We establish a tripled fixed point result for a mixed monotone mapping satisfying nonlinear contractions in ordered generalized metric spaces. Also, some examples are given to support our result.

1. Introduction and Preliminaries

The study of fixed points of mappings satisfying certain contractive conditions has been at the center of rigorous research activity, see [1–3]. The notion of D-metric space is a generalization of usual metric spaces and it is introduced by Dhage [4–7]. Recently, Mustafa and Sims [8, 9] have shown that most of the results concerning Dhage’s D-metric spaces are invalid. In [8, 9], they introduced an improved version of the generalized metric space structure which they called G-metric spaces. For more results on G-metric spaces, one can refer to the papers [10–26].

Now, we give some preliminaries and basic definitions which are used throughout the paper. In 2006, Mustafa and Sims [9] introduced the concept of G-metric spaces as follows.

Definition 1.1 (see [<xref ref-type="bibr" rid="B20">9</xref>]).

Let X be a nonempty set, G:X×X×X→ℝ+ be a function satisfying the following properties:

(G1) G(x,y,z)=0 if x=y=z,

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 will define a metric dG on X by
dG(x,y)=G(x,y,y)+G(y,x,x),∀x,y∈X.

Example 1.2.

Let (X,d) be a metric space. The function G:X×X×X→[0,+∞), defined by
G(x,y,z)=max{d(x,y),d(y,z),d(z,x)},
or
G(x,y,z)=d(x,y)+d(y,z)+d(z,x),
for all x,y,z∈X, is a G-metric on X.

Definition 1.3 (see [<xref ref-type="bibr" rid="B20">9</xref>]).

Let (X,G) be a G-metric space, and let (xn) be a sequence of points of X; therefore, we say that (xn) is G-convergent to x∈X if limn,m→+∞G(x,xn,xm)=0, that is, for any ε>0, there exists N∈ℕ such that G(x,xn,xm)<ε, for all n,m≥N. One calls x the limit of the sequence and writes xn→x or limn→+∞xn=x.

Proposition 1.4 (see [<xref ref-type="bibr" rid="B20">9</xref>]).

Let (X,G) be a G-metric space. The following are equivalent:

(xn) is G-convergent to x,

G(xn,xn,x)→0 as n→+∞,

G(xn,x,x)→0 as n→+∞,

G(xn,xm,x)→0 as n,m→+∞.

Definition 1.5 (see [<xref ref-type="bibr" rid="B20">9</xref>]).

Let (X,G) be a G-metric space. A sequence (xn) is called a G-Cauchy sequence if, for any ε>0, there is N∈ℕ such that G(xn,xm,xl)<ε for all m,n,l≥N, that is, G(xn,xm,xl)→0 as n,m,l→+∞.

Proposition 1.6 (see [<xref ref-type="bibr" rid="B20">9</xref>]).

Let (X,G) be a G-metric space. Then the following are equivalent:

the sequence (xn) is G-Cauchy,

for any ε>0, there exists N∈ℕ such that G(xn,xm,xm)<ε, for all m,n≥N.

Definition 1.7 (see [<xref ref-type="bibr" rid="B20">9</xref>]).

A G-metric space (X,G) is called G-complete if every G-Cauchy sequence is G-convergent in (X,G).

Definition 1.8.

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 (xn), (yn), and (zn) converging to x, y, and z, respectively, (F(xn,yn,zn)) is G-convergent to F(x,y,z).

Recently, Berinde and Borcut [27] introduced these definitions.

Definition 1.9.

Let (X,≤) be a partially ordered set and F:X×X×X→X. The mapping F is said to have the mixed monotone property if, for any x,y,z∈X,
x1,x2∈X,x1≤x2⟹F(x1,y,z)≤F(x2,y,z),y1,y2∈X,y1≤y2⟹F(x,y1,z)≥F(x,y2,z),z1,z2∈X,z1≤z2⟹F(x,y,z1)≤F(x,y,z2).

