Abstract
Edge theoretic extended contractions are introduced and coincidence point theorems and common fixed-point theorems are proved for such contraction mappings in a metric space endowed with a graph. As further applications, we have proved the existence of a solution of a nonlinear integral equation of Volterra type and given a suitable example in support of our result.
1. Introduction and Preliminaries
The celebrated Banach contraction principle is a motivation for many fixed-point theorems. It guarantees the existence and uniqueness of solution of various equations arising in mathematics. The initial generalizations of Banach’s result came up in the form of Kannan’s contraction, Chatterjea’s contraction, Reich’s contraction, Ciric’s contraction, Hardy-Roger’s contraction, and Ciric’s quasicontraction. Among these, Ciric’s quasicontraction is the most general form in the sense that any mapping which does not satisfy Ciric’s quasicontraction does not satisfy any of the previously mentioned contractions. Further, these results have been widely investigated and many interesting applications have been found by many authors (see [1–7]). -contraction and fixed-point theorem for -contraction mappings were introduced by Wardowski [8] as a generalisation of the Banach contraction principle.
Definition 1 (see [8]). Consider the collection of functions satisfying the following:
is strictly increasing
If is a sequence, then iff
There exists such that
An operator is an -contraction if we can find such that
Later, the concept of -weak contraction and ordered -contractions was introduced by Wardowski and Van Dung [9] and Durmaz et al. [10], respectively. In , Sawangsup et al. [11] extended the -contraction using a relation theoretic approach which was later generalised by Imdad et al. [12] and Alfaqih et al. [13]. Espinola and Kirk [14] introduced graph theory in fixed-point theory, and Jachymski [15] continued this idea by using different views thereby introducing the -contraction and proved fixed-point theorem for a -contraction mapping. These ideas were further extended and generalised by [16–24].
It is interesting to note that all these contraction conditions ensure the existence of a unique fixed point or common fixed point of the mappings under consideration. However, it is observed that a mapping which possesses nonunique fixed points does not satisfy the above contractions, for if and are any two fixed points of a self-map of a metric space , then and thus, we see that does not satisfy Ciric’s quasicontraction, Wardowski’s -contraction, and Wardowski and Van Dung’s -weak contraction. Thus, these contraction conditions cannot be used to prove the existence of nonunique fixed points of a function defined in a metric space. On the other hand, many equations obtained by modeling various problems of engineering and science need not necessarily have a unique solution. Thus, it becomes meaningful to obtain extended forms of above contractions which will ensure the existence of nonunique fixed points of self-maps defined in a metric space.
Motivated by this fact, in this paper, we have introduced extended -contraction (Jungck-Wardowski contraction), extended -contraction (Ciric-Wardowski contraction), and extended -contraction (Ciric-Wardowski quasicontraction) and established fixed-point theorems which will ensure the existence of nonunique fixed points of a self-map and coincidence points of a pair of self-maps, respectively, in a metric space endowed with a graph. As an application of our result, we have also proven the existence of solution of a nonlinear integral equation of Volterra type.
Throughout this paper, we consider the metric space to be endowed with the graph , , and ; .
Definition 2 (see [15]). A sequence is edge-preserving if for all .
Definition 3. Let . A sequence is -edge-preserving if for all .
Definition 4. is edge-preserving if implies .
Definition 5. is -edge-preserving if for all , implies .
Definition 6 (see [15]). is edge-complete if every edge-preserving Cauchy sequence in converges to some point in .
Definition 7 (see [15]). is edge-continuous at if implies for any edge-preserving sequence . If is edge-continuous at all , then is an edge-continuous mapping.
Definition 8. Let and . We say that is -edge continuous at if implies for any edge-preserving sequence . If is -edge continuous at all , then is an -edge continuous mapping.
Definition 9. is edge-compatible if and only if for any sequence and edge-preserving sequence , implies
We will use the following lemmas taken from [25, 26]:
Lemma 10. (see [25]). Let be a nonempty set and . Then, there exists a subset such that and is one-one.
Lemma 11 (see [26]). Let be a sequence in metric space such that If is not Cauchy in , then there exist and sequences and in such that and the sequences tend to be as
2. Edge Theoretic Extended Contractions
Let be the collection of all nondecreasing continuous functions .
