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Tripled coincidence point results for generalized contractions in ordered generalized metric spaces
Fixed Point Theory and Applications volume 2012, Article number: 101 (2012)
Abstract
In this paper, we establish some tripled coincidence point results for a mixed g-monotone mapping F : X3 → X satisfying (ψ, ϕ)-contractions in ordered generalized metric spaces. Also, an application and some examples are given to support our results.
Introduction and preliminaries
Banach contraction principle is one of the core subject that has been studied. It has so many different generalizations with different approaches. One of the remarkable generalizations, known as Φ-contraction, was given by Boyd and Wong [1] in 1969. In 1997, Alber and Guerre-Delabriere [2], introduced the notion of a weak ϕ-contraction which generalizes Boyd and Wong results, so Banach's result. Very recently, inspired from the notion of weak ϕ-contractions, a new concept of ( ψ, ϕ)-contractions was introduced (see e.g. [3–7]).
Mustafa and Sims [8] introduced the notion of generalized metric spaces or simply G-metric spaces as a generalization of the concept of a metric space. Based on the concept of G-metric space, Mustafa et al. [9–11] proved several fixed point theorems for mapping satisfying different contractive condition. The study of common fixed point was initiated by Abbas and Rhoades [12]. The first result for contractive mappings in ordered G-metric spaces was obtained by Saadati et al. [13]. Bhashkar and Lakshmikantham [14] introduced the concept of a coupled fixed point of a mapping F : X × X → X and proved some coupled fixed point theorems in ordered metric spaces. Some authors obtained some interesting coupled fixed point theorems in G-metric spaces (see e.g. [15–18]). For more results on G-metric spaces, one can refer to the papers [9–13, 15–28].
Throughout the paper, ℕ* is the set of positive integers. In 2004, Mustafa and Sims [8] introduced the concept of G-metric spaces as follows:
Definition 1. (see [8]). Let X be a non-empty set, G : X × X × X → ℝ+ be a function satisfying the following properties
(G1) G(x, y, z) = 0 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 y ≠ z,
(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 (rectangle inequality).
Then the function G is called a generalized metric, or, more specially, a G-metric on X, and the pair (X, G) is called a G-metric space.
Every G-metric on X defines a metric d G on X by
Example 2. Let (X, d) be a metric space. The function G : X × X × X → [0, +∞), defined by
or
for all x, y, z ∈ X, is a G-metric on X.
Definition 3. (see [8]). Let (X, G) be a G-metric space, and let {x n } be a sequence of points of X, therefore, we say that {x n } is G-convergent to x ∈ X if , that is, for any ε > 0, there exists N ∈ ℕ such that G(x, x n , x m ) < ε, for all n, m ≥ N. We call x the limit of the sequence and write x n → x or .
Proposition 4. (see [8]). Let (X,G) be a G-metric space. The following are equivalent:
-
(1)
{x n } is G-convergent 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 5. (see [8]). Let (X, G) be a G-metric space. A sequence {x n } is is called a G-Cauchy sequence if, for any ε > 0, there is N ∈ ℕ such that G(x n , x m , x l ) < ε for all m, n, l ≥ N, i.e., G(x n , x m , x l ) → 0 as n, m, l → ∞.
Proposition 6. (see [8]). Let (X, G) be a G-metric space. Then the following are equivalent:
-
(1)
the sequence {x n } is G-Cauchy,
-
(2)
for any ε > 0, there exists N ∈ ℕ such that G(x n , x m , x m ) < ε, for all m, n ≥ N.
Definition 7. (see [8]). A G-metric space (X, G) is called G-complete if every G-Cauchy sequence is G-convergent in (X, G).
Definition 8. Let (X, G) be a G-metric space. A mapping F : X × X × X → X is said to be continuous if for any three G-convergent sequences {x n }, {y n } and {z n } converging to x, y and z respectively, {F(x n , y n , z n )} is G-convergent to F(x, y, z).
Following the paper of Berinde and Borcut [29], Aydi, Karapınar and Postolache [30] introduced the following definitions:
Definition 9. (see [30]). Let (X, ≤) be a partially ordered set and F : X × X × X → X, g : X → X. The mapping F is said to has the mixed g-monotone property if for any x, y, z ∈ X
Definition 10. (see [30]). Let F : X × X × X → X and g : X → X. An element (x, y, z) is called a tripled coincidence point of F and g if
The point (gx, gy, gz) is called a point of coincidence of F and g.
Definition 11. (see [30]). Let F : X × X × X → X and g : X → X. An element (x, y, z) is called a tripled common fixed point of F and g if
Definition 12. (see [30]). Let X be a non-empty set. Let F: X × X × X → X and g: X → X are such that
whenever x, y, z ∈ X, then F and g are said to be commutative.
