Abstract
In this article, we study coupled coincidence and coupled common fixed point theorems in ordered generalized metric spaces for nonlinear contraction condition related to a pair of altering distance functions. Our results generalize and modify several comparable results in the literature.
2000 MSC: 54H25; 47H10; 54E50.
Keywords:
ordered set; coupled coincidence point; coupled common fixed point; generalized metric space; altering distance function; weakly contractive condition; contraction of integral type1 Introduction
Fixed points of mappings in ordered metric space are of great use in many mathematical problems in applied and pure mathematics. The first result in this direction was obtained by Ran and Reurings [1], in this study the authors presented some applications of their obtained results to matrix equations. In [2,3], Nieto and López extended the result of Ran and Reurings [1] for nondecreasing mappings and applied their result to get a unique solution for a first order differential equation. While Agarwal et al. [4] and O'Regan and Petrutel [5] studied some results for a generalized contractions in ordered metric spaces. Bhaskar and Lakshmikantham [6] introduced the notion of a coupled fixed point of a mapping F from X × X into X. They established some coupled fixed point results and applied their results to the study of existence and uniqueness of solution for a periodic boundary value problem. Lakshmikantham and Ćirić [7] introduced the concept of coupled coincidence point and proved coupled coincidence and coupled common fixed point results for mappings F from X × X into X and g from X into X satisfying nonlinear contraction in ordered metric space. For the detailed survey on coupled fixed point results in ordered metric spaces, topological spaces, and fuzzy normed spaces, we refer the reader to [624].
On the other hand, in [25], Mustafa and Sims introduced a new structure of generalized metric spaces called Gmetric spaces. In [2632], some fixed point theorems for mappings satisfying different contractive conditions in such spaces were obtained. Abbas et al. [33] proved some coupled common fixed point results in two generalized metric spaces. While Shatanawi [34] established some coupled fixed point results in Gmetric spaces. Saadati et al. [35] established some fixed point in generalized ordered metric space. Recently, Choudhury and Maity [36] initiated the study of coupled fixed point in generalized ordered metric spaces.
In this article, we derive coupled coincidence and coupled common fixed point theorems in generalized ordered metric spaces for nonlinear contraction condition related to a pair of altering distance functions.
2 Basic concepts
Khan et al. [37] introduced the concept of altering distance function.
Definition 2.1. A function ϕ : [0, + ∞) → [0, + ∞) is called an altering distance function if the following properties are satisfied:
(1) ϕ is continuous and nondecreasing,
(2) ϕ (t) = 0 if and only if t = 0.
For more details on the following definitions and results, we refer the reader to Mustafa and Sims [25].
Definition 2.2. Let X be a nonempty set and let G : X × X × X → ℝ^{+ }be a function satisfying the following properties:
(G1) G(x, y, z) = 0 if and only if x = y = z,
(G2) 0 < G(x, x, y) for all x, y ∈ X with x ≠ y,
(G3) G(x, x, y) ≤ G(x, y, z) for all x, y, z ∈ X with z ≠ y,
(G4) G(x, y, z) = G(x, z, y) = G(y, z, x) = ...(: symmetry in all three variables),
(G5) G(x, y, z) ≤ G(x, a, a) + G(a, y, z) for all x, y, z, a ∈ X.
Then the function G is called a generalized metric or, more specifically, a Gmetric on X and the pair (X, G) is called a Gmetric space.
Definition 2.3. Let (X, G) be a Gmetric space and (x_{n}) be a sequence in X. We say that (x_{n}) is Gconvergent to a point x ∈ X or (x_{n}) Gconverges to x if, for any ε > 0, there exists k ∈ ℕ such that G(x, x_{n}, x_{m}) < ε for all m, n ≥ k, that is, . In this case, we write x_{n }→ x or .
Proposition 2.1. Let (X, G) be a Gmetric space. Then the following are equivalent:
(1) (x_{n}) is Gconvergent to x.
(2) G(x_{n}, x_{n}, x) → 0 as n → + ∞.
(3) G(x_{n}, x, x) → 0 as n → + ∞.
(4) G(x_{n}, x_{m}, x) → 0 as n, m → + ∞.
