Existence of tripled coincidence points in ordered b-metric spaces and an application to a system of integral equations

(1)

R E S E A R C H

Open Access

Existence of tripled coincidence points in

ordered

b-metric spaces and an application

to a system of integral equations

Vahid Parvaneh

1*

_{, Jamal Rezaei Roshan}

2

_{and Stojan Radenovi´c}

3

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

Gilan-E-Gharb Branch, Islamic Azad University, Gilan-E-Gharb, Iran Full list of author information is available at the end of the article

Abstract

In this paper, tripled coincidence points of mappings satisfying some nonlinear contractive conditions in the framework of partially orderedb-metric spaces are obtained. Our results extend the results of Berinde and Borcut (Nonlinear Anal. 74:4889-4897, 2011) and Borcut (Appl. Math. Comput. 218:7339-7346, 2012) from the context of ordered metric spaces to the setting of orderedb-metric spaces. Moreover, some examples of the main result are given. Finally, some tripled coincidence point results for mappings satisfying some contractive conditions of integral type in complete partially orderedb-metric spaces are deduced. Also, an application is given to support our results.

MSC: Primary 47H10; secondary 54H25

Keywords: b-metric space; partially ordered set; tripled ﬁxed point

1 Introduction and preliminaries

Existence of coupled fixed points in partially ordered metric spaces was first investigated in  by Guo and Lakshmikantham [], and then in [, ]. Further results in this direction under weak contraction conditions in different metric spaces were proved in,e.g., [–].

Recently, Berinde and Borcut [] introduced a new concept of a tripled fixed point and obtained some tripled fixed point theorems for contractive type mappings in partially or-dered metric spaces. For a survey of tripled fixed point theorems and related topics, we refer the reader to [–].

Deﬁnition .[, ] Let (X,) be a partially ordered set,f :X_→_X _and_g_:_X_→_X_. . An element(x,y,z)∈X_{is called a tripled ﬁxed point of}_f _if_f_(x,_y,_{z) =}_x_,

f(y,x,y) =yandf(z,y,x) =z.

. An element(x,y,z)∈X_{is called a tripled coincidence point of the mappings}_f _and

giff(x,y,z) =gx,f(y,x,y) =gyandf(z,y,x) =gz.

. An element(x,y,z)∈X_{is called a tripled common ﬁxed point of}_f _and_g_if

x=g(x) =f(x,y,z),y=g(y) =f(y,x,y)andz=g(z) =f(z,y,x).

. We say thatf has the mixedg-monotone property iff(x,y,z)isg-nondecreasing in

x,g-nonincreasing inyandg-nondecreasing inz, that is, if for anyx,y,z∈X,

x,x∈X, gxgx ⇒ f(x,y,z)f(x,y,z),

y,y∈X, gygy ⇒ f(x,y,z)f(x,y,z)

(2)

and

z,z∈X, gzgz ⇒ f(x,y,z)f(x,y,z).

Deﬁnition .[] LetX be a nonempty set. We say that the mappingsf :X_→_X _and

g:X→Xcommute ifg(f(x,y,z)) =f(gx,gy,gz) for allx,y,z∈X.

In [], Berinde and Borcut proved the following result and formulated it as Theorems  and .

Theorem .[] Let(X,)be a partially ordered set and suppose there is a metric d on X such that(X,d)is a complete metric space.Let F:X→X be a mapping having the mixed monotone property on X.Assume that there exist constants j,k,l∈[, )with j+k+l<  for which

dF(x,y,z),F(u,v,w)≤jd(x,u) +kd(y,v) +ld(z,w)

for all x,y,z,u,v,w∈X with xu, yv and zw.Suppose either F is continuous or (X,d,)is regular.If there exist x,y,z∈X such that xF(x,y,z),yF(y,x,y)

and zF(z,y,x),then there exist x,y,z∈X such that F(x,y,z) =x,F(y,x,y) =y and

F(z,y,x) =z.

In [], Borcut and Berinde proved the following result and formulated it as Theorem .

Theorem . Let(X,)be a partially ordered set and suppose there is a metric d on X such that(X,d)is a complete metric space.Let F:X_→_{X and g}_:_X_→_{X be such that F}

has the mixed g-monotone property on X.Assume that there exist constants j,k,l∈[, ) with j+k+l< such that

dF(x,y,z),F(u,v,w)≤jd(gx,gu) +kd(gy,gv) +ld(gz,gw)

for all x,y,z,u,v,w∈X with gxgu,gygv and gzgw.Suppose that F(X₎_⊆_g(X),_g

is continuous and commutes with F and also suppose either F is continuous or(X,d,)is regular.If there exist x,y,z∈X such that xF(x,y,z),yF(y,x,y)and z

F(z,y,x),then there exist x,y,z∈X such that F(x,y,z) =gx,F(y,x,y) =gy and F(z,y,x) =

gz.

Notice that Theorem . follows from Theorem . by takingg=iX(the identity map). In [], Borcut obtained the following.