Khan et al. [31] introduced the concept of altering distance function as follows:
Definition 13. (altering distance function, [31]) The function ψ : [0, + ∞) → [0, + ∞) is called an altering distance function if the following properties are satisfied:
-
(1)
ψ is continuous and non-decreasing,
-
(2)
ψ(t) = 0 if and only if t = 0.
Let Ψ be the set of altering distances. Again, we denote by Φ the set of functions ϕ : [0, +∞) → [0, +∞) such that
-
(i)
ϕ is lower-continuous and non-decreasing,
-
(ii)
ϕ(t) = 0 if and only if t = 0.
The notion of a fixed point of N-order was first introduced by Samet and Vetro [32]. Later, Berinde and Borcut [29] proved some tripled fixed point results (N = 3) in partially ordered metric spaces (see also [33–36]). In this paper, we establish tripled coincidence point results for mappings F : X3 → X and g : X → X involving nonlinear contractions in the setting of ordered G-metric spaces. Also, we present an application and some examples in support of our results.
Main results
Before stating our results, we give the following useful lemma.
Lemma 14. Consider three non-negative real sequences {a n }, {b n } and {c n }. Suppose there exists α ≥ 0 such that
Then, .
Proof. First, we have c n ≤ max{a n , b n , c n }, then
For all n ∈ ℕ, we have
which implies that . Having in mind that
so it follows that
By (3) and (4), we get that . □
The aim of this paper is to prove the following theorem.
Theorem 15. Let (X, ≤) be a partially ordered set and (X, G) be a G-metric space such that (X, G) is G-complete. Let F : X3 → X and g : X → X. Assume there exist ψ ∈ Ψ and ϕ ∈ Φ such that for x, y, z, a, b, c, u, v, w ∈ X, with gx ≥ ga ≥ gu, gy ≤ gb ≤ gv and gz ≥ gc ≥ gw, we have
Assume that F and g satisfy the following conditions:
-
(1)
F(X3) ⊆ g(X),
-
(2)
F has the mixed g-monotone property,
-
(3)
F is continuous,
-
(4)
g is continuous and commutes with F.
Suppose there exist x0, y0, z0 ∈ X such that gx0 ≤ F(x0, y0, z0), gy0 ≥ F(y0, x0, y0) and gz0 ≤ F(z0, y0, x0), then F and g have a tripled coincidence point in X, i.e., there exist x, y, z ∈ X such that
Proof. Suppose x0, y0, z0 ∈ X are such that gx0 ≤ F(x0, y0, z0), gy0 ≥ F(y0, x0, y0), and gz0 ≤ F(z0, y0, x0). Since F(X3) ⊆ g(X), we can choose gx1 = F(x0, y0, z0), gy1 = F(y0, x0, y0) and gz1 = F(z0, y0, x0). Then gx0 ≤ gx1, gy0 ≥ gy1 and gz0 ≤ gz1. Similarly, define gx2 = F(x1, y1, z1), gy2 = F(y1, x1, y1) and gz2 = F(z1, y1, x1). Since F has the mixed g-monotone property, we have gx0 ≤ gx1 ≤ gx2, gy2 ≤ gy1 ≤ gy0 and gz0 ≤ gz1 ≤ gz2. Continuing this process, we can construct three sequences {x n }, {y n } and {z n } in X such that
and
If, for some integer n0, we have , then , and ; i.e., is a tripled coincidence point of F and g. Thus we shall assume that (gx n +1, gy n +1, gz n +1) ≠ (gx n , gy n , gz n ) for all n ∈ ℕ; i.e., we assume that either gx n +1 ≠ gx n or gy n +1 ≠ gy n or gz n +1 ≠ gz n . For any n ∈ ℕ*, we have from (5)
and
Since ψ:[0,+∞) → [0, +∞) is a non-decreasing function, for a, b, c ∈ [0,+∞), we have ψ(max{a, b, c}) = max{ψ(a), ψ(b), ψ(c)}. Then, from (6), (7), and (8), it follows that
The fact that ψ is non-decreasing yields that
Thus, {max{G(gx n +1, gx n , gx n ),G(gy n , gy n , gy n +1),G(gz n +1, gz n , gz n )}} is a positive non-increasing sequence. Hence there exists r ≥ 0 such that
Having in mind that ϕ is non-decreasing, then
so from (6)-(8), we get that
On the other hand,
so by (10), the real sequence {max{G(gx n , gx n -1, gx n -1),G(gy n -1, gy n -1, gy n )}} is bounded. Thus, there exists a real number r1 with 0 ≤ r1 ≤ r and subsequences {x n ( k )} of {x n } and {y n ( k )} of {y n } such that
We rewrite (12)
Letting k → +∞ in (15), having in mind (10), (14), the continuity of ψ and the lower semi-continuity of ϕ, we obtain
which implies that ϕ(r1) = 0, and using a property of ϕ, we find r1 = 0. Thanks to Lemma 14 together with (10) and (14), it yields that
For any k ∈ ℕ, we rewrite (8) as
Again, letting k → +∞ in (17), having in mind (10), (16) and by the properties of ψ, ϕ, we obtain
which gives that ϕ(r) = 0, so r = 0, i.e., by (10),
Our next step is to show that {gx n }, {gy n } and {gz n } are G-Cauchy sequences.
