Coincidence points and common fixed points for three mappings with integral type implicit contraction conditions on ordered metric spaces

Adv. Inequal. Appl. 2019, 2019:11 https://doi.org/10.28919/aia/4060 ISSN: 2050-7461

COINCIDENCE POINTS AND COMMON FIXED POINTS FOR THREE MAPPINGS WITH INTEGRAL TYPE IMPLICIT CONTRACTION CONDITIONS

ON ORDERED METRIC SPACES

YONGJIE PIAO*

Department of Mathematics, Yanbian University, Yanji 133002, China

Copyright c2019 Yongjie Piao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.In this paper, by using the known classΨ∗of 5-dimensional functions determined by the known classes

{C,Φu}, we discuss the existence problems of coincidence points for three mappings of integral type with

semi-implicit contraction conditions and obtain a common fixed point theorem for two mappings on ordered metric

spaces. Finally, we give a sufficient condition under which there exists a unique common fixed point.

Keywords:classC; classΨ∗; classΦu; semi-implicit; sub-additive; common fixed point.

2010 AMS Subject Classification:47H05; 47H10; 54E40; 54H25.

1. Introduction and preliminaries

Throughout this paper, we assume that_R+= [0,+∞)and

Φ= {φ :φ : R+ →R+|φ is Lebesgue integral, summable on each compact subset of

R+ and R0εφ(t)dt>0 for eachε>0}

The famous Banach’s contraction principle is as follows:

∗_{Corresponding author}

E-mail address: [email protected]

Received March 18, 2019

Theorem 1.1([1])Let f be a self mapping on a complete metric space(X,d)satisfying

d(f x,f y)≤cd(x,y), ∀x,y∈X, (1.1)

wherec∈[0,1)is a constant. Then f has a unique fixed point ˆx∈X such that limn→∞fnx=xˆ

for eachx∈X.

It is known that the Banach’s contraction principle has a lot of generalizations and various

applications in many directions, see, for examples, [2-12] and the references cited therein.

Es-pecially, in 1962, Rakotch[13] extended the Banach’s contraction principle with replacing the contraction constantcin (1.1) by a contraction functionγ and obtained the next theorem:

Theorem 1.2([13])Let f be a self-mapping on a complete metric space(X,d)satisfying

d(f x,f y)≤γ(d(x,y))d(x,y), ∀x,y∈X, (1.2)

whereγ:_R+→[0,1)is a monotonically decreasing function. Then f has a unique fixed point

ˆ

x∈X such that limn→∞fnx=xˆfor eachx∈X.

In 2002, Branciari[14]gave an integral version of Theorem 1.1 as follows:

Theorem 1.3([14])Let f be a self-mapping on a complete metric space(X,d)satisfying

Z d(f x,f y)

0

φ(t)dt ≤c

Z d(x,y)

0

φ(t)dt, ∀x,y∈X, (1.3)

where c∈(0,1) is a constant and φ ∈Φ. Then f has a unique fixed point ˆx∈X such that

limn→∞fnx=xˆfor eachx∈X.

The further generalizations of Branciari’ theorem were given with replacing the contraction

constantcin (1.3) by two or three contraction functions in [15-16].

The following concept of classC of functionsF :[0,∞)2→_Rwas introduced in [17-18].

Definition 1.1([17-18])A continuous mappingF :[0,∞)2→_Ris calledC-classfunction if it

satisfies following axioms:

(i)F(s,t)≤s;

(ii)F(s,t) =simplies that eithers=0 ort=0 for alls,t∈[0,∞).

Note for someF we have thatF(0,0) =0.

Definition 1.2([18]) Let Φu be a set of all functions ϕ :R+ →R+ satisfying the following

(i)ϕ is continuous;

(ii)ϕ(t)>0 for all t >0 andϕ(0)≥0.

Definition 1.3([18])Letψ∈Ψ∗if and only ifψ:R+5→R+is a continuous and nondecreasing

function about the 4th and 5th variables and the following conditions hold:

(i) there existsF1∈C,ϕ1∈Φu,such thatu≤ψ(v,u,v,0,u+v)impliesu≤F1(v,ϕ1(v)); (ii) there existsF₂∈C,ϕ2∈Φu,such thatu≤ψ(v,v,u,u+v,0)impliesu≤F2(v,ϕ2(v)); (iii)ψ(t,0,0,t,t)<t,ψ(0,t,0,0,t)<t andψ(0,0,t,t,0)<tfor allt>0.