Theorem .[, Corollary ] Let(X,)be a partially ordered set and suppose there is a metric d onX such that(X,d)is a complete metric space.Let f :X_→_X_{and g}_:_X_→_X

be such that f has the g-mixed monotone property.Assume that there exists k∈[, )such that

df(x,y,z),f(u,v,w)≤kmaxd(gx,gu),d(gy,gv),d(gz,gw)

for all x,y,z,u,v,w∈X with gxgu, gygv and gzgw.Suppose f(X₎_⊆_g(_X_),_{g is}

(3)

(a) f is continuous,or

(b) X has the following properties:

(i) if a non-decreasing sequencexn→x,thenxnxfor alln; (ii) if a non-increasing sequenceyn→y,thenynyfor alln.

If there exist x,y,z ∈X such that gxf(x,y,z), gyf(y,x,z) and gz

f(z,y,x),then f and g have a tripled coincidence point.

The concept of ab-metric space was introduced by Czerwik in []. Since then, several papers have been published on the ﬁxed point theory of various classes of single-valued and multi-valued operators inb-metric spaces (see,e.g., [–]).

Consistent with [] and [], the following deﬁnitions and results will be needed in the sequel.

Definition .[] LetXbe a (nonempty) set ands≥ be a given real number. A function d:X×X→ +_{is a}_{b-metric if, for all}_x,_y,_z_∈_X_{, the following conditions are satisfied:} (b) d(x,y) = iffx=y,

(b) d(x,y) =d(y,x),

(b) d(x,z)≤s[d(x,y) +d(y,z)].

The pair (X,d) is called ab-metric space.

It should be noted that the class ofb-metric spaces is eﬀectively larger than that of metric spaces since ab-metric is a metric whens= , and there areb-metric spaces which are not metric spaces. Here, we present an easy example of this kind (see also [, p.]).

Example .[] Let (X,d) be a metric space andρ(x,y) = (d(x,y))p_{, where}_p_{>  is a real} number. Thenρis ab-metric withs= p–. However, (X,ρ) is not necessarily a metric space.

For example, letX be the set of real numbers and letd(x,y) =|x–y|be the usual Eu-clidean metric. Thenρ(x,y) = (x–y) _{is a}_{b-metric on}_with_s_{= , but is not a metric}

on.

Also, the following example of ab-metric space was given in [].

Example .[] LetX be the set of Lebesgue measurable functions on [, ] such that





f(x)dx<∞.

DeﬁneD:X×X→[,∞) by

D(f,g) =





f(x) –g(x)dx.

As (_|f(x)–g(x)|_dx)/_{is a metric on}_X_{, then, from the previous example,}_D_{is a}_b-metric

(4)

The purpose of this paper is to obtain some tripled coincidence point theorems for two mappings satisfying a (ψ,ϕ)-contractive condition in orderedb-metric spaces. Our results extend, unify and generalize the comparable results in [, , ] from the context of ordered metric spaces to the setup of orderedb-metric spaces.

We also need the following deﬁnitions.

Deﬁnition .[] Let (X,d) be ab-metric space. Then a sequence{xn}inX is called:

(a) b-convergent if there existsx∈Xsuch thatd(xn,x)→asn→ ∞. In this case, we writelim_n→∞xn=x.

(b) b-Cauchy ifd(xn,xm)→asn,m→ ∞.

Proposition .(See [, Remark .]) In a b-metric space(X,d),the following assertions hold:

(p) Ab-convergent sequence has a unique limit. (p) Eachb-convergent sequence isb-Cauchy.

(p) In general,ab-metric is not continuous(see also an example in[]).

Deﬁnition .[] Let (X,d) and (X,d) be twob-metric spaces.

() The space(X,d)isb-complete if everyb-Cauchy sequence inXb-converges. () A functionf :X→Xisb-continuous at a pointx∈X if it isb-sequentially

continuous atx, that is, whenever{xn}isb-convergent tox,{f(xn)}isb-convergent

tof(x).

Deﬁnition . Let (X,d) be ab-metric space. Mappingsf :X→Xandg:X→Xare called compatible if

lim

n→∞d

gf(xn,yn,zn),f(gxn,gyn,gzn)= ,

lim

n→∞d

gf(yn,xn,yn),f(gyn,gxn,gyn)= ,

and

lim

n→∞d

gf(zn,yn,xn),f(gzn,gyn,gxn)= 

hold whenever{xn},{yn}and{zn}are sequences inXsuch that

lim

n→∞f(xn,yn,zn) =n→∞lim gxn,

lim

n→∞f(yn,xn,yn) =n→∞lim gyn,

and

lim

n→∞f(zn,yn,xn) =n→∞lim gzn.

Deﬁnition . LetX be a nonempty set. Then (X,d,) is called a partially ordered b-metric space ifdis ab-metric on a partially ordered set (X,).

(5)

(i) if a non-decreasing sequencexn→x, thenxnxfor alln, (ii) if a non-increasing sequenceyn→y, thenynyfor alln.

The notion of an altering distance function was introduced by Khanet al.[] as follows.

Deﬁnition .[] The functionψ: [,∞)→[,∞) is called an altering distance func-tion if the following properties are satisﬁed:

. ψis continuous and strictly increasing. . ψ(t) = if and only ift= .

2 Main results

We use the following simple lemma in proving our main results.

Lemma . Let(X,d,)be an ordered b-metric space(with the parameter s)and let f : X_→_X _{and g}_:_X_→_X_.

