weakly contractive mappings in ordered G-metric spaces - Core

Computers and Mathematics with Applications 63 (2012) 298–309

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Coupled fixed point results for (ψ, φ)-weakly contractive mappings in ordered G-metric spaces Hassen Aydi a , Mihai Postolache b,∗ , Wasfi Shatanawi c a

Université de Sousse, Institut Supérieur d’Informatique et des Technologies de Communication de Hammam Sousse, Route GP1-4011 Hammam Sousse, Tunisie

b

University Politehnica of Bucharest, Faculty of Applied Sciences, 313 Splaiul Independenţei, Romania

c

Department of Mathematics, Hashemite University, P.O. Box 150459, Zarqa 13115, Jordan

article

info

Article history: Received 31 May 2011 Received in revised form 16 October 2011 Accepted 14 November 2011

abstract In this paper, we establish coupled coincidence and common coupled fixed point theorems for (ψ, φ)-weakly contractive mappings in ordered G-metric spaces. Presented theorems extend, generalize and improve many existing results in the literature. An example is given. © 2011 Elsevier Ltd. All rights reserved.

Keywords: G-metric space Ordered set Coupled coincidence point Common coupled fixed point Mixed monotone property

1. Previous definitions and results The fixed point theorems in partially ordered metric spaces play a major role to prove the existence and uniqueness of solutions for some differential and integral equations. Thus, the attraction of fixed point theorems to a large number of mathematicians is understandable. One of the most interesting fixed point theorems in ordered metric spaces was investigated by Ran and Reurings [1]. Ran and Reurings [1] applied their result to linear and nonlinear matrix equations. Then, many authors obtained several interesting results in ordered metric spaces (see [2–19]). Bhaskar and Lakshmikantham [20] initiated the study of a coupled fixed point in ordered metric spaces and applied their results to prove the existence and uniqueness of solutions for a periodic boundary value problem. For more works in coupled and coincidence point theorems, we refer the reader to [21–28]. Some authors generalized the concept of metric spaces in different ways. Mustafa and Sims [29] introduced the notion of G-metric space in which the real number is assigned to every triplet of an arbitrary set as a generalization of the notion of metric spaces. Based on the notion of G-metric spaces, Mustafa et al. [30–33] obtained some fixed point theorems for mappings satisfying various contractive conditions. Abbas and Rhoades [34] initiated the study of common fixed point in G-metric spaces, while Saadati et al. [35] studied some fixed point theorems in partially ordered G-metric spaces. For more results in G-metric spaces, we refer the reader to [36–39]. In 2010, Abbas et al. [40] introduced the concepts of w and w ∗ -compatible mappings. Abbas et al. [41] utilized the concept of w and w ∗ -compatibility to prove an interesting uniqueness theorem of coupled fixed point in G-metric spaces. For more results of coupled fixed point in G-metric spaces, see [42–44].

∗

Corresponding author. Tel.: +40 0722 798 417. E-mail addresses: [email protected], [email protected] (M. Postolache).

0898-1221/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2011.11.022

H. Aydi et al. / Computers and Mathematics with Applications 63 (2012) 298–309

299

In the sequel, the letters R, R+ and N will denote the set of all real numbers, the set of all nonnegative real numbers and the set of all natural numbers, respectively. Consistent of Mustafa and Sims [29], the following definitions and results will be needed in the sequel. Definition 1.1 ([29]). Let X be a nonempty set, and let G: X × X × X → R+ be a function satisfying the following properties: (G1) (G2) (G3) (G4) (G5)

G(x, y, z ) = 0 if x = y = z, 0 < G(x, x, y) for all x, y ∈ X , with x ̸= y, G(x, x, y) ≤ G(x, y, z ) for all x, y, z ∈ X , with y ̸= z, G(x, y, z ) = G(x, z , y) = G(y, z , x) = · · ·, (symmetry in all three variables), 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 specifically, a G-metric on X . The pair (X , G) is called G-metric space. Definition 1.2 ([29]). Let (X , G) be a G-metric space and let (xn ) be a sequence of points of X . A point x ∈ X is said to be the limit of the sequence (xn ), if limn,m→+∞ G(x, xn , xm ) = 0. We say that the sequence (xn ) is G-convergent to x, or (xn ) G-converges to x. Thus, xn → x in a G-metric space (X , G) if for any ε > 0 there exists a natural number k, such that G(x, xn , xm ) < ε for all m, n ≥ k. Proposition 1.1 ([29]). Let (X , G) be a G-metric space. Then 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 → +∞.

(1) (2) (3) (4)

Definition 1.3 ([29]). Let (X , G) be a G-metric space. A sequence (xn ) is called a G-Cauchy sequence, if for any ε > 0, there exists k ∈ N such that G(xn , xm , xl ) < ε for all m, n, l ≥ k, that is G(xn , xm , xl ) → 0 as n, m, l → +∞. Proposition 1.2 ([29]). Let (X , G) be a G-metric space. Then the following are equivalent: (1) the sequence (xn ) is G-Cauchy, (2) for any ε > 0 there exists k ∈ N such that G(xn , xm , xm ) < ε for all m, n ≥ k. Proposition 1.3 ([29]). Let (X , G) be a G-metric space. Then, f : X → X is G-continuous at x ∈ X if and only if it is G-sequentially continuous at x, that is, whenever (xn ) is G-convergent to x, (f (xn )) is G-convergent to f (x). Proposition 1.4 ([29]). Let (X , G) be a G-metric space. Then, the function G(x, y, z ) is jointly continuous in all three of its variables. Definition 1.4 ([29]). A G-metric space (X , G) is called G-complete if every G-Cauchy sequence is G-convergent in (X , G). Definition 1.5 ([43]). Let (X , G) be a G-metric space. A mapping F : X × X → X is said to be continuous if for any two G-convergent sequences (xn ) and (yn ) converging to x and y respectively, (F (xn , yn )) is G-convergent to F (x, y). The following definition was introduced by Bhaskar and Lakshmikantham in [20]. Definition 1.6 ([20]). Let (X , ≤) be a partially ordered set and F : X × X → X . Then the map F is said to have mixed monotone property if F (x, y) is monotone non-decreasing in x and is monotone non-increasing in y, that is, x1 ≤ x2

implies F (x1 , y) ≤ F (x2 , y) for all y ∈ X

y1 ≤ y2

implies F (x, y2 ) ≤ F (x, y1 ) for all x ∈ X .