Definition 2.4. Let (X, G) be a Gmetric space and (x_{n}) be a sequence in X. We say that (x_{n}) is a GCauchy sequence if, for any ε > 0, there exists k ∈ ℕ such that G(x_{n}, x_{m}, x_{l}) < ε for all n, m, l ≥ k, that is, G(x_{n}, x_{m}, x_{l}) → 0 as n, m, l → +∞.
Proposition 2.2. Let (X, G) be a Gmetric space. Then the following are equivalent:
(1) The sequence (x_{n}) is a GCauchy sequence.
(2) For any ε > 0, there exists k ∈ ℕ such that G(x_{n}, x_{m}, x_{m}) < ε for all n, m ≥ k.
Definition 2.5. Let (X, G) and (X', G') be two Gmetric spaces. We say that a function f : (X, G) → (X', G') is Gcontinuous at a point a ∈ X if and only if, for any ε > 0, there exists δ > 0 such that
A function f is Gcontinuous on X if and only if it is Gcontinuous at every point a ∈ X.
Proposition 2.3. Let (X, G) be a Gmetric space. Then the function G is jointly continuous in all three of its variables.
We give some examples of Gmetric spaces.
Example 2.1. Let (ℝ, d) be the usual metric space. Define a function G_{s}:ℝ × ℝ × ℝ → ℝ by
for all x, y, z ∈ ℝ. Then it is clear that (ℝ, G_{s}) is a Gmetric space.
Example 2.2. Let X = {a, b}. Define a function G: X × X × X ℝ by
and extend G to X × X × X by using the symmetry in the variables. Then it is clear that (X, G) is a Gmetric space.
Definition 2.6. A Gmetric space (X, G) is said to be Gcomplete if every GCauchy sequence in (X, G) is Gconvergent in (X, G).
For more details about the following definitions, we refer the reader to [6,7].
Definition 2.7. Let X be a nonempty set and F : X × X → X be a given mapping. An element (x, y) ∈ X × X is called a coupled fixed point of F if F(x, y) = x and F(y, x) = y.
Definition 2.8. Let (X, ≤) be a partially ordered set. A mapping F : X × X → X is said to have the mixed monotone property if F(x,y) is monotone nondecreasing in x and is monotone nonincreasing in y, that is, for any x, y ∈ X,
and
Lakshmikantham and Ćirić [7] introduced the concept of a gmixed monotone mapping.
Definition 2.9. Let (X, ≤) be a partially ordered set, F : X × X → X and g : X → X be mappings. The mapping F is said to have the mixed gmonotone property if F(x, y) is monotone gnondecreasing in x and is monotone gnonincreasing in y, that is, for any x, y ∈ X,
and
Definition 2.10. Let X be a nonempty set, F : X × X → X and g : X → X be mappings. An element (x, y) ∈ X × X is called a coupled coincidence point of F and g if F(x, y) = gx and F(y, x) = gy.
Definition 2.11. Let X be a nonempty set, F : X × X → X and g : X → X be mappings. An element (x, y) ∈ X × X is called a coupled common fixed point of F and g if F(x, y) = gx = x and F(y, x) = gy = y.
Definition 2.12. Let X be a nonempty set, F : X × X → X and g: X → X be mappings. We say that F and g are commutative if g(F(x, y)) = F(gx, gy) for all x, y ∈ X.
Definition 2.13. Let X be a nonempty set, F : X × X → X and g : X → X be mappings. Then F and g are said to be weak* compatible (or w*compatible) if g(F(x, x)) = F(gx, gx) whenever g(x) = F(x, x).
3 Main results
The following is the first result.
Theorem 3.1. Let (X, ≤) be a partially ordered set and (X, G) be a complete Gmetric space. Let F : X × X → X and g : X → X be continuous mappings such that F has the mixed gmonotone property and g commutes with F. Assume that there are altering distance functions ψ and ϕ such that
for all x, y, u, v, w, z ∈ X with gw ≤ gu ≤ gx and gy ≤ gv ≤ gz. Also, suppose that F(X × X) ⊆ g(X). If there exist x_{0}, y_{0 }∈ X such that gx_{0 }≤ F(x_{0}, y_{0}) and F(y_{0}, x_{0}) ≤ gy_{0}, then F and g have a coupled coincidence point.
