# Coincidence Points of Weaker Contractions in Partially Ordered Metric Spaces

Authors: Kuo-Ching Jen; Ing-Jer Lin; Chi-Ming Chen
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We prove new coincidence point theorems for the(φ,ψ,ϕ,ξ)-contractions and generalized Meir-Keeler-type α-ψ-contractions in partially ordered metric spaces. Our results generalize many recent coincidence point theorems in the literature.


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## Body

## 1. Introduction and Preliminaries

Throughout this paper, byℝ+, we denote the set of all nonnegative real numbers, while ℕ is the set of all natural numbers. Let (X,d) be a metric space, D a subset of X, and f:D→X a map. We say f is contractive if there exists α∈[0,1) such that for all x,y∈D,
(1)d(fx,fy)≤α·d(x,y).
The well-known Banach's fixed point theorem asserts that if D=X, f is contractive and (X,d) is complete, then f has a unique fixed point in X. It is well known that the Banach contraction principle [1] is a very useful and classical tool in nonlinear analysis. Also, this principle has many generalizations. For instance, a mapping f:X→X is called a quasicontraction if there exists k<1 such that
(2)d(fx,fy)≤k·max{d(x,y),d(x,fx),ffffffffd(y,fy),d(x,fy),d(y,fx)},
for any x,y∈X. In 1974, Ćirić [2] introduced these maps and proved an existence and uniqueness fixed point theorem.Recently, Eslamian and Abkar proved the following theorem.Theorem 1 (see [3]).
Let(X,d) be a complete metric space and f:X→X be such that
(3)ψ(d(fx,fy))≤α(d(x,y))-β(d(x,y)),dddddddddddddddddddforeachx,y∈X,
where ψ,α,β:ℝ+→ℝ+ are as follows: ψ is continuous and nondecreasing, α is continuous, β is lower semicontinuous, and
(4)ψ(t)-α(t)+β(t)>0∀t>0,ψ(t)=0ifft=0,α(0)=β(0)=0.
Then f has a fixed point in X.Recently, fixed point theory has developed rapidly in partially ordered metric spaces (e.g., [4–8]).In 2012, Choudhury and Kundu [9] proved the following coincidence theorem as a generalization of Theorem 1.Theorem 2 (see [9]).
Let(X,⊑) be a partially ordered set and suppose that there exists a metric d in X such that (X,d) is a complete metric space and f,g:X→X be such that fX⊂gX, f is g-nondecreasing, gX is closed, and
(5)ψ(d(fx,fy))≤α(d(gx,gy))-β(d(gx,gy)),fffffffffffffforeachx,y∈Xsuchthatgx⊑gy,
where ψ,α,β:ℝ+→ℝ+ are such that ψ is continuous and nondecreasing, α is continuous, β is lower semicontinuous, and
(6)ψ(t)-α(t)+β(t)>0∀t>0,ψ(t)=0ifft=0,α(0)=β(0)=0.
Also, if any nondecreasing sequence {xn} in X converges to ν, then we assume that
(7)xn⊑ν∀n∈ℕ.
If there exists x0∈X with gx0⊑fx0, then f and g have a coincidence point in X.In this paper, we prove new coincidence point theorems for the(φ,ψ,ϕ,ξ)-contractions and generalized Meir-Keeler-type α-ψ-contractions in partially ordered metric spaces. Our results generalize many recent coincidence point theorems in the literature.


## 2. Main Results

We start with the following definition.Definition 3 (g-nondecreasing mapping [4]).
