#### Abstract

We define some notions of contraction mappings in -metric space endowed with a graph and subsequently establish some fixed point results for such classes of contractions. According to the applications of our results, we obtain fixed point theorems for cyclic operators and an existence theorem for the solution of an integral equation.

#### 1. Introduction and Preliminaries

The study of -metric spaces was initiated in some works of Bakhtin, Heinonen, Bourbaki, and Czerwik [1–4]. Afterwards, several articles which deal with fixed point theorems for single-valued and multivalued functions in -metric space appeared [1–8].

*Definition 1 (see [1, 4]). *Let be a set, and let be a given real number. A function is said to be a -metric on , and the pair is called a -metric space if, for all , (d1) if and only if , (d2), (d3).

Note that the class of -metric spaces contains the class of metric spaces.

*Example 2. *Let with , where . Then, is a -metric on with .

A sequence in a -metric space is said to be convergent if and only if there exists such that as . In this case, we write . A sequence in a -metric space is said to be Cauchy if and only if as . A -metric space is complete if every Cauchy sequence in converges. In general, a -metric is not continuous.

The famous Banach contraction principle [9] infers that every contraction on a complete metric space has a unique fixed point. Recently, Jachymski [10] introduced the notion of Banach -contraction to generalize Banach contraction principle as follows. Let be a metric space, let be the diagonal of the Cartesian product , and let be a directed graph such that the set of its vertices coincides with and the set of its edges contains all loops; that is, . Assume that has no parallel edges. A mapping is called a Banach -contraction if (i)* * for all , (ii) , such that for all . A mapping is known as Picard operator [11] if has a unique fixed point and for all .

Various generalizations of Banach's principle have been obtained by weakening contractive conditions. In this context, Matkowski [12] introduced class of -contractions in metric fixed point theory, and subsequently further study was developed in this setting by different authors when underlying space was taken to be a partially ordered set (see, e.g., [13, 14]).

Let . Consider the following properties:, for all , for ,, for all , converges for all . It is easily seen that and imply and and imply .

We recall that a function satisfying and is said to be a comparison function. A function satisfying and is known as -comparison function.

Any -comparison function is a comparison function, but converse may not be true. For example, ; is a comparison function but not a -comparison function. On the other hand, define ; and ; , and then is a -comparison function. For details on contractions, we refer the readers to [15, 16].

Berinde [17] took further step to investigate contractions when the framework was taken to be a -metric space, and for some technical reasons, he had to introduce the notion of -comparison function; in particular, he obtained some estimations for rate of convergence [17]. For other related results, see also [5, 7, 17–21].

*Definition 3. *Let be a fixed real number. A function is known as -comparison function if it satisfies , and the following holds: converges for all .

The concept of -comparison function coincides with comparison function when . Let be a -metric space with coefficient , and then ; with is a -comparison function.

#### 2. Main Results

Throughout this section, let be a -metric space with coefficient , and is the diagonal of the Cartesian product . is a directed graph such that the set of its vertices coincides with , and the set of its edges contains all loops; that is, . Assume that has no parallel edges. We assign to each edge having vertices and a unique element . Now, we introduce the following definition.

*Definition 4. *One says that a mapping is a - contraction if for all :
where is a comparison function.

*Remark 5. *Note that a Banach -contraction is a - contraction.

*Example 6. *Any constant mapping is a - contraction for any graph with .

*Example 7. *Any self-mapping on is trivially a - contraction, where .

*Example 8. *Let , and define by . Then, is a -metric on with . Further, , for all . Then, is a - contraction with and . Note that is not a metric on .

*Definition 9. *Two sequences and in are said to be equivalent if , and if each of them is a Cauchy sequence, then they are called Cauchy equivalent.

As a consequence of Definition 9, we get the following lemma.

*Remark 10. *Let and be equivalent sequences in . If converges to , then also converges to and vice versa. is a Cauchy sequence whenever is a Cauchy sequence and vice versa.

Now, we recollect some preliminaries from graph theory which we need for the sequel. Let , and let be a directed graph. By letter , we denote the undirected graph obtained from by ignoring the direction of edges. If and are vertices in a graph , then a path in from to of length is a sequence of vertices such that , and for . A graph is called connected if there is a path between any two vertices. is weakly connected if is connected. For a graph such that is symmetric and is a vertex in , the subgraph consisting of all edges and vertices which are contained in some path beginning at is known as a component of containing . So that , where is the equivalence class of a relation defined on by the rule: if there is a path in from to . Clearly, is connected. A graph is known as -graph in [22] if for any sequence in with and for ; then there exists a subsequence of such that for .