and

Inspired by Definition 1.6, Lakshmikantham and Ćirić [21] introduced the concept of a g-mixed monotone mapping. Definition 1.7 ([21]). Let (X , ≤) be a partially ordered set and F : X × X → X . Then, the map F is said to have mixed g-monotone property if F (x, y) is monotone g-non-decreasing in x and is monotone g-non-increasing in y, that is, gx1 ≤ gx2

implies F (x1 , y) ≤ F (x2 , y) for all y ∈ X

gy1 ≤ gy2

implies F (x, y2 ) ≤ F (x, y1 ) for all x ∈ X .

and

300


Definition 1.8 ([20]). An element (x, y) ∈ X × X is called a coupled fixed point for the mapping F : X × X → X if F (x, y) = x and F (y, x) = y. Definition 1.9 ([21]). An element (x, y) ∈ X × X is called a coupled coincidence point of the mappings F : X × X → X and g: X → X if F (x, y) = gx and F (y, x) = gy. Definition 1.10 ([21]). Let X be a non-empty set. We say that the mappings F : X × X → X and g: X → X are commutative if gF (x, y) = F (gx, gy), for all x, y ∈ X . Abbas et al. [40] introduced the concept of w and w ∗ -compatible mappings and utilized this concept to prove an interesting uniqueness theorem of a coupled fixed point for mappings F and g in cone metric spaces. Definition 1.11 ([40]). Mappings F : X × X → X and g : X → X are called

(W1 ) w -compatible if g (F (x, y)) = F (gx, gy) whenever gx = F (x, y) and gy = F (y, x); (W2 ) w ∗ -compatible if g (F (x, x)) = F (gx, gx) whenever gx = F (x, x). The purpose of this paper is to obtain some coupled coincidence point theorems in ordered G-metric spaces satisfying some contractive conditions. 2. Main results Let (X , G) be a G-metric space and F : X × X → X and g: X → X be two mappings. Throughout this section, we let M (x, u, s, y, v, t ) := max{G(gx, gu, gs), G(gy, g v, gt ), G(F (x, y), gu, gs), G(F (y, x), g v, gt )}

(2.1)

unless otherwise stated. Also, set

Ψ = {ψ | ψ : [0, ∞) → [0, ∞) is continuous and non-decreasing with ψ(t ) = 0 iff t = 0} and

Φ = {ϕ | ϕ : [0, ∞) → [0, ∞) is lower semi continuous, ϕ(t ) > 0 for all t > 0, ϕ(0) = 0}. For some works on the class of Ψ or the class of Φ , we refer the reader to [45,46]. According to Shatanawi et al. [47], mappings F and g are called (ψ, φ)-weakly contractive on X if there exist ψ ∈ Ψ and φ ∈ Φ , such that

ψ(G(F (x, y), F (u, v), F (s, t ))) ≤ ψ(M (x, u, s, y, v, t )) − φ(M (x, u, s, y, v, t ))

(2.2)

for all x, y, u, v, s, t ∈ X Our first result is the following. Theorem 2.1. Let (X , ≤) be partially ordered set and (X , G) be complete G-metric space. Let F and g be (ψ, φ)-weakly contractive mappings on X , with gx ≤ gu ≤ gs and gt ≤ g v ≤ gy, or gs ≤ gu ≤ gx and gy ≤ g v ≤ gt. Assume that F and g satisfy the following conditions: (1) (2) (3) (4)

F (X × X ) ⊆ g (X ), F has the mixed g-monotone property, F is continuous, g is continuous and commutes with F . If there exist x0 , y0 ∈ X such that gx0 ≤ F (x0 , y0 ) and F (y0 , x0 ) ≤ gy0 , then F and g have a coupled coincidence point.

Proof. Let x0 and y0 in X such that gx0 ≤ F (x0 , y0 ) and F (y0 , x0 ) ≤ gy0 . Since F (X × X ) ⊆ g (X ), we can choose x1 , y1 ∈ X such that gx1 = F (x0 , y0 ) and gy1 = F (y0 , x0 ). Again, since F (X × X ) ⊆ g (X ), we can choose x2 , y2 ∈ X such that gx2 = F (x1 , y1 ) and gy2 = F (y1 , x1 ). Since F has the mixed g-monotone property, we have gx0 ≤ gx1 ≤ gx2 and gy2 ≤ gy1 ≤ gy0 . Continuing this process, we can construct two sequences (xn ) and (yn ) in X such that gxn = F (xn−1 , yn−1 ) ≤ gxn+1 = F (xn , yn ) and gyn+1 = F (yn , xn ) ≤ gyn = F (yn−1 , xn−1 ). If for some integer n, we have (gxn+1 , gyn+1 ) = (gxn , gyn ), then F (xn , yn ) = gxn and F (yn , xn ) = gyn , that is F and g have a coincidence point. From now on, we assume (gxn+1 , gyn+1 ) ̸= (gxn , gyn ) for all n ∈ N∪{0}, that is, we assume that either gxn+1 = F (xn , yn ) ̸= gxn or gyn+1 = F (yn , xn ) ̸= gyn . Since gxn−1 ≤ gxn and gyn ≤ gyn−1 hold for all n ∈ N, by inequality (2.2), we have

ψ(G(gxn , gxn+1 , gxn+1 )) = ψ(G(F (xn−1 , yn−1 ), F (xn , yn ), F (xn , yn ))) ≤ ψ(M (xn−1 , xn , xn , yn−1 , yn , yn )) − φ(M (xn−1 , xn , xn , yn−1 , yn , yn )) ≤ ψ(M (xn−1 , xn , xn , yn−1 , yn , yn )),