(a)If a relationis deﬁned onXby

XU ⇐⇒ xu∧yv∧zw, X= (x,y,z),U= (u,v,w)∈X, and a mapping D:X_×_X_→+_{is given by}

D(X,U) =maxd(x,u),d(y,v),d(z,w), X= (x,y,z),U= (u,v,w)∈X,

then(X_,_D,₎_{is an ordered b-metric space}_{(with the same parameter s).}_{The space}₍_X_,_D)

is b-complete iﬀ(X,d)is b-complete.

(b)If the mapping f has the g-mixed monotone property,then the mapping F:X_→_X

given by

FX=f(x,y,z),f(y,x,y),f(z,y,x), X= (x,y,z)∈X, is G-nondecreasing w.r.t.,i.e.,

GXGU ⇒ FXFU,

where G:X_→_X_{is deﬁned by}

GX= (gx,gy,gz), X= (x,y,z)∈X.

(c)If f is continuous from(X_,_D)_to₍_X_,_d),_{then F is continuous in}₍_X_,_D).

(d)If f and g are compatible,then F and G are compatible.

Let (X,d,) be an orderedb-metric space,f :X→X andg:X →X. In the rest of this paper, unless otherwise stated, for allx,y,z,u,v,w∈X, let

Mf(x,y,z,u,v,w)

(6)

and

Mg(x,y,z,u,v,w) =max

d(gx,gu),d(gy,gv),d(gz,gw).

Now, the main result is presented as follows.

Theorem . Let(X,d,)be a partially ordered b-metric space with the parameter s> , and let f :X_→_X _{and g}_:_X_→_X _{be such that f}₍_X₎_⊆_g(_X_)._{Assume that}

ψsεMf(x,y,z,u,v,w)≤ψMg(x,y,z,u,v,w)–ϕMg(x,y,z,u,v,w) () for all x,y,z,u,v,w∈Xwith gxgu,gygv and gzgw,or gugx,gvgy and gwgz, whereψ,ϕ: [,∞)→[,∞)are altering distance functions andε> .

Assume also that

() f has the mixedg-monotone property; () gisb-continuous and compatible withf.

Also,suppose that either

(a) f isb-continuous and(X,d)isb-complete,or (b) (X,d)is regular and(g(X),d)isb-complete.

If there exist x,y,z∈X such that gxf(x,y,z), gyf(y,x,y) and gz

f(z,y,x),then f and g have a tripled coincidence point inX.

Proof LetDbe theb-metric and be the partial order onX _{deﬁned in Lemma ..}

Also, deﬁne the mappingsF,G:X_→_X_by_FX_{= (f}_(x,_y,_z),_f_(y,_x,_y),_f_(z,_y,_{x)) and}_GX₌

(gx,gy,gz),X= (x,y,z) as in Lemma .. Then (X,D,) is an orderedb-metric space (with the same parametersasX) andFis aG-nondecreasing mapping on it such thatF(X)⊆ G(X). Moreover, the contractive condition () implies that

ψsεD(FX,FU)≤ψD(GX,GU)–ϕD(GX,GU) () holds for allX,U∈X_{such that}_GX_and_GU_are_{-comparable. Since}_ϕ_{has non-negative}

values andψis strictly increasing, () implies that

D(FX,FU)≤ 

sεD(GX,GU), ()

where  < /sε_{< /s}_{for all}_X,_U_∈_X _{such that}_GX _and_GU _are_{-comparable. We will}

prove in the next lemma that under these circumstances, it follows thatFandGhave a coincidence point X= (x,¯ y,¯ z)¯ ∈X _{which is obviously a tripled coincidence point of}_f

andg.

The following lemma is an ‘ordered variant’ of the basic result of Czerwik [] (adapted for two mappings).

Lemma . Let(X,d,)be a partially ordered b-metric space and let f and g be two self-mappings onX.Assume that there existsλ∈[,

s)such that

(7)

for all x,y∈X with gxgy or gxgy.Let the following conditions hold:

(i) f isg-nondecreasing with respect toandf(X)⊆g(X); (ii) there existsx∈Xsuch thatgxfx;

(iii) f andgare continuous and compatible and(X,d)is complete,or (iii) (X,d,)is regular and one off(X)org(X)is complete.

Then f and g have a coincidence point inX.

Proof Because offX⊆gXand (ii), we can deﬁne a Jungck sequence by yn=fxn=gxn+,

for alln= , , , . . . .

It can be proved by induction thatynyn+for alln. Ifyn=yn+for somen, thenxn+is

a coincidence point off andg. Hence, we suppose thatyn=yn+for alln. It can be proved

in a standard way (see,e.g., [, Lemma .]) that{yn}is a Cauchy sequence. Suppose ﬁrst that (iii) holds. Then there exists

lim

n→∞fxn=n→∞limgxn=z∈X.

Further, sincef andgare continuous and compatible, we get that

lim

n→∞fgxn=fz, n→∞lim gfxn=gz

and

lim

n→∞d(fgxn,gfxn) = .

We will show thatfz=gz. Indeed, we have

d(fz,gz) ≤ s d(fz,fgxn) +d(fgxn,gz)

=sd(fz,fgxn) +sd(fgxn,gz)

≤ sd(fz,fgxn) +s d(fgxn,gfxn) +d(gfxn,gz)

→s· +s· +s· =  ()

asn→ ∞, and it follows thatfz=gz. It means thatf andghave a coincidence point. In the case (iii), it follows that

lim

n→∞fxn=n→∞limgxn=gu

for someu∈X. Because of regularity, we havegxngu. Applying () withx=xnand y=u, we have

d(fxn,fu)≤λd(gxn,gu)→ (n→ ∞).