Proof. Let x_{0}, y_{0 }∈ X such that gx_{0 }≤ F(x_{0}, y_{0}) and F(y_{0}, x_{0}) ≤ gy_{0}. Since we have F(X × X) ⊆ g(X), we can choose x_{1}, y_{1 }∈ X such that gx_{1 }= F(x_{0}, y_{0}) and gy_{1 }= F(y_{0}, x_{0}). Again, since F(X × X) ⊆ g(X), we can choose x_{2}, y_{2 }∈ X such that gx_{2 }= F(x_{1}, y_{1}) and gy_{2 }= F(y_{1}, x_{1}). Since F has the mixed gmonotone property, we have gx_{0 }≤ gx_{1 }≤ gx_{2 }and gy_{2 }≤ gy_{1 }≤ gy_{0}. Continuing this process, we can construct two sequences (x_{n}) and (y_{n}) in X such that
and
If, for some integer n, we have (gx_{n+1}, gy_{n+1}) = (gx_{n}, gy_{n}), then F(x_{n}, y_{n}) = gx_{n }and F(y_{n}, x_{n}) = gy_{n}, that is, (x_{n}, y_{n}) is a coincidence point of F and g. So, from now on, we assume that (gx_{n+1}, gy_{n+1}) ≠ (gx_{n}, gy_{n}) for all n ∈ ℕ, that is, we assume that either gx_{n+1 }≠ gx_{n }or gy_{n+1 }≠ gy_{n}.
We complete the proof with the following steps.
Step 1: We show that
For each n ∈ ℕ, using the inequality (1), we obtain
Since ψ is a nondecreasing function, we get
On the other hand, we have
Since ψ is a nondecreasing function, we get
Thus, by (4) and (6), we have
Thus (max{G(gx_{n1}, gx_{n}, gx_{n}), G(gy_{n1}, gy_{n}, gy_{n})}) is a nonnegative decreasing sequence. Hence, there exists r ≥ 0 such that
Now, we show that r = 0. Since ϕ : [0, + ∞) → [0, + ∞) is a nondecreasing function, then, for any a, b ∈ [0, + ∞), we have ψ(max{a, b}) = max{ψ(a), ψ(b)}. Thus, by (3)) and (5), we have
Letting n → +∞ in the above inequality and using the continuity of ψ, we get
Hence ϕ(r) = 0. Thus r = 0 and (2) holds.
Step 2: We show that (gx_{n}) and (gy_{n}) are GCauchy sequences. Assume that (x_{n}) or (y_{n}) is not a GCauchy sequence, that is,
or
This means that there exists ϵ > 0 for which we can find subsequences of integers (m(k)) and (n(k)) with n(k) > m(k) > k such that
Further, corresponding to m(k) we can choose n(k) in such a way that it is the smallest integer with n(k) > m(k) and satisfying (7). Then we have
Thus, by (G_{5}) and (8), we have
Thus, by (2), we have
Similarly, we have
Thus, by (9) and (10), we have
Using (7), we get
Now, using the inequality (1), we obtain
and
Thus, by (12) and (13), we get
Letting k → +∞ in the above inequality and using (11) and the fact that ψ and ϕ are continuous, we get
Hence ϕ(ϵ) = 0 and so ϵ = 0, which is a contradiction. Therefore, (gx_{n}) and (gy_{n}) are GCauchy sequences.
Step 3: The existence of a coupled coincidence point. Since (gx_{n}) and (gy_{n}) are GCauchy sequences in a complete Gmetric space (X, G), there exist x, y ∈ X such that (gx_{n}) and (gyn) are Gconvergent to points x and y, respectively, that is,
and
Then, by (14), (15) and the continuity of g, we have
and
Therefore, (g(gx_{n})) is Gconvergent to gx and (g(gy_{n})) is Gconvergent to gy. Since F and g commute, we get
and
Using the continuity of F and letting n → +∞ in (18) and (19), we get gx = F(x, y) and gy = F(y, x). This implies that (x, y) is a coupled coincidence point of F and g. This completes the proof.
Tacking g = I_{X }(: the identity mapping) in Theorem 3.1., we obtain the following coupled fixed point result.
