1. Introduction and Preliminaries
In 1922, Banach [
1] published a renowned fixed point theorem that initiated the metric fixed point theory. Indeed, Banach’s result represents an abstraction of the method of a successive approximation that has been used to find a solution to certain differential equations. In this regard, it is quite reasonable to connect the foundation of the fixed point theory to the earlier than Banach, such as, Liouville [
2], Poincaré [
3], or Picard [
4]. The iteration, used in the proof of Banach [
1], is called Picard iteration. On the other hand, the formulation of Banach is artwork: In a complete metric space
, every contraction (that is,
for all
and for some
) possesses a unique fixed point. This art-piece has been improved, generalized and developed in several directions.
Among all, we restrict ourselves to investigation of a fixed point of certain mapping that forms a contraction at a point for some iteration the mapping power-contraction. This trend was initiated by Bryant [
5]: In a complete metric space
, every
m-iterative contraction (that is,
for all
, for some
and for some
) possess a unique fixed point. Notice that in this theorem the mentioned function is not necessarily continuous. This is the main advantages of this trend when it is compared with the classical Banach’s fixed point or other fixed point results dealing with the linear contractions.
In what follows we recall the initial result, Sehgal [
6], regarding an investigation of a fixed point theorem for mappings with a contractive iterate.
Theorem 1 ([6]). Let be a complete metric space, F a continuous self-mapping of that satisfies the condition that there exists a real number q, such that for each there exists a positive integer such that for each , Then T has a unique fixed point in .
We first underline differences between the results of Sehgal [
6] and Bryant [
5]. Although Sehgal [
6] assumed continuity of the quoted mapping above, in the result of Bryant [
5] the continuity assumption is not required. In this aspect, Bryant’s fixed point theorem seems more general than Seghal’s fixed point theorem. On the other hand, in the results of Seghal’s, for each
, there is a positive integer
such that
forms a contraction. In Bryant’s fixed point theorem, there is a positive integer
m that does not vary according to given
. On the contrary, for any
, we have
. Consequently, the results of Seghal is a genuine generalization of Bryant’s fixed point theorem. It was understood later that the continuity assumptions in the result of Seghal [
6] is superfluous [
7].
Next, we recollect the result of Kincses-Totik [
8] which besides removing the continuity assumption extends the contractive condition using an auxiliary function defined below.
Let be the class of all monotonically decreasing function such that for all For a non-empty set , the expression denotes the set of all fixed point of . Note that is a singleton if and only if F has a unique fixed point on .
Theorem 2 ([8]). Let F be a selfmapping on a complete metric space . Suppose that there exists and for each there exists a positive integer such thatfor each with . Then, there is such that . We consider the following simple examples to indicate our motivation:
Example 1. Let be defined by Under the standard Euclidean metric, it is clear that F is not continuous and does not form a contraction. On the other hand, for any , for it is a contraction. More precisely, for all and hence is a contraction on [0,1].
This example shows that Theorem 2 is genuine extension of the several existing results, including the results of Seghal [
6] and consequently Banach [
1]. Indeed, the fixed point theorems of Seghal [
6] and Banach [
1] are not applicable since the considered mapping is not continuous.
Following this, a number of authors deepen the research by considering an iteration of the mapping, see e.g., [
6,
7,
8,
9,
10,
11,
12,
13,
14,
15]. In this paper, inspired from the renowned results of Kincses-Totik [
8], we propose the most general form of fixed point problems that are called “contractions at a point”.
2. Main Results
Now, we state and prove the main result of the paper.
Theorem 3. Let F be a selfmapping of a metric space and . If for each there exists a positive integer such thatfor each with , wherethen there is such that . Proof. Let be an arbitrary point in and a positive integer. Starting from we will build inductively the sequence by , .
The proof is composed of five steps.
Step 1. We assert that the set is bounded. Let and such that with and .
Using the triangle inequality and considering (
3), we have
Let
and
where
We distinguish two situations:
- A.
Suppose that . Then,
if
, returning in (
5), we have
if
, returning in (
5), we have
which means
- B.
