1. Introduction and Preliminaries
The Banach’s contraction principle [
1] first appeared in explicit form in 1922, where it was used to establish the existence of a solution for an integral equation. Since then, because of its simplicity, usefulness and constructiveness, it has become a very popular and a fundamental tool in solving existence problems arising not only in pure and applied mathematics but also in many branches of sciences, engineering, social sciences, economics and medical sciences. One of the most common generalizations of Banach’s contraction is the 1971’s Ćirić contraction [
2] (also see [
3]), in that he considered all possible six values
,
and
by combining
for all
and
T (self mapping on a metric space
). Later, in 1982, Istrǎţescu [
4] introduced the class of convex contractions in metric space, where he considered seven values
,
,
,
,
,
and
for all
. Further, he showed with example (see Example 1.3, [
4]) that
T is in the class of convex contraction but it is not a contraction. Recently, some researchers studied on generalization of such class of mappings in the setting of various spaces (for example, Alghamdi et al. [
5], Ghorbanian et al. [
6], Latif et al. [
7], Miandaragh et al. [
8], Miculescu [
9], etc.). Khan et al. [
10], introduced the notion of generalized convex contraction mapping of type-2 by extending the generalized convex contraction (respectively, generalized convex contraction of order-2) of Miandaragh et al. [
8] and the convex contraction mapping of type-2 of Istrǎţescu [
4]. Very recently, Khan et al. [
11], discussed the notions of
-convex contraction (respectively
-contraction) and asymptotically
-regular (respectively
-regular) sequence and, showed that
-convex contraction reduces to two-sided convex contraction [
4]. Further, they have also shown with examples that the notions of asymptotically
T-regular and
-regular sequences are independent to each other.
Generalizing the Banach contraction principle, Wardowski [
12] introduced the notion of
F-contraction and proved a new fixed point theorem concerning
F-contractions.
Definition 1. [12] Let be a mapping satisfying the following: F is strictly increasing, i.e., for all such that
For each sequence of positive numbers if and only if
There exists such that .
We denote , the set of all functions satisfying the above definition.
Definition 2. [12] A mapping is said to be an F-contraction on if there exist and such that , Example 1. [12] The following functions are in . - (i)
- (ii)
- (iii)
- (iv)
.
By using
F-contraction, Wardowski [
12] proved a fixed point theorem which generalizes Banach’s contraction principle in a different way than in the known results from the literature.
Theorem 1. [12] Let be a complete metric space and be an F-contraction. Then, we have - (i)
T has a unique fixed point
- (ii)
For all , the sequence is convergent to .
Definition 3. ([2,3]) Let be a metric space and be a mapping. Then, T is said to be orbitally continuous on X if implies that . Let be a mapping on a non-empty set X. We denote .
Definition 4. [13] Let be a self mapping on a non-empty set X and be a mapping, we say that T is an α-admissible if , implies that . Obviously, may or may not be symmetric and (i.e., symmetric) if and only if .
Definition 5. [14] Let and . We say that T is said to be a triangular α-admissible if implies ,
and imply for all . (see for more examples Karapinar et al. [14]).
Example 2. Let . Define and by for all and Then, T is α-admissible as implies that for and for all .
Definition 6. [15] Let T be an α-admissible mapping on a non-empty set X. We say that X has the property if for each , there exists such that and . Definition 7. An α-admissible mapping T is said to be an -admissible, if for each , we have . If , we say that T is vacuously -admissible.
The above definition is used by some authors without its nomenclature concerning the uniqueness of fixed point (for examples Alsulami et al. [
16], Khan et al. [
10], etc.).
Example 3. Let . Define and by for all and Then, T is α-admissible. Since T has no fixed point, i.e., , so T is vacuously -admissible.
Example 4. Let . Define and by for all and Obviously, T is α-admissible and . Then, T is -admissible
Example 5. Let . Define and by for all and Obviously, T is α-admissible and . Note that T is not an -admissible, since for .
In the next section, we extend the notion of convex contraction [
4] to an
-
F-convex contraction and prove a fixed point theorem in the setting of metric space.
3. Fixed Point Results of an --Contraction
We prove the following lemma which will be used in the sequel.
Lemma 1. Let be a metric space and be an -convex contraction satisfying the conditions:
- (i)
T is α-admissible;
- (ii)
there exists such that .
Define a sequence in X by for all , then , whenever or for .
Proof. Following the same steps as in Lemma 1, the last paragraph was replaced with the following: Therefore, and hence . By a similar argument, we obtain continuing in these way, we arrive at whenever or for . □
Proof. Let
be such that
and
is a sequence defined by
for all
. Since
T is
-admissible,
implies that
. One can obtain inductively that
for all
. Assume that
for all
. Then,
for all
. Setting
. From (
2), taking
and
, we obtain
Since
F is strictly increasing and
, by (
3) and (
4), we obtain
If
, then (
5) gives
This is a contradiction. It follows that
Since
and
F is strictly increasing, it follows thatx
Again, from (
2) taking with
and
, we obtain
By (
3) and (
6), we obtain
If
, then we obtain
This is again a contradiction. It follows that
Therefore, . Continuing in this process, one can prove inductively that is a strictly non-increasing sequence in X. ☐
Theorem 2. Let be a complete metric space and be an α-F-convex contraction satisfying the following conditions:
- (i)
T is α-admissible;
- (ii)
there exists such that
- (iii)
T is continuous or, orbitally continuous on X.
Then, T has a fixed point in X. Further, if T is -admissible, then T has a unique fixed point . Moreover, for any if for all , then .