Corollary 3.1. Let (X, ≤) be a partially ordered set and (X, G) be a complete Gmetric space. Let F : X × X → X be a continuous mapping satisfying the mixed monotone property. Assume that there exist the altering distance functions ψ and ϕ such that
for all x, y, u, v, w, z ∈ X with w ≤ u ≤ x and y ≤v ≤ z. If there exist x_{0}, y_{0 }∈ X such that x_{0 }≤ F(x_{0}, y_{0}) and F(y_{0}, x_{0}) ≤ y_{0}, then F has a coupled fixed point.
Now, we derive coupled coincidence point results without the continuity hypothesis of the mappings F, g and the commutativity hypothesis of F, g. However, we consider the additional assumption on the partially ordered set (X, ≤).
We need the following definition.
Definition 3.1. Let (X, ≤) be a partially ordered set and G be a Gmetric on X. We say that (X, G, ≤) is regular if the following conditions hold:
(1) if a nondecreasing sequence (x_{n}) is such that x_{n }→ x, then x_{n }≤ x for all n ∈ ℕ,
(2) if a nonincreasing sequence (y_{n}) is such that y_{n }→ y, then y ≤ y_{n }for all n ∈ ℕ.
The following is the second result.
Theorem 3.2. Let (X, ≤) be a partially ordered set and G be a Gmetric on × such that (X, G, ≤) is regular. Assume that there exist the altering distance functions ψ, ϕ and mappings F : X × X → X and g: X → X such that
for all x, y, u, v, w, z ∈ X with gw ≤ gu ≤ gx and gy ≤ gv ≤ gz. Suppose also that (g(X), G) is Gcomplete, F has the mixed gmonotone property and F(X × X) ⊆ g(X). If there exist x_{0}, y_{0 }∈ X such that gx_{0 }≤ F(x_{0}, y_{0}) and F(y_{0}, x_{0}) ≤ gy_{0}, then F and g have a coupled coincidence point.
Proof. Following Steps 1 and 2 in the proof of Theorem 3.1., we know that (gx_{n}) and (gy_{n}) are GCauchy sequences in g(X) with gx_{n }≤ gx_{n+1 }and gy_{n }≥ gy_{n+1 }for all n ∈ ℕ. Since (g(X), G) is Gcomplete, there exist x, y ∈ X such that gx_{n }→ gx and gy_{n }→ gy. Since (X, G, ≤) is regular, we have gx_{n }≤ gx and gy ≤ gy_{n }for all n ∈ ℕ. Thus we have
Letting n → +∞ in the above inequality and using the continuity of ψ and ϕ, we obtain ψ(G(F(x, y),gx, gx)) = 0, which implies that G(F(x, y), gx, gx) = 0. Therefore, F(x, y) = gx.
Similarly, one can show that F(y, x) = gy. Thus (x, y) is a coupled coincidence point of F and g, this completes the proof.
Tacking g = I_{X }in Theorem 3.2., we obtain the following result.
Corollary 3.2. Let (X, ≤) be a partially ordered set and G be a Gmetric on X such that (X, G, ≤) is regular and (X, G) is Gcomplete. Assume that there exist the altering distance functions ψ, ϕ and a mapping
F : X × X → X having the mixed monotone property such that
for all x, y, u, v, w, z ∈ X with w ≤ u ≤ x and y ≤ v ≤ z. If there exist x_{0}, y_{0 }∈ X such that x_{0 }≤ F(x_{0}, y_{0}) and F(y_{0}, x_{0}) ≤ y_{0}, then F has a coupled fixed point.
Now, we prove the existence and uniqueness theorem of a coupled common fixed point. If (X, ≤) is a partially ordered set, we endow the product set X × X with the partial order defined by
Theorem 3.3. In addition to the hypotheses of Theorem 3.1., suppose that, for any (x, y), (x*, y*) ∈ X × X, there exists (u, v) ∈ X × X such that (F(u, v), F(v, u)) is comparable with (F(x, y), F(y, x)) and (F(x*, y*), F(y*, x*)). Then F and g have a unique coupled common fixed point, that is, there exists a unique (x, y) ∈ X × X such that x = gx = F(x, y) and y = gy = F(y, x).