It follows that d(fxn,fu)→ when n→ ∞, that is,fxn→fu. Hence,f andg have a

(8)

Let

M(x,y,z,u,v,w) =maxd(x,u),d(y,v),d(z,w).

Takingg=i_X(the identity mapping onX) in Theorem ., we obtain the following tripled ﬁxed point result.

Corollary . Let(X,d,)be a b-complete partially ordered b-metric space and let f : X_→_X _{be a mapping having the mixed monotone property.}_{Assume that}

ψsε_M

f(x,y,z,u,v,w)

≤ψM(x,y,z,u,v,w)–ϕM(x,y,z,u,v,w), ()

for all x,y,z,u,v,w∈X with xu,yv and zw,or ux, vy and wz,where

ψ,ϕ: [,∞)→[,∞)are altering distance functions andε> . Also,suppose that either

(a) f isb-continuous,or (b) (X,d)is regular.

If there exist x,y,z ∈X such that x f(x,y,z), y f(y,x,y) and z

f(z,y,x),then f has a tripled ﬁxed point inX.

Takingψ(t) =tandϕ(t) = t

+t for allt∈[,∞), in Corollary ., we obtain the following tripled ﬁxed point result.

Corollary . Let(X,d,)be a b-complete partially ordered b-metric space and let f : X_→_X _{be a mapping having the mixed monotone property.}_{Assume that}

sεMf(x,y,z,u,v,w)≤ M(x,y,z,u,v,w)

 +M(x,y,z,u,v,w) ()

for someε> and all x,y,z,u,v,w∈X with xu,yv and zw,or ux,vy and wz.

Also,suppose that either

Remark . . Let in Theorem .,

Mf(x,y,z,u,v,w) =df(x,y,z),f(u,v,w).

Then the contractive condition () reduces to the following:

ψsε_d_f_(x,_y,_z),_f_(u,_v,_w)_≤_ψ_M

g(x,y,z,u,v,w)

–ϕMg(x,y,z,u,v,w)

, ()

which appeared in [] in the context ofG-metric spaces.

(9)

a. We obtain more general tripled coincidence point theorems, because whenf andg

satisfy condition (), then they also satisfy ().

b. The technique of the proof is essentially simpler than the one used in [], that is, we need not use Lemma  from [].

. We can replace the contractive condition () by the following:

ψsε_M

f(x,y,z,u,v,w)

≤ψMg(x,y,z,u,v,w)

–ϕMg(x,y,z,u,v,w)

, ()

where

M_f(x,y,z,u,v,w)

=  d

f(x,y,z),f(u,v,w)+df(y,x,y),f(v,u,v)+df(z,y,x),f(w,v,u).

The following corollary can be deduced from our previously obtained results.

Corollary . Let(X,d,)be a partially ordered b-complete b-metric space with s> . Let f :X_→_X_{be a mapping with the mixed monotone property such that}

ψsε_M

f(x,y,z,u,v,w)

≤ψ

d(x,u) +d(y,v) +d(z,w) 

–ϕmaxd(x,u),d(y,v),d(z,w), ()

for someε> and all x,y,z,u,v,w∈X with xu,yv and zw,or ux,vy and wz.Also,suppose that either

(a) f isb-continuous,or (b) (X,d,)is regular.

Proof Iff satisﬁes (), thenf satisﬁes (). Hence, the result follows from Corollary ..

In Theorem ., if we takeg=i_X,ψ(t) =tandϕ(t) = ( –k)tfor allt∈[,∞), where k∈[, ), we obtain the following result.

Corollary . Let(X,d,)be a partially ordered b-complete b-metric space with s> . Let f :X_→_X_{be a mapping having the mixed monotone property and}

maxdf(x,y,z),f(u,v,w),df(y,x,y),f(v,u,v),df(z,y,x),f(w,v,u)

≤ k

sεmax

d(x,u),d(y,v),d(z,w),

for some k∈[, ),ε> and all x,y,z,u,v,w∈X with xu,yv and zw,or ux, vy and wz.Also,suppose that either

(10)

Corollary . Let(X,d,)be a partially ordered b-complete b-metric space with s> . Let f :X_→_X_{be a mapping with the mixed monotone property such that}

maxdf(x,y,z),f(u,v,w),df(y,x,y),f(v,u,v),df(z,y,x),f(w,v,u)

≤ k

sε d(x,u) +d(y,v) +d(z,w)

()

for some k∈[, ),ε> and all x,y,z,u,v,w∈X with xu,yv and zw,or ux, vy and wz.Also,suppose that either

Proof Iff satisﬁes (), thenf satisﬁes the contractive condition of Corollary ..

In the following theorem, we give a suﬃcient condition for the uniqueness of the com-mon tripled ﬁxed point (see also [, , ]).