(2.3)


301

where M (xn−1 , xn , xn , yn−1 , yn , yn ) = max{G(gxn−1 , gxn , gxn ), G(gyn−1 , gyn , gyn ), G(F (xn−1 , yn−1 ), gxn , gxn ), G(F (yn−1 , xn−1 ), gyn , gyn )}

= max{G(gxn−1 , gxn , gxn ), G(gyn−1 , gyn , gyn ), G(gxn , gxn , gxn ), G(gyn , gyn , gyn )} = max{G(gxn−1 , gxn , gxn ), G(gyn−1 , gyn , gyn )}. Thus we have

ψ(G(gxn , gxn+1 , gxn+1 )) ≤ ψ(max{G(gxn−1 , gxn , gxn ), G(gyn−1 , gyn , gyn )}) − φ(max{G(gxn−1 , gxn , gxn ), G(gyn−1 , gyn , gyn )}) ≤ ψ(max{G(gxn−1 , gxn , gxn ), G(gyn−1 , gyn , gyn )}).

(2.4)

Since ψ is a non-decreasing function, we get G(gxn , gxn+1 , gxn+1 ) ≤ max{G(gxn−1 , gxn , gxn ), G(gyn−1 , gyn , gyn )}.

(2.5)

Similarly, since gyn ≤ gyn−1 and gxn−1 ≤ gxn hold for all n ∈ N, by inequality (2.2), we have

ψ(G(gyn , gyn+1 , gyn+1 )) = ψ(G(F (yn−1 , xn−1 ), F (yn , xn ), F (yn , xn ))) ≤ ψ(M (yn−1 , yn , yn , xn−1 , xn , xn )) − φ(M (yn−1 , yn , yn , xn−1 , xn , xn )) ≤ ψ(M (yn−1 , yn , yn , xn−1 , xn , xn )), where M (yn−1 , yn , yn , xn−1 , xn , xn ) = max{G(gyn−1 , gyn , gyn ), G(gxn−1 , gxn , gxn ), G(F (yn−1 , xn−1 ), gyn , gyn ), G(F (xn−1 , yn−1 ), gxn , gxn )} = max{G(gyn−1 , gyn , gyn ), G(gxn−1 , gxn , gxn ), G(gyn , gyn , gyn ), G(gxn , gxn , gxn )} = max{G(gyn−1 , gyn , gxn ), G(gxn−1 , gxn , gxn )}. Thus, we have

ψ(G(gyn , gyn+1 , gyn+1 )) ≤ ψ(max{G(gyn−1 , gyn , gyn ), G(gxn−1 , gxn , gxn )}) − φ(max{G(gyn−1 , gyn , gyn ), G(gxn−1 , gxn , gxn )}) ≤ ψ(max{G(gyn−1 , gyn , gyn ), G(gxn−1 , gxn , gxn )}).

(2.6)

Since ψ is a non-decreasing function, we get G(gyn , gyn+1 , gyn+1 ) ≤ max{G(gxn−1 , gxn , gxn ), G(gyn−1 , gyn , gyn )}.

(2.7)

By (2.5) and (2.7), we have max{G(gxn , gxn+1 , gxn+1 ), G(gyn , gyn+1 , gyn+1 )} ≤ max{G(gxn−1 , gxn , gxn ), G(gyn−1 , gyn , gyn )}. Thus (max{G(gxn−1 , gxn , gxn ), G(gyn−1 , gyn , gyn )}) is a positive non-increasing sequence. Hence, there exists r ≥ 0 such that limn→+∞ max{G(gxn−1 , gxn , gxn ), G(gyn−1 , gyn , gyn )} = r. Now, we show that r = 0. Since ψ : [0, +∞) → [0, +∞) is a non-decreasing function, for a, b ∈ [0, +∞), we have ψ(max{a, b}) = max{ψ(a), ψ(b)}. Thus, by (2.4) and (2.6),

ψ(max{G(gxn , xn+1 , gxn+1 ), G(gyn , gyn+1 , gyn+1 )}) = max{ψ(G(gxn , gxn+1 , gxn+1 )), ψ(G(gyn , gyn+1 , gyn+1 ))} ≤ ψ(max{G(gxn−1 , gxn , gxn ), G(gyn−1 , gyn , gyn )}) − φ(max{G(gxn−1 , gxn , gxn ), G(gyn−1 , gyn , gyn )}). Letting n → +∞ in the above inequality, and using the continuity of ψ and the lower semi-continuity of φ , we get ψ(r ) ≤ ψ(r ) − φ(r ). Hence φ(r ) = 0, that is, r = 0. Therefore lim max{G(gxn−1 , gxn , gxn ), G(gyn−1 , gyn , gyn )} = 0.

n→+∞

Our next step is to show that (gxn ) and (gyn ) are G-Cauchy sequences. Assume the contrary, that is, (gxn ) or (gyn ) is not a G-Cauchy sequence, that is, lim

n,m→+∞

G(gxm , gxn , gxn ) ̸= 0

or

lim

n,m→+∞

G(gym , gyn , gyn ) ̸= 0.

(2.8)

302


This means that there exists ϵ > 0 for which we can find subsequences of integers (mk ) and (nk ) with nk > mk > k such that max{G(gxmk , gxnk , gxnk ), G(gymk , gynk , gynk )} ≥ ϵ.

(2.9)

Further, corresponding to mk we can choose nk in such a way that it is the smallest integer with nk > mk and satisfying (2.9). Then max{G(gxmk , gxnk −1 , gxnk −1 ), G(gymk , gynk −1 , gynk −1 )} < ϵ.