Assume the contrary, i.e., {gx n }, {gy n } or {gz n } is not a G-Cauchy sequence, i.e.,
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 (19). Then
By (G5) and (20), we have
Thus, by (18) we obtain
Similarly, we have
Again by (G 5) and (20), we have
Letting k → +∞ and using (18), we get
Using (19) and (24)-(26), we have
Now, using inequality (5) we obtain
and
We deduce from (28)-(30) that
On the other hand, since
then from (27),
Therefore, the real sequence is bounded. Thus, up to a subsequence (still denoted the same), there exists ε1 with 0 ≤ ε1 ≤ ε such that
Inserting this in (31) and using (27), (33) together with the properties of ψ, ϕ, we get that
which leads to ϕ(ε1) = 0, so ε1 = 0. By this and (27), due to Lemma 14, we obtain
Combining this to (19) and (26), we find
Letting k → ∞ in (30) and using (27), we deduce
i.e., ε = 0, it is a contradiction. We conclude that {gx n }, {gy n } and {gz n } are G-Cauchy sequences in the G-metric space (X, G), which is G-complete. Then, there are x, y, z ∈ X such that {gx n }, {gy n } and {gz n } are respectively G-convergent to x, y and z, i.e., from Proposition 4, we have
From (34)-(36) and the continuity of g, we get thanks to Proposition 8
Since gx n +1 = F(x n , y n , z n ), gy n +1 = F(y n , x n , y n ) and gz n +1 = F(z n , y n , x n ), so the commutativity of F and g yields that
Now we show that F(x, y, z) = gx, F(y, x, y) = gy and F(z, y, x) = gz.
The mapping F is continuous, so since the sequences {gx n }, {gy n } and {gz n } are, respectively, G-convergent to x, y and z, hence using Definition 8, the sequence {F(gx n , gy n , gz n )} is G-convergent to F(x, y, z). Therefore, from (40), {g(gx n +1)} is G-convergent to F(x, y, z). By uniqueness of the limit, from (37) we have F(x, y, z) = gx.
Similarly, one finds F(y, x, y) = gy and F(z, y, x) = gz, and this makes end to the proof. □
Corollary 16. Let (X, ≤) be a partially ordered set and (X, G) be a G-metric space such that (X, G) is G-complete. Let F : X3 → X and g : X → X. Assume there exists k ∈ [0,1) such that for x, y, z, a, b, c, u, v, w ∈ X, with gx ≥ ga ≥ gu, gy ≤ gb ≤ gv and gz ≥ gc ≥ gw, we have
Assume that F and g satisfy the following conditions:
-
(1)
F(X3) ⊂ g(X),
-
(2)
F has the mixed g-monotone property,
-
(3)
F is continuous,
-
(4)
g is continuous and commutes with F.
Suppose there exist x0, y0, z0 ∈ X such that gx0 ≤ F(x0, y0, z0), gy 0 ≥ F(y0, x0, y0) and gz0 ≤ F(z0, y0, x0), then F and g have a tripled coincidence point in X, i.e., there exist x, y, z ∈ X such that
Proof. If follows by taking ψ(t) = t and ϕ(t) = (1 - k)t for all t ≥ 0. □
Corollary 17. Let (X, ≤) be a partially ordered set and (X, G) be a G-metric space such that (X, G) is G-complete. Let F : X3 → X and g : X → X. Assume there exists k ∈ [0,1) such that for x, y, z, a, b, c, u, v, w ∈ X, with gx ≥ ga ≥ gu, gy ≤ gb ≤ gv and gz ≥ gc ≥ gw, we have
Assume that F and g satisfy the following conditions:
-
(1)
F(X3) ⊆g(X),
-
(2)
F has the mixed g-monotone property,
-
(3)
F is continuous,
-
(4)
g is continuous and commutes with F.