Theorem . In addition to the hypotheses of Theorem ., suppose that f and g commute and that for all (x,y,z) and (x∗,y∗,z∗)∈X_, _{there exists} _(u,_v,_w)_∈_X _such

that (f(u,v,w),f(v,u,v),f(w,v,u)) is comparable with (f(x,y,z),f(y,x,y),f(z,y,x)) and (f(x∗,y∗,z∗),f(y∗,x∗,y∗),f(z∗,y∗,x∗)).Then f and g have a unique common tripled ﬁxed point.

Proof We shall use the notation as in the proof of Theorem .. It was proved in this theorem that the set of tripled coincidence points off andg,i.e., the set of coincidence points ofF andGinX_{, is nonempty. We shall show that if}_X_and_X∗ _{are coincidence}

points ofFandG, that is,GX=FXandGX∗=FX∗, thenGX=GX∗.

Choose an element U= (u,v,w)∈X _{such that}_FU_{= (f}_(u,_v,_w),_f_(v,_u,_v),_f_(w,_v,_{u)) is}

comparable withFXandFX∗. LetU=Uand chooseU∈Xso thatGU=FU. Then

we can inductively deﬁne a sequence{GUn}such thatGUn+=FUn. SinceGXandGU

are-comparable, we may assume thatGXGU. Using the mathematical induction, it

is easy to prove thatGXGUnfor alln≥. Applying (), one obtains that

ψsεD(GX,GUn+)

=ψsεD(FX,FUn)

≤ψD(GX,GUn)–ϕD(GX,GUn)

≤ψD(GX,GUn)

. ()

From the properties ofψ, we deduce that the sequence{D(GX,GUn)}is non-increasing. Hence, if we proceed as in Theorem ., we can show that

lim

n→∞D(GX,GUn) = ,

(11)

Similarly, we can show that{GUn}isb-convergent toGX∗. Since the limit is unique, it follows thatGX=GX∗.

SinceGX=FX, by commutativity off andg, we haveGGX=GFX=FGX. LetGX=A. Then GA=FA. Thus, Ais another coincidence point off andg. ThenA=GX =GA. Therefore,A= (a,b,c) is a tripled common ﬁxed point off andg.

To prove the uniqueness, assume thatPis another common ﬁxed point ofFandG. Then P=GP=FPand alsoGP=GA. Thus,P=GP=GA=A. Hence, the tripled common ﬁxed

point is unique.

3 Examples

The following examples support our results.

Example . LetX = (–∞,∞) be endowed with the usual ordering and the complete b-metricd(x,y) = (x–y), wheres= . Deﬁnef:X→Xandg:X→Xas

f(x,y,z) = 

√(x–y+z), g(x) =x

√

.

Letψ,ϕ: [,∞)→[,∞) be deﬁned byψ(t) =√tandϕ(t) =

⎧ ⎨ ⎩

t

, t≤,

√ t

 , t> .

Now, we have

ψsdf(x,y,z),f(u,v,w)=



 (x–y+z) – (u–v+w)



=

  (x

√

 –u√) + (v√ –y√) + (z√ –w√)

≤

  (x

√

 –u√)_{+ (y}√_{ –}_v√_)_{+ (z}√_{ –}_w√_)

≤



max(x√ –u√)_{, (y}√_{ –}_v√_)_{, (z}√_{ –}_w√_)

=ψmaxd(gx,gu),d(gy,gv),d(gz,gw)

–ϕmaxd(gx,gu),d(gy,gv),d(gz,gw).

Analogously, we can show that

ψsdf(y,x,y),f(v,u,v)≤ψmaxd(gx,gu),d(gy,gv),d(gz,gw) –ϕmaxd(gx,gu),d(gy,gv),d(gz,gw)

and

ψsdf(z,y,x),f(w,v,u)≤ψmaxd(gx,gu),d(gy,gv),d(gz,gw) –ϕmaxd(gx,gu),d(gy,gv),d(gz,gw).

Thus,

(12)

Hence, all of the conditions of Theorem . are satisﬁed (withε= ). Moreover, (, , ) is a tripled coincidence point off andg.

Example . LetX=be endowed with the usual order and theb-metricd(x,y) = (x– y)_with_s_{= . Consider the mapping}_f _:_X_→_X_{given by}

f(x,y,z) = x–y+z  ,

and functionsψ,ϕ: [, +∞)→[, +∞) deﬁned asψ(t) =tandϕ(t) =_t. Takeε=  in Corollary .. The contractive condition () is satisﬁed since

ψsdf(x,y,z),f(u,v,w)

= 

x–y+z  –

u–v+w 



= 

 (x–u) + (v–y) + (z–w)



≤ 

 (x–u)

_{+ (y}_–_v)_{+ (z}_–_w)

≤ 

max

(x–u), (y–v), (z–w)

=ψmaxd(x,u),d(y,v),d(z,w)–ϕmaxd(x,u),d(y,v),d(z,w).

It follows thatf has a tripled ﬁxed point (which is (, , )).

Note that if instead of theb-metricdwe try to use the standard metricρ(x,y) =|x–y| (with all other data unchanged), the conclusion cannot be obtained. Indeed, the inequality

ψρf(x,y,z),f(u,v,w)≤ψmaxρ(x,u),ρ(y,v),ρ(z,w) –ϕmaxρ(x,u),ρ(y,v),ρ(z,w)

does not hold since forx= ,y=z=u=v=w=  it reduces to_ ≤_ .