Definition 1.10.

Let F:X×X×X→X. An element (x,y,z) is called a tripled fixed point of F if
F(x,y,z)=x,F(y,x,y)=y,F(z,y,x)=z.

Very recently, Berinde and Borcut [28] proved some tripled coincidence theorems for contractive type mappings in partially ordered metric spaces. Also, Samet and Vetro [29] introduced the notion of fixed point of N-order as natural extension of that of coupled fixed point and established some new coupled fixed point theorems in complete metric spaces, using a new concept of F-invariant set.

Berinde and Borcut [27] proved the following theorem.

Theorem 1.11.

Let (X,≤,d) be a partially ordered set and suppose there is a metric d on X such that (X,d) is a complete metric space. Suppose F:X×X×X→X such that F has the mixed monotone property and
d(F(x,y,z),F(u,v,w))≤jd(x,u)+kd(y,v)+ld(z,w),
for any x,y,z∈X for which x≤u, v≤y and z≤w. Suppose either F is continuous or X has the following properties:

if a nondecreasing sequence xn→x, then xn≤x for all n,

if a nonincreasing sequence yn→y, then y≤yn for all n,

if a nondecreasing sequence zn→z, then zn≤z for all n.

If there exist x0,y0,z0∈X such that x0≤F(x0,y0,z0), y0≥F(y0,x0,z0), and z0≤F(z0,y0,x0), then there exist x,y,z∈X such that
F(x,y,z)=x,F(y,x,y)=y,F(z,y,x)=z,
that is, F has a tripled fixed point.

In this paper, we establish a tripled fixed point result for a mapping having a mixed monotone property in G-metric spaces. Also, we give some examples to illustrate our result.

2. Main Results

Let Φ be the set of all non-decreasing functions ϕ:[0,+∞)→[0,+∞) such that limn→+∞ϕn(t)=0 for all t>0. If ϕ∈Φ, then following Matkowski [30], we have

ϕ(t)<t for all t>0,

ϕ(0)=0.

The aim of this paper is to prove the following theorem.

Theorem 2.1.

Let (X,≤) be partially ordered set and (X,G) a G-metric space. Let F:X3→X be a continuous mapping having the mixed monotone property on X. Assume there exists ϕ∈Φ such that for x,y,z,a,b,c,u,v,w∈X, with x≥a≥u, y≤b≤v, and z≥c≥w, one has
G(F(x,y,z),F(a,b,c),F(u,v,w))≤ϕ(max{G(x,a,u),G(y,b,v),G(z,c,w)}).
If there exist x0,y0,z0∈X such that x0≤F(x0,y0,z0), y0≥F(y0,x0,y0), and z0≤F(z0,y0,x0), then F has a tripled fixed point in X, that is, there exist x,y,z∈X such that
F(x,y,z)=x,F(y,x,y)=y,F(z,y,x)=z.

Proof.

