Urysohn integral equations approach by common fixed points in complex-valued metric spaces
Wutiphol Sintunavarat
Yeol Je Cho
Poom Kumam
Recently, the complex-valued metric spaces which are more general than the metric spaces were first introduced by Azam et al. (Numer. Funct. Anal. Optim. 32:243-253, 2011). They also established the existence of fixed point theorems under the contraction condition in these spaces. The aim of this paper is to introduce the concepts of a C-Cauchy sequence and C-complete in complex-valued metric spaces and establish the existence of common fixed point theorems in C-complete complex-valued metric spaces. Furthermore, we apply our result to obtain the existence theorem for a common solution of the Urysohn integral equations a a b b
-
x(t) =
x(t) =
K1(t, s, x(s)) ds + g(t),
K2(t, s, x(s)) ds + h(t),
where t [a, b] R, x, g, h C([a, b], Rn) and K1, K2 : [a, b] [a, b] Rn Rn.
MSC: 47H09; 47H10
1 Introduction
The study of metric spaces has played a vital role in many branches of pure and applied
sciences. We can find useful applications of metric spaces in mathematics, biology, medicine,
physics and computer science (see []). Several mathematicians improved, generalized
and extended the concept of metric spaces to vector-valued metric spaces of Perov [],
G-metric spaces of Mustafa and Sims [], cone metric spaces of Huang and Zhang [],
modular metric spaces of Chistyakov [], partial metric spaces of Matthews [] and others.
Since Banach [] introduced his contraction principle in complete metric spaces in ,
this field of fixed point theory has been rapidly growing. It has been very useful in many
fields such as optimization problems, control theory, differential equations, economics
and many others. A number of papers in this field have been dedicated to the
improvement and generalization of Banachs contraction mapping principle in many spaces and
ways (see []).
Recently, Azam et al. [] introduced a new space, the so-called complex-valued metric
space, and established a fixed point theorem for some type of contraction mappings as
follows.
Theorem . (Azam et al. []) Let (X, d) be a complete complex-valued metric space and
S, T : X X be two mappings. If S and T satisfy
for all x, y X, where , are nonnegative reals with + < , then S and T have a unique
common fixed point in X.
Theorem . of Azam et al. in [] is an essential tool in the complex-valued metric space
to claim the existence of a common fixed point for some mappings. However, it is most
interesting to find another new auxiliary tool to claim the existence of a common fixed
point. Some other works related to the results in a complex-valued metric space are [,
].
In this paper, we introduce the concept of a C-Cauchy sequence and C-complete in
complex-valued metric spaces and also prove some common fixed point theorems for
new generalized contraction mappings in C-complete complex-valued metric spaces.
On the other hand, integral equations arise naturally from many applications in
describing numerous real world problems. These equations have been studied by many authors.
Existence theorems for the Urysohn integral equations can be obtained applying various
fixed point principles.
As applications, we show the existence of a common solution of the following system of
Urysohn integral equations by using our common fixed point results:
x(t) =
x(t) =
K t, s, x(s) ds + g(t),
K t, s, x(s) ds + h(t),
2 Preliminaries
In this section, we discuss some background of the complex-valued metric spaces of Azam
et al. in [] and give some notions for our results. Also, some essential lemmas which are
useful for our results are given.
Let C be the set of complex numbers. For z, z C, we will define a partial order on
C as follows:
Re(z) Re(z) and
Im(z) Im(z).
We note that z z if one of the following holds:
(C) Re(z) = Re(z) and Im(z) = Im(z);
(C) Re(z) < Re(z) and Im(z) = Im(z);
(C) Re(z) = Re(z) and Im(z) < Im(z);
(C) Re(z) < Re(z) and Im(z) < Im(z).
It obvious that if a, b R such that a b, then az bz for all z C.
In particular, we write z z if z = z and one of (C), (C) and (C) is satisfied, and
we write z z if only (C) is satisfied. The following are well known:
z z
z z.
Definition . [] Let X be a nonempty set. Suppose that the mapping d : X X C
satisfies the following conditions:
(d) d(x, y) for all x, y X and d(x, y) = if and only if x = y;
(d) d(x, y) = d(y, x) for all x, y X;
(d) d(x, y) d(x, z) + d(z, y) for all x, y, z X.
Then d is called a complex-valued metric on X and (X, d) is called a complex-valued metric
space.
Example . Let X = C. Define the mapping d : X X C by
d(z, z) = |x x| + |y y|i,
Example . Let X = X X, where
X = z C : Re(z) , Im(z) =
X = z C : Re(z) = , Im(z) .
Example . Let X = X X, where
X = z C : Re(z) , Im(z) =
X = z C : Re(z) = , Im(z) .
Define the mapping d : X X C by
d(z, z) =
x + y + i(x + y), z X, z X,
x + y + i(x + y), z X, z X,
Define the mapping d : X X C by
d(z, z) =
z, z X,
z, z X,
x + y + i( x + y), z (...truncated)