Abstract
We define the notion of an integral G-contraction for mappings on metric spaces and establish some fixed point theorems for such mappings. Our results generalize and unify some recent results by Jachymski, Branciari and those contained therein. As an application, we obtain a result for cyclic operators. Moreover, we provide an example to show that our results are substantial improvements of some known results in literature.
MSC:47H10, 54H25.
Similar content being viewed by others
1 Introduction
Branciari [1] generalized the Banach contraction principle by proving the existence of a unique fixed point of a mapping on a complete metric space satisfying a general contractive condition of integral type. Afterwards, many authors undertook further investigations in this direction (see, e.g., [2–6]). Ran and Reurings [7] initiated the study of fixed points of mappings on partially ordered metric spaces. A number of interesting fixed point theorems have been obtained by different authors for this setting; see, for example, [7–11]. Jachymski [12] used the platform of graph theory instead of partial ordering and unified the results given by authors [7, 8, 11]. He showed that a mapping on a complete metric space still has a fixed point provided the mapping satisfies the contraction condition for pairs of points which form edges in the graph. Subsequently, Beg et al. [13] established a multivalued version of the main result of Jachymski [12]. Aydi et al. [14] studied fixed point theorems for weakly G-contraction mappings in G-metric spaces. Later on, Bojor [15] obtained some results in such settings by weakening the condition of Banach G-contractivity and introducing some new type of connectivity of a graph.
In this paper, motivated by the work of Jachymski [12] and Branciari [1], we introduce two new contraction conditions for mappings on complete metric spaces and, using these contractive conditions, obtain some fixed point theorems. Our results generalize and unify some results by the above mentioned authors.
2 Preliminaries
Let be a partially ordered set. A mapping is said to be nonincreasing if , . A mapping f is said to be nondecreasing if , . A mapping f from a metric space into is called a Picard operator (PO) [11] if f has a unique fixed point and for all . Two sequences and in a metric space are said to be equivalent if . Moreover, if each of them is Cauchy, then these are called Cauchy equivalent. A mapping from a metric space into is called orbitally continuous if for all and any sequence of positive integers, implies as .
Let be a directed graph. By we denote the graph obtained from G by reversing the direction of edges, and by letter we denote the undirected graph obtained from G by ignoring the direction of edges. It will be more convenient to treat as a directed graph for which the set of its edges is symmetric, i.e., . If x and y are vertices in a graph G, then a path in G from x to y of length l is a sequence of vertices such that , and for . A graph G is called connected if there is a path between any two vertices. G is weakly connected if is connected. For a graph G such that is symmetric and x is a vertex in G, the subgraph consisting of all edges and vertices which are contained in some path beginning at x is called component of G containing x. In this case , where is the equivalence class of a relation R defined on by the rule: if there is a path in G from y to z. Clearly, is connected. A graph G is known as a -graph in X [16] if for any sequence in X with and for , there exists a subsequence of such that for .
Subsequently, in this paper, X is a complete metric space with metric d, and Δ is the diagonal of the Cartesian product . G is a directed graph such that the set of its vertices coincides with X, and the set of its edges contains all loops, i.e., . Assume that G has no parallel edges. We may treat G as a weighted graph by assigning to each edge the distance between its vertices. A mapping is called orbitally G-continuous [12] if for all and any sequence of positive integers, and , imply .
We state, for convenience, the following definition and result.
Definition 2.1 [[12], Definition 2.1]
A mapping is called a Banach G-contraction or simply a G-contraction if f preserves edges of G, i.e.,
and f decreases weights of edges of G in the following way:
Let Φ denote the class of all mappings which are Lebesgue integrable, summable on each compact subset of , nonnegative and for each , .
Theorem 2.2 [[1], Theorem 2.1]
Let be a complete metric space, , and let be a mapping such that for each ,
where . Then f has a unique fixed point such that for each , .
3 Main results
We begin this section, motivated by Jachymski [12] and Branciari [1], by introducing the following definition.
Definition 3.1 A mapping is called an integral G-contraction if f preserves edges (see (2.1)) and
for some and .
Remark 3.2 Note that if satisfies (2.3), then f is an integral -contraction where . Moreover, every Banach G-contraction is an integral G-contraction (take ), but the converse may not hold.
Proposition 3.3 Let be an integral G-contraction with contraction constant and , then:
-
(i)
f is both an integral -contraction and an integral -contraction with the same contraction constant and ϕ.