Suppose x0,y0,z0∈X are such that x0≤F(x0,y0,z0), y0≥F(y0,x0,y0), and z0≤F(z0,y0,x0). Define x1=F(x0,y0,z0), y1=F(y0,x0,y0), and z1=F(z0,y0,x0). Then x0≤x1, y0≥y1, and z0≤z1. Again, define x2=F(x1,y1,z1), y2=F(y1,x1,y1), and z2=F(z1,y1,x1). Since F has the mixed monotone property, we have x0≤x1≤x2, y2≤y1≤y0, and z0≤z1≤z2. Continuing this process, we can construct three sequences (xn), (yn), and (zn) in X such that
xn=F(xn-1,yn-1,zn-1)≤xn+1=F(xn,yn,zn),yn+1=F(yn,xn,yn)≤yn=F(yn-1,xn-1,yn-1),zn=F(zn-1,yn-1,xn-1)≤zn+1=F(zn,yn,xn).
If, for some integer n, we have (xn+1,yn+1,zn+1)=(xn,yn,zn), then F(xn,yn,zn)=xn, F(yn,xn,yn)=yn, and F(zn,yn,xn)=zn; that is, (xn,yn,zn) is a tripled fixed point of F. Thus we will assume that (xn+1,yn+1,zn+1)≠(xn,yn,zn) for all n∈ℕ; that is, we assume that either xn+1≠xn or yn+1≠yn or zn+1≠zn. For any n∈ℕ*, we have from (2.1)
G(xn+1,xn,xn)∶=G(F(xn,yn,zn),F(xn-1,yn-1,zn-1),F(xn-1,yn-1,zn-1))≤ϕ(max{G(xn,xn-1,xn-1),G(yn,yn-1,yn-1),G(zn,zn-1,zn-1)}),G(yn+1,yn,yn)∶=G(F(yn,xn,yn),F(yn-1,xn-1,yn-1),F(yn-1,xn-1,yn-1))≤ϕ(max{G(yn,yn-1,yn-1),G(xn,xn-1,xn-1)})≤ϕ(max{G(yn,yn-1,yn-1),G(xn,xn-1,xn-1),G(zn,zn-1,zn-1)}),G(zn+1,zn,zn)∶=G(F(zn,yn,xn),F(zn-1,yn-1,xn-1),F(zn-1,yn-1,xn-1))≤ϕ(max{G(zn,zn-1,zn-1),G(yn,yn-1,yn-1),G(xn,xn-1,xn-1)}).
From (2.4), it follows that
max{G(xn+1,xn,xn),G(yn,yn,yn+1),G(zn+1,zn,zn)}≤ϕ(max{G(xn,xn-1,xn-1),G(yn,yn-1,yn-1),G(zn,zn-1,zn-1)}).
By repeating (2.5) n-times and using the fact that ϕ is non-decreasing, we get that
max{G(xn+1,xn,xn),G(yn+1,yn,yn),G(zn+1,zn,zn)}≤ϕ(max{G(xn,xn-1,xn-1),G(yn,yn-1,yn-1),G(zn,zn-1,zn-1)})≤ϕ2(max{G(xn-1,xn-2,xn-2),G(yn-1,yn-2,yn-2),G(zn-1,zn-2,zn-2)})⋮≤ϕn(max{G(x1,x0,x0),G(y1,y0,y0),G(z1,z0,z0)}).

Now, we shill show that (xn) is a G-Cauchy sequence in X. Let ϵ>0. Since
limn→+∞ϕn(max{G(x1,x0,x0),G(y1,y0,y0),G(z1,z0,z0)})=0,
and ϵ>ϕ(ϵ), there exists n0∈ℕ such that
ϕn(max{G(x1,x0,x0),G(y1,y0,y0),G(z1,z0,z0)})<ϵ-ϕ(ϵ)∀n≥n0.
By (2.6), this implies that
max{G(xn+1,xn,xn),G(yn+1,yn,yn),G(zn+1,zn,zn)}<ϵ-ϕ(ϵ)∀n≥n0.
For m,n∈ℕ, we prove by induction on m that
max{G(xn,xn,xm),G(yn,yn,ym),G(zn,zn,zm)}<ϵ∀m≥n≥n0.
Since ϵ-ϕ(ϵ)≤ϵ, then by using (2.9) and the property (G4), we conclude that (2.10) holds when m=n+1. Now suppose that (2.10) holds for m=k. For m=k+1, we have
G(xn,xn,xk+1)≤G(xn,xn,xn+1)+G(xn+1,xn+1,xk+1)<ϵ-ϕ(ϵ)+G(F(xn,yn,zn),F(xn,yn,zn),F(xk,yk,zk))≤ϵ-ϕ(ϵ)+ϕ(max{G(xn,xn,xk),G(yn,yn,yk),G(zn,zn,zk)})≤ϵ-ϕ(ϵ)+ϕ(ϵ)=ϵ.
Similarly, we show that
G(yn,yn,yk+1)<ϵ,G(zn,zn,zk+1)<ϵ.
Hence, we have
max{G(xn,xn,xk+1),G(yn,yn,yk+1),G(zn,zn,zk+1)}<ϵ.
Thus (2.10) holds for all m≥n≥n0. Hence (xn), (yn), and (zn) are G-Cauchy sequences in X. Since X is a G-complete metric space, there exist x,y,z∈X such that (xn), (yn), and (zn) converge to x, y, and z, respectively. Finally, we show that (x,y,z) is a tripled fixed point of F. Since F is continuous and (xn,yn,zn)→(x,y,z), we have xn+1=F(xn,yn,zn)→F(x,y,z). By the uniqueness of limit, we get that x=F(x,y,z). Similarly, we show that y=F(y,x,y) and z=F(z,y,x). So (x,y,z) is a tripled fixed point of F.