Proposition 11. *Let be a - contraction, where is a comparison function; then *(i)* is a **-** contraction and a **-** contraction as well,*(ii)* is **-invariant, and ** is a **-** contraction provided that ** is such that **. *

*Proof. *(i) By using (d2) (Definition 1) it can be easily proved.

(ii) Let . Then, there is a path between and . Since is a - contraction, then for all . Thus, .

Suppose . Then, , since is a - contraction. But is invariant, so we conclude that . Condition (2) is satisfied automatically as is a subgraph of .

From now on, we assume that coefficient of -comparison function is at least as large as the coefficient of -metric .

Lemma 12. *Let be a - contraction, where is a -comparison function. Then, given any and , two sequences and are equivalent.*

*Proof. *Let , and let ; then there exists a path in from to with , , and . From Proposition 11, is a - contraction. So,
for all and . Hence,
We observe that is a path in from to . Using (d3) Definition 1 and (4), we have
Letting , we obtain .

Proposition 13. *Let be a - contraction, where is a -comparison function. Suppose that there is in such that . Then, is a Cauchy sequence in .*

*Proof. *Since , let be a path from to . Then using the same arguments as in Lemma 12, we arrive at
Let , and then from above inequality; it follows for
Denoting for each
relation (7) becomes
since is a -comparison function, so that for each ,
Then, corresponding to each , there is a real number such that
In view of (11), relation (9) gives as . Which shows that is a Cauchy sequence in .

*Definition 14. *Let , and let , and the sequence in is such that with for . One says that a graph is -graph if there exists a subsequence and a natural number such that for all [23]. One says that a graph is -graph if for ; then , where ; is a path from to in .

Obviously every -graph is a -graph for any self-mapping on , but converse may not hold as shown in the following.

*Example 15. *Let with respect to -metric . Consider a graph consisting of and , *∪*, , , . Note that is not a -graph as . Define as . Then, is a -graph, since .

*Example 16. *Let with respect to -metric , and let be identity map on . Consider a graph consisting of and
since as . We note that is a -graph, but as . Thus, is not an -graph.

*Example 17. *Let with respect to -metric , and let be identity map on . Consider a graph consisting of and
since as . Clearly is not a -graph, but it is easy to verify that is an -graph.

Above examples show that for a given notions, -graph and -graph are independent even if is identity map.

Theorem 18. *Let be a complete -metric space, and let be a - contraction, where is -comparison function. Assume that is continuous and there is in for which is an edge in . Then, the following assertions hold. *(1)*If is a -graph, then has a unique fixed point , and for any , . Further, if is weakly connected, then is Picard operator. *(2)*If is weakly connected -graph, then has a unique fixed point and for any , . *

*Proof. *(1) It follows from Proposition 13 that is a Cauchy sequence in . Since is complete, there exists such that . Since , for all , and is a graphs, there exists a subsequence of and such that for all . Observe that is a path in . Therefore, . From (2), we get
Letting , we obtain , as is continuous. Since is a subsequence of , we conclude that . Finally, if , it follows from Lemma 12 that .

(2) Let be weakly connected -graph. From Proposition 13, , and then as . Now, for each , let be a path from to with and in ; then
Letting , the above inequality yields . Let be arbitrary; then from Lemma 12 and Remark 10, it is easily seen that .

The following example shows that the condition of or -graph in the hypothesis of Theorem 18 can not be dropped.

*Example 19. *Let , let , and let for all and . Then, is a complete -metric space with . Further, is continuous, and is a - contraction (with ), where and ; . Note that is weakly connected, but has no fixed point in . Observe that is not a -graph because the sequence for and , but it does not contain any subsequence such that . Also, we note that for any fixed , as .

*Definition 20. *Let be a -metric space. A mapping is called orbitally continuous if for all and any sequence of positive integers, implies as . A mapping is called orbitally -continuous if for all and any sequence of positive integers, and for all imply .

Theorem 21. *Let be a complete -metric space, and let be a - contraction, where is a -comparison function. Assume that is continuous, is orbitally -continuous, and there is in for which is an edge in . Then, has a fixed point . Moreover, for any , .*

*Proof. *It follows from Proposition 13 that is a Cauchy sequence in . Since is complete, there exists such that . Since for all , is orbitally -continuous. Therefore, continuity of implies that . Let be arbitrary; then it follows from Lemma 12 that .