Let(X,⊑) be a partially ordered set and f,g:X→X. Then f is said to be g-nondecreasing if, for x,y∈X,
(8)gx⊑gy⟹fx⊑fy.In the sequel, we denote byΨ the class of functions ψ:ℝ+5→ℝ+ satisfying the following conditions:  (ψ1)
ψ is an increasing, continuous function in each coordinate,(ψ2)
for allt∈ℝ+, ψ(t,t,t,0,2t)≤t, ψ(t,t,t,2t,0)≤t, ψ(0,0,t,t,0)≤t, and ψ(t,0,0,t,t)≤t,(ψ3)
ψ(t1,t2,t3,t4,t5)=0 if and only if t1=t2=t3=t4=t5=0.Next, we denote byΦ the class of functions ϕ:ℝ+→ℝ+ satisfying the following conditions: (ϕ1)
ϕ is a continuous function and monotone nondecreasing;(ϕ2)
ϕ(t)>0 for t>0 and ϕ(0)=0;(ϕ3)
ϕ is subadditive, that is, ϕ(t1+t2)≤ϕ(t1)+ϕ(t2) for all t1,t2>0.And, we denote the following sets of functions:(9)Θ={φ:ℝ+⟶ℝ+suchthatφiscontinuous},Ξ={ξ:ℝ+⟶ℝ+suchthatξislowercontinuous}.LetX be a nonempty set and (X,⊑) be a partially ordered set endowed with a metric d. Then, the triple (X,⊑,d) is called a partially ordered metric space.We now state the(φ,ψ,ϕ,ξ)-contraction and the main fixed point theorem for the (φ,ψ,ϕ,ξ)-contraction in partially ordered metric spaces, as follows.Definition 4.
Let(X,⊑,d) be a partially ordered complete metric space, and let f,g:X→X. Then the pair (f,g) is called a (φ,ψ,ϕ,ξ)-contraction if the following inequality holds:
(10)φ(d(fx,fy))ddd≤ψ(ϕ(d(gx,gy)),ϕ(d(gx,fx)),dddddddϕ(d(gy,fy)),ϕ(d(gx,fy)),ϕ(d(gy,fx)))ddddd-ξ(max{d(gx,gy),d(gx,fx),d(gy,fy)}),
for all x,y∈X with gx⊑gy, where φ∈Θ, ψ∈Ψ, ϕ∈Φ and ξ∈Ξ.We now state the main fixed point theorem for the(φ,ψ,ϕ,ξ)-contraction in partially ordered metric spaces, as follows.Theorem 5.
Let(X,⊑,d) be a partially ordered complete metric space, and let f,g:X→X be such that fX⊂gX, f is g-nondecreasing and gX is closed. Suppose the pair (f,g) is a (φ,ψ,ϕ,ξ)-contraction, and
(11)φ(t)-ϕ(t)+ξ(t)>0∀t>0,φ(t)=0ifft=0,ϕ(0)=ξ(0)=0.
Also, if any nondecreasing sequence {xn} in X converges to ν, then we assume that
(12)xn⊑ν∀n∈ℕ.
If there exists x0∈X with gx0⊑fx0, then f and g have a coincidence point in X.Proof.
SincefX⊂gX and there exists x0∈X with gx0⊑fx0, we can choose x1∈X such that gx1=fx0. Since f is g-nondecreasing, we have fx0⊑fx1. In this process, we construct the sequence {xn} recursively as
(13)fxn=gxn+1∀n∈ℕ.
Thus, we also conclude that
(14)gx0⊑fx0=gx1⊑fx1=gx2⊑⋯⊑fxn-1=gxn⊑fxn=gxn+1⊑⋯.
If any two consecutive terms in (14) are equal, then the conclusion of the theorem follows. So we may assume that
(15)d(fxn-1,fxn)≠0,∀n∈ℕ.
Now, we claim that d(fxn,fxn+1)≤d(fxn-1,fxn) for all n∈ℕ. If not, we assume that d(fxn-1,fxn)<d(fxn,fxn+1) for some n∈ℕ, substituting x=xn and y=xn+1 in (10) and using the definition of the function ψ, we have
(16)ψ(ϕ(d(gxn,gxn+1)),ϕ(d(gxn,fxn)),ϕ(d(gxn+1,fxn+1)),ϕ(d(gxn,fxn+1)),ϕ(d(gxn+1,fxn)))=ψ(ϕ(d(fxn-1,fxn)),ϕ(d(fxn-1,fxn)),ϕ(d(fxn,fxn+1)),ϕ(d(fxn-1,fxn+1)),ϕ(d(fxn,fxn)))≤ψ(ϕ(d(fxn,fxn+1)),ϕ(d(fxn,fxn+1)),ϕ(d(fxn,fxn+1)),2ϕ(d(fxn,fxn+1)),ϕ(0))≤ϕ(d(fxn,fxn+1)),ξ(max{d(gxn,gxn+1),d(gxn,fxn),ffffsfffd(gxn+1,fxn+1)})=ξ(max{d(fxn-1,fxn),d(fxn-1,fxn),fffffffffffd(fxn,fxn+1)})=ξ(d(fxn,fxn+1)),
and hence
(17)φ(d(fxn,fxn+1))≤ϕ(d(fxn,fxn+1))-ξ(d(fxn,fxn+1)).