Corollary 2.2.

Let (X,≤) be partially ordered set and (X,G) a G-metric space. Let F:X3→X be a continuous mapping having the mixed monotone property on X. Suppose that there exists k∈[0,1) such that for x,y,z,a,b,c,u,v,w∈X, with x≥a≥u, y≤b≤v, and z≥c≥w one has
G(F(x,y,z),F(a,b,c),F(u,v,w))≤kmax{G(x,a,u),G(y,b,v),G(z,c,w)}.
If there exist x0,y0,z0∈X such that x0≤F(x0,y0,z0), y0≥F(y0,x0,y0), and z0≤F(z0,y0,x0), then F has a tripled fixed point in X, that is, there exist x,y,z∈X such that
F(x,y,z)=x,F(y,x,y)=y,F(z,y,x)=z.

Proof.

It follows from Theorem 2.1 by taking ϕ(t)=kt.

Corollary 2.3.

Let (X,≤) be partially ordered set and (X,G) be a G-metric space.

Let F:X3→X be a continuous mapping having the mixed monotone property on X. Suppose that there exists k∈[0,1) such that for x,y,z,a,b,c,u,v,w∈X, with x≥a≥u, y≤b≤v, and z≥c≥w one has
G(F(x,y,z),F(a,b,c),F(u,v,w))≤k3(G(x,a,u)+G(y,b,v)+G(z,c,w)).
If there exist x0,y0,z0∈X such that x0≤F(x0,y0,z0), y0≥F(y0,x0,y0), and z0≤F(z0,y0,x0), then F has a tripled fixed point in X, that is, there exist x,y,z∈X such that
F(x,y,z)=x,F(y,x,y)=y,F(z,y,x)=z.

Proof.

Note that
G(x,a,u)+G(y,b,v)+G(z,c,w)≤3max{G(x,a,u),G(y,b,v),G(z,c,w)}.
Then, the proof follows from Corollary 2.2.

By adding an additional hypothesis, the continuity of F in Theorem 2.1 can be dropped.

Theorem 2.4.

Let (X,≤) be a partially ordered set and (X,d) a complete metric space. Let F:X×X×X→X be a mapping having the mixed monotone property. Assume that there exists ϕ∈Φ such that
G(F(x,y,z),F(a,b,c),F(u,v,w))≤ϕ(max{G(x,a,u),G(y,b,v),G(z,c,w)})
for all x,y,z,a,b,c,u,v,w∈X with x≥a≥u, y≤b≤v, and z≥c≥w. Assume also that X has the following properties:

if a nondecreasing sequence xn→x, then xn≤x for all n∈ℕ,

if a nonincreasing sequence yn→y, then yn≥y for all n∈ℕ.

If there exist x0,y0,z0∈X such that x0≤F(x0,y0,z0), y0≥F(y0,x0,y0), and z0≤F(z0,y0,x0), then F has a tripled fixed point.Proof.