Slightly strengthening the continuity condition on , our next theorem deals with the graph which may fail to have the property that there is in for which is an edge in .

Theorem 22. *Let be a complete -metric space, and let be a - contraction, where is a -comparison function. Assume that is continuous, is orbitally continuous, and there is in for which . Then, for any , , where is a fixed point of .*

*Proof. *It follows from Proposition 13 that is a Cauchy sequence in . Since is complete, there exists such that . Since is orbitally continuous, then which yields . Let be arbitrary; then from Lemma 12, .

*Remark 23. *In addition to the hypothesis of Theorems 21 and 22, if we assume that is weakly connected, then will become Picard operator [11] on .

*Remark 24. *Theorem 18 generalizes/extends claims and of [10, Theorem 3.2] and [7, Theorem ]. Theorem 21 generalizes claims and of [10, Theorem 3.3]. Theorem 22 generalizes claims and of [10, Theorem 3.4], and thus generalizes extends results of Nieto and Rodríguez-López [24, Theorems 2.1 and 2.3], Petrusel and Rus [11, Theorem 4.3], and Ran and Reurings [25, Theorem 2.1]. We mention here that Theorem 18 can not be improved using comparison function instead of -comparison function (see, Gwóźdź-Łukawska and Jachymski [26, Example 2]).

We observe that Theorem 22 can be used to extend famous fixed point theorem of Edelstein to the case of -metric space. We need to define notion of -chainable property for -metric space.

*Definition 25. *A -metric space is said to be -chainable, for some , if for there exist with such that for .

Corollary 26. *Let be a complete -chainable -metric space. Assume that is continuous, and there exists a -comparison function such that satisfying,
**
for all . Then, is a Picard operator.*

*Proof. *Consider a graph consisting of and if and only if . Since is -chainable, is weakly connected. Let , and from (16), we have
Then, . Therefore, in view of (16), is - contraction. Further, (16) implies that is continuous. Now, the conclusion follows by using Theorem 22.

Now, we establish a fixed point theorem using a general contractive condition.

Theorem 27. *Let be a complete -metric space, and let be a -graph in such that and is an edge-preserving mapping. Assume that is continuous and there exist with and for all **
If there is in for which is an edge in , then has a fixed point in .*

*Proof. *Since is edge-preserving, then for all . From (18) and using (d3), it follows that
On rearranging,
Repeating iteratively, we have
For and using (d3) Definition 1, we have
Since , then is a Cauchy sequence in . By completeness of , the sequence converges to . Since is a -graph, there exists a subsequence and a natural number such that for all . Using (18) for all , we have
Since the -metric is continuous and , letting inequality (23) yields . Also note that is a path in and hence in , therefore .

We note that Theorem 27 does not guarantee the uniqueness of fixed point, but this can be accomplished under some assumptions as in the following theorem.

Theorem 28. *In addition to the hypothesis of Theorem 27, one further assumes that if for the same set of and for any two fixed points such that and . Then, has a unique fixed point.*

*Proof. *Let and be two fixed points of , and then there exists such that . By induction, we have for all . From (18), we have
On rearranging,
Continuing recursively, (25) gives

Since , then . Similarly, one can show that , which by using (d3) Definition 1 infers that .

Suppose that is a partially ordered set. Consider graph consisting of , and coincides with . We note that if a self-mapping is monotone with respect to the order, then, for graph , it is obvious that is edge-preserving, or equivalently we can say that maps comparable elements onto comparable elements.

Following corollaries are the direct consequences of Theorem 28.

Corollary 29. *Let be a complete metric space, where is partially ordered set with respect to . Let be nondecreasing (or nonincreasing) with respect to . Assume that there exists with such that,
**
for all comparable . If the following conditions hold: *(i)*there exists such that , *(ii)*for nondecreasing (or nonincreasing) sequence , there exists a subsequence such that , for all .**Then, has a fixed point. Moreover, if for all there exists such that and , then the fixed point is unique.*

Corollary 30. *Let be a complete metric space, where is partially ordered set with respect to . Let be nondecreasing (or nonincreasing) with respect to . Assume that there exists a constant such that
**
for all comparable . If the following conditions hold: *(i)*there exists such that , *(ii)*for nondecreasing (or nonincreasing) sequence , there exists a subsequence such that , for all .**Then, has a fixed point. Moreover, if for all there exists such that and , then the fixed point is unique.*