-
(ii)
is f-invariant and is an integral -contraction provided that there exists some such that .
Proof (i) is a consequence of symmetry of d.
(ii) Let . Then there is a path between x and in . Since f is an integral G-contraction, then , . Thus .
Suppose that , then as f is an integral G-contraction. But is f invariant, so we conclude that . Furthermore, (3.1) is satisfied automatically because is a subgraph of G. □
Lemma 3.4 Let be an integral G-contraction and , then
Proof Let and , then there exists such that , and for all . By Proposition 3.3 it follows that and
holds for all and . Denote for all . If for some , then it follows from (3.3) that for all with . Therefore, in this case, . Now, assume that . We claim that is a non-increasing sequence. Otherwise, there exists such that . Now, using the properties of ϕ, it follows from (3.3) that
Since, , this yields a contradiction. Therefore, . Let , then it follows from (3.3) that
which implies and it further implies that , a contradiction. Thus, , , in both cases. From the triangular inequality, we have , and letting gives . □
Now we define a subclass of integral G-contractions. We call this a class of sub-integral G-contractions. Let us denote by Ω the class of all mappings satisfying following:
for every . Note that every constant function belongs to the class Ω.
Example 3.5 Define by
It is easy to see that , satisfy (3.5) and thus belong to the class Ω.
Definition 3.6 We say that an integral G-contraction is a sub-integral G-contraction if .
Lemma 3.7 Let be a sub-integral G-contraction and . Then there exists such that
Proof Let and , then there exists such that , and for all . Since , using the triangular inequality, it follows that
Moreover,
since for all and . From (3.7) and (3.8) we get
where . □
Definition 3.8 Let , and the sequence in X be such that with for . We say that the graph G is a -graph if there exists a subsequence such that for .
Obviously, every -graph is a -graph for any self-mapping f on X, but the converse may not hold as shown in the following.
Example 3.9 Let with respect to the usual metric . Consider the graph G consisting of and . Note that G is not a -graph as . Define as . Then G is a -graph since for each .
Theorem 3.10 Let be a sub-integral G-contraction. Assume that
-
(i)
,
-
(ii)
G is a -graph.
Then, for any , is a Picard operator. Further, if G is weakly connected, then f is a Picard operator.
Proof Let , then . Let , using Lemma 3.7, we have
It follows that is a Cauchy sequence in X. Therefore, . Let y be another element in , then it follows from Lemma 3.4 that , too. Next, we show that t is a fixed point of f. Since and for all and G is a -graph, then there exists a subsequence of such that for all . Therefore, is a path in G and so in from to t, thus . From (3.1), we get
letting , we have , which implies that . This shows that is a Picard operator. Moreover, if G is weakly connected, then f is a Picard operator since . □
Corollary 3.11 Let be a complete metric space endowed with a graph G such that G is -graph. Then the following statements are equivalent:
-
(1)
G is weakly connected.
-
(2)
Every sub-integral G-contraction f on X is a Picard operator provided that .
Proof : It is immediate from Theorem 3.10.
: On the contrary, suppose that G is not weakly connected, then is disconnected, i.e., there exists such that and . Let , we construct a self-mapping f by (as in [[12], Theorem 3.1]):
Let , then , which implies hence , since G contains all loops and further (3.1) is trivially satisfied (take ). But and are two fixed points of f contradicting the fact that f has a unique fixed point. □
Theorem 3.12 Let be a sub-integral G-contraction. Assume that f is orbitally G-continuous and . Then, for any and , where t is a fixed point of f. Further, if G is weakly connected, then f is a Picard operator.
Proof Let , then the arguments used in the proof of Theorem 3.10 imply that is a Cauchy sequence. Therefore, . Since for all and f is orbitally G-continuous, therefore . Note that if y is another element from , then it follows from Lemma 3.4 that . Finally, if G is weakly connected, then , which yields that f is a Picard operator. □
Remark 3.13 Theorem 3.12 generalizes claims 20 & 30 of [[12], Theorem 3.3].
Theorem 3.14 Let be a sub-integral G-contraction. Assume that f is orbitally continuous and if there exists some such that , then, for , , where t is a fixed point of f. Further, if G is weakly connected, then f is a Picard operator.