Proof. Let be such that and define a sequence by for all . If , i.e., for some , then is a fixed point of T.
Now, we assume that for all . Then, for all . Since T is -admissible, implies that . Therefore, one can obtain inductively that for all . Setting .
Now from (
2), taking
and
, where
, we obtain
Since
F is strictly increasing and
T is
-admissible, by (
3), we obtain
If
, then we obtain
This is a contradiction. Therefore,
By Lemma 1, we obtain:
whenever
or
for
.
Therefore, by
with (
8), we obtain
From
, there exists
such that
Also, from (
7), we obtain
where
or
for
.
Letting
in (
11) together with (
9) and (
10), we obtain
Now, it arises in the following cases.
Case-(i): If
n is even and
, then from (
12), we obtain
Case-(ii): If
n is odd and
, then from (
12), we obtain
It may be observed from the above cases that, there exists
such that
Now, we show that
is a Cauchy sequence. For all
, we obtain
Since
is convergent, taking
, we get
. This shows that
is a Cauchy sequence in
X. Since
X is complete, there exists
such that
. Now we prove that
z is a fixed point of
T. Suppose
T is continuous, then
This shows that z is a fixed point of T.
Again, we suppose that T is orbitally continuous on X, then as . Since is complete this implies that . Therefore, .
Further, we suppose that
T is
-admissible, it follows that for all
, we have
. From (
2) and (
3), we obtain
Since
and using
F is strictly increasing, we obtain
This is a contradiction. Therefore, T has a unique fixed point in X. ☐
One can verify the validity of Theorem 1 with Examples 7 and 8 with proper choose of and p.
Corollary 1. Let be a complete metric space and be a function. Suppose that be a self mapping satisfying the following conditions:
- (i)
for all where ; - (ii)
T is α-admissible;
- (iii)
there exists such that ;
- (iv)
T is continuous or, orbitally continuous on X.
Then, T has a fixed point in X. Further, if T is an -admissible, then T has a unique fixed point . Moreover, for any if for all , then .
Proof. Setting
. Obviously,
. Taking natural logarithm on both sides of (
16), we obtain
which implies that
for all
with
where
. This shows that
T is
-
F-convex contraction with
. Thus, all the conditions of Theorem 2 are satisfied and hence,
T has a unique fixed point in
X. ☐
The following is proved in Khan et al. ([
10], [Theorem 2.2]) by extending convex contraction of type-2 (Istrǎţescu [
4]) to generalized convex contraction of type-2 in the setting of
b-metric space .
Corollary 2. Let be a complete metric space and be a function. Suppose that be a self mapping satisfying the following conditions:
- (i)
for all where , such that ; - (ii)
T is α-admissible;
- (iii)
there exists such that ;
- (iv)
T is continuous or, orbitally continuous on X.
Then, T has a fixed point in X. Further, if T is -admissible, then T has a unique fixed point . Moreover, for any if for all , then .
Proof. Setting
. Obviously,
. For all
with
, we obtain
where
. Therefore, by above Corollary 1,
T has a unique fixed point in
X. ☐
Corollary 3. Let T be a continuous mapping on a complete metric space into itself. If there exists satisfying the following inequalityfor all , then T has a unique fixed point in X. 4. Application
In this section, we apply our result to establish an existence theorem forthe non-linear Fredholm integral equation and give a numerical example to validate the application of our obtained result.
Let be a set of all real continuous functions on equipped with metric for all . Then, is a complete metric space.
Now, we consider the non-linear Fredholm integral equation
where
. Assume that
and
continuous, where
is a given function in
X.
Theorem 3. Suppose that is a metric space equipped with metric for all and be a continuous operator on X defined by If there exists such that for all with and satisfying the following inequalitythen the integral operator defined by (18) has a unique solution and for each , for all , we have . Proof. We define
such that
for all
. Therefore,
T is
-admissible. Setting
such that
,
. Let
and define a sequence
in
X by
for all
. By (18), we obtain
We show that
T is
-
F-convex contraction on
. Using (18) and (19), we obtain
Taking maximum on both sides for all
, we obtain
Now, taking natural logarithm on both sides, we obtain
Thus, we obtain
where
. This shows that
T is
-
F-convex contraction with
for all
with
. Since
T is
-admissible and
is complete metric space. Therefore, the iteration scheme
converges to some point
i.e.,
. By continuity of
T, one can prove that
T has a fixed point, i.e.,
. Consequently,
. Also, for all
,
follows that
T is
-admissible. Thus, all the conditions of Theorem 2 are satisfied and hence, the integral operator
T defined by (
18) has a unique solution
. ☐
The following example shows the existence of unique solution of an integral operator satisfying all the hypothesis in Theorem 3, however, one can also check that the following example does not satisfy F-contraction.
Example 9. Let be a set of all continuous functions defined on equipped with metric for all . Let be the operator defined by Letting and . Then, (21) becomes Then, (i) and are continuous,
(ii) for all ,
(iii) for all ,
(iv) For all with and and using (21) and (22), we obtain Taking maximum on both sides for all , we obtain From the above, one can verify that the integral operator T is not an F-contraction. Also, we have Taking maximum on both sides for all , we obtainwhere Setting by for all and such that , . Therefore, we obtain Taking natural logarithm on both sides, we obtainthat is, This shows that T is α-F-convex contraction with for all . Thus, all the conditions of Theorem 3 are satisfied and therefore, the integral Equation (21) has a unique solution. One can verify that is the exact solution of the equation (21). Using the iteration scheme (20), (23) becomes Letting be an initial solution. Putting , successively in (24), we obtain Therefore, is the unique solution.