On some fixed-point theorems for ψ-contraction on metric space involving a graph
© Öztürk and Girgin; licensee Springer. 2014
Received: 2 October 2013
Accepted: 2 January 2014
Published: 24 January 2014
In this paper, we introduce the -contraction and the -graphic contraction in a metric space by using a graph. We explain some conditions for a mapping which is a -contraction to have a unique fixed point and also we give conditions as regards the existence of a fixed point for -graphic contraction by applying the connectivity of the graph in both cases. Moreover, we give examples to show that our results are a substantial improvement of some known results in the literature.
The metric fixed-point theory has been researched extensively in the past two decades such as in a metric space endowed with a partial ordering, and many results appeared giving sufficient conditions for a mapping to be a Picard operator. For these concepts have been given two main theorems, which are the Banach Contraction Principle and the Knaster-Tarski Theorem .
Recently Jachymski  and Gwóźdź-Lukawska and Jachymski  have given an interesting concept in fixed-point theory with some general structures by using the context of metric spaces endowed with a graph. Jachymski  has proved some generalizations of the Banach Contraction Principle to mappings on a metric space endowed with a graph and also has presented its applications to the Kelisky-Rivlin Theorem on iterates of the Bernstein operators on the space . Afterwards different contractions have been studied by various authors. In  the contraction principle for set-valued mappings, in [5–7] Kannan type, Reich type contractions, and φ-contractions have been investigated, respectively. Some new fixed-point results for graphic contractions on a complete metric space with a graph have been presented in ; also they gave a particular case of almost contractions.
In this paper, motivated by the work of Jachymski  and Petruşel , we introduce new contractions for the mappings on complete metric space and prove some fixed-point theorems. Our results generalize and unify some results by the above-mentioned authors.
2 Basic facts and definitions
Let be a metric space and Δ denote the diagonal of the Cartesian product . Let G be a directed graph such that the set of its vertices coincides with X, and the set of its edges contains all loops; that is, . Assume that G has no parallel edges, so one can identify G with the pair .
Some basic notations related to connectivity of graphs can be found in .
the set of all fixed points of f.
Definition 1 
Definition 2 
- (i)if f preserves edges of G;(3)
- (ii)there exists such that(4)
for all .
Definition 3 
Definition 4 
if and only if ;
for every , if and only if ;
for every , .
3 -Contraction and related fixed-point theorems
We establish some fixed-point theorems in metric space with a graph by defining the -contraction.
f preserves edges of G, i.e. , ;
- (ii)f decreases the weight of edges of G, that is, there exists such that
for all .
Lemma 1 If is a -contraction, then f is both a -contraction and a -contraction.
Proof The proof can be obtained by the symmetry of d and the definition of the -contraction. □
for all and .
So it qualifies to set . □
Lemma 3 Let be a complete metric space endowed with a graph G and be a -contraction for which there exists such that . Let be the component of containing . Then is f-invariant and is a -contraction. Furthermore, , and the sequences and are Cauchy equivalent.
Proof The proof of this lemma can obtained by using similar arguments as given in . So we omit the proof. □
The following result shows that there is a close relation between convergence of an iteration sequence which can be obtained by using a -contraction mapping and connectivity of the graph.
G is weakly connected;
for given , the sequences and are Cauchy equivalent;
⇒ (iii) Let f be a -contraction and . By (ii), and are equivalent, which yields .
- (iii)⇒ (ii) Suppose, to the contrary, G is not weakly connected, that is, is disconnected. Let . Then the sets and both are nonempty. Let and define
Obviously, . We show f is a -contraction. Let . Then , so either or . Hence in both cases , so as , and . Thereby, f is a -contraction having two fixed points which violates the assumption. □
The following result is an easy consequence of Theorem 1.
G is weakly connected;
there is such that , for all .
Now, we give an example of f being a -contraction and this example shows that we could not add that is a fixed point of f in Corollary 1.
Then f is a -contraction where .
Proof It can be easily seen that G is a weakly connected graph and f is a -contraction where . It is a fact that , for all but f has no fixed point. □
For any mapping which satisfies the condition of Corollary 1 to have a fixed point we need to add condition (6), which is given in the following theorem.
f has a unique fixed point iff there exists such that .
For any , is a Picard operator.
If and G is weakly connected, then f is a Picard operator.
If , then is a weakly Picard operator.
If , then f is a weakly Picard operator.
for all . By taking the limit as , we deduce . Thereby, is a Picard operator. Also, we conclude that f is a Picard operator, when , since there is weakly connectedness of G.
(vi) is obvious from (iv). For proof of (vii), if then and so holds. Thus f is a weakly Picard operator because of (vi).
which gives . Thus, f is injective and this is the desired result. Finally, one can see that (ii) and (iii) are easy consequences of (i). □
G is weakly connected;
every -contraction such that , for some , is a Picard operator;
for any -contraction, .
⇒ (iii): Let be a -contraction. If is empty, so is , because is a subset of . If is nonempty, then by (ii), is singleton. In these two cases, .
⇒ (i): This implication follows from Theorem 1. □
Remark 1 In the above results by taking , we obtain Corollary 3.2, which is given in .
4 -Graphic contraction and fixed-point theorems
Now, we define -graphic contraction and give some results and examples.
implies (f is edge preserving);
- (ii)there exists a with constants such that
for all , where .
Firstly, we give the following lemmas which can be proved as in the above section.
Lemma 4 If is a -graphic contraction, then f is both a -graphic contraction and a -graphic contraction.
for all , where .
Lemma 6 Suppose that is a -graphic contraction. Then for each , there exists such that the sequence converges to as .
for all . Therefore, and so ; consequently using the property of ψ we have . Then we say that is a Cauchy sequence. By the completeness of X, there exists such that converges as . □
Lemma 7 The self-mapping f is a -graphic contraction for which there exists such that . Then the set invariant with respect to f and is a -graphic contraction, where is the component of containing .
Proof Let x be an element in . Then there exist in from to x, i.e., and for . Since f is a -graphic contraction we get for . So we have a path from to fx. Therefore since . Consequently is invariant with respect to f.
is a path in from to fy such that . Furthermore, f is a -graphic contraction because and f is a -graphic contraction. □
if and only if .
If and G is weakly connected, then f is a weakly Picard operator.
For any , we see that is a weakly Picard operator.
Proof We begin with the statement (iii). Let ; by Lemma 6, there exists such that . Since , then for every . Now assume that . (A similar deduction can be made if .) By condition (9), there is a subsequence of such that for each . A path in G can be formed by using the points and hence . Since f is orbitally G-continuous, we see that is a fixed point for .
To prove (i), using (iii) we have if . Suppose that . By using the assumption that , we immediately obtain . Hence (i) holds.
For proving (ii) let . If we use weak connectivity of G, we have and by applying (iii) we obtain the desired result. □
The next example illustrates that f must be orbitally G-continuous in order to obtain statements which are given in Theorem 3.
Then G is weakly connected, is nonempty and f is a -graphic contraction where , but it is not orbitally G-continuous. Thus, f does not have a fixed point.
Remark 2 In Theorem 3, by replacing the condition that the triple satisfies (9) and f is orbitally G-continuous with the mapping f is orbitally continuous, we have the above result, too.
The following example demonstrates that the -graphic contraction is more general than the -contraction.
Then G is weakly connected and is nonempty and f is a -graphic contraction with which is not a -contraction.
which is a contradiction since . Thus, f is not -contraction. □
Remark 3 In Theorem 3, if we take , then we get Theorem 2.1, which is given in .
The authors are grateful to the reviewers for their careful reviews and useful comments.
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