Recently, Nieto and Rodriguez-Lopez 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 w-distance 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 8 9 10 11 12 13 14 15 .

The aim of this article is to use the concept of w-distance 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 w-distance 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 semi-continuous;

(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 real-valued function f defined in a metric space X is said to be lower semi-continuous 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 w-distance 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, {fⁿ (x)} converges to x*.

(3) The operator f is called orbitally U-continuous 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 φⁿ (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.

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 w-distance 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 subsequence of {f ⁿ (x ₀)} such that for 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 _n-1, p_n } = p _n-1, then (3.1) follows. Otherwise, max{p _n-1, 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, {fⁿ (x ₀)} is a Cauchy sequence in X. Since X is complete, there exists x* ∈ X such that fⁿ (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 semi-continuity of p(fⁿ (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 semi-continuity of p(fⁿ (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 w-distance 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 subsequence of {fⁿ (x ₀)} such that for 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 w-distance 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 subsequence of {f ⁿ (x ₀)} such that for 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 (x0)) ∈ X _≤ and so

for any n ∈ N. Thus, by Lemma 1.2, fⁿ (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 fⁿ (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 w-distance 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 subsequence of {fⁿ (x ₀)} such that for 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 w-distance 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 w-distance 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 subsequence of {fⁿ (x ₀)} such that for 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 w-distance 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 subsequence of {fⁿ (x ₀)} such that for 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.

The authors declare that they have no competing interests.

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.