Example 1. Some examples of function belonging to the class are Let and be the collection of all continuous functions satisfying the following: (i) or implies (ii) and implies
Some examples of function are as follows:
Example 2. (i), for some (ii)(iii)(iv)(v)(vi)(vii)Let be the family of all functions which satisfy the following conditions:
is strictly increasing
iff
for some
Example 3. Some examples of elements of are
Definition 12. A pair of mappings is an -extended -contraction pair if we can find , , , and such that for all ,
Definition 13. A pair of mappings is an -extended -contraction pair if we can find , , , and such that for all , where
Definition 14. A pair of mappings is an -extended -contraction pair provided that there is a , , , and such that for all , where
Definition 15. In Definitions 12, 13, and 14, if conditions (7), (8), and (10) are satisfied only for all with , then the pair is an -extended -edge contraction, -extended -edge contraction, and -extended -edge contraction, respectively.
Definition 16. is a -extended -edge contraction if we can find , , and such that for all with ,
Definition 17. A pair of mappings is a -extended -edge contraction if we can find , , and such that for all with and , is as in (9).
If in the above definitions, then is an -extended -contraction mapping, -extended -contraction mapping, -extended -edge contraction mapping, and -extended -edge contraction mapping, respectively.
Property (). The space is said to have if for any edge-preserving sequence such that ; there exists a subsequence of such that for all
Example 4. Let , , and for all . Then, at and , does not satisfy the conditions of Ciric’s quasicontraction, Wardowski’s -contraction, and Wardowski and Van Dung’s - weak contraction. However, is an -extended -contraction with , as shown below:
Let be defined by
and .
Case 1. . Clearly, Then, we have or
Case 2. and . Note that in this case, .
Example 5. Let , , , and be given by Let be defined by and be defined by . Then,
Hence, for any and , (13) is satisfied and thus is a -extended -edge contraction and -extended -edge contraction. However, the pair () is neither an -extended -edge contraction pair nor an -extended -contraction pair. If we take to be the identity mapping, then is a -extended -edge contraction mapping and -extended -edge contraction mapping. However, again is none of Wardowski’s -contraction, Wardowski and Van Dung’s - weak contraction, and Ciric’s quasicontraction.
3. Main Results
We start by proving the following main theorems:
Theorem 18. Suppose be endowed with a graph satisfying transitivity property, and the following conditions hold for .
for some
is -edge preserving
is an -extended -edge contraction pair of mappings
There exists an edge-complete subset of for which
One of the following conditions holds:
(i) is -edge continuous(ii) and are continuous(iii) satisfies Then, the pair has a coincidence point.
Proof. In view of the assumption (a), we have . If , then is a coincidence point of , i.e., , and there is nothing to prove. Assume that ; then, since , there exists such that .
Similarly, there is such that with and consequently . Inductively, one can construct a sequence such that with
and consequently, as is -edge preserving,
Now, if for some , then is a coincidence point and we are done. Assume that , for all . On using (21), (22), (23), and condition , we have
Now,
Thus, we get
i.e.,
Since is nondecreasing, we get . This further means that as If , we obtain from (27) that which is a contradiction. Hence, Suppose the sequence is not a Cauchy sequence. By Lemma 11, there exist and sequences and in such that such that the sequences and tend to be as By (27) we get
which is a contradiction. So sequence is a Cauchy sequence.
By (21) and (22), is an edge-preserving Cauchy sequence in , and since is edge-complete, there exists such that . As , there exists such that . Hence, on using (21), we obtain
Now, suppose condition is true. Using (22) and (30), we obtain
Suppose condition () is true. By Lemma 10, there is for which and is one-one. Consider the function given by
As is one-one and , is well-defined. Since and are continuous, is also continuous by condition of the hypothesis . Thus, we have and . Therefore,
Suppose condition () is true; that is, satisfied . Since , it follows that is -preserving (due to (22)) and (by (30)) and so we have a subsequence such that
Using (35) and condition of the hypothesis, we have
Now, let .
If is finite, then has a subsequence such that for all . Also, . Thus, we have
Letting , we obtain and . Thus, we get
which is a contradiction. Hence, is not finite. Thus, is infinite and so has a subsequence such that for all . Thus, . As (by (30)), we get .
Theorem 19. If, in addition to hypothesis - of Theorem 18, we assume the following: (i)For all ,(ii)One of or is one-one(iii) and are weakly compatiblethen has a unique common fixed point.
