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On some fixed-point theorems for ψ-contraction on metric space involving a graph
Journal of Inequalities and Applications volume 2014, Article number: 39 (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 .
The conversion of a graph G is denoted by and this is a graph obtained from G by reversing the direction of the edges. Hence
By we denote the undirected graph obtained from G by omitting the direction of the edges. Indeed, it is more convenient to treat as a directed graph for which the set of its edges is symmetric, and under this convention, we have
A subgraph of a graph G is a graph H such that and . Let x and y be vertices in a graph G. A path from x to y of length N () is a sequence of distinct vertices such that , and for . The number edges in G forming the path is called the length of the path. A graph G is connected if there is a path between any two vertices. If a graph G is not connected then it is called disconnected and its different paths are called the components of G. Every component of G is a subgraph of it. Furthermore, G is weakly connected if is connected. Let be the component of G which consists of all edges and vertices contained in some path in G beginning at x. Suppose that G is such that is symmetric; then where denotes the equivalence class of relations ℜ defined on by the rule
Some basic notations related to connectivity of graphs can be found in .
If is an operator, then we denote by
the set of all fixed points of f.
Definition 1 
A mapping is a Banach G-contraction or simply G-contraction if f preserves edges of G;
for all , and f decreases weights of edges of G: for all there exists such that
Definition 2 
The mapping is a G-graphic contraction
if f preserves edges of G;(3)
for all ;
there exists such that(4)
for all .
Definition 3 
A mapping is called orbitally continuous if for all and any sequence of positive integers,
Definition 4 
A mapping is called orbitally G-continuous if for all and any sequence of positive integers,
Definition 5 Let us define the class which satisfies the following conditions:
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.
Definition 6 We say that a mapping is a -contraction if the following hold;
f preserves edges of G, i.e. , ;
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. □
Lemma 2 Let be a -contraction with constant ; for a given and , there exists such that
Proof Let and . Then there is a path in from x to y, which means , , and for . By Lemma 1, f is a -contraction. With an easy induction, we have and
for all and .
Hence using the triangle inequality, we get
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.
Theorem 1 Let be a metric space endowed with a graph G and be a -contraction, then the following statements are equivalent:
G is weakly connected;
for given , the sequences and are Cauchy equivalent;
Proof (i) ⇒ (ii) Let f be a -contraction and . By hypothesis, , so . By Lemma 2, we get
for all . Hence
and if we use a standard argument, then is obtained as a Cauchy sequence. Since also , Lemma 2 leads to . Therefore, and are equivalent. Clearly, because is a Cauchy sequence, so is .
⇒ (iii) Let f be a -contraction and . By (ii), and are equivalent, which yields .
⇒ (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.
Corollary 1 Let be a complete metric space endowed with a graph G and be a -contraction, then the following statements are equivalent:
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.
Example 1 Let be endowed with the usual metric. Take
and as follows:
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.
Theorem 2 Let be a complete metric space and the triple have the following condition:
Let be a -contraction, and . Then the following statements hold.
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.
Proof Initially, we prove the items (iv) and (v). Take and then , so by Lemma 3, if , then and are Cauchy equivalent. Since X is complete, converges to some . It is obvious that . Then by using induction we get
for all , since . By (6), there is a subsequence such that for all . If we use (7), we conclude that is a path in G and also in from x to , and this means that . Since f is a -contraction we have
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).
Let us define a mapping to prove (i): for all . It is sufficient to show that is a bijection. Because , we deduce and then . Beside, if , then by (iv), , which implies and so ρ is a surjective mapping. We show that f is injective. Take which are such that , then and so, by (i),
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). □
Corollary 2 Let be complete metric space and obey condition (6). The following are equivalent:
G is weakly connected;
every -contraction such that , for some , is a Picard operator;
for any -contraction, .
Proof (i) ⇒ (ii): This can be obtained directly from Theorem 2(v).
⇒ (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.
Definition 7 Let be a metric space and G be a graph. The mapping is called a -graphic contraction if the following conditions hold:
implies (f is edge preserving);
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.
Lemma 5 Let be a -graphic contraction with constant . Then, given , there exists such that
for all , where .
Lemma 6 Suppose that is a -graphic contraction. Then for each , there exists such that the sequence converges to as .
Proof Take an arbitrary element x in . By Lemma 5, we obtain
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.
Take ; then there is a path in from to y such that . Also let be a path in from to . Then we realize
is a path in from to fy such that . Furthermore, f is a -graphic contraction because and f is a -graphic contraction. □
Theorem 3 Let be a complete metric space and let the triple have the following condition:
Let be a -graphic contraction and f is orbitally G-continuous. Then the following statements hold:
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.
Example 2 Let be endowed with the usual metric. Consider
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.
Example 3 Let be endowed with the usual metric. Take
and as follows:
Then G is weakly connected and is nonempty and f is a -graphic contraction with which is not a -contraction.
Proof It is clear that G is weakly connected, , and with simple calculations it can be easily seen that f is a -graphic contraction. Take
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 .
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The authors are grateful to the reviewers for their careful reviews and useful comments.
The authors declare that they have no competing interests.
All authors contributed equally in writing this article. All authors read and approved the final manuscript.
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Öztürk, M., Girgin, E. On some fixed-point theorems for ψ-contraction on metric space involving a graph. J Inequal Appl 2014, 39 (2014). https://doi.org/10.1186/1029-242X-2014-39
- connected graph
- fixed point
- metric space
- ψ-type contraction