Example . LetX={(a, ,a) :a∈[, +∞)} ∪ {(,a, ) :a∈[, +∞)} ⊂R_{with the order}

be deﬁned as

(a,b,c)(a,b,c) ⇐⇒ a≤a, b≤b, c≤c.

Letdbe given as

d(x,y) =max|a–a|,|b–b|,|c–c|

,

wherex= (a,b,c) andy= (a,b,c). Clearly, (X,d) is a completeb-metric space with

s= .

Letg:X→X andf :X→X be deﬁned as follows: f(x,y,z) =x,

(13)

Letψ,ϕ: [,∞)→[,∞) be two arbitrary altering distance functions.

According to the order deﬁned onXand the deﬁnition ofg, we see that for any element x∈X,gxis comparable only with itself.

By a careful computation, it is easy to see that all of the conditions of Theorem . (case (a)) are satisﬁed. Finally, Theorem . guarantees the existence of a tripled coincidence point forf andg,i.e., the point ((, , ), (, , ), (, , )).

4 Applications

In this section, we obtain some tripled coincidence point theorems for a mapping satisfy-ing a contractive condition of integral type in a complete orderedb-metric space.

We denote bythe set of all functionsμ: [, +∞)→[, +∞) verifying the following conditions:

(I) μis a positive Lebesgue integrable mapping on each compact subset of[, +∞); (II) for allε> ,_εμ(r)dr> .

Corollary . Replace the contractive condition()of Theorem.by the following: There exists aμ∈such that

ψ(sεMf(x,y,z,u,v,w))



μ(r)dr≤

ψ(Mg(x,y,z,u,v,w))



μ(r)dr–

ϕ(Mg(x,y,z,u,v,w))



μ(r)dr. ()

Let the other conditions of Theorem.be satisﬁed.Then f and g have a tripled coinci-dence point.

Proof Consider the function(x) =_xμ(r)dr. Then () becomes

ψsεMf(x,y,z,u,v,w)≤ψMg(x,y,z,u,v,w)–ϕMg(x,y,z,u,v,w). Takingψ=◦ψandϕ=◦ϕand applying Theorem ., we obtain the proof (it is easy

to verify thatψandϕare altering distance functions).

Corollary . Substitute the contractive condition()of Theorem.by the following: There exists aμ∈such that

ψ sε_M

f(x,y,z,u,v,w)



μ(r)dr

≤ψ

Mg(x,y,z,u,v,w)



μ(r)dr

–ϕ

Mg(x,y,z,u,v,w)



μ(r)dr

. ()

Let the other conditions of Theorem.be satisﬁed.Then f and g have a tripled coinci-dence point.

Proof Again, as in Corollary ., deﬁne the function(x) =_xφ(r)dr. Then () reduces to

ψsεMf(x,y,z,u,v,w)≤ψMg(x,y,z,u,v,w)–ϕMg(x,y,z,u,v,w). Now, if we deﬁneψ=ψ◦ andϕ=ϕ◦ and apply Theorem ., then the proof is

(14)

Corollary . Replace the contractive condition()of Theorem.by the following: There exists aμ∈such that

ψ

ψ(sεMf(x,y,z,u,v,w))



μ(r)dr

≤ψ

ψ(Mg(x,y,z,u,v,w))



μ(r)dr

–ϕ

ϕ(Mg(x,y,z,u,v,w))



μ(r)dr

()

for altering distance functionsψ,ψ,ϕandϕ.If the other conditions of Theorem.are

satisﬁed,then f and g have a tripled coincidence point.

Similar to [], letNbe a ﬁxed positive integer. Let{μi}≤i≤N be a family ofNfunctions which belong to. For allt≥, we deﬁne

I(t) =

t



μ(r)dr,

I(t) = It



μ(r)dr=

t

μ(r)dr



μ(r)dr,

I(t) = It



μ(r)dr=

t

μ(r)dr  μ(r)dr



μ(r)dr,

· · ·,

IN(t) =

I(N–)t



μN(r)dr.

We have the following result.

Corollary . Replace the inequality()of Theorem.by the following condition:

ψINsεMf(x,y,z,u,v,w)≤ψINMg(x,y,z,u,v,w)–ϕINMg(x,y,z,u,v,w). ()

Let the other conditions of Theorem.be satisﬁed.Then f and g have a tripled coincidence point.

Proof Considerˆ =ψ◦INandˆ =ϕ◦IN. Then the above inequality becomes

ˆ

sεMf(x,y,z,u,v,w)≤ ˆMg(x,y,z,u,v,w)–ˆMg(x,y,z,u,v,w).

Applying Theorem ., we obtain the desired result (it is easy to verify thatˆ andˆ are

altering distance functions).

Another consequence of the main theorem is the following result.

(15)

There existμ,μ∈such that sε_Mf₍_x_,_y_,_z_,_u_,_v_,_w₎



μ(r)dr≤

Mg(x,y,z,u,v,w)



μ(r)dr–

Mg(x,y,z,u,v,w)



μ(r)dr.

Let the other conditions of Theorem.be satisﬁed.Then f and g have a tripled coincidence point.

Proof It is clear that the functiont→_tμi(r)dris an altering distance function.

5 Existence of a solution for a system of integral equations

Motivated by the work in [], we study the existence of solutions for a system of nonlinear integral equations using the results proved in the previous sections.

