Some fixed-point theorems for mixed monotone operators in partially ordered probabilistic metric spaces
© Wu; licensee Springer. 2014
Received: 9 October 2013
Accepted: 24 January 2014
Published: 24 February 2014
In this paper, some fixed-point theorems for nonlinear contractive operators in partially ordered Menger probabilistic metric spaces are proved. A new extension theorem of the probabilistic versions of Boyd and Wong’s nonlinear contraction theorem is presented. As a consequence, our main results improve and generalize some recent coupled fixed-point theorems and coincidence-point theorems in (Ćirić, Nonlinear Anal. 72:2009-2018, 2010; Jachymski, Nonlinear Anal., 73:2199-2203, 2010; Ćirić, Agarwal and Samet, Fixed Point Theory Appl. 2011:56, 2011).
1 Introduction and preliminaries
It is well known that the probabilistic version of the classical Banach contraction principle was proved in 1972 by Sehgal and Bharucha-Reid . In 2010, a truthful probabilistic version of the Banach fixed-point principle for general nonlinear contractions was presented by Ljubomir Ćirić . Unfortunately, there is a counterexample  to the Ćirić’s key lemma. Meanwhile, Jacek Jachymski  established a corrected probabilistic version of the Banach fixed-point principle for general nonlinear contractions. Also, the fixed-point theorems in probabilistic metric spaces for other contraction mappings were investigated by many authors, see [4–27] the references therein.
In this paper, we try to extend this probabilistic version theorem to the partially ordered Menger probabilistic metric spaces and establish some fixed-point theorems for monotone operators. Also, we show a sufficient and necessary condition for the uniqueness of the fixed point for a class of monotony operators. As a consequence, our main results improved and extended some recent coupled fixed-point theorems and coincidence-point theorems for mixed monotone mappings in the literature [2–4].
At this stage, we recall some well-known definitions and results in the theory of partially ordered set and probabilistic metric spaces which are used later on in the paper. For more details, we refer the reader to [8, 20].
Let be a partially ordered set, the subset is said to be a totally ordered subset if either or holds for all . We say the elements x and y are comparable if either or holds. It is said that the triple is a partially (totally) ordered complete metric space if is a partially (totally) ordered set and is a complete metric space. It is said that the operator is non-decreasing monotone with respect to the order ≤ if for any with we have . Let Φ denote all the functions which satisfy and for all .
Definition 1.1 (Bhaskar and Lakshmikantham )
Definition 1.2 (Bhaskar and Lakshmikantham )
An element is said to be a coupled fixed point of the mapping if and .
Definition 1.3 
Remark 1.1 If F has mixed h-monotone property, then F has mixed monotone property.
Definition 1.4 
A function is called a distribution function if it is non-decreasing and left-continuous with . If in addition , then f is called a distance distribution function. Furthermore, a distance distribution function f satisfying is called a Menger distance distribution function.
The set of all Menger distance distribution functions is denoted by .
Definition 1.5 
△ is associative and commutative,
△ is continuous,
for all ;
whenever and , for each .
Among the important examples of a T-norm we mention the following two T-norms: and . The T-norm is the strongest T-norm, that is, for every T-norm △.
Definition 1.6 
Obviously, for any and .
Definition 1.7 
A Menger probabilistic metric space (abbreviated, Menger PM space) is a triple where X is a nonempty set, △ is a continuous T-norm and F is a mapping from into such that, if denotes the value of F at the pair , the following conditions hold:
(PM1) for all if and only if .
(PM2) for all and .
(PM3) for all and every , .
Definition 1.8 
A sequence in X is said to converge to a point x in X (written as ) if for every and , there is an integer such that , for all . The sequence is said to be a Cauchy sequence if for each and , there is an integer such that , for all . A Menger PM space is said to be complete if every Cauchy sequence in X converges to a point of X.
2 Main results
This implies that for all . Thus . □
then for all .
Proof If , then the result of Lemma 2.2 trivially holds.
If , then . Thus for all . By the proof of Lemma 2.1, we see that for all . Thus for all . Then for all . The proof is complete. □
Lemma 2.3 If is a complete Menger PM space, then is also a complete Menger PM space.