Ansari and Piao et al[18]discussed and obtained several common fixed point theorems for two mappings of integral type with semi-implicit contractive conditions determined by functions

F∈C andϕ∈Φuandψ∈Ψ∗in metric spaces. These results generalized and improved many

common fixed point theorems for mappings with integral type.

The aim of this paper is to discuss the existence problems of coincidence points and common

fixed points for three and two mappings respectively with integral type and semi-implicit

con-tractive conditions on ordered metric spaces, and finally give a sufficient condition under which

there exists a unique common fixed point. The obtained results further extent and improve the

corresponding various known conclusions.

To do this, we give some known definitions and lemmas.

Lemma 1.1([19])Let(X,d)be a metric space and{x_n}a sequence inXsuch thatd(x_n,x_n+1)→ 0 asn→∞. If{xn}is not a Cauchy sequence, then there exist anε>0 and sequences of positive

integers{m(k)}and{n(k)}withm(k)>n(k)>kfor eachk∈_Nsuch thatd(x_m(k),x_n(k))≥ε,

d(x_m(k)−1,x_n(k))<ε and the following result hold

lim k→∞

d(x_m(k)−1,x_n(k)−1) = lim k→∞

d(x_m(k),x_n(k)) = lim k→∞

d(x_m(k)−1,x_n(k)) =ε.

Remark 1.1Under the conditions of Lemma 1.9, we easily obtian the following result:

lim k→∞

d(x_m(k),x_n(k)+1) = lim k→∞

d(x_m(k)+1,x_n(k)) =lim k→∞

d(x_m(k)+1,x_n(k)+1) =lim k→∞

d(x_m(k),x_n(k)−1) =ε.

Definition 1.4LetX be a nonempty set. Then(X,d,)is called an ordered metric space if

(i)(X,d)is a metric space;

Definition 1.5([20-21]) Let (X,) be a partially ordered set and f,g,h:X →X three given

mappings such that f X ⊂hX. We say that(f,g)is partially weakly increasing with respect to

hif and only if for allx∈X andy∈h−1(f x), we have

f xgy.

Definition 1.6([21]) Let(X,d,)be an ordered metric space. We say thatX is regular if the

following hypothesis holds: if {x_n}is a non-decreasing sequence in X with respect tosuch

thatx_n→xasn→∞, thenxnxfor alln∈N; equivalently, if{yn}is a non-increasing sequence

inX with respect tosuch thaty_n→yasn→∞, thenynyfor alln∈N.

Lemma 1.2([15])Letφ ∈Φand{rn}n∈Nbe a nonnegative sequence with limn→∞rn=a. Then

lim n→∞

Z r_n

0

φ(t)dt =

Z a

0

φ(t)dt.

Lemma 1.3([15])Letφ ∈Φand{rn}n∈Nbe a nonnegative sequence. Then

lim n→∞

Z r_n

0

φ(t)dt =0⇐⇒ lim

n→∞

r_n=0.

Definition 1.7([22])φ ∈Φis called to be sub-additive if, for alla,b∈_R+,

Z a+b

0

φ(t)dt ≤

Z a

0

φ(t)dt+

Z b

0

φ(t)dt.

Definition 1.8φ ∈Φis called to be strictly increasing about integral type if, for anyx,y∈[0,∞)

withx<y,

Z x

0

φ(t)dt <

Z y

0

φ(t)dt.

It is easy to check that the functionφ :_R+→_R+,φ(t) = ₁₊1_t for eacht∈_R+is a sub-additive

and strictly increasing function about integral type.

Definition 1.9 LetX 6= /0 and f,g,T :X →X three mappings. If there existv,u∈X such that

f v=gv=T v=u, then vis called a coincidence point of {f,g,T} and uis called a point of

coincidence of{f,g,T}.

LetC(f,g,T)be the set of all points of coincidence of{f,g,T}.

2. Coincidence points and Common fixed points

Theorem 2.1Let(X,d,)be an ordered metric space such thatX is regular andT,f,g:X→X

three mappings. Suppose that for eachx,y∈X withT xandTybeing comparable,

Z d(f x,gy)

0

φ(t)dt

≤ψ

Z d(T x,Ty)

0

φ(t)dt,

Z d(T x,f x)

0

φ(t)dt,

Z d(Ty,gy)

0

φ(t)dt,

Z d(T x,gy)

0

φ(t)dt,

Z d(Ty,f x)

0

φ(t)dt

,

(2.1)

whereφ ∈Φandψ ∈Ψ∗. If

(i)φ is sub-additive and strictly increasing about the integral type;

(ii) f X∪gX ⊂T X;

(iii)(f,g)and(g,f)are both partially weakly increasing with respect toT;

(iv) one of{T X,f X,gX}is complete.

