems for a class of mapping F : X ×X ×X → X
with mixed monotone property in partially ordered metric spaces and satisfying the
following weak contractive condition.
Definition 2.2.1. Let (X, ≤) be a partially ordered set and F : X × X × X → X.
The mapping F is said to have the mixed monotone property if for any x, y, z ∈ X
x
1
, x
2
∈ X, x
1
≤ x
2
⇒ F (x
1
, y, z) ≤ F (x
2
, y, z),
y
1
, y
2
∈ X, y
1
≤ y

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.

Let (X, ≤) be a partially ordered set. Then, we define a partial order on X
3
in the
following way:
For (x, y, z), (u, v, w) ∈ X
3
then
(x, y, z) ≤ (u, v, w ) if and only if x ≤ u, y ≥ v and z ≤ w.
We say that (x, y, z) and (u, v, w) are comparable if
(x, y, z) ≤ (u, v, w) or (u, v , w) ≤ (x, y, z).
Also, we say that (x, y, z) is equal to (u, v, w) if and only if x = u, y = v and z = w.
Definition 2.2.4. Let X be a uniform space. A mapping T : X → X is said to be
ICS if T is injective, continuous and has the property: for every sequence {x
n
} in X,
if sequence {T x
n
} is convergent then {x
n
} is also convergent.
Theorem 2.2.5. Let (X, ≤) be a partially ordered set and P = {d
α
(x, y) : α ∈ I} be
a family of pseudometrics on X such that (X, P) is a Hausdorff sequentially complete
uniform space. Let T : X → X is an ICS mapping and F : X
3
→ X be a mapping
having the mixed monotone property on X. Suppose that
16
1) For every α ∈ I there exists φ

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

j
n
(α)
(t) : n = 0, 1, . . .} ≤
φ
α
(t) for all t > 0 and
φ
α
(t)
t
is non-decreasing on (0, +∞).
3) There are x
0
, y
0
, z
0
∈ X such that x
0
≤ F (x
0
, y
0
, z
0
), y
0
≥ F (y
0

, 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
)


, d
j
n
(α)

T y
0
, T F (y
0
, x
0
, y
0
)

,
d
j
n
(α)

T z
0
, T F (z
0
, y
0
, x
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

0
≤ y
0
then F has a unique fixed point, that is, there exists x ∈ X such that F (x, x, x) = x.
We also gave some examples to illustrate theorems in Sections 2.1, 2.2.
2.3 Applications to nonlinear integral equations
The first, we apply the coupled fixed point theorems proved in the Section 2.1 to
investigate the existence of a unique solution to nonlinear integral equations.
Let us consider the following integral equations
x(t) = h(t) +

∆(t)
0

K
1
(t, s) + K
2
(t, s)

f(s, x(s)) + g(s, x(s))

ds, (2.49)
where K
1
, K
2
∈ C

R

+
× 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.

B2) For each compact subset K ⊂ R
+
, there exist λ, µ ≥ 0 and φ
K
∈ Φ
1
such that
for all x, y ∈ R, x ≥ y and for all t ∈ K, we have
0 ≤ f (t, x) − f(t, y) ≤ λφ
K

x −y
2

, −µφ
K

x −y
2

≤ g(t, x) −g(t, y) ≤ 0
and
max(λ, µ) sup
t∈K

∆(t)
0

K
1

(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)

t
is
non-decreasing and φ
∆
n
(K)
(t) ≤ φ
K
(t) for all n and for all t ≥ 0.
Definition 2.3.2. An element (α, β) ∈ C

R
+
, R

× C

R
+
, R

is a coupled lower and
upper solution of the integral equation (2.49) if α(t) ≤ β(t) and
α(t) ≤ h(t) +

∆(t)
0
K
1
(t, s)


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)

0
K
2
(t, s)

f(s, α(s)) + g(s, β(s))

ds.
Theorem 2.3.3. Consider the integral equation (2.49) with K
1
, K
2
∈ C

R
+
×R
+
, R

,
f, g ∈ C

R
+
×R, R

, h ∈ C

R

+
, 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

s, x(s)

+ g

s, x(s)

+ h

s, x(s)


ds,
(2.50)
where K
1
, K
2
, K
3
∈ C

R
+
×R
+
, R


, 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.

C2) For each compact subset K ⊂ R
+
, there exist the positive numbers λ, µ, η ≥ 0
and φ
K
∈ Φ
1
such that for all x, y ∈ R, x ≥ y and for all t ∈ K, we have 0 ≤ f(t, x) −
f(t, y) ≤ λφ
K

x − y

, −µφ
K

x − y

≤ g(t, x) − g(t, y) ≤ 0, 0 ≤ h(t, x) − h(t, y) ≤
ηφ
K

x −y

and max(λ, µ, η) sup
t∈K

∆(t)
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

such that
φ
K
(t)
t
is
non-decreasing and φ
∆
n
(K)
(t) ≤ φ
K
(t) for all n ∈ N and for all t ≥ 0.
Definition 2.3.5. An element (α, β, γ) ∈ C

R
+
, R

×C

R
+
, R

×C

R
+
, R


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)

0
K
2
(t, s)

f

s, β(s)

+ g

s, α(s)

+ h

s, β(s)


ds
+

∆(t)
0
K
3
(t, s)

f

s, γ(s)


+ 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)



ds
+

∆(t)
0
K
2
(t, s)

f

s, α(s)

+ g

s, β(s)

+ h

s, α(s)


ds
+

∆(t)
0
K

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)


+ g

s, β(s)

+ h

s, α(s)


ds
+

∆(t)
0
K
2
(t, s)

f

s, β(s)

+ g

s, γ(s)

+ h

s, β(s)


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

s, γ
(
s
)


ds.
Theorem 2.3.6. Consider the integral equation (2.50) with K
1
, K
2
, K
3
∈ C

R
+
×
R
+
, R

, f, g, h ∈ C

R
+
× R, R

, k ∈ C


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.