Since φ(t)-ϕ(t)+ξ(t)>0 for all t>0, we have that d(fxn,fxn+1)=0, which contradicts to (15). Therefore, we conclude that
(18)d(fxn,fxn+1)≤d(fxn-1,xn)∀n∈ℕ.
From above argument, we also have that for each n∈ℕ(19)φ(d(fxn,fxn+1))≤ϕ(d(fxn-1,fxn))-ξ(d(fxn-1,fxn)).
It follows (18) that the sequence {d(fxn,fxn+1)} is monotone decreasing, it must converge to some η≥0. Taking limit as n→∞ in (19) and using the continuities of φ and ϕ and the lower semicontinuity of ξ, we get
(20)φ(η)≤ϕ(η)-ξ(η),
which implies that η=0. So we conclude that
(21)limn→∞d(fxn,fxn+1)=0.
We next claim that{fxn} is a Cauchy sequence, that is, for every ε>0, there exists n∈ℕ such that if p,q≥n, then d(fxp,fxq)<ε.
Suppose the above statement is false. Then there existsϵ>0 such that for any n∈ℕ, there are pn,qn∈ℕ with pn>qn≥n satisfying
(22)d(fxqn,fxpn)≥ϵ.
Further, corresponding to qn≥n, we can choose pn in such a way that it is the smallest integer with pn>qn≥n and d(fxqn,fxpn)≥ϵ. Therefore d(fxqn,fxpn-1)<ϵ. Now we have that for all n∈ℕ(23)ϵ≤d(fxpn,fxqn)≤d(fxpn,fxpn-1)+d(fxpn-1,fxqn)<d(fxpn,fxpn-1)+ϵ.
Letting n→∞, then we get
(24)limn→∞d(fxpn,fxqn)=ϵ.
On the other hand, we have
(25)d(fxpn,fxqn)≤d(fxpn,fxpn-1)+d(fxpn-1,fxqn-1)+d(fxqn-1,fxqn),d(fxpn-1,fxqn-1)≤d(fxpn-1,fxpn)+d(fxpn,fxqn)+d(fxqn,fxqn-1).
Letting n→∞, then we get
(26)limn→∞d(fxpn-1,fxqn-1)=ϵ.
By (14), we have that the elements gxpn and gxqn are comparable. Substituting x=xpn and y=xqn in (10), we have that for all n∈ℕ,
(27)ψ(ϕ(d(gxpn,gxqn)),ϕ(d(gxpn,fxpn)),ϕ(d(gxqn,fxqn)),ϕ(d(gxpn,fxqn)),ϕ(d(gxqn,fxpn)))≤ψ(ϕ(d(fxpn-1,fxqn-1)),ϕ(d(fxpn-1,fxpn)),dddfdϕ(d(fxqn-1,fxqn)),ϕ(d(fxpn-1,fxqn)),dddfdϕ(d(fxqn-1,fxpn)))≤ψ(ϕ(d(fxpn-1,fxqn-1)),ϕ(d(fxpn-1,fxpn)),dddddϕ(d(fxqn-1,fxqn)),ϕ(d(fxpn-1,fxpn))hddddh+ϕ(d(fxpn,fxqn)),ϕ(d(fxqn-1,fxqn))dddsdd+ϕ(d(fxqn,fxpn))),M(xpn,xqn)=max{d(gxpn,gxqn),dddddddddddddddd(gxpn,fxpn),d(gxqn,fxqn)}ddddddsddd=max{d(fxpn-1,fxqn-1),d(fxpn-1,fxpn),dddddddddddddddd(fxqn-1,fxqn)}.