(2.10)

By (G5) and (2.10), we have G(gxmk , gxnk , gxnk ) ≤ G(gxmk , gxnk −1 , gxnk −1 ) + G(gxnk −1 , gxnk , gxnk )

< ϵ + G(gxnk −1 , gxnk , gxnk ). Thus, by (2.8) we obtain lim G(gxmk , gxnk , gxnk ) ≤ lim G(gxmk , gxnk −1 , gxnk −1 ) ≤ ϵ. k→+∞

k→+∞

(2.11)

Similarly, we have lim G(gymk , gynk , gynk ) ≤ lim G(gymk , gynk −1 , gynk −1 ) ≤ ϵ.

k→+∞

k→+∞

(2.12)

Again by (G5) and (2.10), we have G(gxmk , gxnk , gxnk ) ≤ G(gxmk , gxmk −1 , gxmk −1 ) + G(gxmk −1 , gxnk −1 , gxnk −1 ) + G(gxnk −1 , gxnk , gxnk )

≤ G(gxmk , gxmk −1 , gxmk −1 ) + G(gxmk −1 , gxmk , gxmk ) + G(gxmk , gxnk −1 , gxnk −1 ) + G(gxnk −1 , gxnk , gxnk ) < G(gxmk , gxmk −1 , gxmk −1 ) + G(gxmk −1 , gxmk , gxmk ) + ϵ + G(gxnk −1 , gxnk , gxnk ). Letting k → +∞ and using (2.8), we get lim G(gxmk , gxnk , gxnk ) ≤ lim G(gxmk −1 , gxnk −1 , gxnk −1 ) ≤ ϵ.

k→+∞

k→+∞

(2.13)

Similarly, we have lim G(gymk , gynk , gynk ) ≤ lim G(gymk −1 , gynk −1 , gynk −1 ) ≤ ϵ.

k→+∞

k→+∞

(2.14)

Using (2.9) and (2.11)-(2.14), we have lim max{G(gxmk , gxnk , gxnk ), G(gymk , gynk , gynk )}

k→+∞

= lim max{G(gxmk , gxnk −1 , gxnk −1 ), G(gxmk −1 , gxnk −1 , gxnk −1 ), k→+∞

G(gymk , gynk −1 , gynk −1 ), G(gymk −1 , gynk −1 , gynk −1 )} = ϵ. Since M (xmk −1 , xnk −1 , xnk −1 , ymk −1 , ynk −1 , ynk −1 )

= max{G(gxmk −1 , gxnk −1 , gxnk −1 ), G(gymk −1 , gynk −1 , gynk −1 ), G(F (xmk −1 , ymk −1 ), gxnk −1 , gxnk −1 ), G(F (ymk −1 , xmk −1 ), gynk −1 , gynk −1 )} = max{G(gxmk −1 , gxnk −1 , gxnk −1 ), G(gymk −1 , gynk −1 , gynk −1 ), G(gxmk , gxnk −1 , gxnk −1 ), G(gymk , gynk −1 , gynk −1 )}, we have lim max{G(gxmk , gxnk , gxnk ), G(gymk , gynk , gynk )}

k→+∞

= lim M (xmk −1 , xnk −1 , xnk −1 , ymk −1 , ynk −1 , ynk −1 ) = ϵ. k→+∞

(2.15)

Now, using inequality (2.2) we obtain

ψ(G(gxmk , gxnk , gxnk )) = ψ(G(F (xmk −1 , ymk −1 ), F (xnk −1 , ynk −1 ), F (xnk −1 , ynk −1 ))) ≤ ψ(M (xmk −1 , xnk −1 , xnk −1 , ymk −1 , ynk −1 , ynk −1 )) − φ(M (xmk −1 , xnk −1 , xnk −1 , ymk −1 , ynk −1 , ynk −1 ))

(2.16)


303

and

ψ(G(gymk , gynk , gynk )) = ψ(G(F (ymk −1 , xmk −1 ), F (ynk −1 , xnk −1 ), F (ynk −1 , xnk −1 ))) ≤ ψ(M (ymk −1 , ynk −1 , ynk −1 , xmk −1 , xnk −1 , xnk −1 )) − φ(M (ymk −1 , ynk −1 , ynk −1 , xmk −1 , xnk −1 , xnk −1 )) = ψ(M (xmk −1 , xnk −1 , xnk −1 , ymk −1 , ynk −1 , ynk −1 )) − φ(M (xmk −1 , xnk −1 , xnk −1 , ymk −1 , ynk −1 , ynk −1 )).

(2.17)

By (2.16) and (2.17), we get

ψ(max{G(gxmk , gxnk , gxnk ), G(gymk , gynk , gynk )}) = max{ψ(G(gxmk , gxnk , gxnk )), ψ(G(gymk , gynk , gynk ))} ≤ ψ(M (xmk −1 , xnk −1 , xnk −1 , ymk −1 , ynk −1 , ynk −1 )) − φ(M (xmk −1 , xnk −1 , xnk −1 , ymk −1 , ynk −1 , ynk −1 )). Letting k → +∞ and using (2.15) and the fact that ψ is continuous and φ is lower semi-continuous, we get ψ(ϵ) ≤ ψ(ϵ) − φ(ϵ). Hence, φ(ϵ) = 0, therefore ϵ = 0, which is a contradiction. This means that (gxn ) and (gyn ) are G-Cauchy sequences in the G-metric space (X , G), which is complete. Then, there are x, y ∈ X such that (gxn ) and (gyn ) are respectively G-convergent to x and y. From Proposition 1.2, we have lim G(gxn , gxn , x) = lim G(gxn , x, x) = 0,

(2.18)

lim G(gyn , gyn , y) = lim G(gyn , y, y) = 0.

(2.19)

n→+∞ n→+∞

n→+∞

n→+∞

From (2.18), (2.19) and the continuity of g, we have lim G(g (gxn ), g (gxn ), gx) = lim G(g (gxn ), gx, gx) = 0,

(2.20)

lim G(g (gyn ), g (gyn ), gy) = lim G(g (gyn ), gy, gy) = 0.

(2.21)

n→+∞

n→+∞

n→+∞

n→+∞

Since gxn+1 = F (xn , yn ) and gyn+1 = F (yn , xn ), the commutativity of F and g yields that g (gxn+1 ) = g (F (xn , yn )) = F (gxn , gyn )

(2.22)

g (gyn+1 ) = g (F (yn , xn )) = F (gyn , gxn ).