Consider the integral equations in the following system.

x(t) =P(t) +

T



S(t,r)fr,x(r)+kr,y(r)+hr,z(r)dr,

y(t) =P(t) +

T



S(t,r) fr,y(r)+kr,x(r)+hr,y(r)dr,

z(t) =P(t) +

T



S(t,r)fr,z(r)+kr,y(r)+hr,x(r)dr.

()

We will consider the system () under the following assumptions:

(i) f,k,h: [,T]× → are continuous; (ii) P: [,T]→ is continuous;

(iii) S: [,T]× →[,∞)is continuous; (iv) there existsq> such that for allx,y∈ ,

≤f(r,y) –f(r,x)≤q(y–x), ≤k(r,x) –k(r,y)≤q(y–x),

and

≤h(r,y) –h(r,x)≤q(y–x);

(v)

p–qp max

t∈[,T]

T



S(t,r)dr

p < ;

(vi) there exist continuous functionsα,β,γ : [,T]→ such that

α(t)≤P(t) +

T



S(t,r) fr,α(r)+kr,β(r)+hr,γ(r)dr,

β(t)≥P(t) +

T



(16)

and

γ(t)≤P(t) +

T



S(t,r) fr,γ(r)+kr,β(r)+hr,α(r)dr.

We consider the spaceX =C([,T],) of continuous functions deﬁned on [,T] en-dowed with theb-metric given by

d(u,v) = max

t∈[,T]

u(t) –v(t)p

for allu,v∈X, wheres= p–_and_p_≥_{. We endow}_X_{with the partial order}_{given by}

xy ⇐⇒ x(t)≤y(t) for allt∈[,T].

It is known that (X,d,) is regular []. Our result is the following.

Theorem . Under assumptions(i)-(vi),the system()has a solution inX_,_where_X₌

C([,T],).

Proof As in [], we consider the operatorsF:X_→_X _and_g_:_X_→_X _{deﬁned by}

F(x,x,x)(t) =P(t) +

T



S(t,r) fr,x(r)

+kr,x(r)

+hr,x(r)

dr,

andg(x) =xfor allt∈[,T],x,x,x,x∈X.

Fhas the mixed monotone property (see [, Theorem ]).

Letx,y,z,u,v,w∈X, withxu,yvandzw. SinceF has the mixed monotone property, we have

F(u,v,w)F(x,y,z). On the other hand,

dF(x,y,z),F(u,v,w)= max

t∈[,T]

F(x,y,z)(t) –F(u,v,w)(t)p.

Note that for allt∈[,T], from (iv) and the fact that for alla,b,c≥, (a+b+c)p_≤ p–_ap_{+ }p–_bp_{+ }p–_cp_{, we have}

F(x,y,z)(t) –F(u,v,w)(t)p

=

T



S(t,r) fr,x(r)–fr,u(r)dr

+

T



S(t,r)kr,y(r)–kr,v(r)dr

+

T



(17)

≤

T



S(t,r) fr,x(r)–fr,u(r)dr

+

T



S(t,r) kr,y(r)–kr,v(r)dr

+

T



S(t,r) hr,z(r)–hr,w(r)dr

p

≤

p–

T



S(t,r)fr,x(r)–fr,u(r)dr p

+ p–

T



S(t,r)kr,y(r)–kr,v(r)dr p

+ p–

T



S(t,r)hr,z(r)–hr,w(r)dr p

≤p–

T



S(t,r)fr,x(r)–fr,u(r)dr

p

+

T



S(t,r) kr,y(r)–kr,v(r)dr

p

+

T



S(t,r) hr,y(r)–hr,v(r)dr

p

≤p–qp

max

r∈[,T]

x(r) –u(r)p+

max

r∈[,T]

y(r) –v(r)p

+max

r∈[,T]

z(r) –w(r)p T



S(t,r)dr

p

= p–qp

max

r∈[,T]

x(r) –u(r)p+ max

r∈[,T]

y(r) –v(r)p

+ max

r∈[,T]

z(r) –w(r)p

T



S(t,r)dr

p .

Thus,

max

t∈[,T]

F(x,y,z)(t) –F(u,v,w)(t)p

≤p–qp d(x,u) +d(y,v) +d(z,w)max

t∈[,T]

T



S(t,r)dr

p

≤p–qpmaxd(x,u),d(y,v),d(z,w)max

t∈[,T]

T



S(t,r)dr

p

. ()

Repeating this idea, using the deﬁnition of theb-metricd, we get

max

t∈[,T]

F(y,x,y)(t) –F(v,u,v)(t)p

≤p–qp d(y,v) +d(x,u) +d(y,v)max

t∈[,T]

T



S(t,r)dr

p

≤p–qpmaxd(y,v),d(x,u)max

t∈[,T]

T



S(t,r)dr

p

t∈[,T]

T



S(t,r)dr

p

(18)

and

max

t∈[,T]

F(z,y,x)(t) –F(w,v,u)(t)p

≤p–qp d(x,u) +d(y,v) +d(z,w)max

t∈[,T]

T



S(t,r)dr

p

t∈[,T]

T



S(t,r)dr

p

. ()

Hence, from the above three inequalities, we have

maxdF(x,y,z),F(u,v,w),dF(y,x,y),F(v,u,v),dF(z,y,x),F(w,v,u)

≤p–qp max

t∈[,T]

T



S(t,r)dr

p

maxd(x,u),d(y,v),d(z,w)

≤p–qpmaxt∈[,T](

T

 |S(t,r)|dr)

p p– max

d(x,u),d(y,v),d(z,w).