Proof. In view of Theorem 18, the set is nonempty. Let . If , then we have , and hence, as one of and is one-one. Otherwise, using condition (39), we obtain
which is a contradiction. So the coincidence point of and is unique.
Let be the unique coincidence point of and , and let such that . As and are weakly compatible, we have . Thus, is a coincidence point of and . By the uniqueness of the coincidence point, we conclude ; that is, is a common fixed point of the pair which is indeed unique. as the coincidence point of and is unique.
Remark 20. If we replace condition of Theorem 18 with the following alternate condition:
There exists a subset of such that and is edge-complete
is an edge-compatible pair
and are edge-continuous
the conclusions of Theorems 18 and 19 still hold.
Proof. Clearly, is an edge-preserving Cauchy sequence in , and by edge-completeness of , we get such that and then, by (21), we have Using the edge continuity of and , we also have Then, by edge-compatibility of and , we get Finally from (44), (45), and (43), we get Hence, is a coincidence point of the pair .
Remark 21. Since every -extended contraction mapping is a -extended contraction, the conclusions of Theorems 18 and 19 remain true for an edge theoretic -extended -contraction pair of mappings also.
On setting in Theorem 18, we deduce the following corresponding fixed-point result.
Theorem 22. Let be a metric space endowed with a directed graph and . Assume that the following conditions are fulfilled: (a)There exists such that (b) is edge-preserving(c) is a -extended -edge contraction mapping(d) There exists a subset of such that and is edge-complete One of the following conditions is satisfied: (i) is edge-continuous(ii) satisfies Then, has a fixed point.
Example 6. Let , , , and be as in Example 5. Then, we have the following: (1)(2) is -edge-preserving. In fact, we see that implies either or . If , then and . If , then and . If , then and (3) is a -extended -edge contraction mapping(4)(5) is -edge-continuousThus, all conditions of Theorem 18 are satisfied and is a coincidence point of and . Moreover, we see that and satisfy conditions (i), (ii) ( is one-one), and (iii) of Theorem 19, and is the unique common fixed point of and .
Remark 23 (an open problem). Prove Theorems 18, 19, and 22 for -extended -contraction mappings.
4. Application to Nonlinear Integral Equations
Consider the Banach space of all continuous functions equipped with norm
Define a metric on by for all . Then, is a complete metric space.
In this section, we show the applicability of Theorem 19 by investigating the existence and uniqueness of a solution for the following nonlinear integral equation of Volterra type: where , , and .
Definition 24. A lower solution for (48) is a function such that
Definition 25. An upper solution for (48) is a function such that Consider the operator defined by Then, is a fixed point of the operator if and only if it is a solution of the integral equation (48).
Let
Theorem 26. Assume that and are nondecreasing in the third variable, for all , and the following conditions hold:
There exists such that
for all , with and . If (48) has a lower solution, e.g., , then a solution exists for the integral equation (48).
Proof. Consider the graph in , with edges given by For any , we have (for all ) which shows that . Thus, is edge-preserving. Now, for all and , we have Taking the supremum, we get or or That is, Thus, inequality (13) is satisfied with and , so that . Also, by Definition 24, we have . Therefore, all the assumptions of Theorem 22 are satisfied, and thus, problem (48) has a solution.
Theorem 27. Assume that is nonincreasing in the third variable and there exists such that for all and . Then, the existence of an upper solution of the integral equation (48) ensures the existence of a solution of (48).
Proof. Define set of edges on by Now, following the steps of the proof of Theorem 26 with an analogous procedure, one can check that all the hypotheses of Theorem 22 are validated, and thus, Theorem 22 ensures the existence of a unique solution of the integral equation (48).
We now furnish a numerical example to validate the hypothesis of Theorem 27.
Example 7. Consider the function defined by . We show that this function is an upper solution in for the following integral equation:
Finally, we see that is the unique solution of (63).
Proof. Define the operator as Now, set , , , , and . We observe the following: (i)Both the functions and are nondecreasing in the third variable(ii)By actual computation, we have(iii) so that is an upper solution for (63)(iv)The following inequalities hold true for all (see Figures 1 and 2):
Furthermore, using the nondecreasing function , we have
Similarly, for all , we have
Hence, all the conditions of Theorem 27 are satisfied. It is evident that the integral equation (63) has a unique solution defined by .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.