(2.23)

By using (2.22), (2.23) and the continuity of F , we get {g (gxn+1 )} is G-convergent to F (x, y) and {g (gyn+1 )} is G-convergent to F (y, x). By uniqueness of the limit, we have F (x, y) = gx and F (y, x) = gy, and this ends the proof. Theorem 2.2. Let (X , ≤) be a partially ordered set and G be a G-metric on X . Let the mappings F and g be (ψ, φ)-weakly contractive, with gx ≤ gu ≤ gs and gt ≤ g v ≤ gy, or gs ≤ gu ≤ gx and gy ≤ g v ≤ gt. Suppose that (g (X ), G) is complete, F has the mixed g-monotone property and F (X × X ) ⊆ g (X ). Also, assume the following: (i) if a non-decreasing sequence xn → x, then xn ≤ x for all n, (ii) if a non-increasing sequence yn → y, then y ≤ yn for all n. If there exist x0 , y0 ∈ X such that gx0 ≤ F (x0 , y0 ) and F (y0 , x0 ) ≤ gy0 , then F and g have a coupled coincidence point. Proof. Proceeding exactly as in Theorem 2.1, we have that (gxn ) and (gyn ) are Cauchy sequences in the complete G-metric space (g (X ), G). Then, there exist x, y ∈ X such that gxn → gx and gyn → gy; that is, lim G(gxn , gx, gx) = lim G(gyn , gy, gy) = 0.

n→+∞

n→+∞

Since (gxn ) is non-decreasing and (gyn ) is non-increasing, by (i) and (ii), we have gxn ≤ gx and gy ≤ gyn for all n ≥ 0. By (2.2)

ψ(G(gxn+1 , F (x, y), F (x, y))) = ψ(G(F (xn , yn ), F (x, y), F (x, y))) ≤ ψ(M (xn , x, x, yn , y, y)) − φ(M (xn , x, x, yn , y, y)), where M (xn , x, x, yn , y, y) = max{G(gxn , gx, gx), G(gyn , gy, gy), G(F (xn , yn ), gx, gx), G(F (yn , xn ), gy, gy)}

= max{G(gxn , gx, gx), G(gyn , gy, gy), G(gxn+1 , gx, gx), G(gyn+1 , gy, gy)}.

(2.24)

304


Letting n → +∞, we get from (2.24), limn→+∞ M (xn , x, x, yn , y, y) = 0. Thus, using the properties of ψ and φ and letting n → +∞,

ψ(lim sup G(gxn+1 , F (x, y), F (x, y))) = lim sup ψ(G(F (xn , yn ), F (x, y), F (x, y))) n→+∞

n→+∞

≤ lim sup ψ(M (xn , x, x, yn , y, y)) − lim inf φ(M (xn , x, x, yn , y, y)) n→+∞

n→+∞

≤ ψ(lim sup M (xn , x, x, yn , y, y)) − φ(lim inf M (xn , x, x, yn , y, y)) n→+∞

n→+∞

= 0,

(2.25)

which gives us lim supn→+∞ G(gxn+1 , F (x, y), F (x, y)) = 0. So lim G(gxn+1 , F (x, y), F (x, y)) = 0.

(2.26)

n→+∞

On the other hand, by condition (G5) we have G(gx, F (x, y), F (x, y)) ≤ G(gx, gxn+1 , gxn+1 ) + G(gxn+1 , F (x, y), F (x, y)). Letting n → +∞, from (2.24) and (2.26), we have G(gx, F (x, y), F (x, y)) = 0. So gx = F (x, y). Similarly, by (2.2)

ψ(G(gyn+1 , F (y, x), F (y, x))) = ψ(G(F (yn , xn ), F (y, x), F (y, x))) ≤ ψ(M (yn , y, y, xn , x, x)) − φ(M (yn , y, y, xn , x, x)), where M (yn , y, y, xn , x, x) = max{G(gyn , gy, gy), G(gxn , gx, gx), G(F (yn , xn ), gy, gy), G(F (xn , yn ), gx, gx)}

= max{G(gyn , gy, gy), G(gxn , gx, gx), G(gyn+1 , gy, gy), G(gxn+1 , gx, gx)}. Letting n → +∞, we get limn→+∞ M (yn , y, y, xn , x, x) = 0. Thus, letting n → +∞,

ψ(lim sup G(gyn+1 , F (y, x), F (y, x))) = lim sup ψ(G(F (yn , xn ), F (y, x), F (y, x))) n→+∞

n→+∞

≤ lim sup ψ(M (yn , y, y, xn , x, x)) − lim inf φ(M (yn , y, y, xn , x, x)) n→+∞

n→+∞

≤ ψ(lim sup M (yn , y, y, xn , x, x)) − φ(lim inf M (yn , y, y, xn , x, x)) n→+∞

n→+∞

= 0,

(2.27)

and hence lim G(gyn+1 , F (y, x), F (y, x)) = 0.

n→+∞

(2.28)

Again by using (G5), we have G(gy, F (y, x), F (y, x)) ≤ G(gy, gyn+1 , gyn+1 ) + G(gyn+1 , F (y, x), F (y, x)). Letting n → +∞, from (2.24) and (2.28), we have G(gy, F (y, x), F (y, x)) = 0, so gy = F (y, x). Thus, (x, y) is a coupled coincidence point of mappings F and g. Now, putting g = IX (the identity map of X ) in the previous results, we obtain Corollary 2.1. Let (X , ≤) be a partially ordered set and G be a G-metric on X . Let F : X × X → X be a function satisfying (2.2) (with g = IX ) for all x, y, u, v, s, t ∈ X with x ≤ u ≤ s and t ≤ v ≤ y, or s ≤ u ≤ x and y ≤ v ≤ t. Assume that (X , G) is complete and F has the mixed monotone property. Also suppose either 1. F is continuous or 2. X has the following: (i) if a non-decreasing sequence {xn } → x, then xn ≤ x for all n; (ii) if a non-increasing sequence {yn } → y, then y ≤ yn for all n. If there exist x0 , y0 ∈ X such that x0 ≤ F (x0 , y0 ) and F (y0 , x0 ) ≤ y0 , then F has a coupled fixed point. Corollary 2.2. Let (X , ≤) be a partially ordered set and G be a G-metric on X . Let F : X × X → X and g: X → X be two mappings such that F (X × X ) ⊆ g (X ) and F has the mixed g-monotone property. Assume there exists h ∈ [0, 1) such that G(F (x, y), F (u, v), F (s, t )) ≤ h max{G(gx, gu, gs), G(gy, g v, gt ), G(F (x, y), gu, gs), G(F (y, x), g v, gt )}