Proof. From Theorem 3.1., the set of coupled coincidence points is nonempty. We shall show that if (x, y) and (x*, y*) are coupled coincidence points, then
By the assumption, there exists (u, v) ∈ X × X such that (F(u, v), F(v, u)) is comparable to (F(x, y), F(y, x)) and (F(x*, y*), F(y*, x*)). Without the restriction to the generality, we can assume that (F(x, y), F(y, x)) ≤ (F(u, v), F(v, u)) and (F(x*, y*), F(y*, x*)) ≤ (F(u, v), F(v, u)). Put u_{0 }= u, v_{0 }= v and choose u_{1}, v_{1 }∈ X so that gu_{1 }= F(u_{0}, v_{0}) and gv_{1 }= F(v_{0}, u_{0}). As in the proof of Theorem 3.1., we can inductively define the sequences (u_{n}) and (v_{n}) such that
Further, set and, by the same way, define the sequences (x_{n}), (y_{n}) and . Since (gx, gy) = (F(x, y), F(y, x)) = (gx_{1}, gy_{1}) and (F(u, v), F(v, u)) = (gu_{1}, gv_{1}) are comparable, gx ≤ gu_{1 }and gv_{1 }≤ gy. One can show, by induction, that
for all n ∈ ℕ. From (1), we have
and
Hence it follows that
Since ψ is nondecreasing, it follows that (max{G(gx, gx, gu_{n}),G(gy, gy, gv_{n})}) is a decreasing sequence.
Hence there exists a nonnegative real number r such that
Using (21) and letting n → +∞ in the above inequality, we get
Therefore, ϕ(r) = 0 and hence r = 0. Thus
Similarly, we can show that
Thus, by (G_{5}), (22), and (23), we have, as n → +∞,
and
Hence gx = gx* and gy = gy*. Thus we proved (20).
On the other hand, since gx = F(x, y) and gy = F(y, x), by commutativity of F and g, we have
Denote gx = z and gy = w. Then, from (24), it follows that
Thus (z, w) is a coupled coincidence point. Then, from (20) with x* = z and y* = w, it follows that gz = gx and gw = gy, that is,
Thus, from (25) and (26), we have z = gz = F(z, w) and w = gw = F(w, z). Therefore, (z, w) is a coupled common fixed point of F and g.
To prove the uniqueness of the point (z, w), assume that (s, t) is another coupled common fixed point of F and g. Then we have
Since the pair (s, t) is a coupled coincidence point of F and g, we have gs = gx = z and gt = gy = w. Thus s = gs = gz = z and t = gt = gw = w. Hence, the coupled fixed point is unique. this completes the proof.
Now, we present coupled coincidence and coupled common fixed point results for mappings satisfying contractions of integral type. Denote by Λ the set of functions α : [0, +∞) → [0, + ∞) satisfying the following hypotheses:
(h1) α is a Lebesgue integrable mapping on each compact subset of [0, + ∞),
Finally, we give the following results.
Theorem 3.4. Let (X, ≤) be a partially ordered set and (X, G) be a complete Gmetric space. Let F : X × X → X and g : X → X be continuous mappings such that F has the mixed gmonotone property and g commutes with F. Assume that there exist α, β ∈ Λ such that
for all x, y, u, v, w, z ∈ X with gw ≤ gu ≤ gx and gy ≤ gv ≤ gz. Also, suppose that F(X × X) ⊆ g(X). If there exist x_{0}, y_{0 }∈ X such that gx_{0 }≤ F(x_{0}, y_{0}) and F(y_{0}, x_{0}) ≤ gy_{0}, then F and g have a coupled coincidence point.
Proof. We consider the functions ψ, ϕ : [0, +∞) → [0, +∞) defined by
for all t ≥ 0. It is clear that ψ and ϕ are altering distance functions. Then the results follow immediately from Theorem 3.1.. This completes the proof.
Corollary 3.3. Let (X, ≤) be a partially ordered set and (X, G) be a complete Gmetric space. Let F : X × X → X be a continuous mappings satisfying the mixed monotone property. Assume that there exist α, β ∈ Λ such that
for all x, y, u, v, w, z ∈ X with w ≤ u ≤ x and y ≤ v ≤ z. If there exist x_{0}, y_{0 }∈ X such that x_{0 }≤ F(x_{0}, y_{0}) and F(y_{0}, x_{0}) ≤ y_{0}, then F has a coupled fixed point.
