1. Introduction
The Banach contraction principle (BCP) [
1] is one of the famous results in fixed point theory which has attracted many authors. Many extensions and generalizations have been appeared in literature by weakening the topology itself of the space or by considering different contractive conditions (for single and valued mappings). For more details, see ([
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23]).
Definition 1. Given a mapping on a metric space .
- (a)
Such Y is a C-contraction if there is such that for all , [24] - (b)
Such Y is a K-contraction if there is such that for all , [25] - (c)
Such Y is a Reich contraction if there are q, r and with such that for all ,
Denote by the set of functions satisfying the following assertions:
is non-decreasing;
for each , if and only if ;
there are
and
so that
for all .
By we denote the class of functions without condition .
Theorem 1. ([26, Corollary 2.1]) Let be a self-mapping on a complete metric space . Suppose there are and so that Then T has a unique fixed point.
Note that the BCP comes immediately from Theorem 1. Motivated by [
26], Hussain et al. [
27] gave sufficient conditions for the existence of a fixed point of a class of generalized contractive mappings via a control function
in the setting of complete metric spaces and
b-complete
b-metric spaces. Denote by
the set of functions
verifying
,
and
. On the other hand, when considering
as a metric space and
(that is, the condition
is omitted from
), Jiang et al. [
28] proved that
defines itself a metric on
X (see Lemma 1 in [
28]) and proved that the results in [
27] are not generalizations of Ćirić, Chatterjea, Kannan, and Reich results.
In this paper, we more restrict the conditions on the control function . For this, denote by the set of functions so that
is continuous and strictly increasing;
for each , if and only if .
Let
be a metric space. For
(that is, without the condition
), note that
does not define a metric on
X (we can not ensure the triangular inequality for a metric). Consequently, we are not in same direction as Jiang et al. [
28]. Even for such restricted control function
, we also obtain a real generalization of the Banach contraction principle. In fact, we will complete the work of Hussain et al. [
27]. We refer the readers to Theorem 3 of [
16].
2. Main Results
Definition 2. Let be a self-mapping on a metric space . Such Y is said to be a -contraction, whenever there are and with such that the following holds:for all . As a new generalization of the BCP, we have
Theorem 2. Each -contraction mapping on a complete metric space has a unique fixed point.
Proof. Let be arbitrary. Define by , . If there is for some N, nothing is to prove. We assume that for each .
If for some
N, we have
then in view of
, we get that
Therefore,
which is a contradiction with respect to (
3).
Consequently, for all
,
which yields that
According to
, we get
In order to show that
is a Cauchy sequence, suppose the contrary, i.e., there is
for which we can find
and
so that
In view of (
5) and (
7), we get
On the other hand, we have
Using now
and (
5)–(
8), we have
This implies that
which is a contradiction. Thus,
is a Cauchy sequence. The completeness of
X implies that there is
so that
as
On the other hand,
Taking
and using
and (
5), we have
We deduce that , so is a fixed point.
Let there are two points
which are two different fixed points of
. So,
We deduce that , so is a fixed point.
Let
be two distinct fixed points of
. We have
which is a contradiction. So,
has a unique fixed point. □
Remark 1. In Theorem 2, we can substitute the continuity of θ by the continuity of
By setting , we have
Corollary 1. Let be a mapping on a complete metric space such that the following holds:for all , where and so that . Then Y has a unique fixed point. Remark 2. Taking in the Corollary 1, we get Theorem 2.6 of [27]. Taking in Theorem 1, we get Theorem 2.8 of [27]. Setting in Theorem 2, we have
Corollary 2. Let be a complete metric space and let be such that the following holds:for all , where and such that . Then Y has a unique fixed point. Remark 3 ([
12])
. Other examples of functions in the set arefor all . By setting , we have
Corollary 3. Let be a continuous mapping on a complete metric space . Suppose that there are with such that the following holds:for all . Then there is a unique fixed point of Y. Corollary 4. Let be a continuous mapping on a complete metric space . Suppose that there are with such that the following holds:for all . Then there is a unique fixed point of Y. Corollary 5. Let be a continuous mapping on a complete metric space . Suppose that there are with such that the following holds:for all . Then Y has a unique fixed point. Example 1. Let be endowed with the metric for all . Define and bywhere α ( is the positive solution of the equation Take Choose and for .
Let . We have the following cases:
Case 1: . According to the mean value Theorem for on the interval , there is some such thatwherebecause that , for each , and for each . Case 2: and . Here,for all . Using the mean value Theorem on the function on the interval , we have Case 3: and . It is similar to case 2.
Case 4: . Here, one writes Hence, Y is a -contraction. Thus all the conditions of Theorem 2 hold and Y has a fixed point (.
3. Weak-JS Contractive Conditions
Let be the class of functions satisfying the following properties:
is continuous;
;
or each , iff .
Remark 4. It is clear that belongs to Φ. Other examples are and .
Definition 3. Let be a metric space and let Y be a self-mapping on .
We say that Y is a weakly JS-contraction if for all with , we havewhere and . Theorem 3. Let be a complete metric space. Let Y be a self-mapping on X so that
- (i)
Y is a weakly JS-contraction;
- (ii)
Y is continuous.
Then Y has a unique fixed point.
Proof. Let
be arbitrary. Define
by
Without loss of generality, assume that
for each
. Since
is a weakly JS-contraction, we derive
So, we deduce that is decreasing, and so there is so such . We will prove that .
We claim that is a Cauchy sequence.
We argue by contradiction, i.e., there is
for which there are
and
of
so that
From (
16) and using the triangular inequality, we get
Taking
, and using (
15), we get
As
, we may apply (
10) to get that
Now, taking
and using
, (
17) and (
18), we have
This implies that
which is a contradiction with respect to (
16).
Thus, is a Cauchy sequence in the complete metric space , so there is some such that .
Now, since is continuous, we get that as . That is, Thus, has a fixed point.
Let
so that
Consider
Thus,
which is a contradiction. Hence,
. □
One can obtain many other contractive conditions by substituting suitable values of
and
in (
10).
Taking for all and , we obtain the JS-contractive condition.
Without the continuity assumption of , we have
Theorem 4. Let be a complete metric space. Let be a mapping. Suppose thatfor all , where and . Then Y has a unique fixed point. Proof. For
, let
be defined by
for
. Note that there is
such that
Hence, we get that
. Thus, we have
which by (
20), implies that
. □
Example 2. Let . Take the metricfor all . Define , and byand . Note that for all , one has . Now, for all , we have Thus, Y is a weakly JS-contraction. All hypotheses of Theorem 3 are verified, so Y has a unique fixed point, which is, .
4. Application to Nonlinear Integral Equations
Consider the following nonlinear integral equation
where
,
(the set of continuous functions from
to
),
and
are given functions.
Theorem 5. Assume that
- (i)
is continuous and there is so that for arbitrary function f with - (ii)
there is so thatfor all and .
Then (22) has a unique solution. Proof. Let
. Define the metric
d on
X by
. Then
is a complete metric space. Consider
by
. Let
and
. We have
Thus
is a
-contraction. All the conditions of Theorem 2 hold, and so
has a unique fixed point, that is, (
22) has a unique solution. □
5. Conclusions
In this paper, we restricted the conditions on the control function
(with respect to the ones given in [
27,
28]) and we obtained a real generalization of the Banach contraction principle (BCP). We also initiated a weakly JS-contractive condition that generalizes its corresponding of Jleli and Samet [
26], and we provided some related fixed point results.