Fixed points for G-contractions on uniform spaces endowed with a graph
© Aghanians et al.; licensee Springer 2012
Received: 25 April 2012
Accepted: 28 September 2012
Published: 17 October 2012
In this paper, we generalize some main results of (Jachymski in Proc. Am. Math. Soc. 136:1359-1373, 2008) from metric to uniform spaces endowed with an ℰ-distance and a graph using a new type of contractions by employing a class of nondecreasing functions.
1 Introduction and preliminaries
In 2004, the concepts of ℰ-distance and S-completeness were introduced for uniform spaces in . Recently in 2008, Jachymski  proved some fixed point results in metric spaces endowed with a graph and generalized simultaneously the Banach contraction principle from metric and partially ordered metric spaces. In 2010, Bojor  introduced -contractions and generalized Jachymski’s results. Finally, Nicolae et al.  presented some fixed point results for a new type of contractions using orbits and also for G-asymptotic contractions in metric spaces endowed with a graph.
The aim of this paper is to generalize Theorems 3.1, 3.2, 3.3 and 3.4 and Proposition 3.1 of Jachymski  from metric to uniform spaces endowed with a graph and to present a new type of contractive mappings. The reader interested in (ordered) uniform fixed point theorems may consult the references [5–7].
Following Willard , a uniformity on a nonempty set X is a nonempty family of subsets of satisfying the following conditions:
(U1) Each contains the diagonal ;
(U2) is closed under finite intersections;
(U3) For each , the set is a member of ;
(U4) For each , there exists a member V of such that whenever for all ;
(U5) contains the supersets of its elements.
Given a uniformity on a nonempty set X, the pair , simply denoted by X, is called a uniform space.
A uniformity on a nonempty set X is separating if the intersection of all members of is equal to . In this case, X is called a separated uniform space.
We are now ready to recall the concepts of ℰ-distance and p-completeness introduced by Aamri and El Moutawakil .
Definition 1 ()
for each , there exists a positive number δ such that whenever and for all ;
- (ii)p satisfies the triangular inequality, i.e.,
A sequence in a uniform space X equipped with an ℰ-distance p is said to be p-convergent to a point , denoted by , if as , and p-Cauchy if as . The uniform space X is said to be p-complete if each p-Cauchy sequence in X is p-convergent to some point of X.
The next lemma shows that in a separated uniform space every sequence is p-convergent to at most one point. The proof is straightforward, and hence it is omitted here.
Lemma 1 ()
Let be a sequence in a separated uniform space X equipped with an ℰ-distance p, and . If and , then . In particular, if for some , then .
Example 1 Suppose that the set is endowed with the trivial uniformity, that is, . Putting for all , it is seen that p is an ℰ-distance on X and each sequence (and even each net) in X is p-convergent only to zero, but clearly, this space is not separated. Therefore, the converse of Lemma 1 is not true in general.
We next review some basic notions of graph theory in relation to uniform spaces that we need in the sequel. For more details on the theory of graphs, see, e.g., .
Let X be a uniform space equipped with an ℰ-distance p and consider a directed graph G with and , that is, contains all loops. Suppose further that G has no parallel edges. The graph G may be considered a weighted graph by assigning the nonnegative number to each edge in .
The undirected graph obtained from G by ignoring the directions of the edges of G is denoted by . Indeed, can be treated as a directed graph for which the set is a symmetric subset of , namely .
If x and y are two vertices in a graph G, then a path in G from x to y is a finite sequence consisting of vertices of G such that , , and for , where . A graph G is said to be connected if there exists a path in G between each two vertices of G, and weakly connected if the graph is connected.
By a subgraph of G, we mean a graph H satisfying and such that contains the vertices of all edges of .
It is clear that the graph is connected for all .
2 Main results
Throughout this section, we assume that X is a uniform space that is endowed with an ℰ-distance p and a directed graph G with and unless stated otherwise.
We denote by the set of all fixed points for a self-map f on X, and further by Φ the class of all nondecreasing functions φ from into .
Following Jachymski , we introduce -contractions on a uniform space endowed with an ℰ-distance and a graph.
Definition 2 Let f be a self-map on X and φ be a function in Φ. Then f is called a -contraction if
(C1) the edges of G are preserved by f, i.e., implies for all ;
for all with .
If p is a metric on X, then we call f a -contraction, and we call f a -contraction if (1) holds for all .
We now give some examples of -contractions.
Example 2 If for some , since contains all loops, it follows that the constant mapping is a -contraction for any . In particular, for all if and only if each constant mapping on X is a -contraction for some .
Example 3 Each -contraction is a -contraction, where is the complete graph with , that is, .
Then Condition (C1) means that f is nondecreasing with respect to ⪯, and Condition (C2) means that f is an order -contraction, i.e., (1) holds for all with .
In the next example, we construct a self-map f that fails to be a -contraction for any , whereas f is a -contraction for some ℰ-distance p and some .
Clearly, and an easy argument shows that Conditions (C1) and (C2) are satisfied. Thus, f is a -contraction.
Therefore, f is not a -contraction for any weakly connected graph G (with and ) and any function . Now let be equipped with the trivial uniformity . Then defined by for all is an E-distance on , and the mapping f is a -contraction for any function .
the same argument shows that f fails to be a -contraction for any .
Then f is a -contraction if we define by the rule .
Remark 1 It is worth mentioning that Conditions (C1) and (C2) are independent of each other. For instance, the identity mapping on preserves the edges of , but there is no for which the contractive condition (1) holds. Conversely, setting for all , it is seen that f is an order -contraction for the constant function but f fails to be nondecreasing.