Following proof of Theorem 2.1 step by step, we construct three G-Cauchy sequences (xn), (yn), and (zn) in X with
x1≤x2≤⋯≤xn≤⋯,y1≥y2≥⋯≥yn≥⋯,z1≤z2≤⋯≤zn≤⋯
such that xn→x∈X, yn→y∈X, and zn→z∈X. By the hypotheses on X, we have xn≤x, yn≥y, and zn≤z for all n∈ℕ. If for some n≥0, xn=x, yn=y, and zn=z, then
x=xn≤xn+1≤x=xn,y=yn≥yn+1≤y=yn,z=zn≤zn+1≤z=zn,
which implies that xn=xn+1=F(xn,yn,zn), yn=yn+1=F(yn,xn,yn), and zn=zn+1=F(zn,yn,xn); that is, (xn,yn,zn) is a tripled fixed point of F. Now, assume that, for all n≥0, (xn,yn,zn)≠(x,y,z). Thus, for each n≥0,
max{G(x,x,xn),G(y,y,yn),G(z,z,zn)}>0.
From (2.19), we have
G(F(x,y,z),F(x,y,z),xn+1)∶=G(F(x,y,z),F(x,y,z),F(xn,yn,zn))≤ϕ(max{G(x,x,xn),G(y,y,yn),G(z,z,zn)}),G(yn+1,F(y,x,y),F(y,x,y))∶=G(F(yn,xn,yn),F(y,x,y),F(y,x,y))≤ϕ(max{G(yn,y,y),G(xn,x,x)})G(F(z,y,x),F(z,y,x),zn+1)∶=G(F(z,y,x),F(z,y,x),F(zn,yn,xn))≤ϕ(max{G(x,x,xn),G(y,y,yn),G(z,z,zn)}).
Letting n→+∞ in (2.23) and using (2.22) in the fact that ϕ(t)<t for all t>0, it follows that x=F(x,y,z), y=F(y,x,y), and z=F(z,y,x). Hence (x,y,z) is a tripled fixed point of F.

Now we give some examples illustrating our results.

Example 2.5.

Take X=[0,+∞) endowed with the complete G-metric:
G(x,y,z)=max{|x-y|,|x-z|,|y-z|},
for all x,y,z∈X. Set k=1/2 and F:X3→X defined by F(x,y,z)=(1/6)x. The mapping F has the mixed monotone property. We have
G(F(x,y,z),F(a,b,c),F(u,v,w))=16G(x,a,u)≤k3max{G(x,a,u),G(y,b,v),G(z,c,w)}
for all x≥a≥u, y≤b≤v, and z≥c≥w, that is, (2.14) holds. Take x0=y0=z0=0, then all the hypotheses of Corollary 2.2 are verified, and (0,0,0) is the unique tripled fixed point of F.

Example 2.6.

As in Example 2.5, take X=[0,+∞) and
G(x,y,z)=max{|x-y|,|x-z|,|y-z|},
for all x,y,z∈X. Set k=1/2 and F:X3→X defined by F(x,y,z)=(1/36)(6x-6y+6z+5). The mapping F has the mixed monotone property. For all x≥a≥u, y≤b≤v, and z≥c≥w, we have
G(F(x,y,z),F(a,b,c),F(u,v,w))=16(|x-u|+|y-v|+|z-w|)=16(G(x,a,u)+G(y,b,v)+G(z,c,w))=k3(G(x,a,u)+G(y,b,v)+G(z,c,w)),
that is, (2.16) holds. Take x0=y0=z0=1/6, then all the hypotheses of Corollary 2.3 hold, and (1/6,1/6,1/6) is the unique tripled fixed point of F.

Remark 2.7.

In our main results (Theorems 2.1 and 2.4), the considered contractions are of nonlinear type. Then, inequality (2.1) does not reduce to any metric inequality with the metric dG (this metric is given by (1.1)). Hence our theorems do not reduce to fixed point problems in the corresponding metric space (X,dG).

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