ThenC(T,f,g)6=/0, that is, there existu,v∈Xsuch thatT v= f v=gv=u∈T X. Furthermore,

eitherC(T,f,g)is singleton or any different two elements inC(T,f,g)are not comparable.

Proof.Let x₀∈X be an arbitrary point. Since f x₀∈ f X ⊂T X, there exists x₁∈X such that

f x₀ =T x₁; Since gx₁ ∈gX ⊂T X, there exists x₂∈X such that gx₁ =T x₂. Continuing this process, we can construct two sequences{x_n}and{y_n}inX as follows

y_2n= f x_2n=T x_2n+1,y2n+1=gx2n+1=T x2n+2,∀n=0,1,2,· · ·. (2.2)

From (2.2), we obtain

x2n+1∈T−1(f x2n),x2n+2∈T−1(gx2n+1),∀n=0,1,2,· · ·,

hence using (iii), we have

f x_2ngx_2n+1, gx_2n+1 f x_2n+2,∀n=0,1,2,· · ·,

that is,

y2n= f x2ngx2n+1=y2n+1 f x2n+2=y2n+2,∀n=0,1,2,· · ·.

Therefore,

Letd_n=d(y_n,y_n+1)for alln=0,1,2,· · ·.SinceT xnT xn+1for alln∈N∪ {0}, using (2.1) and (i), we have

Z d_2n

0

φ(t)dt =

Z d(y_2n,y_2n+1)

0

φ(t)dt =

Z d(f x_2n,gx_2n+1)

0

φ(t)dt

≤ψ

Z d(T x2n,T x2n+1)

0

φ(t)dt,

Z d(T x₂n,f x_2n)

0

φ(t)dt,

Z d(T x_2n+1,gx_2n+1)

0

φ(t)dt,

Z d(T x_2n,gx_2n+1)

0

φ(t)dt,

Z d(T x_2n+1,f x_2n)

0

φ(t)dt

=ψ

Z d_2n−1

0

φ(t)dt,

Z d_2n−1

0

φ(t)dt,

Z d_2n

0

φ(t)dt,

Z d(y_2n−1,y_2n+1)

0

φ(t)dt,0

≤ψ

Z d_2n−1

0

φ(t)dt,

Z d_2n−1

0

φ(t)dt,

Z d_2n

0

φ(t)dt,

Z d_2n−1+d_2n+1

0

φ(t)dt,0

≤ψ

Z d2n−1

0

φ(t)dt,

Z d_2n−1

0

φ(t)dt,

Z d_2n

0

φ(t)dt,

Z d_2n−1

0

φ(t)dt+

Z d_2n

0

φ(t)dt,0

.

So by the condition (ii) ofψ ∈Ψ∗, there existF2∈C andϕ2∈Φusuch that Z d_2n

0

φ(t)dt ≤F2

Z d2n−1

0

φ(t)dt,ϕ2( Z d_2n−1

0

φ(t)dt)

. (2.4)

Similarly,

Z d_2n+1

0

φ(t)dt =

Z d(y_2n+1,y_2n+2)

0

φ(t)dt =

Z d(f x_2n+2,gx_2n+1)

0

φ(t)dt

≤ψ

Z d(T x_2n+2,T x_2n+1)

0

φ(t)dt,

Z d(T x_2n+2,f x_2n+2)

0

φ(t)dt,

Z d(T x_2n+1,gx_2n+1)

0

φ(t)dt,

Z d(T x_2n+2,gx_2n+1)

0

φ(t)dt,

Z d(T x_2n+1,f x_2n+2)

0

φ(t)dt

=ψ

Z d_2n

0

φ(t)dt,

Z d_2n+1

0

φ(t)dt,

Z d_2n

0

φ(t)dt,0,

Z d(y_2n,y_2n+2)

0

φ(t)dt

≤ψ

Z d2n

0

φ(t)dt,

Z d_2n+1

0

φ(t)dt,

Z d_2n

0

φ(t)dt,0,

Z d_2n+d_2n+1

0

φ(t)dt

≤ψ

Z d_2n

0

φ(t)dt,

Z d_2n+1

0

φ(t)dt,

Z d_2n

0

φ(t)dt,0,

Z d_2n

0

φ(t)dt+

Z d_2n+1

0

φ(t)dt

.

