# Common Fixed Points of Generalized Cyclic Meir-Keeler-Type Contractions in Partially Ordered Metric Spaces

Authors: Chiming Chen; Ing-Jer Lin
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The purpose of this paper is to prove some common point theorems
for the generalized cyclic Meir-Keeler-type (α,φ,A,B)-contraction in partially ordered metric spaces. Our results generalize many recent common
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, let D be a subset of X, and let f:D→X be a map. We say that 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 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),d(y,fy),≤k·maxxxd(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.The following definitions and results will be needed in the sequel. LetA and B be two nonempty subsets of a metric space (X,d). A mapping f:A∪B→A∪B is called a cyclic map if f(A)⊆B and f(B)⊆A. In 2003, Kirk et al. [3, 4] proved the following fixed point theorem.Theorem 1 (see [3, 4]).
LetA and B be two nonempty closed subsets of a complete metric space (X,d), and suppose that f:A∪B→A∪B satisfies(i)
f(A)⊂B and f(B)⊂A,(ii)
d(fx,fy)≤k·d(x,y) for all x∈A, y∈B, and k∈(0,1).
ThenA∩B is nonempty, and f has a unique fixed point in A∩B.Recently, many authors proved some fixed point theorems for cyclic maps satisfying various contractive conditions (see, [5–20]).LetX be a nonempty set, and let (X,⊑) be a partially ordered set endowed with a metric d. Then, the triple (X,⊑,d) is called a partially ordered metric space. Two elements x,y∈X are said to be comparable if either x⊑y or y⊑x holds. Altun et al. [21] introduced the notion of weakly increasing mappings and proved some existing theorems.Definition 2 (see [21]).
Let(X,⊑) be a partially ordered set and f,g:X→X. Then f,g are said to be weakly increasing if fx⊑gfx and gx⊑fgx for all x∈X.And the following definition was introduced in [22].Definition 3 (see [22]).
Let(X,⊑) be a partially ordered set, let A,B be closed subsets of X with X=A∪B, and let f,g:X→X. Then the pair (f,g) is said to be (A,B)-weakly increasing if fx⊑gfx for all x∈A and gx⊑fgx for all x∈B.In this paper, we introduce the new notion of generalized cyclic Meir-Keeler-type(α,ψ,A,B)-contraction. The purpose of this paper is to prove some common point theorems for the generalized cyclic Meir-Keeler-type (α,ψ,A,B)-contraction in partially ordered metric spaces. Our results generalize many recent common point theorems in the literature.


## 2. Main Results

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;(ψ2)
ψ(t1,t2,t3,t4,t5)=0 if and only if t1=t2=t3=t4=t5=0.We start with the following definition.Definition 4 (see [23]).
Letf:X→X be a self-mapping of a set X and α:X×X→ℝ+. Then f is called α-admissible if
(3)x,y∈X,α(x,y)≥1⟹α(fx,fy)≥1.Definition 5.
LetA,B be two nonempty subsets of a set X with X=A∪B, let f:A→B, g:B→A with f(A)⊂B and g(B)⊂A, and let α:X×X→ℝ+. Then the pair (f,g) is called α-admissible if the following conditions hold:  (1)
α(fx,fx)≥1, ∀x∈A⇒α(gfx,gfx)≥1,(2)
α(gy,gy)≥1, ∀y∈B⇒α(fgy,fgy)≥1.In 1969, Meir and Keeler [24] introduced the following notion of Meir-Keeler-type contraction in a metric space (X,d).Definition 6.
Letting(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
(4)η≤d(x,y)<η+γ⟹d(fx,fy)<η.We now state the new notions of generalized cyclic Meir-Keeler-type (ψ,A,B)-contractions and generalized Meir-Keeler-type (α,ψ,A,B)-contractions in partially ordered metric spaces as follows.Definition 7.
Let(X,⊑,d) be a partially ordered metric space, let A,B be two nonempty subsets of X with X=A∪B, and let f:A→B, g:B→A with f(A)⊂B and g(B)⊂A. Then the pair (f,g) is called a generalized cyclic Meir-Keeler-type (ψ,A,B)-contraction; if for any comparable elements x,y∈X with x∈A and y∈B, we have that for each η>0 there exists δ>0 such that
(5)η≤ψ(d(x,y),d(x,fx),d(y,gy),d(x,gy),d(y,fx))<η+δ⟹d(fx,gy)<η,
where ψ∈Ψ.Definition 8.