Proof Let be such that , then using the same arguments as in the proof of Theorem 3.10, is Cauchy and thus . Moreover, , since f is orbitally continuous. Note that if y is another element from , then it follows from Lemma 3.4 that . If G is weakly connected, then . This yields that f is a Picard operator. □
Remark 3.15 Theorem 3.14 generalizes claims 20 & 30 of [[12], Theorem 3.4] and thus generalizes and extends the results of Nieto and Rodrýguez-López [[8], Theorems 2.1 and 2.3], Petrusel and Rus [[11], Theorem 4.3] and Ran and Reurings [[7], Theorem 2.1].
Corollary 3.16 Let be a complete metric space endowed with a graph G. Then the following statements are equivalent:
-
(1)
G is weakly connected.
-
(2)
Every sub-integral G-contraction f on X is a Picard operator provided that f is orbitally continuous.
Proof (1) ⇒(2) is obvious from Theorem 3.14. Note that the example constructed in Corollary 3.11 is orbitally continuous. Hence, (2) ⇒ (1). □
Remark 3.17 Corollary 3.16 generalizes claims 20 & 30 of [[12], Corollary 3.3].
Kirk et al. [17] introduced cyclic representations and cyclic contractions and they have been further investigated by many authors (see, e.g., [18–20]). Let X be a nonempty set, m be a positive integer and be nonempty closed subsets of X and be an operator. Then is known as a cyclic representation of X w.r.t. f if
and the operator f is known as a cyclic operator.
Theorem 3.18 Let be a complete metric space. Let m be a positive integer, be nonempty closed subsets of and . Assume that
-
(i)
is a cyclic representation of Y w.r.t. f;
-
(ii)
there exists such that whenever, , , where .
Then f has a unique fixed point and for any .
Proof We note that is a complete metric space. Let us consider a graph G consisting of and . By (i) and (ii) it follows that f is a sub-integral G-contraction. Now let in Y such that for all . Then in view of (3.10), the 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 , it vindicates G is a weakly connected -graph and thus conclusion follows from Theorem 3.10. □
Remark 3.19 Taking , Theorem 3.18 subsumes the main result of [17].
Theorem 3.20 Let be an integral G-contraction. Assume that the following assertions hold:
-
(i)
there exists such that
(3.11)where .
-
(ii)
G is a -graph.
Then, for any , is a Picard operator. Furthermore, if G is weakly connected then f is a Picard operator.
Proof Let , then . Now, it follows from Proposition 3.3(ii) that . Moreover, from Lemma 3.4 we get
We claim that is Cauchy sequence. Otherwise, there exists some in such a way that, for each , there are with satisfying
We may choose sequences , such that corresponding to , natural number is smallest satisfying (3.13). Therefore,
On letting and using (3.12), we get
Moreover, using (3.12) and (3.14), it follows from
and
that
Since , it follows from assertion (ii) that
Letting and using (3.14), (3.15), we get . As , this implies that . Therefore, is a Cauchy sequence in X. The rest of the proof runs on the same lines as the proof of Theorem 3.10. □
Remark 3.21 Theorem 3.20 generalizes [[1], Theorem 2.1].
Remark 3.22 The conclusion of Theorem 3.20 that f is a Picard operator remains valid if we replace assertion (ii) by (ii)′ f is orbital G-continuous or (ii)″ f is orbitally continuous.
Example 3.23 Let be equipped with the usual metric d. Define , by
for all and is any fixed positive integer. Consider the graph G such that and . We observe that (2.1) holds. Moreover, , so that (3.1) is equivalent to
Next we show that (3.17) is satisfied for .
Case i. Let , then (3.1) is trivially satisfied.
Case ii. Let ; for , (3.1) is trivially satisfied. Let ; for
and
From inequality (3.17) we need to show that
or, equivalently,
Since for all and , thus inequality (3.20) is satisfied.
Case iii. Let for , then we have
and
so we need to show that
On rearranging we have
or
By analyzing L.H.S., we see that and for all and for all , which infers that inequality (3.23) is indeed true. Therefore, f is an integral G-contraction with contraction constant . Note that G is a weakly connected -graph and (3.11) also holds for . Thus all the conditions of Theorem 3.20 are satisfied and f is a Picard operator with fixed point 0. Note that f is not a Banach G-contraction since, for ,
By setting in the above example, we have
Therefore one cannot apply Theorem 2.2 [1].