So by the condition (i) ofψ ∈Ψ∗, there existF1∈C andϕ1∈Φusuch that Z d_2n+1

0

φ(t)dt ≤F1 Z d_2n

0

φ(t)dt,ϕ1( Z d_2n

0

φ(t)dt)

. (2.5)

Combing (2.4)-(2.5) and using the property ofF1andF2, we have Z d_n+1

0

φ(t)dt ≤

Z d_n

0

If there existsn₀∈_Nsuch thatd_n0+1>dn0, then by the strictly increasing property ofφ about integral type, we obtain

Z d_n 0+1

0

φ(t)dt>

Z d_n 0

0

φ(t)dt,

which is a contradiction to (2.6), hence we have

d_n+1≤dn,∀n=0,1,2,· · ·. (2.7)

(2.7) implies that there isu∈_R+ such that limn→∞dn=u. From Lemma 1.2 and (2.5), we

obtain

Z u

0

φ(t)dt = lim n→∞

Z d_2n+1

0

φ(t)dt≤F1(lim n→∞

Z d_2n

0

φ(t)dt,lim n→∞

ϕ1( Z d_2n

0

φ(t)dt))

=F1( Z u

0

φ(t)dt,ϕ1( Z u

0

φ(t)dt)).

SoRu

0 φ(t)dt =0 orϕ1( Ru

0φ(t)dt) =0 by the property ofF1, which implies that Ru

0φ(t)dt =0. Therefore,u=0, i.e.,

lim n→∞

d(xn,xn+1) = lim n→∞

dn=0. (2.8)

We claim that{y_n}is Cauchy. Otherwise, by Lemma 1.1 and Remark 1.1 and (2.8), there

existsε>0 such that for eachk∈N, there existm(k),n(k)∈Nwithm(k)>n(k)such that the

parity ofm(k)andn(k)is different and the following result holds

lim k→∞

d(y_m(k),y_n(k)) = lim k→∞

d(y_m(k),y_n(k)−1) =lim k→∞

d(y_m(k)−1,y_n(k)−1) = lim k→∞

d(y_m(k)−1,y_n(k)) =ε.

Ifm(k)is even andn(k)is odd, then by Lemma 1.2, (2.1), (2.8), (2.9) and (iii) of Definition

1.3,

0<

Z ε

0

φ(t)dt

=lim k→∞

Z d(y_m(k),y_n(k))

0

φ(t)dt

=lim k→∞

Z d(f x_m(k),gx_n(k))

0

φ(t)dt

≤lim k→∞

ψ

Z d(T x_m(k),T x_n(k))

0

φ(t)dt,

Z d(T x_m(k),f x_m(k))

0

φ(t)dt,

Z d(T x_n(k),gx_n(k))

0

φ(t)dt,

Z d(T x_m(k),gx_n(k))

0

φ(t)dt,

Z d(f x_m(k),T x_n(k))

0

φ(t)dt

=lim k→∞

ψ

Z d(y_m(k)−1,y_n(k)−1)

0

φ(t)dt,

Z d(y_m(k)−1,y_m(k))

0

φ(t)dt,

Z d(y_n(k)−1,y_n(k))

0

φ(t)dt,

Z d(x_m(k)−1,y_n(k))

0

φ(t)dt,

Z d(y_m(k),y_n(k)−1)

0

φ(t)dt

=ψ

Z ε

0

φ(t)dt,0,0,

Z _ε

0

φ(t)dt,

Z _ε

0

φ(t)dt

<

Z ε

0

φ(t)dt.

This is a contradiction. Similarly, we obtain the same contradiction for the case thatm(k)is odd

andn(k)is even. Hence,{yn}is a Cauchy sequence.

IfT X is complete, theny_n∈T X for alln∈_N∪ {0}implies that there existu,v∈X such that

y_n→u=T vasn→∞.

If f Xis complete, theny_2n= f x_2n∈ f Xfor alln=0,1,2,· · · implies that there existu,v₁∈X

such thaty_2n→u= f v₁ asn→∞, therefore there existsv∈X such thaty2n→u= f v1=T v by the condition f X⊂T X. On the other hand, (2.8) and

d(y_2n+1,u)≤d((y_2n+1,y_2n) +d(y_2n,u)

imply thaty_2n+1→u= f v1asn→∞, henceyn→u=T vasn→∞.