Proof. Tacking g = I_{X }in Theorem 3.3., we obtain Corollary 3.3..
Putting β(s) = (1  k)α(s) with k ∈ [0,1) in Theorem 3.3., we obtain the following result.
Corollary 3.4. Let (X, ≤) be a partially ordered set and (X, G) be a complete Gmetric space. Let F : X × X → X and g: X → X be continuous mappings such that F has the mixed gmonotone property and g commutes with F. Assume that there exist α ∈ Λ and k ∈ [0, 1) such that
for all x, y, u, v, w, z ∈ X with gw ≤ gu ≤ gx and gy ≤ gv ≤ gz. Also, suppose that F(X × X) ⊆ g(X).
If there exist x_{0}, y_{0 }∈ X such that gx_{0 }≤ F(x_{0}, y_{0}) and F(y_{0}, x_{0}) ≤ gy_{0}, then F and g have a coupled coincidence point.
Tacking α(s) = 1 in Corollary 3.4., we obtain the following result.
Corollary 3.5. Let (X, ≤) be a partially ordered set and (X, G) be a complete Gmetric space. Let F : X × X → X and g : X → X be continuous mappings such that F has the mixed gmonotone property and g commutes with F. Assume that there exists k ∈ [0, 1) such that
for all x, y, u, v, w, z ∈ X with gw ≤ gu ≤ gx and gy ≤ gv ≤ gz. Also, suppose that F(X × X) ⊆ g(X). If there exist x_{0}, y_{0 }∈ X such that gx_{0 }≤ F(x_{0}, y_{0}) and F(y_{0}, x_{0}) ≤ gy_{0}, then F and g have a coupled coincidence point.
Corollary 3.6. Let (X, ≤) be a partially ordered set and (X, G) be a complete Gmetric space. Let F : X × X → X and g : X → X be continuous mappings such that F has the mixed gmonotone property and g commutes with F. Assume that there exist nonnegative real numbers a, b with a + b ∈ [0,1) such that
for all x, y, u, v, w, z ∈ X with gw ≤ gu ≤ gx and gy ≤ gv ≤ gz. Also, suppose that F(X × X) ⊆ g(X). If there exist x_{0}, y_{0 }∈ X such that gx_{0 }≤ F(x_{0}, y_{0}) and F(y_{0}, x_{0}) ≤ gy_{0}, then F and g have a coupled coincidence point.
Proof. We have
for all x, y, u, v, w, z ∈ X with gw ≤ gu ≤ gx and gy ≤ gv ≤ gz. Then Corollary 3.6. follows from Corollary 3.5..
Remark 3.1. Note that similar results can be deduced from Theorems 3.2. and 3.3..
Remark 3.2. (1) Theorem 3.1 in [36] is a special case of Theorem 3.1..
(2) Theorem 3.2 in [36] is a special case of Theorem 3.2..
Example 3.1. Let X = 0,1, 2, 3,... and G : X × X × X → R^{+ }be defined as follows:
Then (X, G) is a complete Gmetric space [36]. Let a partial order ≼ on X be defined as follows: For x, y ∈ X, x ≼ y holds if x > y and 3 divides (x  y) and 3 ≼ 1 and 0 ≼ 1 hold. Let F : X × X → X be defined as follows:
Let w ≼ u ≼ x ≼ y ≼ v ≼ z hold, then equivalently, we have w ≥ u ≥ x ≥ y ≥ v ≥ z. Then F(x, y) = F(u, v) = F(w, z) = 1. Let for t ≥ 0 and k ∈ [0,1) and let g(x) = x for x ∈ X. Thus lefthand side of (1) is G(1, 1,1) = 0 and hence (1) is satisfied. Then with x_{0 }= 81 and y_{0 }= 0 the Theorem 3.2. is applicable to this example. It may be observed that in this example the coupled fixed point is not unique. Hence, (0,0) and (1,0) are two coupled fixed point of F.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
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.
Acknowledgements
YJC was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 20110021821).
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