Suppose there exist x0, y0, z0 ∈ X such that gx0 ≤ F(x0, y0, z0), gy0 ≥ F(y0, x0, y0) and gz0 ≤ F(z0, y0, x0), then F and g have a tripled coincidence point in X, i.e., there exist x, y, z ∈ X such that
Proof. It suffices to remark that
□
In the next theorem, we omit the continuity hypothesis of F. We need the following definition.
Definition 18. Let (X, ≼) be a partially ordered set and G be a G-metric on X. We say that (X, G, ≤) is regular if the following conditions hold:
-
(i)
if a non-decreasing sequence {x n } is such that x n → x, then x n ≤ x for all n,
-
(ii)
if a non-increasing sequence {y n } is such that y n → y, then y ≤ y n for all n.
Theorem 19. Let (X, ≤) be a partially ordered set and (X, G) be a G-metric space. Let F : X3 → X and g: X → X. Assume there exist ψ ∈ Ψ and ϕ ∈ Φ such that for x, y, z, a, b, c, u, v, w ∈ X, with gx ≥ ga ≥ gu, gy ≤ gb ≤ gv and gz ≥ gc ≥ gw, we have
Assume that (X, G, ≤) is regular. Suppose that (g(X),G) is G-complete, F has the mixed g-monotone property and F(X × X × X) ⊆ g(X). Also, assume there exist x0, y0, z0 ∈ X such that gx0 ≤ F(x0, y0, z0), gy0 ≥ F(y0, x0, y0) and gz 0 ≤ F(z0, y0, x0), then F and g have a tripled coincidence point in X, i.e., there exist x, y, z ∈ X such that
Proof. Proceeding exactly as in Theorem 15, we have that {gx n }, {gy n } and {gz n } are G-Cauchy sequences in the G-complete G-metric space (g(X), G). Then, there exist x, y, z ∈ X such that gx n → gx, gy n → gy and gz n → gz. Since {gx n } and {gz n } are non-decreasing and {gy n } is non-increasing, using the regularity of (X, G, ≤), we have gx n ≤ gx, gz n ≤ gz and gy ≤ gy n for all n ≥ 0. Using (5), we get
Letting n → +∞ in the above inequality, we obtain that
which implies that G(F(x, y, z), gx, gx) = 0, i.e., gx = F(x, y, z).
Similarly, one can show that gy = F(y, x, y) and gz = F(z, y, x). Thus we proved that (x, y, z) is a tripled coincidence point of F and g. □
Similarly, we can state the following corollary.
Corollary 20. Let (X, ≤) be a partially ordered set and (X, G) be a G-metric space. Let F : X3 → X and g: X → X. Assume there exists k ∈ [0, 1) such that for x, y, z, a, b, c, u, v, w ∈ X, with gx ≥ ga ≥ gu, gy ≤ gb ≤ gv and gz ≥ gc ≥ gw, we have
Assume that (X, G, ≤) is regular. Suppose that (g(X),G) is G-complete, F has the mixed g-monotone property and F(X × X × X) ⊆ g(X). Also, assume there exist x0, y0, z0 ∈ X such that gx0 ≤ F(x0, y0, z0), gy 0 ≥ F(y0, x0, y0) and gz0 ≤ F(z0, y0, x0), then F and g have a tripled coincidence point in X, i.e., there exist x, y, z ∈ X such that
Corollary 21. Let (X, ≤) be a partially ordered set and (X, G) be a G-metric space. Assume that (X, G, ≤) is regular. Let F : X3 → X and g : X → X. Assume there exists k ∈ [0, 1) such that for x, y, z, a, b, c, u, v, w ∈ X, with gx ≥ ga ≥ gu, gy ≤ gb ≤ gv and gz ≥ gc > gw, we have
Suppose that (g(X),G) is G-complete, F has the mixed g-monotone property and F(X × X × X) ⊆ g(X). Also, assume there exist x0, y0, z0 ∈ X such that gx0 ≤ F(x0, y0, z0), gy0 ≥ F(y0, x0, y0) and gz0 < F(z0, y0, x0), then F and g have a tripled coincidence point in X, i.e., there exist x, y, z ∈ X such that
Remark 22. Other corollaries could be derived from Theorems 15 and 19 by taking g = Id x .
Now, form previous obtained results, we will deduce some tripled coincidence point results for mappings satisfying a contraction of integral type in G-metric space. Let us introduce first some notations.