By above argument and using inequality (10), we can conclude that
(28)φ(ϵ)≤ψ(ϕ(ϵ),0,0,ϕ(ϵ),ϕ(ϵ))-ξ(ϵ)≤ϕ(ϵ)-ξ(ϵ),
which implies that ϵ=0, a contradiction. Therefore, the sequence {fxn} is a Cauchy sequence.
SinceX is complete and gX is closed, there exists ν∈X such that
(29)limn→∞gxn=limn→∞fxn=gν.
Later, we prove that ν is a coincidence point of f and g. From (14) and (29), we deduce that
(30)gxn⊑gν,∀n∈ℕ.
Substituting x=xn and y=ν in (10), we have that
(31)φ(d(fxn,fν))dd≤ψ(ϕ(d(gxn,gν)),ϕ(d(gxn,fxn)),ddddddϕ(d(gν,fν)),ϕ(d(gxn,fν)),ddddddϕ(d(gν,fxn)))dddd-ξ(max{d(gxn,gν),d(gxn,fxn),d(gν,fν)}).
Taking n→∞ in the above inequality, we have
(32)φ(d(gν,fν))≤ψ(0,0,ϕ(d(gν,fν)),ϕ(d(gν,fν)),0)-ξ(d(gν,fν))≤ϕ(d(gν,fν))-ξ(d(gν,fν)),
which implies that d(gν,fν)=0, that is, gν=fν. So we complete the proof.We give the following example to illustrate Theorem5.Example 6.
LetX=[0,1]. We define a partial order “⊑” on X as x⊑y if and only if x≥y for all x,y∈X. We take the usual metric d(x,y)=|x-y| for all x,y∈X. Let f,g:X→X be defined as
(33)f(x)=116x2,g(x)=14x2.
Let φ,ϕ,ξ:ℝ+→ℝ+ be defined as
(34)φ(t)=ϕ(t)=t,ξ(t)=t8∀t∈[0,1],
and let ψ:ℝ+5→ℝ+ denote
(35)ψ(t1,t2,t3,t4,t5)=12·max{t1,t2,t3,t42,t52}.
Without loss of generality, we assume that x>y and verity inequality (10).
For allx,y∈[0,1] with x>y, we have
(36)φ(d(fx,fy))=116(x2-y2),ϕ(d(gx,gy))=14(x2-y2),ϕ(d(gx,fx))=14x2-116x2=316x2,ϕ(d(gy,fy))=14y2-116y2=316y2,ϕ(d(gx,fy))=14x2-116y2>316x2,ϕ(d(fx,gy))=|116x2-14y2|,ξ(maxd(gx,gy),d(gx,fx),d(gy,fy))={14(x2-y2),ifx>2y,316x2,ifx≤2y,ψ(ϕ(d(gx,gy)),ϕ(d(gx,fx)),ϕ(d(gy,fy)),ϕ(d(gx,fy)),ϕ(d(fx,gy)))=18x2-132y2.
Therefore, inequality (10) is satisfied and all the conditions of Theorem 5 are satisfied, and we obtained that 0 is a coincidence point of f and g.Applying Definition4, Theorem 5, and Example 6, if we let
(37)ψ(ϕ(d(gx,gy)),ϕ(d(gx,fx)),ϕ(d(gy,fy)),ϕ(d(gx,fy)),ϕ(d(gy,fx)))=max{12ϕ(d(fx,gy)ϕ(d(gx,gy)),ϕ(d(gx,fx)),ϕ(d(gy,fy)),dddddd12ϕ(d(gx,fy)),12ϕ(d(fx,gy)},
we are easy to get the following theorem.Theorem 7.
Let(X,⊑,d) be a partially ordered complete metric space, and let f,g:X→X be such that fX⊂gX, f is g-nondecreasing, gX is closed, and
(38)φ(d(fx,fy))≤max{(d(gx,fy))ϕ(d(gx,gy)),ϕ(d(gx,fx)),fffffffffffdffffϕ(d(gy,fy)),ϕ(d(gx,fy)),ψ(d(fx,gy))}dddd-ξ(max{d(gx,gy),d(gx,fx),d(gy,fy)}),
for all x,y∈X such that gx⊑gy, where φ∈Θ, ψ∈Ψ, ϕ∈Φ and ξ∈Ξ, and
(39)φ(t)-ϕ(t)+ξ(t)>0∀t>0,φ(t)=0ifft=0,ϕ(0)=ξ(0)=0.
