Abstract
In this article, we consider ordered metric spaces concerning generalized distance and prove some fixed point theorems in these spaces. Our results generalize, improve, and simplify the proof of the previous results given by some authors.
Mathematics Subject Classification (2000)
47H10, 54H25
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1. Introduction and Preliminary
Recently, Nieto and RodriguezLopez [1, 2], Ran and Reurins [3], Petrusel and Rus [4] presented some new results in partially ordered metric spaces. Their main idea was to combine the ideas of iterative technique in the contractive mapping with these in monotone technique.
Recently, Kada et al. [5, 6] in 1996 introduced the concept of wdistance in a metric space and prove some fixed point theorems. For the study of fixed point theorem concerning generalized distance followed in other articles, see [5, 7–15].
The aim of this article is to use the concept of wdistance to generalize the fixed point theorems in partially ordered metric spaces. Our results not only generalize some fixed point theorems, but also improve and simplify the previous results.
In the sequel, we state some definitions and a lemma which we will use in our main results.
Definition 1.1. ([5, 8, 10]) Let (X, d) be a metric space. Then, a function p : X × X → [0, ∞) is called a wdistance on X if the following conditions are satisfied:

(a)
p(x, z) ≤ p(x, y) + p(y, z) for any x, y, z ∈ X;

(b)
for any x ∈ X, p(x, .) : X → [0, ∞) is lower semicontinuous;

(c)
for any ε > 0, there exists δ > 0 such that p(x, z) ≤ δ and p(z, y) ≤ δ imply d(x, y) ≤ ε.
We know that a realvalued function f defined in a metric space X is said to be lower semicontinuous at a point x_{0} ∈ X if either or , whenever x_{ n } ∈ X for each n ∈ N and x_{ n } → x_{0}.
Lemma 1.2. ([5, 7]) Let (X, d) be a metric space and p be a wdistance on X. Let {x_{ n } }, {y_{ n } } be sequences in X, {α_{ n } }, {β_{ n } } be sequences in [0, ∞) converging to zero and let x, y, z ∈ X. Then, the following conditions hold:

(1)
If p(x_{ n } , y) ≤ α_{ n } and p(x_{ n } , z) ≤ β_{ n } for any n ∈ N, then y = z. In particular, if p(x.y) = 0 and p(x, z) = 0, then y = z;

(2)
If p(x_{ n } , y_{ n } ) ≤ α_{ n } and p(x_{ n } , z) ≤ β_{ n } for any n ∈ N, then d(y_{ n } , z) → 0;

(3)
If p(x_{ n } , x_{ m } ) ≤ α_{ n } for any n, m ∈ N with m > n, then {x_{ n } } is a Cauchy sequence;

(4)
If p(y, x_{ n } ) ≤ α_{ n } for any n ∈ N, then {x_{ n } } is a Cauchy sequence.
Let f : X → X be an operator:

(1)
I(f) is the set of all nonempty invariant subsets of f, i.e., I(f) = {Y ⊂ X : f(Y ) ⊂ Y } and F_{ f } = {x ∈ X : x = f(x)}.

(2)
The operator f is called Picard operator (briefly, PO) if there exists x* ∈ X such that F_{ f } = {x*} and, for all x ∈ X, {f^{n} (x)} converges to x*.

(3)
The operator f is called orbitally Ucontinuous for any U ⊂ X × X if the following condition holds:
For any x ∈ X, as i → ∞ and for any i ∈ N imply that as i → ∞.

(4)
Let (X, ≤) be a partially ordered set. Then,
and [x, y]_{≤} = {z ∈ X : x ≤ z ≤ y}, where x, y ∈ X and x ≤ y.

(5)
If g : Y → Y is an operator, then the Cartesian product of f and g is the mapping f × g : X × Y → X × Y defined by (f × g)(x, y) = (f(x), g(y)) for all (x, y) ∈ X × Y.

(6)
φ : R _{+} → R _{+} is said to be a comparison function if it is increasing and φ^{n} (t) → 0 as n → ∞. As a consequence, we also have φ (t) < t for any t > 0, φ (0) = 0, and φ is right continuous at 0.
2. Main Results
Now, we give the main results of this article.
Theorem 2.1. Let (X, d, ≤) be an ordered metric space and f : X → X be an operator. Let p be a wdistance on (X, d) and suppose that

(a)
X _{≤} ∈ I(f × f );

(b)
there exists x _{0} ∈ X such that (x _{0}, f (x _{0})) ∈ X _{≤} ;

(c)
(c _{1}) f is orbitally continuous or
(c_{2}) f is orbitally X_{≤}continuous and there exists a subsequenceof {f ^{n} (x_{0})} such thatfor any k ∈ N ;