We denote by Γ the set of functions α: [0, +∞) → [0, +∞) satisfying the following conditions:
-
(i)
α is a Lebesgue integrable mapping on each compact subset of [0, +∞),
-
(ii)
for all ε > 0, we have
Let N ∈ ℕ* be fixed. Let {α i }1 ≤ i ≤ N be a family of N functions that belong to Γ. For all t ≥ 0, we denote (I i )i =1,..., N as follows:
We have the following result.
Theorem 23. Let (X, ≤) be a partially ordered set and (X, G) be a G-metric space such that (X, G) is G-complete. Let F : X3 → X and g : X → X. Assume there exist ψ ∈ Ψ and ϕ ∈ Φ such that for x, y, z, a, b, c, u, v, w ∈ X, wth gx ≥ ga ≥ gu, gy ≤ gb ≤ gv and gz ≥ gc ≥ gw, we have
Assume that F and g satisfy the following conditions:
-
(1)
F(X3) ⊆ g(X),
-
(2)
F has the mixed g-monotone property,
-
(3)
F is continuous,
-
(4)
g is continuous and commutes with F.
Suppose there exist x0, y0, z0 ∈ X such that gx0 ≤ F(x0, y0, z0), gy0 ≥ F(y0, x0, y0) and gz0 ≤ F(z0, y0, x0), then F and g have a tripled coincidence point in X, i.e., there exist x, y, z ∈ X such that
Proof. Take
It is easy to show that and . From (46), we have
Now, applying Theorem 15, we obtain the desired result. □
Similarly, we have
Theorem 24. Let (X, ≤) be partially ordered set and (X, G) be a G-metric space. Let F : X3 → X and g: X → X. Assume there exist ψ ∈ Ψ and ϕ ∈ Φ such that for x, y, z, a, b, c, u, v, w ∈ X, with gx ≥ ga ≥ gu, gy ≤ gb ≤ gv and gz ≥ gc ≥ gw, we have
Assume that (X, G, ≤) is regular. Suppose that (g(X),G) is G-complete, F has the mixed g-monotone property and F(X × X × X) ⊆ g(X). Also, assume there exist x0, y0, z0 ∈ X such that gx0 ≤ F(x0, y0, z0), gy0 ≥ F(y0, x0, y0) and gz0 ≤ F(z0, y0, x0), then F and g have a tripled coincidence point in X, i.e., there exist x, y, z ∈ X such that
Application to integral equations
In this section, we study the existence of solutions to nonlinear integral equations using the results proved in section "Main results".
Consider the integral equations in the following system
We will analyze the system (48) under the following assumptions:
-
(i)
f, k, h: [0, T] × ℝ → ℝ are continuous,
-
(ii)
p : [0, T] → ℝ is continuous,
-
(iii)
S: [0, T] × ℝ → [0, ∞) is continuous,
-
(iv)
there exists q > 0 such that for all x, y ∈ ℝ, y ≥ x,
-
(v)
We suppose that
-
(vi)
There exist continuous functions α, β, γ : [0, T] → ℝ such that
We consider the space X = C([0,T],ℝ) of continuous functions defined on [0,T] endowed with the (G-complete) G-metric given by
We endow X with the partial ordered ≤ given by: x, y ∈ X, x ≤ y ⇔ x(t) ≤ y(t) for all t ∈ [0, T].
On the other hand, we may adjust as in [37] to prove that (X, G, ≤) is regular.
Our result is the following.
Theorem 25. Under assumptions (i)-(vi), the system (48) has a solution in X3 = (C([0, T], ℝ))3.
Proof. We consider the operators F : X3 → X and g : X → X defined by
for all x1, x2, x3, x ∈ X.
First, we will prove that F has the mixed monotone property (since g is the identity on X).
In fact, for x1 ≤ y1 and t ∈ [0, T], we have
Taking into account that x1(t) ≤ y1(t) for all t ∈ [0, T], so by (iv), f(s, y1 (s)) ≥ f(s, x1 (s)). Then F(y1, x2, x3)(t) ≥ F(x1, x2, x3)(t) for all t ∈ [0,T],i.e.,
Similarly, for x2 ≤ y2 and t ∈ [0, T], we have
Having x2(t) ≤ y2(t), so by (iv), k(s, x2(s)) ≥ k(s, y2(s)). Then F(x1, x2, x3)(t) ≥ F(x1, y2, x3)(t) for all t ∈ [0,T], i.e.,
Now, for x3 ≤ y3 and t ∈ [0, T], we have
Taking into account that x3(t) ≤ y3(t) for all t ∈ [0, T], so by (iv), h(s, x3(s)) ≥ h(s, y3(s)). Then F(x1, x2, x3)(t) ≥ F(x1, x2, y3)(t) for all t ∈ [0, T], i.e.,
Therefore, F has the mixed monotone property.