(2.29)


305

for all x, y, u, v, s, t ∈ X with gx ≤ gu ≤ gs and gt ≤ g v ≤ gy, or gs ≤ gu ≤ gx and gy ≤ g v ≤ gt. Also suppose either 1. F and g are continuous, (X , G) is complete and g commutes with F , or 2. (g (X ), G) is complete and X has the following: (i) if a non-decreasing sequence {xn } → x, then xn ≤ x for all n; (ii) if a non-increasing sequence {yn } → y, then y ≤ yn for all n. If there exist x0 , y0 ∈ X such that gx0 ≤ F (x0 , y0 ) and F (y0 , x0 ) ≤ gy0 , then F and g have a coupled coincidence point. Proof. Define ψ, φ : [0, +∞) → [0, +∞), ψ(t ) = t , result follows from Theorem 2.1 or Theorem 2.2.

φ(t ) = (1 − h)t where h ∈ [0, 1). Then, (2.29) holds. Hence the

Corollary 2.3. Let (X , ≤) be a partially ordered set and G be a G-metric on X . Let F : X × X → X and g: X → X be two mappings such that F (X × X ) ⊆ g (X ) and F has the mixed g-monotone property and G(F (x, y), F (u, v), F (s, t )) ≤ a1 G(gx, gu, gs) + a2 G(gy, g v, gt ) + a3 G(F (x, y), gu, gs) + a4 G(F (y, x), g v, gt )

(2.30)

for all x, y, u, v, s, t ∈ X with gx ≤ gu ≤ gs and gt ≤ g v ≤ gy, or gs ≤ gu ≤ gx and gy ≤ g v ≤ gt where a1 , a2 , a3 and a4 are non-negative real numbers with a1 + a2 + a3 + a4 ∈ [0, 1). Also suppose either 1. F and g are continuous, (X , G) is complete and g commutes with F , or 2. (g (X ), G) is complete and X has the following properties: (i) if a non-decreasing sequence {xn } → x, then xn ≤ x for all n; (ii) if a non-increasing sequence {yn } → y, then y ≤ yn for all n. If there exist x0 , y0 ∈ X such that gx0 ≤ F (x0 , y0 ) and F (y0 , x0 ) ≤ gy0 , then F and g have a coupled coincidence point. Proof. Note that a1 G(gx, gu, gs) + a2 G(gy, g v, gt ) + a3 G(F (x, y), gu, gs) + a4 G(F (y, x), g v, gt )

≤ (a1 + a2 + a3 + a4 ) max{G(gx, gu, gs), G(gy, g v, gt ), G(F (x, y), gu, gs), G(F (y, x), g v, gt )}. The proof follows from Corollary 2.2.

Theorem 2.3. Under the hypotheses of Theorem 2.2, suppose that gy0 ≤ gx0 . Then, it follows gx = F (x, y) = F (y, x) = gy. Moreover, if F and g are w -compatible, then F and g have a coupled coincidence point of the form (u, u). Proof. If gy0 ≤ gx0 , then gy ≤ gyn ≤ gy0 ≤ gx0 ≤ gxn ≤ gx for all n ∈ N. Thus, if gx ̸= gy (and then G(gx, gx, gy) ̸= 0 and G(gy, gy, gx) ̸= 0), hence by inequality (2.2), we have

ψ(G(gy, gx, gx)) = ψ(G(F (y, x), F (x, y), F (x, y))) ≤ ψ(M (y, x, x, x, y, y)) − φ(M (y, x, x, x, y, y)), where M (y, x, x, x, y, y) = max{G(gy, gx, gx), G(gx, gy, gy), G(F (y, x), gx, gx), G(F (x, y), gy, gy)}

= max{G(gy, gx, gx), G(gx, gy, gy), G(gy, gx, gx), G(gx, gy, gy)} = max{G(gy, gx, gx), G(gx, gy, gy)}. Hence

ψ(G(gx, gy, gy)) ≤ ψ(max{G(gx, gy, gy), G(gy, gx, gx)}) − φ(max{G(gx, gy, gy), G(gy, gx, gx)}).

(2.31)

Since gy ≤ gx, hence using the same idea we have

ψ(G(gx, gx, gy)) = ψ(F (x, y), F (x, y), F (y, x)) ≤ ψ(max{G(gx, gy, gy), G(gy, gx, gx)}) − φ(max{G(gx, gy, gy), G(gy, gx, gx)}).

(2.32)

From (2.31) and (2.32), we have

ψ(max{G(gx, gx, gy), G(gx, gy, gy)}) = max{ψ(G(gx, gx, gy)), ψ(G(gx, gy, gy))} ≤ ψ(max{G(gx, gy, gy), G(gy, gx, gx)}) − φ(max{G(gx, gy, gy), G(gy, gx, gx)}). Thus, we obtain φ(max{G(gx, gy, gy), G(gy, gx, gx)}) = 0. Therefore G(gx, gy, gy) = 0 and G(gy, gx, gx) = 0, a contradiction. Hence gx = gy, that is gx = F (x, y) = F (y, x) = gy. Now, let u = gx = gy. Since F and g are w -compatible, then gu = g (gx) = g (F (x, y)) = F (gx, gy) = F (u, u). Thus, F and g have a a coupled coincidence point of the form (u, u). Let (X , ≤) be a partially ordered set. We endow the product set X × X with the partial order given by for all

(x, y), (u, v) ∈ X × X ,

(x, y) ≤ (u, v) ⇐⇒ x ≤ u, v ≤ y.