(d)
there exists a comparison function φ : R _{+} → R _{+} such that
for all (x, y) ∈ X_{≤}, where

(e)
the metric d is complete.
Then F_{ f } ≠ ∅.
Proof. If f(x_{0}) = x_{0}, then the proof is completed. Let x_{0} ∈ X be such that (x_{0}, f (x_{0})) ∈ X_{≤}. By (a), since (f × f )(X_{≤}) ⊂ X_{≤}, we have (f × f )(x_{0}, f (x_{ o } )) ∈ X_{≤} and so (f(x_{0}), f^{2}(x_{ o } )) ∈ X_{≤}.
Continuing this process, we obtain
for any n ∈ N.
Now, we show that
for any n ∈ N. Let p_{0} = p(x_{0}, f (x_{0})) and p_{ n } = p(f ^{n} (x_{0}), f^{n+1}(x_{0})) for any n ∈ N. Then we have
for any n ∈ N. If max{p_{n1}, p_{ n } } = p_{n1}, then (3.1) follows. Otherwise, max{p_{n1}, p_{ n } } = p_{ n } Then, by (3.2), we have p_{ n } ≤ φ(p_{ n } ) ≤ p_{ n } and so p_{ n } = 0 and (3.1) follows. By induction, we obtain
or, equivalently,
for any n ∈ N, Now, we have
as n → ∞.
Similarly, we have
as n → ∞ and so, by induction, we obtain
as n → ∞ for any k > 0. Therefore, {f^{n} (x_{0})} is a Cauchy sequence in X. Since X is complete, there exists x* ∈ X such that f^{n} (x_{0}) → x* as n → ∞.
Now, we show that x* is a fixed point. If (c_{1}) holds, then f^{n+1}(x_{0}) → f (x*) and, by lower semicontinuity of p(f^{n} (x_{0}), ·), we have
and α_{ n } , β_{ n } → 0 as n → ∞. Thus, by (3.3) and Lemma 1.2, we conclude that f (x*) = x*.
Now, suppose that (c_{2}) holds. Since converges to x* and f is X_{≤}orbitally continuous, it follows that converges to f (x*). Similarly, by lower semicontinuity of p(f^{n} (x_{0}), ·), we conclude that f (x*) = x*. This completes the proof. □
Corollary 2.2. Let (X, d, ≤) be an ordered metric space and f : X → X be an operator.
Let p be a wdistance on (X, d) and suppose that

(a)
X _{≤} ∈ I(f × f );

(b)
there exists x _{0} ∈ X such that (x _{0}, f (x _{0})) ∈ X _{≤} ;

(c)
(c _{1})) f is orbitally continuous or
(c_{2}) f is orbitally X_{≤}continuous and there exists a subsequenceof {f^{n} (x_{0})} such thatfor any k ∈ N ;

(d)
and there is a comparison function φ : R _{+} → R _{+} such that
for any (x, y) ∈ X_{≤}, where

(e)
the metric d is complete;

(f)
if (x, y) ∈ X _{≤} and (y, z) ∈ X _{≤} .vskip 1 mm
Then, F_{ f } ≠ ∅.
Theorem 2.3. Let (X, d, ≤) be an ordered metric space and f : X → X be an operator.
Let p be a wdistance on (X, d) and suppose that

(a)
X _{≤} ∈ I(f × f );

(b)
There exists x _{0} ∈ X such that (x _{0}, f (x _{0})) ∈ X _{≤} ;

(c)
(c _{1}) f is orbitally continuous or
(c_{2}) f is orbitally X_{≤}continuous and there exists a subsequenceof {f^{n}(x_{0})} such thatfor any k ∈ N ;

(d)
there is a comparison function φ : R _{+} → R _{+} such that
for any (x, y) ∈ X_{≤}, where

(e)
the metric d is complete;

(f)
if x, y ∈ X with (x, y) ∉ X _{≤}, then there exists c(x, y) ∈ X such that (x, c(x, y)) ∈ X _{≤} and (y, c(x, y)) ∈ X _{≤} ^{.} .
Then, f is PO.
Proof. According to Theorem 2.1, there exists x* ∈ X such that f(x*) = x*. Take x ∈ X.
If (x, x_{0}) ∈ X_{≤}, then (f ^{n} (x), f ^{n} (x 0)) ∈ X_{≤} and so
for any n ∈ N. Thus, by Lemma 1.2, f^{n} (x) → x* as n → ∞.
If (x, x_{0}) ∉ X_{≤}, then there exists z ∈ X such that (x, z) ∈ X_{≤} and (x_{0}, z) ∈ X_{≤} and so
for any n ∈ N. Thus, by Lemma 1.2, we have f^{n} (z) → x* as n → ∞. Also, since (x, z) ∈ X_{≤}, we have f ^{n} (z) → x* as n → ∞. Consequently, f ^{n} (x) → x* as n → ∞.
Now, if there exist y ∈ X such that f(y) = y, then
and so, by Lemma 2.1, y = x*, i.e., F_{ f } = {x*}. This completes the proof. □
Corollary 2.4. Let (X, d, ≤) be an ordered metric space and f : X → X be an operator.
Let p be a wdistance on (X, d) and suppose that