In what follows we estimate the quantity G(F(x, y, z), F(a, b, c), F(u, v, w)) for all x, y, z, a, b, c, u, v, w ∈ X, with x ≥ a ≥ u, y ≤ b ≤ v and z ≥ c ≥ w. Since F has the mixed monotone property, we have
We obtain
Note that for all t ∈ [0, T], from (iv), we have
Thus,
Repeating this idea, we may get using definition of the the G-metric G
From (v), we have . This proves that the operator F satisfies the contractive condition appearing in Corollary 20.
Let α, β, γ be the functions appearing in assumption (vi), then by (vi), we get
Applying Corollary 20, we deduce the existence of x1, x2, x3 ∈ X such that
i.e., (x1, x2, x3) is a solution of the system (48). □
Examples
In this section, we state two examples to support the usability of our results. Before we present our first example we worth to mention the following remark.
Remark 26. All our results still valid if (u, v, w) = (a, b, c).
Example 27. Let X = [0, 1] with usual order. Define G : X × X × X → X by
Define F: X × X × X → X by
Also, define ψ, ϕ : [0, +∞) → [0, +∞) by ψ(t) = t and . Then:
-
a.
(X, G, ≤) is a G-complete regular G-metric space.
-
b.
For x, y, z, u, v, w ∈ X with x ≥ u ≥ u, y ≤ v ≤ v and z ≥ w ≥ w, we have
-
c.
F has the mixed monotone property.
Proof. To prove (b), given x, y, z, u, v, w ∈ X with x ≥ u, y ≤ v and z ≥ w. Then:
Case 1: y > min{x, z} and v ≥ min{u, w}. Here, we have
Case 2: y ≥ min{x, z} and u ≥ w ≥ v. Here, we have y ≤ v ≤ w ≤ u ≤ x and y ≤ v ≤ w ≤ z. Hence y = v = w = u = x or y = v = w = z. Therefore
Case 3: y ≥ min{x, z} and w ≥ u ≥ v. Here, we have y ≤ v ≤ u ≤ w ≤ z and y ≤ v ≤ u ≤ x. Thus y = v = u = w = z or y = v = u = x. Therefore
Case 4: x ≥ z ≥ y and v ≥ min{u, w}.
Suppose w ≤ v, then w - y ≤ v - y and hence
Then
Suppose v < w, then u ≤ v < w and hence u ≤ v ≤ w ≤ z ≤ x. So
Therefore,
Case 5: z ≥ x ≥ y and v ≥ min{u, w}.
Suppose u ≤ v, then u - y ≤ v - y and hence
Therefore,
Suppose v < u, then w ≤ v < u and hence w ≤ v <u ≤ x ≤ z. So
Therefore,
Case 6: x ≥ z ≥ y and u ≥ w ≥ v. Here, we have
Case 7: x ≥ z ≥ y and w ≥ u ≥ v. Here, we have y ≤ v ≤ u ≤ w ≤ z ≤ x. Thus,
Case 8: z ≥ x ≥ y and u ≥ w ≥ v. Here, we have y ≤ v ≤ w ≤ u ≤ x ≤ z. Therefore, we have
Case 9: z ≥ x ≥ y, w ≥ u ≥ v. Here, we have y ≤ v ≤ u ≤ w ≤ z. Therefore, we have
To prove (c), let x, y, z ∈ X. To show that F(x, y, z) is monotone non-decreasing in x, let x1, x2 ∈ X with x1 ≤ x2. If y ≥ min{x1, z}, then F(x1, y, z) = 0 ≤ F(x2, y, z).
If y < min{x1, z}, then
Therefore, F(x, y, z) is monotone non-decreasing in x. Similarly, we may show that F(x, y, z) is monotone non-decreasing in z and monotone non-increasing in y. Thus, by Theorem 19 and Remark 26, F has a tripled fixed point. Here, (0, 0,0) is the unique tripled fixed point of F.
□
Now, we state our second example to support the usability of our results for non-symmetric G-metric spaces.
Example 28. Let X = {0, 1, 2, 3,...}. Define G : X × X × X → X by
F: X × X × X → X by
and g: X → X by gx = x2. Also, define ψ, ϕ: [0, +∞) → [0, +∞) by ψ(t) = t2 and ϕ(t) = t. Then
-
a.
(X, G, ≤) is a complete nonsymmetric G-metric space.
-
b.
For x, y, z, u, v, w ∈ X with gx ≥ gu ≥ gu, gy ≤ gv ≤ gv and gz ≥ gw ≥ gw, we have
-
c.
F has the mixed g-monotone property.
-
d.