Similarly, we can obtain the same result for the case of gX being complete. Hence we can

Since X is regular, using (2.3) andy_n →u, we obtain y_n u for all n=0,1,2,· · ·, hence

T x_n+1=y_nu=T vfor alln=0,1,2,· · ·. By Lemma 1.2, (2.1), (2.8) andy_n→u, Z d(u,gv)

0

φ(t)dt

=lim n→∞

Z d(y_2n,gv)

0

φ(t)dt

=lim n→∞

Z d(f x_2n,gv)

0

φ(t)dt

≤limψ

Z d(T x_2n,T v)

0

φ(t)dt,

Z d(T x_2n,f x_2n)

0

φ(t)dt,

Z d(T v,gv)

0

φ(t)dt,

Z d(T x_2n,gv)

0

φ(t)dt,

Z d(T v,f x_2n)

0

φ(t)dt

=limψ

Z d(y_2n−1,u)

0

φ(t)dt,

Z d(y_2n−1,y_2n)

0

φ(t)dt,

Z d(u,gv)

0

φ(t)dt,

Z d(y_2n−1,gv)

0

φ(t)dt,

Z d(u,y_2n)

0

φ(t)dt

=ψ

0,0,

Z d(u,gv)

0

φ(t)dt,

Z d(u,gv)

0

φ(t)dt,0

.

HenceRd(u,gv)

0 φ(t)dt =0 by the condition (iii) of Definition 1.3, thereforeu=gv. Similarly, we can obtainu= f v, henceu=T v= f v=gv. This completeC(f,g,T)6= /0.

If C(f,g,T) is not singleton and there exist two different element u and u1 in C(f,g,T) such that they are comparable, then there exists v,v₁∈ X such that u= f v= gv=T v and

u1= f v1=gv1=T v1andT vandT v1are comparable, hence by (2.1), Z d(u,u₁)

0

φ(t)dt

=

Z d(f v,gv₁)

0

φ(t)dt

≤ψ

Z d(T v,T v₁)

0

φ(t)dt,

Z d(T v,f v)

0

φ(t)dt,

Z d(T v₁,gv₁)

0

φ(t)dt,

Z d(T v,gv₁)

0

φ(t)dt,

Z d(T v₁,f v)

0

φ(t)dt

=ψ

Z d(u,u₁)

0

φ(t)dt,0,0,

Z d(u,u₁)

0

φ(t)dt,

Z d(u,u₁)

0

φ(t)dt

.

HenceRd(u,u1)

0 φ(t) =0 by (iii) of Definition 1.3, which impliesu=u1. This is a contradiction, hence any different two elements inC(f,g,T)are not comparable.

Theorem 2.2Let(X,d,)be an ordered metric space such thatX is regular and f,g:X →X

two mappings. Suppose that for eachx,y∈X withxandybeing comparable,

Z d(f x,gy)

0

φ(t)dt

≤ψ

Z d(x,y)

0

φ(t)dt,

Z d(x,f x)

0

φ(t)dt,

Z d(y,gy)

0

φ(t)dt,

Z d(x,gy)

0

φ(t)dt,

Z d(y,f x)

0

φ(t)dt

,

(2.10)

whereφ ∈Φandψ ∈Ψ∗. If

(i)φ is sub-additive and strictly increasing about the integral type;

(ii)xgxfor allx∈ f X andx f xfor allx∈gX;

(iii) f X orgX is complete.

Then f,ghave a common fixed pointv, that is, there existv∈X such thatv= f v=gv.

Proof. LetT =1X, then f x∈1X−1(f x) and gx∈1

−1

X (gx)and (ii) imply that (f,g) and(g,f)

are both partially weakly increasing with respect to 1X, hence there exists v∈ X such that

f v=gv=T v=vby Theorem 2.1, hencevis a common fixed point of{f,g,T}.

Now, we give a sufficient condition under which there exists a unique common fixed point in

Theorem 2.2

Theorem 2.3 Suppose that all of the conditions of Theorem 2.2 are satisfied. Furthermore, if

the following conditions hold:

(iv) for eachx,y∈X, there existsz∈ f X∪gX such that{z,x}and{z,y}are both comparable

pair respectively;

(v) for any two elementsu,v∈X withuv, fnuvandgnuvfor alln∈_N;

(vi) ψ ∈Ψ∗ satisfies that there exist F3∈C and ϕ3∈Φu such that u≤ψ(v,0,u+v,u,v)

impliesu≤F₃(v,ϕ3(v))andψ is also non-decreasing about the 3th variable; (vii) f gX=g f X.

Then f andghave a unique common fixed point.