(a)
if x, y ∈ X with (x, y)X _{≤} there exists c(x, y) ∈ X such that (x, c(x, y)) ∈ X _{≤} and (y, c(x, y)) ∈ X _{≤} ;

(b)
X _{≤} ∈ I(f × f ) ;

(c)
There exists x _{0} ∈ X such that (x _{0}, f (x _{0})) ∈ X _{≤} ;

(d)
(d _{1}) f is orbitally continuous or
(d_{2}) f is orbitally X_{≤}continuous and there exists a subsequenceof {f^{n} (x_{0})} such thatfor any k ∈ N ;

(e)
there is a comparison function φ : R _{+} → R _{+} such that
for any (x, y) ∈ X_{≤}, where

(f)
the metric d is complete,
Then, f is PO.
Corollary 2.5. Let (X, d, ≤) be an ordered metric space and f : X → X be an operator.
Let p be a wdistance on (X, d) and suppose that

(a)
if x, y ∈ X with (x, y)X _{≤}, then there exists c(x, y) ∈ X such that (x, c(x, y)) ∈ X _{≤} and (y, c(x, y)) ∈ X _{≤} ;

(b)
if (x, y) ∈ X _{≤} and (y, z) ∈ X _{≤}, then (x, z) ∈ X _{≤} ;

(c)
f is orbitally continuous (iv) there is a comparison function φ : R _{+} → R _{+} such that
for any (x, y) ∈ X_{≤}, where

(d)
the metric d is complete,
Then, f is PO.
Corollary 2.6. Let (X, d, ≤) be an ordered metric space and f : X → X be an operator.
Let p be a wdistance on (X, d) and suppose that

(a)
if x, y ∈ X with (x, y)X _{≤}, then there exists c(x, y) ∈ X such that (x, c(x, y)) ∈ X _{≤} and (y, c(x, y)) ∈ X _{≤} ;

(b)
X _{≤} ∈I(f × f ) ;

(c)
there exists x _{0} ∈ X such that (x _{0}, f (x _{0})) ∈ X _{≤} ;

(d)
if (x, y) ∈ X _{≤} and (y, z) ∈ X _{≤}, then (x, z) ∈ X _{≤} ;

(e)
(e _{1}) f is orbitally continuous or
(e_{2}) f is orbitally X_{≤}continuous and there exists a subsequenceof {f^{n} (x_{0})} such thatfor any k ∈ N ;

(f)
there is a comparison function φ : R _{+} → R _{+} such that
for any (x, y) ∈ X_{≤}, where

(g)
the metric d is complete,
Then, f is PO.
Corollary 2.7. Let (X, d, ≤) be an ordered metric space and f : X → X be an operator.
Let p be a wdistance on (X, d) and suppose that

(a)
if x, y ∈ X with (x, y)X _{≤}, then there exists c(x, y) ∈ X such that (x, c(x, y)) ∈ X _{≤} and (y, c(x, y)) ∈ X _{≤} ;

(b)
f is increasing or decreasing;

(c)
there exists x _{0} ∈ X such that (x _{0}, f (x _{0})) ∈ X _{≤} ;

(d)
(d _{1}) f is orbitally continuous or
(d_{2}) f is orbitally X_{≤}continuous and there exists a subsequenceof {f^{n} (x_{0})} such thatfor any k ∈ N ;

(e)
there is a comparison function φ : R _{+} → R _{+} such that
for any (x, y) ∈ X_{≤}, where

(f)
the metric d is complete,
Then, f is PO.
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Acknowledgements
The authors would like to thank the referees and area editor Professor Simeon Reich for giving useful suggestions and comments for the improvement of this article. Y. J. Cho was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF2008313C00050).
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All authors carried out the proof. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.
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Graily, E., Vaezpour, S.M., Saadati, R. et al. Generalization of fixed point theorems in ordered metric spaces concerning generalized distance. Fixed Point Theory Appl 2011, 30 (2011). https://doi.org/10.1186/16871812201130
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DOI: https://doi.org/10.1186/16871812201130