Let(X,⊑,d) be a partially ordered metric space, let A,B be two nonempty subsets of X with X=A∪B, and let f:A→B, g:B→A with f(A)⊂B and g(B)⊂A, and α:X×X→ℝ+. Then (f,g) is called a generalized cyclic Meir-Keeler-type (α,ψ,A,B)-contraction if the following conditions hold:  (1)
the pair(f,g) is α-admissible;(2)
for any comparable elementsx,y∈X with x∈A and y∈B, we have that for each η>0 there exists δ>0 such that
(6)η≤ψ(d(x,y),d(x,fx),d(y,gy),d(x,gy),d(y,fx))η<η+δ⟹α(fx,fx)α(gy,gy)d(fx,gy)<η,where ψ∈Ψ.Remark 9.
Note that iff is a generalized cyclic Meir-Keeler-type (α,ψ,A,B)-contraction, then we have that for any comparable elements x,y∈X with x∈A and y∈B,
(7)α(fx,fx)α(gy,gy)d(fx,gy)≤ψ(d(x,y),d(x,fx),d(y,gy),d(x,gy),d(y,fx)).
Further, if
(8)ψ(d(x,y),d(x,fx),d(y,gy),d(x,gy),d(y,fx))=0,
then d(fx,gy)=0.
On the other hand, if(9)ψ(d(x,y),d(x,fx),d(y,gy),d(x,gy),d(y,fx))>0,
then
(10)α(fx,fx)α(gy,gy)d(fx,gy)<ψ(d(x,y),d(x,fx),d(y,gy),d(x,gy),d(y,fx)).We now state our first main result for the generalized cyclic Meir-Keeler-type(α,ψ,A,B)-contraction as follows.Theorem 10.
Let(X,⊑,d) be a partially ordered complete metric space, let A,B be nonempty closed subsets of X with X=A∪B, let α:X×X→ℝ+, and let f,g:X→X be two mappings such that the pair (f,g) is a generalized cyclic Meir-Keeler-type (α,ψ,A,B)-contraction and (A,B)-weakly increasing. Suppose that the following conditions hold:  (i)
f or g is continuous;(ii)
there existsx0∈A with α(fx0,fx0)≥1;(iii)
ifα(xn,xn)≥1 for all n∈ℕ and limn→∞xn=ν, then α(fν,fν)≥1 and α(gν,gν)≥1.
Thenf and g have a common fixed point in X.Proof.
By (ii), there existsx0∈X with α(fx0,fx0)≥1. Since f(A)⊂B and the pair (f,g) is α-admissible, there exists x1∈B such that
(11)x1=fx0,α(gx1,gx1)=α(gfx0,gfx0)≥1.
Since g(B)⊂A and the pair (f,g) is α-admissible, there exists x2∈A such that
(12)x2=gx1,α(fx2,fx2)=α(fgx1,fgx1)≥1.
Continuing this process, we construct the sequence {xn} in X such that
(13)x2n+1=fx2n,x2n+2=gx2n+1,x2n∈A,x2n+1∈B,
and for all n∈ℕ∪{0},
(14)α(x2n+1,x2n+1)=α(fx2n,fx2n)≥1,α(x2n+2,x2n+2)=α(gx2n+1,gx2n+1)≥1.
Since the pair (f,g) is (A,B)-weakly increasing, we have that
(15)x1=fx0⊑gfx0=gx1=x2⊑fgx1=fx2=x3⊑⋯,
and so we conclude that for all n∈ℕ∪{0},
(16)gfx2n=gx2n+1=x2n+2⊑fgx2n+1=fx2n+2=x2n+3.
Step  1. We will show that {xn} is a Cauchy sequence in (X,⊑,d).
Case  1. Suppose that x2n=x2n+1 for some n∈ℕ in the inequality (16). Since x2n and x2n+1 are comparable in X with x2n∈A and x2n+1∈B, by the Remark 9, we have
(17)d(x2n+1,x2n+2)=d(fx2n,gx2n+1)≤α(fx2n,fx2n)α(gx2n+1,gx2n+1)d(fx2n,gx2n+1)≤ψ(d(x2n,x2n+1),d(x2n,fx2n),d(x2n+1,gx2n+1),d(x2n,gx2n+1),d(x2n+1,fx2n))=ψ(d(x2n,x2n+1),d(x2n,x2n+1),d(x2n+1,x2n+2),d(x2n,x2n+2),d(x2n+1,x2n+1))≤ψ(0,0,d(x2n+1,x2n+2),d(x2n+1,x2n+2),0).