Then and . Thus both and are Cauchy sequences in . Following Definition 1.8, it is a standard argument to show that the Cauchy sequence converges to a point of . Thus is a complete Menger PM space. The proof is complete. □
- (i)there is a such that, for all and with ,(1)
there exists an such that ;
either if (a) G is a continuous operator or (b) if a non-decreasing monotone sequence in X tends to , then for all n.
Then the operator G has a fixed point in X.
If there exists such that , then and is a fixed point of G. Then the result of Theorem 2.1 trivially holds.
This means for all .
Thus (2) holds for .
This implies that the sequence is a Cauchy sequence in X. Then by the completeness of X, there is such that .
Then the operator G has a fixed point in X.
Letting n go to infinity, we have . It follows from Lemma 2.1 that . Then the operator G has a fixed point in X. The proof of Theorem 2.1 is complete. □
Let , then we have the following.
G is a continuous operator;
if a monotone sequence in X tends to , then and are comparable for all n.
Then the operator G has a fixed point in X if and only if . Furthermore, if D is a totally ordered nonempty subset, then the operator G has a unique fixed point in X
Proof ⇒: It is easy to see that all the fixed points of G fall in the set D. Thus if the operator G has a fixed point in X, then .
Following a similar proof as of Theorem 2.1, we conclude that the operator G has a fixed point in X.
On the other hand, by and the monotony of , we see that . Thus . It follows from Lemma 2.1 that . The proof of Theorem 2.2 is complete. □
if non-decreasing sequence tends to x, then for all n,
if non-increasing sequence tends to y, then for all n.
If there exist such that and , then A has a coupled point, that is, there exist such that and .
Then a coupled point of A is a fixed point of G and vice versa.
If , then and . Noting the mixed monotone property of A, we see that and , then . Thus G is a non-decreasing operator with respect to the order ⪯ on .
Also, there exists an such that .
If a non-decreasing monotone sequence in tends to , then , that is, and . Thus is non-decreasing sequence tending to x and a non-increasing sequence tending to y. Thus and for all n. This implies . Obviously, the continuity of A implies the continuity of G.
Following Theorem 2.1, we see that A has a coupled point, that is, there exist such that and . □
for all for which and and all . Suppose also that , is closed and
if is a non-decreasing sequence with in
then for all n holds,
if is a non-increasing sequence with in
then for all n holds.
If there exist such that and , then A and h have a coupled coincidence point, that is, there exist such that and .
for , .
The proof is complete. □
Remark 2.1 Corollary 2.2 improves and generalizes Theorem 7 in .
Following similar arguments as in the proof of Theorems 2.1 and 2.2, we can deduce the next result. We omit the details of the proof.
Then the operator G has a unique fixed point in X.
Corollary 2.3 (, Theorem 1)
Then the operator G has a unique fixed point in X, and, for any , .
we conclude from Theorem 2.3 that the operator G has a unique fixed point in X, and, for any , . □
Corollary 2.4 (, Theorem 13)
where is a monotonically decreasing function. Then the operator G has a unique fixed point in X.
This is a contradiction. Thus . This implies that . Then the Corollary 2.4 is a consequence of Theorem 2.3 or Corollary 2.3. □
In this paper, we establish some fixed-point theorems for monotony operators and extend the probabilistic version of the Banach fixed-point principle for general nonlinear contractions to the partially ordered Menger probabilistic metric spaces. Also, we show a sufficient and necessary condition to the existence of the fixed point for a class of monotone operators. As a consequence, our main results improved and extended some recent coupled fixed-point theorems and coincidence-point theorems for the mixed monotone mappings in the literature [2–4].
Finally, we mention two possible applications of our results. One is to the theory of fuzzy metric spaces. Since the difference between Menger PM space and fuzzy metric spaces lies in the different metric, some coincidence-point theorems and fixed-point theorems in fuzzy metric spaces can be obtained under some suitable restrictions. Another possible application of our results is to the theory of random operator equations.
The authors are grateful to the reviewers for their valuable comments and suggestions. This work was partly supported by National Natural Science Foundation of China (11301039).
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