306


Theorem 2.4. Under the hypotheses of Theorem 2.1, suppose in addition that for every (x, y) and (x∗ , y∗ ) in X , 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∗ )). Then, F and g have a unique common coupled fixed point, that is, there exists a unique pair (x, y) ∈ X × X such that x = gx = F (x, y) and y = gy = F (y, x). Proof. From Theorem 2.1, the set of coupled coincidence points is non-empty. We shall show that if (x, y) and (x∗ , y∗ ) are coupled coincidence points, that is, if gx = F (x, y), gy = F (y, x) and gx∗ = F (x∗ , y∗ ), gy∗ = F (y∗ , x∗ ), then gx = gx∗

and gy = gy∗ .

(2.33)

By assumption, there exists a (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∗ )). Put u0 = u, v0 = v and choose u1 , v1 ∈ X so that gu1 = F (u0 , v0 ) and g v1 = F (v0 , u0 ). Then, similarly as in the proof of Theorem 2.1, we can inductively define sequences {gun } and {g vn } such that gun+1 = F (un , vn ) and

g vn+1 = F (vn , un ).

Further, set x0 = x, y0 = y, x0 = x∗ , y∗0 = y∗ , and on the same way, define the sequences {gxn }, {gyn } and {gx∗n }, {gy∗n }. Since ∗

(gx, gy) = (F (x, y), F (y, x)) = (gx1 , gy1 ) and

(F (u, v), F (v, u)) = (gu1 , g v1 ) are comparable, then gx ≤ gu1 and g v1 ≤ gy. One can show by induction that gx ≤ gun

and g vn ≤ gy

for all n ∈ N. By inequality (2.2)

ψ(G(gx, gx, gun+1 )) = ψ(G(F (x, y), F (x, y), F (un , vn ))) ≤ ψ(M (x, x, un , y, y, vn )) − φ(M (x, x, un , y, y, vn )), where M (x, x, un , y, y, vn ) = max{G(gx, gx, gun ), G(gy, gy, g vn ), G(F (x, y), gx, gun ), G(F (y, x), gy, g vn )}

= max{G(gx, gx, gun ), G(gy, gy, g vn ), G(gx, gx, gun ), G(gy, gy, g vn )} = max{G(gx, gx, gun ), G(gy, gy, g vn )}. Hence

ψ(G(gx, gx, gun+1 )) ≤ ψ(max{G(gx, gx, gun ), G(gy, gy, g vn )}) − φ(max{G(gx, gx, gun ), G(gy, gy, g vn )}). Since g vn ≤ gy and gx ≤ gun , then adjusting as the precedent idea one finds

ψ(G(gy, gy, g vn+1 )) = ψ(F (y, x), f (y, x), F (vn , un )) ≤ ψ(max{G(gx, gx, gun ), G(gy, gy, g vn )}) − φ(max{G(gx, gx, gun ), G(gy, gy, g vn )}). Therefore,

ψ(max{G(gx, gx, gun+1 ), G(gy, gy, g vn+1 )}) = max{ψ(G(gx, gx, gun+1 )), ψ(G(gy, gy, g vn+1 ))} ≤ ψ(max{G(gx, gx, gun ), G(gy, gy, g vn )}) − φ(max{G(gx, gx, gun ), G(gy, gy, g vn )}). By properties of ψ and φ , we deduce max{G(gx, gx, gun+1 ), G(gy, gy, g vn+1 )} ≤ max{G(gx, gx, gun ), G(gy, gy, g vn )}. Thus (max{G(gx, gx, gun ), G(gy, gy, g vn )}) is a non-negative non-increasing sequence. Hence, adjusting as in Theorem 2.1, one can show that lim max{G(gx, gx, gun ), G(gy, gy, g vn )} = 0.

n→+∞

Thus, lim G(gx, gx, gun ) = lim G(gy, gy, g vn ) = 0.

n→+∞

n→+∞

(2.34)

Thanks to the fact that G(w, s, s) ≤ 2G(w, w, s) for any w, s ∈ X , we find from (2.34) lim G(gx, gun , gun ) = lim G(gy, g vn , g vn ) = 0.

n→+∞

n→+∞

(2.35)


307

Similarly, one can prove that lim G(gx∗ , gx∗ , gun ) = lim G(gy∗ , gy∗ , g vn ) = 0

(2.36)

lim G(gx∗ , gun , gun ) = lim G(gy∗ , g vn , g vn ) = 0.

(2.37)

n→+∞

n→+∞

and n→+∞

n→+∞

By the rectangular inequality and (2.34)-(2.37), we have as n → +∞, G(gx, gx, gx∗ ) ≤ G(gx, gx, gun+1 ) + G(gun+1 , gun+1 , gx∗ ) → 0, and G(gy, gy, gy∗ ) ≤ G(gy, gy, g vn+1 ) + G(g vn+1 , g vn+1 , gy∗ ) → 0. Hence gx = gx∗ and gy = gy∗ . Thus, we proved (2.33). Since gx = F (x, y) and gy = F (y, x), by commutativity of F and g, we have g (gx) = g (F (x, y)) = F (gx, gy) and g (gy) = g (F (y, x)) = F (gy, gx).

(2.38)

Denote gx = z and gy = w . Then, from (2.38) gz = F (z , w) and

g w = F (w, z ).

(2.39)

Thus (z , w) is a coupled coincidence point of F and g. Then, from (2.33) with x = z and y = w , it follows that gz = gx and g w = gy, that is, ∗

gz = z

∗

and g w = w.