But from (v), we have

p–qp max

t∈[,T]

T



S(t,r)dr

p < .

This proves that the operatorFsatisﬁes the contractive condition appearing in Corol-lary . (withε= ).

Letα,β,γ be the functions appearing in assumption (vi). Then by (vi) we get

αF(α,β,γ), βF(β,α,β), γ F(γ,β,α).

Applying Corollary ., we deduce the existence of x,x,x ∈ X such that x =

F(x,x,x),x=F(x,x,x) andx=F(x,x,x).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

JRR, VP and SR have worked together on each section of the paper such as the literature review, results and examples. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Gilan-E-Gharb Branch, Islamic Azad University, Gilan-E-Gharb, Iran.}2_{Department of}

Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran.3_{Faculty of Mechanical Engineering,}

University of Belgrade, Kraljice Marije 16, Beograd, 11120, Serbia.

Acknowledgements

The authors express their gratitude to the referees and Professor Zoran Kadelburg for their helpful suggestions which improved the presentation, in particular the proof of Theorem 2.2.

Received: 4 March 2013 Accepted: 29 April 2013 Published: 16 May 2013

References

1. Guo, D, Lakshmikantham, V: Coupled ﬁxed points of nonlinear operators with applications. Nonlinear Anal.11, 623-632 (1987)

(19)

3. Lakshmikantham, V, Ćirić, L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal.70(12), 4341-4349 (2009)

4. Aydi, H, Postolache, M, Shatanawi, W: Coupled ﬁxed point results for (ψ,ϕ)-weakly contractive mappings in ordered

G-metric spaces. Comput. Math. Appl.63, 298-309 (2012)

5. Luong, NV, Thuan, NX: Coupled ﬁxed points in partially ordered metric spaces and application. Nonlinear Anal.74, 983-992 (2011)

6. Shatanawi, W: Coupled ﬁxed point theorems in generalized metric spaces. Hacet. J. Math. Stat.40, 441-447 (2011) 7. Berinde, V, Borcut, M: Tripled ﬁxed point theorems for contractive type mappings in partially ordered metric spaces.

Nonlinear Anal.74, 4889-4897 (2011)

8. Aydi, H, Karapınar, E, Shatanawi, W: Tripled coincidence point results for generalized contractions in ordered generalized metric spaces. Fixed Point Theory Appl.2012, 101 (2012)

9. Borcut, M: Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces. Appl. Math. Comput.218, 7339-7346 (2012)

10. Borcut, M, Berinde, V: Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces. Appl. Math. Comput.218, 5929-5936 (2012)

11. Choudhury, BS, Karapınar, E, Kundu, A: Tripled coincidence point theorems for nonlinear contractions in partially ordered metric spaces. Int. J. Math. Math. Sci.2012, Article ID 329298 (2012)

12. Czerwik, S: Contraction mappings inb-metric spaces. Acta Math. Inform. Univ. Ostrav.1, 5-11 (1993) 13. Aghajani, A, Abbas, M, Roshan, JR: Common ﬁxed point of generalized weak contractive mappings in partially

orderedb-metric spaces. Math. Slovaca (in press)

14. Akkouchi, M: Common ﬁxed point theorems for two selfmappings of ab-metric space under an implicit relation. Hacet. J. Math. Stat.40(6), 805-810 (2011)

15. Boriceanu, M, Bota, M, Petrusel, A: Multivalued fractals inb-metric spaces. Cent. Eur. J. Math.8(2), 367-377 (2010) 16. Hussain, N, Ðorić, D, Kadelburg, Z, Radenovi ć, S: Suzuki-type fixed point results in metric type spaces. Fixed Point

Theory Appl.2012, 126 (2012)

17. Hussain, N, Shah, MH: KKM mappings in coneb-metric spaces. Comput. Math. Appl.62, 1677-1684 (2011)

18. Jovanović, M, Kadelburg, Z, Radenović, S: Common fixed point results in metric-type spaces. Fixed Point Theory Appl.

2010, Article ID 978121 (2010). doi:10.1155/2010/978121

19. Khamsi, MA, Hussain, N: KKM mappings in metric type spaces. Nonlinear Anal.73(9), 3123-3129 (2010) 20. Singh, SL, Prasad, B: Some coincidence theorems and stability of iterative procedures. Comput. Math. Appl.55,

2512-2520 (2008)

21. Khan, MS, Swaleh, M, Sessa, S: Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc.

30, 1-9 (1984)

22. Nashine, HK, Samet, B: Fixed point results for mappings satisfying (ψ,ϕ)-weakly contractive condition in partially ordered metric spaces. Nonlinear Anal.74, 2201-2209 (2011)

23. Nieto, JJ, Rodriguez-López, R Existence and uniqueness of ﬁxed point in partially ordered sets and applications to ordinary diﬀerential equations. Order22(3), 223-239 (2005)

doi:10.1186/1687-1812-2013-130