Also, if any nondecreasing sequence {xn} in X converges to ν, then one assumes that
(40)xn⊑ν∀n∈ℕ.If there existsx0∈X with gx0⊑fx0, then f and g have a coincidence point in X.In the other research of this paper, we recall the Meir-Keeler-type contraction [10] and α-admissible mapping [11]. In 1969, Meir and Keeler [10] introduced the following notion of Meir-Keeler-type contraction in a metric space (X,d).Definition 8.
Let(X,d) be a metric space, f:X→X. Then f is called a Meir-Keeler-type contraction whenever for each η>0 there exists γ>0 such that
(41)η≤d(x,y)<η+γ⟹d(fx,fy)<η.And, the following definition was introduced in [11].Definition 9.
Letf:X→X be a self-mapping of a set X and α:X×X→ℝ+. Then f is called a α-admissible mapping if
(42)x,y∈X,α(x,y)≥1⟹α(fx,fy)≥1.We introduce the notion ofα-g-admissible mapping, as follows.Definition 10.
Letf,g:X→X be a self-mapping of a set X and α:X×X→ℝ+. Then f is called a α-g-admissible mapping if
(43)x,y∈X,α(gx,gy)≥1⟹α(fx,fy)≥1.We give the following example to illustrate Definition10.Example 11.
LetX=ℝ+ and we define
(44)g(x)=x+12,f(x)=g(x)+1x+1,α(x,y)=x+y.
Then f is a α-g-admissible mapping.We now state the new notions of generalized Meir-Keeler-typeψ-contractions and generalized Meir-Keeler-type α-ψ-contractions in partially ordered complete metric spaces, as follows.Definition 12.
Let(X,⊑,d) be a partially ordered complete metric space, and let f,g:X→X. Then the pair (f,g) is called a generalized Meir-Keeler-type ψ-contraction whenever for each η>0, there exists δ>0 such that
(45)η≤ψ(d(gx,gy),d(gx,fx),d(gy,fy),ffd(gx,fy),d(gy,fx))<η+δ⟹d(fx,fy)<η,
for all x,y∈X with gx⊑gy, where ψ∈Ψ.Definition 13.
Let(X,⊑,d) be a partially ordered complete metric space, let f,g:X→X, and α:X×X→ℝ+. Then (f,g) is called a generalized Meir-Keeler-type α-ψ-contraction if the following conditions hold:  (1)
f is α-g-admissible;(2)
for eachη>0 there exists δ>0 such that
(46)η≤ψ(d(gx,gy),d(gx,fx),d(gy,fy),fffffd(gx,fy),d(gy,fx))<η+δ⟹α(fx,fx)α(gy,gy)d(fx,fy)<η,for all x,y∈X with gx⊑gy, where ψ∈Ψ.Remark 14.
Note that iff is a generalized Meir-Keeler-type α-ψ-contraction, then we have that for all x,y∈X(47)α(fx,fx)α(gy,gy)d(fx,fy)≤ψ(d(gx,gy),d(gx,fx),d(gy,fy),ffffgfd(gx,fy),d(gy,fx)).
Further, if
(48)ψ(d(gx,gy),d(gx,fx),d(gy,fy),ssd(gx,fy),d(gy,fx))=0,
then d(fx,fy)=0.
On the other hand, if(49)ψ(d(gx,gy),d(gx,fx),d(gy,fy),ddd(gx,fy),d(gy,fx))>0,
then
(50)α(fx,fx)α(gy,gy)d(fx,fy)<ψ(d(gx,gy),d(gx,fx),d(gy,fy),fffhfffd(gx,fy),d(gy,fx)).We now state our main result for the generalized Meir-Keeler-typeα-ψ-contraction, as follows.Theorem 15.