F(X × X × X) ⊆ gX.
-
e.
(X, G, ≤) is regular.
Proof. For (a) see Example 3.5 of Choudhury and Maity [17]. To prove (b), given x, y, z, u, v, w ∈ X with gx ≥ gu, gy ≤ gv and gz ≥ gw. Then: □
Case 1: (x ≤ 3 ⋁ z ≤ 3) ⋀ (u ≤ 3 ⋁ w ≤ 3). Here, we have F(x, y, z) = 0 and F(u, v, w) = 0. Thus
Case 2: (x ≤ 3 ⋁ z ≤ 3) ⋀ (u ≥ 4 ⋀ w ≥ 4). Here, x < u or z < w which is impossible because gx ≥ gu and gz ≥ gw.
Case 3: (x ≥ 4 ⋀ z ≥ 4) ⋀ (u ≤ 3 ⋁ w ≤ 3). Here, we have F(x, y, z) = 1 and F(u, v, w) = 0. Thus
Also,
In both cases, we have
Using the fact that if a, b ∈ ℕ with a ≤ b, then a2 - a ≤ b2 - b, we deduce that
Therefore,
Case 4: (x ≥ 3 ⋀ z ≥ 3) ⋀ (u ≥ 3 ⋀ w ≥ 3). Here, we have F(x, y, z) = 1 and F(u, v, w) = 1. Thus,
The proof of (c) and (d) are easy. To prove (e), let {x n } be a non-decreasing sequences in X such that {x n } G-converges to x. Then G(x n , x, x) → 0. By definition of G, we conclude that x n = x for all n except finitely many. Thus x n ≤ x for all n ∈ ℕ. Similarly, we show that if {y n } is a non-increasing sequence in X such that {y n } G-converges to y, then y n ≥ y for all n ∈ ℕ. Thus, (X, G, ≤) is regular.
By Theorem 19 and Remark 26, F and g have a tripled coincidence point in X. Here, (0, 0, 0) is the tripled coincidence point of F and g.
References
Boyd DW, Wong SW: On nonlinear contractions. Proc Am Math Soc 1969, 20: 458–464. 10.1090/S0002-9939-1969-0239559-9
Alber , Ya I, Guerre-Delabriere S: Principle of weakly contractive maps in Hilbert space. In New Results in Operator Theory, Advances and Applications. Volume 98. Edited by: Gohberg, I, Lyubich, Yu. Birkhauser, Basel; 1997:7–22.
Cherichi M, Samet B: Fixed point theorems on ordered gauge spaces with applications to nonlinear integral equations. Fixed Point Theory Appl 2012, 2012: 13. 10.1186/1687-1812-2012-13
Dhutta PN, Choudhury BS: A generalization of contraction principle in metric spaces. Fixed Point Theory Appl 2008. Article ID 406368
Dorić D: Common fixed point for generalized ( φ , ψ )-weak contractions. Appl Math Lett 2009, 22(12):1896–1900. 10.1016/j.aml.2009.08.001
Popescu O: Fixed points for ( ψ, ϕ )-weak contractions. Appl Math Lett 2011, 24: 1–4. 10.1016/j.aml.2010.06.024
Samet B, Vetro C, Vetro P: Fixed point theorems for α - ψ -contractive type mappings. Nonlinear Anal 2012, 75: 2154–2165. 10.1016/j.na.2011.10.014
Mustafa Z, Sims B: A new approach to generalized metric spaces. J Nonlinear Convex Anal 2006, 7(2):289–297.
Mustafa Z: A new structure for generalized metric spaces with applications to fixed point theory. PhD thesis. The University of Newcastle, Callaghan, Australia; 2005.
Mustafa Z, Obiedat H, Awawdeh F: Some fixed point theorem for mapping on complete G -metric spaces. Fixed Point Theory Appl 2008, 2008: 12. Article ID 189870
Mustafa Z, Sims B: Fixed point theorems for contractive mappings in complete G -metric spaces. Fixed Point Theory Appl 2009, 2009: 10. Article ID 917175
Abbas M, Rhoades BE: Common fixed point results for non-commuting mappings without continuity in generalized metric spaces. Appl Math Comput 2009, 215: 262–269. 10.1016/j.amc.2009.04.085
Saadati R, Vaezpour SM, Vetro P, Rhoades BE: Fixed point theorems in generalized partially ordered G- metric spaces. Math Comput Model 2010, 52: 797–801. 10.1016/j.mcm.2010.05.009
Gnana Bhashkar T, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal 2006, 65: 1379–1393. 10.1016/j.na.2005.10.017
Abbas M, Khan AR, Nazir T: Coupled common fixed point results in two generalized metric spaces. Appl Math Comput 2011, 217: 6328–6336. 10.1016/j.amc.2011.01.006
Aydi H, Damjanović B, Samet B, Shatanawi W: Coupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces. Math Comput Model 2011, 54: 2443–2450. 10.1016/j.mcm.2011.05.059
Choudhury BS, Maity P: Coupled fixed point results in generalized metric spaces. Math Comput Model 2011, 54: 73–79. 10.1016/j.mcm.2011.01.036
Shatanawi W: Coupled fixed point theorems in generalized metric spaces. Hacet J Math Stat 2011, 40(3):441–447.