If d(x2n+1,x2n+2)>0, then ψ(0,0,d(x2n+1,x2n+2), d(x2n+1,x2n+2),0)>0. By Remark 9, we get a contradiction. So we conclude that d(x2n+1,x2n+2)=0; that is, x2n+1=x2n+2. Similarly, we may show that x2n+2=x2n+3. Hence {xn} is a constant sequence, and so {xn} is a Cauchy sequence in (X,⊑,d).
Case  2.  Suppose that x2n≠x2n+1 for all n∈ℕ in the inequality (16).
Substep  1. We show that the sequence {d(xn,xn+1):n∈ℕ∪{0}} is decreasing.
Subcase  1. If n is even, then we let n=2m for some m∈ℕ. Since x2m∈A, x2m+1∈B, and x2m, x2m+1 are comparable in X, we have
(18)d(xn+1,xn+2)=d(x2m+1,x2m+2)=d(fx2m,gx2m+1)≤α(fx2m,fx2m)α(gx2m+1,gx2m+1)d(fx2m,gx2m+1)<ψ(d(x2m,x2m+1),d(x2m,fx2m),d(x2m+1,gx2m+1),d(x2m,gx2m+1),d(x2m+1,fx2m))=ψ(d(x2m,x2m+1),d(x2m,x2m+1),d(x2m+1,x2m+2),d(x2m,x2m+2),d(x2m+1,x2m+1))≤ψ(d(x2m,x2m+1),d(x2m,x2m+1),d(x2m+1,x2m+2),d(x2m,x2m+1)+d(x2m+1,x2m+2),0)=ψ(d(xn,xn+1),d(xn,xn+1),d(xn+1,xn+2),d(xn,xn+1)+d(xn+1,xn+2),0).
If d(x2m,x2m+1)<d(x2m+1,x2m+2), then the above inequality becomes
(19)d(x2m+1,x2m+2)<ψ(d(x2m+1,x2m+2),d(x2m+1,x2m+2),d(x2m+1,x2m+2),2d(x2m+1,x2m+2),0)≤d(x2m+1,x2m+2),
which is a contradiction. So we have that
(20)d(x2m+1,x2m+2)≤d(x2m,x2m+1).
Subcase  2. If n is odd, then we let n=2m+1 for some m∈ℕ. Since x2m+2∈A, x2m+3∈B and x2m+2, x2m+3 are comparable in X, we have
(21)d(xn+2,xn+1)=d(x2m+3,x2m+2)=d(fx2m+2,gx2m+1)≤α(fx2m+2,fx2m+2)α(gx2m+1,gx2m+1)×d(fx2m+2,gx2m+1)<ψ(d(x2m+2,x2m+1),d(x2m+2,fx2m+2),d(x2m+1,gx2m+1),d(x2m+2,gx2m+1),d(x2m+1,fx2m+2))=ψ(d(x2m+2,x2m+1),d(x2m+2,x2m+3),d(x2m+1,x2m+2),d(x2m+2,x2m+2),d(x2m+1,x2m+3))≤ψ(d(x2m+2,x2m+1),d(x2m+2,x2m+3),d(x2m+1,x2m+2),0,d(x2m+1,x2m+2)+d(x2m+2,x2m+3))=ψ(d(xn+1,xn),d(xn+1,xn+2),d(xn,xn+1),0,d(xn,xn+1)+d(xn+1,xn+2)).
If d(x2m+1,x2m+2)<d(x2m+2,x2m+3), then the above inequality becomes
(22)d(x2m+2,x2m+3)<ψ(d(x2m+2,x2m+3),d(x2m+2,x2m+3),d(x2m+3,x2m+3),0,2d(x2m+2,x2m+3))≤d(x2m+2,x2m+3),
which is a contradiction. So we have that
(23)d(x2m+2,x2m+3)<d(x2m+1,x2m+2).
From (20) and (23), we conclude that
(24)d(x+1,xn+2)<d(xn+1,xn+1).
From the above argument, we have that the sequence {d(xn,xn+1):n∈ℕ∪{0}} is decreasing, and it must converge to some η≥0; that is,
(25)limn→∞d(xn,xn+1)=η.
Substep  2. We next claim that
(26)limn→∞d(xn,fxn+1)=0.
Notice that η=inf{d(fxn,fxn+1):n∈ℕ∪{0}}. We claim that η=0. Suppose, to the contrary, that η>0.
Ifn is even, by the argument of Subcase 1 and the inequality (25), we have
(27)limn→∞ψ(d(xn,xn+1),d(xn,xn+1),d(xn+1,xn+2),d(xn,xn+1)+d(xn+1,xn+2),0)=η.