Let(X,⊑,d) be a partially ordered complete metric space, let α:X×X→ℝ+ be continuous in each coordinate, and let f,g:X→X be such that fX⊂gX, f is g-nondecreasing, and gX is closed. Suppose the pair (f,g) is a generalized Meir-Keeler-type α-ψ-contraction and the following conditions hold.(i)
If any nondecreasing sequence{xn} in X converges to ν, then we assume that
(51)xn⊑ν∀n∈ℕ.(ii)
There existsx0∈X with gx0⊑fx0 and α(fx0,fx0)≥1.(iii)
Ifα(fxn,fxn)≥1 for all n∈ℕ, then limn→∞α(fxn,fxn)≥1. Then f and g have a coincidence point in X.Proof.
SincefX⊂gX and by (ii), there exists x0∈X with gx0⊑fx0 and α(fx0,fx0)≥1, we can choose x1∈X such that gx1=fx0. Since f is g-nondecreasing, we have fx0⊑fx1. In this process, we construct the sequence {xn} recursively as
(52)fxn=gxn+1∀n∈ℕ.
Thus, we also conclude that
(53)gx0⊑fx0=gx1⊑fx1=gx2⊑⋯⊑fxn-1=gxn⊑fxn=gxn+1⊑⋯.
If any two consecutive terms in (53) are equal, then the conclusion of the theorem follows. So we may assume that
(54)d(fxn-1,fxn)≠0,∀n∈ℕ.
On the other hand, since f is α-g-admissible and α(fx0,fx0)=α(gx1,gx1)≥1, we have
(55)α(fx1,fx1)=α(gx2,gx2)≥1.
By continuing this process, we get
(56)α(fxn,fxn)=α(gxn+1,gxn+1)≥1∀n∈ℕ∪{0}.
By (53), (54), and (56), substituting x=xn and y=xn+1 in (50), we have
(57)d(fxn,fxn+1)≤α(fxn,fxn)α(gxn+1,gxn+1)d(fxn,fxn+1)<ψ(d(gxn,gxn+1),d(gxn,fxn),d(gxn+1,fxn+1),dddddd(gxn,fxn+1),d(gxn+1,fxn))=ψ(d(fxn-1,fxn),d(fxn-1,fxn),d(fxn,fxn+1),ddddddd(fxn-1,fxn+1),d(fxn,fxn)).
If d(fxn-1,fxn)≤d(fxn,fxn+1), then the inequality (57) becomes
(58)d(fxn,fxn+1)<ψ(d(fxn-1,fxn),d(fxn-1,fxn),d(fxn,fxn+1),fffffffd(fxn-1,fxn+1),d(fxn,fxn))≤ψ(d(fxn,fxn+1),d(fxn,fxn+1),d(fxn,fxn+1),fffffffffff2d(fxn,fxn+1),0)≤d(fxn,fxn+1),
which implies a contradiction, and we get that d(fxn,fxn+1)<d(fxn-1,fxn).
From the argument above, we have that the sequence{d(fxn,fxn+1)} is decreasing, and it must converge to some η≥0, that is,
(59)limn→∞d(fxn,fxn+1)=η.
It follow from that (57) and (59), we have
(60)limn→∞ψ(d(fxn-1,fxn),d(fxn-1,fxn),d(fxn,fxn+1),sssd(fxn-1,fxn+1),d(fxn,fxn))=η.
Notice that η=inf{d(fxn,fxn+1):n∈ℕ∪{0}}. We claim that η=0. Suppose, to the contrary, that η>0. Since (f,g) is a generalized Meir-Keeler-type α-ψ-contraction, corresponding to η use, and taking into account the above inequality (60), there exist δ>0 and a natural number k such that
(61)η≤ψ(d(fxk-1,fxk),d(fxk-1,fxk),ssssd(fxk,fxk+1),d(fxk-1,fxk+1),ssssd(fxk,fxk))<η+δ⟹α(fxk,fxk)α(gxk+1,gxk+1)ssssssssssss×d(fxk,fxk+1)<η,
which implies
(62)d(fxk,fxk+1)≤α(fxk,fxk)×α(gxk+1,gxk+1)d(fxk,fxk+1)<η.
So we get a contradiction, since η=inf{d(fxn,fxn+1):n∈ℕ∪{0}}. Thus we have that
(63)limn→∞d(fxn,fxn+1)=0.