Aydi H, Shatanawi W, Vetro C: On generalized weakly G -contraction mapping in G -metric spaces. Comput Math Appl 2011, 62: 4222–4229. 10.1016/j.camwa.2011.10.007
Aydi H, Shatanawi W, Postolache M: Coupled fixed point results for ( ψ, ϕ )-weakly contractive mappings in ordered G -metric spaces. Comput Math Appl 2012, 63: 298–309. 10.1016/j.camwa.2011.11.022
Cho YJe, Rhoades BE, Saadati R, Samet B, Shatanawi W: Nonlinear coupled fixed point theorems in ordered generalized metric spaces with integral type. Fixed Point Theory Appl 2012, 2012: 8. 10.1186/1687-1812-2012-8
Gholizadeh L, Saadati R, Shatanawi W, Vaezpour SM: Contractive mapping in generalized, ordered metric spaces with application in integral equations. Math Probl Eng 2011: 14. Article ID 380784, doi:10.1155/2011/380784
Mustafa Z, Khandaqji M, Shatanawi W: Fixed point results on complete G-metric spaces. Stud Sci Math Hungar 2011, 48: 304–319.
Mustafa Z, Shatanawi W, Bataineh M: Existence of fixed point results in G -metric spaces. Int J Math Math Sci 2009, 2009: 10. Article ID 283028
Rao KPR, Lakshmi KB, Mustafa Z: A unique common fixed point theorem for six maps in G -metric spaces. Int J Nonlinear Anal Appl 2011., 2(2): ISSN: 2008–6822 (electronic)
Shatanawi W: Fixed point theory for contractive mappings satisfying Φ-maps in G -metric spaces. Fixed Point Theory Appl 2010, 2010: 9. Article ID 181650
Shatanawi W: Some fixed point theorems in ordered G -metric spaces and applications. Abstr Appl Anal 2011., 2011: Article ID 126205
Shatanawi W, Abbas M, Nazir T: Common coupled coincidence and coupled fixed point results in two generalized metric spaces. Fixed Point Theory Appl 2011, 2011: 80. 10.1186/1687-1812-2011-80
Berinde V, Borcut M: Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Anal 2011, 74: 4889–4897. 10.1016/j.na.2011.03.032
Aydi H, Karapınar E, Postolache M: Tripled coincidence point theorems for weak ϕ -contractions in partially ordered metric spaces. Fixed Point Theory Appl 2012, 2012: 44. 10.1186/1687-1812-2012-44
Khan MS, Swaleh M, Sessa S: Fixed point theorems by altering distances between the points. Bull Aust Math Soc 1984, 30: 1–9. 10.1017/S0004972700001659
Samet B, Vetro C: Coupled fixed point, f -invariant set and fixed point of N -order. Ann Funct Anal 2010, 1(2):46–56.
Abbas M, Aydi H, Karapınar E: Tripled fixed points of multi-valued nonlinear contraction mappings in partially ordered metric spaces. Abstr Appl Anal 2011., 2011: Article ID 8126904
Aydi H, Karapınar E, Shatanawi W: Tripled fixed point results in generalized metric spaces. J Appl Math 2012., 2012: Article ID 314279
Aydi H, Karapınar E: New Meir-Keeler type tripled fixed point theorems on ordered partial metric spaces. Math Probl Eng 2012., 2012: Article ID 409872
Karapınar E: Couple fixed point theorems For nonlinear contractions in cone metric spaces. Comput Math Appl 2010, 59: 3656–3668. 10.1016/j.camwa.2010.03.062
Nieto JJ, Rodriguez-Lόpez R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Order 2005, 22(3):223–239. 10.1007/s11083-005-9018-5
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Aydi, H., Karapınar, E. & Shatanawi, W. Tripled coincidence point results for generalized contractions in ordered generalized metric spaces. Fixed Point Theory Appl 2012, 101 (2012). https://doi.org/10.1186/1687-1812-2012-101
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DOI: https://doi.org/10.1186/1687-1812-2012-101