Since (f,g) is a generalized cyclic Meir-Keeler-type (α,ψ,A,B)-contraction, corresponding to η use and taking into account the above (27), there exist δ>0 and a natural number k such that
(28)η≤ψ(d(xk,xk+1),d(xk,xk+1),d(xk+1,xk+2),d(xk,xk+1)+d(xk+1,xk+2),0)<η+δ⟹α(fxk,fxk)α(gxk+1,gxk+1)d(fxk,gxk+1)<η,
which implies
(29)d(fxk,gxk+1)≤α(fxk,fxk)α(gxk+1,gxk+1)d(fxk,gxk+1)<η.
So we get a contradiction, since η=inf{d(xn,xn+1):n∈ℕ∪{0}. Thus we have that
(30)limn→∞d(xn,fxn+1)=0.
Ifn is odd, by the argument of Subcase 2 and the inequality (25), we have
(31)limn→∞ψ(d(xn+1,xn),d(xn+1,xn+2),d(xn,xn+1),0,d(xn,xn+1)+d(xn+1,xn+2))=η.
Similarly, we can prove that
(32)limn→∞d(xn,fxn+1)=0.
Substep  3. We show that {xn} is a Cauchy sequence in (X,⊑,d). It is sufficient to show that {x2n} is a Cauchy sequence in (X,⊑,d).
Suppose, to the contrary, that{x2n} is not a Cauchy sequence in (X,⊑,d). Then there exist ϵ>0 and two subsequences {x2m(k)} and {x2n(k)} of {x2n} such that n(k) is the smallest integer for which n(k)>m(k)>k,
(33)d(x2m(k),x2n(k))≥ϵ,d(x2m(k),x2n(k)-2)<ϵ,
and we get
(34)ϵ≤d(x2m(k),x2n(k))≤d(x2m(k),x2n(k)-2)+d(x2n(k)-2,x2n(k)-1)+d(x2n(k)-1,x2n(k))<ϵ+d(x2n(k)-2,x2n(k)-1)+d(x2n(k)-1,x2n(k)).
Letting k→∞ in the above inequality, we get
(35)limn→∞d(x2m(k),x2n(k))=ϵ.
On the other hand, we also obtain that
(36)ϵ≤d(x2m(k),x2n(k))≤d(x2m(k),x2n(k)-1)+d(x2n(k)-1,x2n(k))≤d(x2m(k)+1,x2n(k)-1)+d(x2m(k),x2m(k)+1)+d(x2n(k)-1,x2n(k))≤d(x2m(k),x2n(k)-1)+2d(x2m(k),x2m(k)+1)+d(x2n(k)-1,x2n(k))≤d(x2m(k),x2n(k))+2d(x2m(k),x2m(k)+1)+2d(x2n(k)-1,x2n(k)).
Letting k→∞ in the above inequality, we get
(37)limn→∞d(x2m(k),x2n(k))=limn→∞d(x2m(k),x2n(k)-1)=limn→∞d(x2m(k)+1,x2n(k)-1)=ϵ.
Since
(38)d(x2m(k)+1,x2n(k)-1)≤d(x2m(k)+1,x2n(k))+d(x2n(k),x2n(k)-1)≤d(x2m(k)+1,x2n(k)-1)+2d(x2n(k),x2n(k)-1),
letting k→∞ in the above inequality, we have
(39)limn→∞d(x2m(k)+1,x2n(k))=ϵ.
Since x2m(k)∈A, x2n(k)-1∈B, and x2m(k), x2n(k)-1 are comparable in X, we have
(40)d(x2m(k)+1,x2n(k))=d(fx2m(k),gx2n(k)-1)≤α(fx2m(k),fx2m(k))α(gx2n(k)-1,gx2n(k)-1)×d(fx2m(k),gx2n(k)-1)<ψ(d(x2m(k),x2n(k)-1),d(x2m(k),fx2m(k)),d(x2n(k)-1,gx2n(k)-1),d(x2m(k),gx2n(k)-1),d(x2n(k)-1,fx2m(k)))=ψ(d(x2m(k),x2n(k)-1),d(x2m(k),x2m(k)+1),d(x2n(k)-1,x2n(k)),d(x2m(k),x2n(k)),d(x2n(k)-1,x2m(k)+1)).