Proof.We have proved that f andghave a common fixed pointv∈X in Theorem 2.2. Suppose

Case 1. Ifuandvare comparable, then by (2.10),

Z d(u,v)

0

φ(t)dt

=

Z d(f u,gv)

0

φ(t)dt

≤ψ

Z d(u,v)

0

φ(t)dt,

Z d(u,f u)

0

φ(t)dt,

Z d(v,gv)

0

φ(t)dt,

Z d(u,gv)

0

φ(t)dt,

Z d(v,f u)

0

φ(t)dt

=ψ

Z d(u,v)

0

φ(t)dt,0,0,

Z d(u,v)

0

φ(t)dt,

Z d(v,u)

0

φ(t)dt

hence by (iii) of Definition 1.3,

Z d(u,v)

0

φ(t)dt =0,

thereforeu=v.

Case 2. Ifuand vare not comparable, thenu6=vand there exist w∈ f X∪gX such thatw

anduare comparable andwandvare also comparable by (iv). In this case,w6=uandw6=v.

Without loss of generality, letw∈ f X, thenwgwby (ii) in Theorem 2.2. Sincegw∈g f X=

f gX⊂ f X by (vii),gwggw=g2wby (ii) in Theorem 2.2 again. Repeating this process, we obtain

wgwg2w · · · gnw,∀n=1,2,· · ·.

Ifuw, then using the above conclusion and (v), we obtain that for anyn=1,2,· · ·,

fnuwgwg2w · · · gnw,

hence fnuandgnware comparable for alln=0,1,2,· · ·. By (2.10),

Z d(u,gnw)

0

φ(t)dt

=

Z d(f fn−1u,ggn−1w)

0

φ(t)dt

≤ψ

Z d(fn−1u,gn−1w)

0

φ(t)dt,

Z d(fn−1u,f fn−1u)

0

φ(t)dt,

Z d(gn−1w,ggn−1w)

0

φ(t)dt,

Z d(fn−1u,ggn−1w)

0

φ(t)dt,

Z d(f fn−1u,gn−1w)

0

φ(t)dt

=ψ

Z d(u,gn−1w)

0

φ(t)dt,0,

Z d(gn−1w,gnw)

0

φ(t)dt,

Z d(u,gnw)

0

φ(t)dt,

Z d(u,gn−1w)

0

φ(t)dt

≤ψ

Z d(u,gn−1w)

0

φ(t)dt,0,

Z d(u,gn−1w)

0

φ(t)dt+

Z d(u,gn−1w)

0

φ(t)dt,

Z d(u,gnw)

0

φ(t)dt,

Z d(u,gn−1w)

0

φ(t)dt

hence by (vi), there existF₃∈C andϕ3∈Φusuch that Z d(u,gnw)

0

φ(t)dt ≤F3

Z d(u,gn−1w)

0

φ(t)dt,ϕ3(

Z d(u,gn−1w)

0

φ(t)dt)

. (2.11)

So by (i) in Definition 1.1,

Z d(u,gnw)

0

φ(t)dt ≤

Z d(u,gn−1w)

0

φ(t)dt,

hence by the strictly increasing property about the integral type ofφ, we obtain

d(u,gnw)≤d(u,gn−1w),∀n=1,2,· · ·. (2.12)

From (2.12), we know that {d(u,gnw)}∞

n=1 is a nondecreasing and non-negative real se-quence, hence there existsM(u,w)≥0 such that

lim n→∞

d(u,gnw) =M(u,w). (2.13)

Lettingn→∞in (2.11) and using (2.13), we obtain

Z M(u,w)

0

φ(t)dt ≤F3

Z M(u,w)

0

φ(t)dt,ϕ3(

Z M(u,w)

0

φ(t)dt)

, (2.14)

hence we obtainRM(u,w)

0 φ(t)dt =0 orϕ3(

RM(u,w)

0 φ(t)dt) =0, therefore lim

n→∞

d(u,gnw) =M(u,w) =0.

This implies that

lim n→∞

gnw=u. (2.15)

Ifwu, then sinceu=gu∈gX, we also obtain similarly

gnwu f u f2u · · · fnu, ,∀n=1,2,· · ·,

which shows thatgnwand fnuare comparable for all n=0,1,2,· · ·. Hence similarly, we also

obtain (2.15). Therefore (2.15) always holds for two comparable elements uand w. Sincew

andvare also comparable, we also obtain

lim n→∞

gnw=v. (2.16)

Hence (2.15) and (2.16) implies thatu=v,which is a contradiction. Therefore f andghave a

Conflict of Interests

The authors declare that there is no conflict of interests.

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