Remark 2 Putting φ the constant function in Definition 2, we get the concept of Banach G-contraction with on a metric space , which was introduced by Jachymski .
Definition 3 We say that the sequences and are p-Cauchy equivalent in X if both of them are p-Cauchy and further, as .
Hereafter we assume that φ is an arbitrary fixed function in Φ and is a -contraction unless stated otherwise.
G is weakly connected.
If f is a -contraction, then and are p-Cauchy equivalent for all .
Each -contraction has at most one fixed point in X.
(A) implies (B).
(B) implies (C) provided that X is separated.
(C) implies (A) provided that for all .
- (ii)Let x and y be two fixed points for f. Since, by the hypothesis, and are p-Cauchy equivalent, it follows that
- (iii)Suppose on the contrary that G is not weakly connected. Then there exists an such that both sets and are nonempty. Fix any and define by
where φ is any arbitrary function in Φ. Thus, f is a -contraction. □
Corollary 1 If X is p-complete and G is weakly connected, then there exists an such that for all .
Therefore, . □
Proposition 1 If for some we have , then is f-invariant and is a -contraction. Moreover, if , then and are p-Cauchy equivalent.
Proof Let . Then there exists a path in from to x, i.e., and for each i. So, for each i, that is, is a path in from to fx, and since , there is another path in from to , i.e., , , and for each j. Thus, . Therefore, is f-invariant.
Since , it follows that f is a -contraction.
Moreover, because is weakly connected, Theorem 1 implies that the sequences and are p-Cauchy equivalent for all . □
Following Petruşel and Rus , we introduce the concept of a p-Picard operator.
Definition 4 A self-map f on X is called a p-Picard operator if f has a unique fixed point in X and for all .
Given a self-map f on X, we denote the set by .
Theorem 2 Let X be p-complete, separated and satisfy the following property:
(∗) For each sequence in X, p-convergent to some , if for all , then there exists a subsequence of such that for all .
is a p-Picard operator for each .
If and G is weakly connected, f is a p-Picard operator.
if and only if .
f has a unique fixed point if and only if there exists an such that .
By the property (∗), there exists a subsequence of such that for all .
If G is weakly connected, then , where , and so, by (i), f is a p-Picard operator.
- (iii)Set and define a mapping by
Thus, θ is surjective.
Therefore, . Consequently, θ is injective.
(iv), (v) They are immediate consequences of (iii). □
- (1)Let f and φ be as in [, Example 2]. Since for each , we have
- (2)In essence, the mapping f in [, Example 2] fails to be a -contraction for any . Otherwise, if f is a -contraction for some , then from
which is a contradiction.
Now, uniformize X with the usual uniformity and consider an ℰ-distance p on X by . Then X is separated, p-complete and satisfies the property (∗). Moreover, . Therefore, considering φ as in Example 5, it is seen that f is a -contraction, and so by Theorem 2, f is a p-Picard operator with the unique fixed point zero.
Our next result is a generalization of [, Corollary 3.2].
G is weakly connected;
If f is a -contraction such that for some , then f is a p-Picard operator;
Each -contraction has at most one fixed point in X.
⇒ (iii): Let f be a -contraction. If , then there is nothing to prove. Otherwise, by Theorem 2, . Thus, by the hypothesis, f is a p-Picard operator and so it has a unique fixed point.
⇒ (i): It follows from Theorem 1. □
Following the idea of Jachymski , we define two different types of p-continuity of self-maps on X and then we discuss them.
f is orbitally p-continuous if for each and each sequence of positive integers, implies .
f is graph orbitally p-continuous if for each and each sequence of positive integers with for such that , one has .
It is clear that p-continuity (see [, Definition 2.3]) implies orbital p-continuity, and orbital p-continuity implies graph orbital p-continuity. But the converse of these relations is not true in general as the next example shows.
Next, consider the ℰ-distance on X and the graph G with . Then the self-map f on X defined by the rule if and is graph orbitally p-continuous since for all n implies that is a constant sequence. But setting and , it is seen that , whereas .
for each with , there exists an such that for all .
if and only if there exists an such that .
If G is weakly connected, f is a p-Picard operator.
If there exists an such that , then, by (i), we have . The converse is trivial.
Since G is weakly connected, it follows that for all . So, by (i), there exists an such that for all . Now, similar to the proof of Theorem 2, one can show that is the only fixed point for f, and hence f is a p-Picard operator. □
A generalization of [, Corollary 3.3] is given in the next result.
G is weakly connected;
Each orbitally p-continuous -contraction is a p-Picard operator;
Each orbitally p-continuous -contraction has at most one fixed point in X.
In particular, if is disconnected, then there exists an orbitally p-continuous --contraction that has at least two fixed points in X.
⇒ (iii): It is trivial.
- (iii)⇒ (i): According to the proof of Theorem 1, it suffices to show that the self-map f is orbitally p-continuous. To this end, let and be a sequence of positive integers such that . Then is either the constant sequence or the constant sequence . If the former holds, then . Since , it follows by Lemma 1 that . Therefore,
Otherwise, if the latter holds, a similar argument shows that f is orbitally p-continuous. □
for each , there exists an such that for all .
if and only if .
If and G is weakly connected, f is a p-Picard operator.
If , then, by (i), . Conversely, suppose that . Since , it follows that .
If , since G is weakly connected, we have . Hence, by (i), there exists an such that for all . Now, similar to the proof of Theorem 2, it is seen that is the only fixed point for f. □
Remark 4 In all theorems and corollaries above, setting (), we get the usual (ordered) version of fixed point theorems in (partially ordered) uniform spaces.
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