- Open Access
Fixed point theorems for integral G-contractions
© Samreen and Kamran; licensee Springer. 2013
Received: 23 March 2013
Accepted: 2 May 2013
Published: 6 June 2013
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.
Branciari  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  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  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.  established a multivalued version of the main result of Jachymski . Aydi et al.  studied fixed point theorems for weakly G-contraction mappings in G-metric spaces. Later on, Bojor  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  and Branciari , 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.
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)  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  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  if for all and any sequence of positive integers, and , imply .
We state, for convenience, the following definition and result.
Definition 2.1 [, Definition 2.1]
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 [, Theorem 2.1]
where . Then f has a unique fixed point such that for each , .
3 Main results
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.
f is both an integral -contraction and an integral -contraction with the same contraction constant and ϕ.
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. □
which implies and it further implies that , a contradiction. Thus, , , in both cases. From the triangular inequality, we have , and letting gives . □
for every . Note that every constant function belongs to the class Ω.
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 .
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 .
G is a -graph.
Then, for any , is a Picard operator. Further, if G is weakly connected, then f is a Picard operator.
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 . □
G is weakly connected.
Every sub-integral G-contraction f on X is a Picard operator provided that .
Proof : It is immediate from Theorem 3.10.
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 [, 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 [, Theorem 3.4] and thus generalizes and extends the results of Nieto and Rodrýguez-López [, Theorems 2.1 and 2.3], Petrusel and Rus [, Theorem 4.3] and Ran and Reurings [, Theorem 2.1].
G is weakly connected.
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 [, Corollary 3.3].
and the operator f is known as a cyclic operator.
is a cyclic representation of Y w.r.t. f;
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 .
- (i)there exists such that(3.11)
G is a -graph.
Then, for any , is a Picard operator. Furthermore, if G is weakly connected then f is a Picard operator.
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 [, 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.
Next we show that (3.17) is satisfied for .
Case i. Let , then (3.1) is trivially satisfied.
Since for all and , thus inequality (3.20) is satisfied.
Therefore one cannot apply Theorem 2.2 .
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  and provides partial improvement to the main result of Branciari . 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.
Authors are grateful to referees for their suggestions.
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