We next claim that{fxn} is a Cauchy sequence, that is, for every ε>0, there exists n∈ℕ such that if p,q≥n, then d(fxp,fxq)<ε.
Suppose the above statement is false. Then there existsϵ>0 such that for any n∈ℕ, there are pn,qn∈ℕ with pn>qn≥n satisfying
(64)d(fxqn,fxpn)≥ϵ.
Further, corresponding to qn≥n, we can choose pn in such a way that is it the smallest integer with pn>qn≥n and d(fxqn,fxpn)≥ϵ. Therefore d(fxqn,fxpn-1)<ϵ. Now we have that for all n∈ℕ(65)ϵ≤d(fxpn,fxqn)≤d(fxpn,fxpn-1)+d(fxpn-1,fxqn)<d(fxpn,fxpn-1)+ϵ.
Letting n→∞, then we get
(66)limn→∞d(fxpn,fxqn)=ϵ.
On the other hand, we have
(67)ssssd(fxpn,fxqn)≤d(fxpn,fxpn-1)+d(fxpn-1,fxqn-1)sssssssssssssssssssss+d(fxqn-1,fxqn),d(fxpn-1,fxqn-1)≤d(fxpn-1,fxpn)+d(fxpn,fxqn)sssssssssssssssssssss+d(fxqn,fxqn-1).
Letting n→∞, then we get
(68)limn→∞d(fxpn-1,fxqn-1)=ϵ.
By (53), we have that the elements gxpn and gxqn are comparable. Substituting x=xpn and y=xqn in (50), we have that for all n∈ℕ,
(69)d(fxpn,fxqn)≤α(fxpn,fxpn)α(gxqn,gxqn)d(fxpn,fxqn)<ψ(d(gxpn,gxqn),d(gxpn,fxpn),d(gxqn,fxqn),ffffsffd(gxpn,fxqn),d(gxqn,fxpn))≤ψ(d(fxpn-1,fxqn-1),d(fxpn-1,fxpn),dddsdddd(fxqn-1,fxqn),d(fxpn-1,fxqn),ddddsddd(fxqn-1,fxpn))≤ψ(d(fxpn-1,fxqn-1),d(fxpn-1,fxpn),fffsdffd(fxqn-1,fxqn),d(fxpn-1,fxpn)+d(fxpn,fxqn),fffsddfd(fxqn-1,fxqn)+d(fxqn,fxpn)).
Letting n→∞ in (69), then we get
(70)ϵ<ψ(ϵ,0,0,ϵ,ϵ)≤ϵ,
which implies a contradiction. Thus, {fxn} is a Cauchy sequence.
SinceX is complete and gX is closed, there exists ν∈X such that
(71)limn→∞gxn=limn→∞fxn=gν.
Since α is continuous in each coordinate and by the condition (iii), we have
(72)α(gν,gν)=limn→∞α(fxn,fxn)≥1.
Later, we prove thatν is a coincidence point of f and g. From (53) and (71), we deduce that
(73)gxn⊑gν,∀n∈ℕ.
By (72) and substituting x=xn and y=ν in (50), we have that
(74)d(fxn,fν)≤α(fxn,fxn)α(gν,gν)d(fxn,fν)<ψ(d(gxn,gν),d(gxn,fxn),fffffd(gν,fν),d(gxn,fν),d(gν,fxn)).
Taking n→∞ in the above inequality, we have
(75)d(gν,fν)<ψ(d(gν,gν),d(gν,gν),ffsffd(gν,fν),d(gν,fν),d(gν,gν))≤d(gν,fν).
This implies that gν=fν. So we complete the proof.Apply Theorem15, we are easy to get the following theorem.Theorem 16.
Let(X,⊑,d) be a partially ordered complete metric space, and let f,g:X→X be such that fX⊂gX, f is g-nondecreasing, and gX is closed. Suppose the pair (f,g) is a generalized Meir-Keeler-type ψ-contraction and the following conditions hold.  (i)
If any nondecreasing sequence{xn} in X converges to ν, then we assume that
(76)xn⊑ν∀n∈ℕ.(ii)
There existsx0∈X with gx0⊑fx0. Then f and g have a coincidence point in X.



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*Source: 507805-2013-09-09.xml*