Conclusion
The notion of an integral G-contraction not only generalizes/extends the notion of a Banach G-contraction, but it also improves the integral inequality (2.3). Whereas, the notion of a sub-integral G-contraction generalizes the notion of a Banach G-contraction, but it partially generalizes the integral inequality (2.3). Therefore, Theorem 3.10 generalizes/extends some results of Jachymski [12] and provides partial improvement to the main result of Branciari [1]. A very natural question is bound to be posed: Are the conclusions of Theorems 3.10, 3.12, 3.14 still valid for integral G-contractions? In Theorem 3.20, we have provided a partial answer to this question by imposing condition (3.11). But it remains open to investigate an affirmative answer without the crucial condition of (3.11). Furthermore, Example 3.23 invokes the generality of Theorem 3.20.
References
Branciari A: A fixed point theorem for mappings satisfying a general contractive condition of integral type. Int. J. Math. Math. Sci. 2002, 29: 531–536. 10.1155/S0161171202007524
Aliouche A: A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type. J. Math. Anal. Appl. 2006, 322: 796–802. 10.1016/j.jmaa.2005.09.068
Djoudi A, Aliouche A: Common fixed point theorems of Gregus type for weakly compatible mappings satisfying contractive conditions of integral type. J. Math. Anal. Appl. 2007, 329: 31–45. 10.1016/j.jmaa.2006.06.037
Rhoades BE: Two fixed-point theorems for mappings satisfying a general contractive condition of integral type. Int. J. Math. Math. Sci. 2003, 63: 4007–4013.
Suzuki T: Meir-Keeler contractions of integral type are still Meir-Keeler contractions. Int. J. Math. Math. Sci. 2007., 2007: Article ID 39281
Vijayaraju P, Rhoades BE, Mohanraj R: A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type. Int. J. Math. Math. Sci. 2005, 15: 2359–2364.
Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435–1443. 10.1090/S0002-9939-03-07220-4
Nieto JJ, Rodríguez-López R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–239. 10.1007/s11083-005-9018-5
Nieto JJ, Pouso RL, Rodríguez-López R: Fixed point theorems in ordered abstract spaces. Proc. Am. Math. Soc. 2007, 135: 2505–2517. 10.1090/S0002-9939-07-08729-1
Pathak HK, Tiwari S:Common fixed point and best simultaneous approximations for Ciric type -weak contraction and weak asymptotic contraction. Int. J. Pure Appl. Math. 2010, 62: 291–304.
Petrusel A, Rus IA: Fixed point theorems in ordered L -spaces. Proc. Am. Math. Soc. 2006, 134: 411–418.
Jachymski J: The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 2007, 136: 1359–1373. 10.1090/S0002-9939-07-09110-1
Beg I, Butt AR, Radojević S: The contraction principle for set valued mappings on a metric space with a graph. Comput. Math. Appl. 2010, 60: 1214–1219. 10.1016/j.camwa.2010.06.003
Aydi H, Shatanawi W, Vetro C: On generalized weakly G -contraction mapping in G -metric spaces. Comput. Math. Appl. 2011, 62: 4222–4229. 10.1016/j.camwa.2011.10.007
Bojor F: Fixed point theorems for Reich type contractions on metric spaces with a graph. Nonlinear Anal. 2012, 75: 3895–3901. 10.1016/j.na.2012.02.009
Aleomraninejad SMA, Rezapour S, Shahzad N: Some fixed point results on a metric space with a graph. Topol. Appl. 2012, 159: 659–663. 10.1016/j.topol.2011.10.013
Kirk WA, Srinivasan PS, Veeranmani P: Fixed points for mappings satisfying cyclical contractive condition. Fixed Point Theory 2003, 4(1):79–89.
Karapinar E, Sadarangani K:Fixed point theory for cyclic -contractions. Fixed Point Theory Appl. 2011., 2011: Article ID 69
Alghamdi MA, Petrusel A, Shahzad N: A fixed point theorem for cyclic generalized contractions in metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 122
Petric MA: Some remarks concerning C̀iric̀-Reich-Rus operators. Creat. Math. Inf. 2009, 18: 188–193.
Acknowledgements
Authors are grateful to referees for their suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally in this article.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Samreen, M., Kamran, T. Fixed point theorems for integral G-contractions. Fixed Point Theory Appl 2013, 149 (2013). https://doi.org/10.1186/1687-1812-2013-149
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2013-149