Corollary 31. *Let be a complete metric space, where is partially ordered set with respect to . Let be nondecreasing (or nonincreasing) with respect to . Assume that there exists a constant such that
**
for all comparable . If the following conditions hold: *(i)*there exists such that , *(ii)*for nondecreasing (or nonincreasing) sequence , there exists a subsequence such that , for all .**Then, has a fixed point. Moreover, if for all there exists such that and , then the fixed point is unique.*

*Remark 32. *We note that in Theorem 27, the condition “there is in for which is an edge in ” yields , where is a fixed point of . Consider a graph . For such graph under the assumptions of Theorems 27 and 28, it infers that is a Picard operator. Thus, many standard fixed point theorems can be easily deduced from Theorem 28 as follows.

Corollary 33 (Hardy and Rogers [27]). *Let be a complete metric space, and let . Suppose that there exists constants such that
**
for all , where ; then has a unique fixed point in .*

Corollary 34 (Kannan [28]). *Let be a complete metric space, and let . Suppose that there exists a constants such that
**
for all , where ; then is Picard operator.*

Corollary 35 (Chatterjea [29]). *Let be a complete metric space, and let . Suppose that there exists a constants such that
**
for all , where ; then is Picard operator.*

#### 3. Applications

Let be a nonempty set, let be a positive integer, be nonempty closed subsets of , and let be an operator. Then, is known as cyclic representation of w.r.t. if and operator is known as cyclic operator [30].

Theorem 36. *Let be complete -metric space such that is continuous functional on . Let be positive integer, let be nonempty closed subsets of , let , be a -comparison function, and let . Further, suppose that *(i)* is cyclic representation of w.r.t. ,*(ii)* whenever , where .**Then, has a unique fixed point and for any .*

*Proof. *We note that is complete -metric space. Let us consider a graph consisting of and . By (i), it follows that preserves edges. Now, let in such that for all ; then in view of (33), sequence has infinitely many terms in each so that one can easily extract a subsequence of converging to in each . Since 's are closed, then . Now, it is easy to form a subsequence in some , such that for , and it indicates that is weakly connected -graph, and thus conclusion follows from Theorem 18.

*Remark 37 (see [31, Theorem 2.1(1)]). *It can be deduced from Theorem 36 if is a metric space.

On the same lines as in proof of Theorem 36, we obtain the following consequence of Theorem 28.

Theorem 38. *Let be a complete -metric space such that is continuous functional on . Let be positive integer, let be nonempty closed subsets of , let , and let . Further, suppose that *(i)* is cyclic representation of w.r.t. ,*(ii)*there exist with such that
**
whenever , where .**Then, has a fixed point .*

*Remark 39. *[32, Theorem 7] and [33, Theorem 3.1] can be deduced from Theorem 38, but it does not ensure to be a Picard operator.

*Remark 40. *We note that in proof [32, Theorem 7], the author's argument to prove that is (as assumed in proof of Theorem 36), a -graph is valid only if the terms of sequence are Picard iterations; otherwise it is void. For example, let where , , , and define as and . We see that (33) is satisfied and , but does not contain infinitely many terms of . In the following, we give the corrected argument to prove that is a -graph.

Let in such that for all . Keeping in mind construction of , there exists at least one pair of closed sets for some such that both sets contain infinitely many terms of sequence , since 's are closed so that for some , and thus one can easily extract a subsequence such that holds for .

Now, we establish an existence theorem for the solution of an integral equation as a consequence of our Theorem 18.

Theorem 41. *Consider the integral equation
**
where and is continuous. Assume that*(i)* is nondecreasing for each ,*(ii)*there exists a -comparison function and a continuous function × such that for each and (i.e., for all ),*(iii)*,*(iv)*there exists such that for all . **Then, the integral equation (35) has a unique solution in the set or , for all .*

*Proof. *Let , where , and define a mapping by
Consider a graph consisting of and for all . From property , we observe that the mapping is nondecreasing, thus preserves edges. Furthermore, is a -graph; that is, for every nondecreasing sequence which converges to some ; then for all . Now, for every with , we have
Hence, . From , we have , so that or for all . The conclusion follows from Theorem 18.

Note that Theorem 41 specifies region of solution which invokes the novelty of our result.