Letting k→∞ in the above inequality and using (37) and (39), we get
(41)ϵ=limn→∞d(x2m(k)+1,x2n(k))<ψ(ϵ,0,0,ϵ,ϵ)≤ϵ,
which implies a contradiction. So we get that {xn} is a Cauchy sequence in (X,⊑,d).
Step  2. Finally, we prove the existence of common fixed point of f and g.
Since(X,⊑,d) is complete and {xn} is a Cauchy sequence in (X,⊑,d), there exists ν∈X such that
(42)limn→∞xn=limn→∞x2n=limn→∞x2n-1=ν.
From (42) and since α(xn,xn)≥1 for all n∈ℕ, we have α(fν,fν)≥1 and α(gν,gν)≥1.
Since{x2n} is a sequence in A and A is closed, by (42), we have that ν∈A. Similarly, since {x2n+1} is a sequence in B and B is closed, by (42), we have that ν∈B. We now claim that ν is a common fixed point of f and g. Without loss of generality, we assume that f is continuous, and by (42), we have
(43)x2n+1=fx2n⟶ν,asn→∞.
By the uniqueness of the limit, we have that ν=fν.
Sinceν⊑ν with ν∈A and ν∈B, we have
(44)d(ν,gν)=d(fν,gν)≤α(fν,fν)α(gν,gν)d(fν,gν)<ψ(d(ν,ν),d(ν,fν),d(ν,gν),d(ν,gν),d(ν,fν))=ψ(0,0,d(ν,gν),d(ν,gν),0)≤d(ν,gν).
This implies that ν=gν. So we complete the proof.Applying Theorem10 and if we let α(x,y)=1, then we immediately get the following theorem.Theorem 11.
Let(X,⊑,d) be a partially ordered complete metric space, let A,B be nonempty closed subsets of X with X=A∪B, and let f,g:X→X be two mappings such that the pair (f,g) is a generalized cyclic Meir-Keeler-type (ψ,A,B)-contraction and (A,B)-weakly increasing. If f or g is continuous, then f and g have a common fixed point in X.We next state our second main result for the generalized cyclic Meir-Keeler-type(α,ψ,A,B)-contraction as follows.Theorem 12.
Let(X,⊑,d) be a partially ordered complete metric space, let A,B be nonempty closed subsets of X with X=A∪B, let α:X×X→ℝ+, and let f,g:X→X be two mappings such that the pair (f,g) is a generalized cyclic Meir-Keeler-type (α,ψ,A,B)-contraction and (A,B)-weakly increasing. Suppose that the following conditions hold:  (i)
if{xn} is a nondecreasing sequence in X and limn→∞xn=ν, then xn⊑ν;(ii)
there existsx0∈A with α(fx0,fx0)≥1;(iii)
ifα(xn,xn)≥1 for all n∈ℕ and limn→∞xn=ν, then α(fν,fν)≥1 and α(gν,gν)≥1.
Thenf and g have a common fixed point in X.Proof.
From the same proof’s process of Theorem10, we can construct a nondecreasing sequence {xn} in X with x2n∈A, x2n+1∈B, and xn→ν for some ν∈X. Since xn→ν and A, B are nonempty closed subsets of X, we have x2n→ν, x2n+1→ν, and ν∈A∩B. By the condition (i), we get xn⊑ν for all n∈ℕ.
Sincex2n∈A and ν∈B, we have
(45)d(x2n+1,gν)=d(fx2n,gν)≤α(fx2n,fx2n)α(gν,gν)d(fx2n,gν)<ψ(d(x2n,ν),d(x2n,fx2n),d(ν,gν),d(x2n,gν),d(ν,fx2n))=ψ(d(x2n,ν),d(x2n,x2n+1),d(ν,gν),d(x2n,gν),d(ν,x2n+1)).
Letting n→∞ in the above inequality, we get
(46)d(ν,gν)<ψ(0,0,d(ν,gν),d(ν,gν),0)≤d(ν,gν).
This implies that d(ν,gν)=0; that is, ν=gν. Similarly, we may show that ν=fν. So ν is a common fixed point of f and g.Applying Theorem12, it is easy to get the following theorem.Theorem 13.
Let(X,⊑,d) be a partially ordered complete metric space, let A,B be nonempty closed subsets of X with X=A∪B, and let f,g:X→X be two mappings such that the pair (f,g) is a generalized cyclic Meir-Keeler-type (ψ,A,B)-contraction and (A,B)-weakly increasing. Suppose that the following condition holds:if{xn} is a nondecreasing sequence in X and limn→∞xn=ν, then xn⊑ν.
Thenf and g have a common fixed point in X.



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