Suppose that
. Since
is non-increasing, we have
. Returning in (
5),
if
, we have
if
, we have
or
Let
and
. Then,
Combining (
6), (
7), (
8) and (
9), we obtain
where
or, varying
k
which proves that the set
is bounded.
Step 2. We shall now demonstrate that is a Cauchy sequence.
Let
. Considering
, where
are arbitrary natural numbers such that
and
, we can write
Let
such that
Hence,
and
where
such that
From the boundedness of the set A we know that, in particular, . On the other hand, let and such that , for . Again, we should consider two cases:
If we can find
such that
, then, since
If for any
we have
then, since the function
is non-increasing, we have
Furthermore, for
we found that
which shows us that the sequence
is Cauchy sequence on a complete metric space. From here, there exists
such that
Step 3. We now show
In order to show that
is a fixed point of
F, we firstly need to show that
. We have from (
3)
where
As previous, let
such that
Hence,
and therefore
Continuing and using a method similar to the argument as above, we find that our claim is true, that is
Step 4.. In the following, we demonstrate that
is a fixed point of
. Using the triangle inequality, (
13) and since
Letting
and taking into account (
12) and (
14), we get
which implies
.
Step 5. Uniqueness of fixed point of
Let
such that
. Replacing in (
3)
which is a contradiction. Therefore,
has a unique fixed point. On the other hand, since
, we easily have
which shows that
is a fixed point of
. However, due to uniqueness of a fixed point, we obtain that,
. □
Example 2. Let and for any , where . Let be a mapping defined by Let also be the function . Since for and we have , ,andwhich proves that F is not a contraction. On the other hand, by simple calculation, we getand for any . Thus, for fixed we can find such that for all From Theorem 3, it follows that F has a unique fixed point, .
Example 3. Let , and defined by Then, for any and we have Let also the nonincreasing function such that Then, for every , there is such that for any Furthermore, for choosing , we have for every In conclusion, for every there is such that the inequality (3) holds for every Thus, all the presumptions of Theorem 3 are fulfilled. Accordingly, . On the other hand, since for and ,we deduce that F does not form a contraction. Corollary 1. Let F be a selfmap on a complete metric space . Suppose that there exists and for each there exists so that for each with ,where . Then, there is such that . Sketch of the proof. On account of the fact that , where , we employ Theorem 3. Indeed, it is a special case for taking a special value of .
Corollary 2. Let F be a selfmap on a complete metric space such that for each there exists so that for each with ,where and . Then there is such that . Sketch of the proof. Since for any each
taking
in Theorem 3 we get the stated result.
Corollary 3. Suppose that F is a selfmap on a complete metric space . Presuming also that there exists a . If for each there exists such that for each with ,then there is such that . Sketch of the proof. We shall use the following inequality to derive the desired result from Theorem 3.
where
is defined as in Theorem 3.
Corollary 4. Let F be a selfmap on a complete metric space . Assume that there exists a and for each there exists such that for each with , Then there is such that .
Sketch of the proof. Indeed, this is derived from Corollary 1 by keeping in mind the following inequality:
where
is defined as in Theorem 3.
4. Conclusions
Notice that at the first glance, in the formulation of our main result, there is a great similarity with the well-known results in the literature, such as, Reich [
20] or Hardy-Rogers [
21]. In such results by taking “the maximum” of the distances,
and
. Indeed, almost all metric fixed point results have used these five distances (or some of them) to formulate results. We should underline that in such results, the considered self-mapping
F is necessarily continuous. On the other hand, in our main results, we have two main contributions: The first contribution is to remove the necessity of the continuity of the given mapping. The second main contribution is to get a contraction of mapping for its iteration that depends on the given point. More precisely, suppose for
, we get that
involves a contraction but for
y, distinct from
, we may find
forms a contraction.
By the fact that
for all
, we derive Theorem 2, the main theorem of [
8], as a corollary of Theorem 3. Since
for all
, and by letting
, where
, we deduce the main results of [
6], as a consequence of our main result. Consequently, our main results also cover the renowned Banach-Caccioppoli fixed point theorem, for
in Corollary 4.