(2.40)

From (2.39) and (2.40), we get z = gz = F (z , w) and w = g w = F (w, z ). Therefore, (z , w) is a coupled common fixed point of F and g. To prove the uniqueness, assume that (s, t ) is another coupled common fixed point of F and g. Then s = gs = F (s, t ) and t = gt = F (t , s). Since the pair (s, t ) is a coupled coincidence point of F and g, we have gs = gx and gt = gy. Thus, s = gs = gz = z and t = gt = g w = w . Hence, the coupled fixed point is unique. Theorem 2.5. Under the hypotheses of Theorem 2.2, suppose in addition that for every (x, y) and (x∗ , y∗ ) in X , 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∗ )). If F and g are w-compatible, then F and g have a unique common coupled fixed point of the form (u, u). Proof. By Theorem 2.2, the set of coupled fixed points of F and g is not empty. Let (x, y) be a coupled fixed point of F and g. Now, let (x∗ , y∗ ) be another coupled fixed point of F and g. Then, we follow the proof of Theorem 2.4 step by step, to prove that gx = gx∗

and gy = gy∗ .

(2.41)

Note that if (x, y) is a coupled fixed point of F and g, then (y, x) is also a coupled fixed point of F and g. Thus by (2.41), we have gx = gy. Put u = gx = gy. Since gx = F (x, y), gy = F (y, x) and F and g are w -compatible, we have gu = g (gx) = g (F (x, y)) = F (gx, gy) = F (u, u). Thus (u, u) is a coupled fixed point of F and g. So gu = gx = gy = u and hence we have u = gu = F (u, u). Therefore, (u, u) is a common coupled fixed point of F and g. To prove the uniqueness of the coupled fixed point of F and g, let (v, w) be another coupled fixed point of F and g; that is v = g v = F (v, w) and w = g w = F (w, v). By (2.41), we have gx = gu = g v and gy = gu = g w . Therefore u = v = w . Thus F and g have a unique common coupled fixed point of the form (u, u). Now, we give an example illustrating our obtained results. Example 2.1. Let X = [0, +∞) be endowed with the usual metric, and with the usual order in R. Consider the function G: [0, +∞)3 → [0, +∞),

G(x, y, z ) = max{|x − y|, |x − z |, |y − z |}.

(2.42)

It is known from [29] that (X , G) is a G-metric space. Set F and g as

x − y F: X × X → X,

F (x, y) =

0

8

if x ≥ y if x < y,

g: X → X ,

gx =

1 2

x.

It is clear that F has the mixed g-monotone property. Also, it is obvious that (g (X ), G) is complete and F (X × X ) ⊆ g (X ). Moreover, taking x0 = y0 = 0, then x0 ≤ F (x0 , y0 ) and F (y0 , x0 ) ≤ y0 .

308


Define ψ, φ : [0, +∞) → [0, +∞), and (2.2), we obtain G(F (x, y), F (u, v), F (s, t )) ≤

1 2

ψ(t ) = t ,

φ(t ) = 21 t, and looking at the formulas for ψ and φ , by using (2.1)

M (x, u, s, y, v, t ).

(2.43)

By definition of g, we shall prove that (2.43) holds for all x, y, u, v, s, t ∈ X with x ≤ u ≤ s and t ≤ v ≤ y. For this, we distinguish the following cases: Case 1. x ≥ y. In this case, we have t ≤ v ≤ y ≤ x ≤ u ≤ s, so F (x, y) =

x−y 8

,

F (u, v) =

u−v 8

,

s−t

F (s, t ) =

8

.

By (2.42), we get G(F (x, y), F (u, v), F (s, t )) = max

= =



−

8 2

−

8

s−t 1

u−v

x−y

x−y s−t 8

≤

8

,

s 4

|g (s) − F (x, y)| ≤

− 1 2

8 x−y 16

− =

x−y s−t

,

8 1



2

s 2

8

−

−

x−y

G(F (x, y), gu, gs) ≤

u−v



8



8 1 2

M (x, u, s, y, v, t ),

which is (2.43). Case 2. x < y. We divide the study in two sub-cases: t • If u ≥ v . Here F (x, y) = 0 and F (u, v) = u−v . Then, t ≤ v ≤ u ≤ s, so s ≥ t, that is, F (s, t ) = s− . We have 8 8 G(F (x, y), F (u, v), F (s, t )) = max

= ≤



s−t 8 1 2

u−v s−t s−t 8

≤

s 4

,

=

8 1 2

,

8

−

u−v



8

|g (s) − F (x, y)|

G(F (x, y), gu, gs) ≤

1 2

M (x, u, s, y, v, t ),

that is (2.43) holds. • If u < v . Here, F (x, y) = 0 = F (u, v), the case where s < t is obvious, because we get F (s, t ) = 0. If s ≥ t, we have t F ( s, t ) = s − . Therefore, 8 G(F (x, y), F (u, v), F (s, t )) =

s−t 8

≤

s 4

≤

1 2

G(F (x, y), gu, gs) ≤

1 2

M (x, u, s, y, v, t ),

which is (2.43). Note that (2.43) still holds when s ≤ u ≤ x and y ≤ v ≤ t. On the other hand, X satisfies the conditions (i) and (ii) given in Theorem 2.2. All the required hypotheses of Theorem 2.2 are satisfied. Clearly, F and g have a coupled fixed point, which is (0, 0). 3. Conclusion In this work, we established coupled coincidence and common coupled fixed point theorems for (ψ, φ)-weakly contractive mappings in ordered G-metric spaces. We accompanied our theoretical results by an applied example. Our results are extensions of several results as in relevant items from the reference section of this paper, as well as in the literature in general. Acknowledgments The authors thank the referees for their appreciation, valuable comments and suggestions. References [1] A.C.M. Ran, M.C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004) 1435–1443. [2] J.J. Nieto, R.R. López, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser.) 23 (12) (2007) 2205–2212. [3] Lj. Ćirć, M. Abbas, R. Saadati, N. Hussain, Common fixed points of almost generalized contractive mappings in ordered metric spaces, Appl. Math. Comput. 217 (2011) 5784–5789.


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