20
CHAPTER 3
FIXED POINT THEOREMS IN LOCALLY CONVEX
ALGEBRAS AND APPLICATIONS
In this chapter, we give a fixed point theorem in locally convex algebras and apply
it to investigate the existence of solution for a class of nonlinear integral equations
with unbounded deviations.
3.1 Locally convex algebras
In this section, we introduce some basic knowledge of locally convex algebra.
Throughout this chapter, we consider associative and commutative algebras over the
field K of complex numbers or real numbers.
Definition 3.1.1. Let X be an algebra over K. X is called a topological algebra if
there is a topology τ on X such that
1) (X, τ) is a topological vector space;
2) The multiplication in X is continuous
Definition 3.1.2. Let X be a topological algebra. Then
1) A seminorm p : X → R is called to be submultiplicative if p(xy) ≤ p(x)p(y) for
all x, y ∈ X.
2) A set U ⊂ X is called to be multiplicative if U.U ⊂ U.
Definition 3.1.3. The topological algebra X is called a locally multiplicatively convex
algebra if X has a local basis consisting of multiplicative and convex sets.
Definition 3.1.6. Let X be a locally convex space and T : X → X. Then,
2) T is totally bounded if for any bounded set S of X, T (S) is a totally bounded
set of X.
3) T is completely continuous if it is continuous and totally bounded.
21
3.2 Fixed points in locally convex algebras
Let Φ = {φ
α
: R

+
→ 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

α
(T x −T y) ≤ ψ
α

p
j(α)
(x −y)

,
for all x, y ∈ X and for all α ∈ I, where {ψ
α
}
α∈I
is a subfamily of Ψ.
If ψ
α
(t) = k
α
t for all t ≥ 0, where k
α
is a real number for all α ∈ I, then T is called
Lipschitzian with the family of Lipschitz constants {k
α
}
α∈I
.
Extending the results in uniform spaces and Banach algebras, we obtained following
theorem on local convex algebras.
Theorem 3.2.5. Let X be a locally convex algebra such that topology of X is Hausdorff
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 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

∈ Φ such that
φ
α
(t)
t
is non-decreasing and sup{M
j
k
(α)
ψ
j
k
(α)
(t) : k = 0, 1, 2, . . .} ≤ φ
α
(t) for every
t > 0, where M
α
= sup

p
α

B(x)

: x ∈ S

.
Then, the operator equation x = AxBx has a solution.
Corollary 3.2.7. Let X be a locally convex algebra such that topology of X is Haus-

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

< 1 and sup{M
j
k
(α)
k
j
k
(α)
: k = 0, 1, 2, . . .} ≤ r
α
< 1,
where M
α
= sup

p
α

B(x)

: x ∈ S

.
Then, the operator equation x = AxBx has a solution.
3.3 Applications to nonlinear integral equations
In this section, we apply the previous result to investigate the existence of a
solution to nonlinear integral equations with unbounded deviations.
Let us consider the following integral equation
x(t) = F


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

0
f

s, x(s)

ds

,
(3.3)
where an unknown function x ∈ C(R
+
, R), ∆
i
, τ
j
: R
+
→ R
+
are continuous functions,
in general case, unbounded and, q : R
+
→ R, f : R
+
×R → R are continuous functions,
F : R
+
× R
m+n
→ [0, 1].

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

(t) :
R
+
→ R
+
, l = 1, 2, . . . , n are continuous and ∆
i
(t) ≤ t, τ
l
(t) ≤ t for every t > 0.
D2) The function F : (t, u
1
, u
2
, . . . , u
m
, v
1
, . . . , v
n
) : R
+
× R
m+n
→ [0, 1] is contin-
uous and satisfies the conditions


F (t, u
1

, . . . , 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

1
|, . . . , |v
n
− v
n
|

,
23
where the function Ω(t, x
1
, . . . , x
m
, y
1
, . . . , y
n
) : R
m+n+1
+
→ R
+
is continuous in t,
non-decreasing and continuous in each x
i
, y
l
, Ω(t, ay, . . . , ay, y, . . . , y) < y for every
constant a > 0 and
Ω(t, ay, . . . , ay, y, . . . , y)

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

+
, R).
The following example is an illustration for the Theorem 3.3.
Example 3.3.3 Consider the following nonlinear functional integral equation
x(t) =
1
2 +


x

τ(t)




te
−t
2
+

t
0
se
−s
2

1+x
2
(s)


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.