Some fixed point theorems for \((\alpha,\psi)\)-rational type contractive mappings

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R E S E A R C H

Open Access

Some ﬁxed point theorems for

(α

,

ψ

)

-rational type contractive mappings

Hamed H Alsulami

1,2*

_{, Sumit Chandok}

3

_{, Mohamed-Aziz Taoudi}

4,5

_{and ˙Inci M Erhan}

6

*_{Correspondence:}

[email protected] 1_{Nonlinear Analysis and Applied}

Mathematics Research Group (NAAM), Jeddah, Saudi Arabia 2_{King Abdulaziz University, Jeddah,}

Saudi Arabia

Full list of author information is available at the end of the article

Abstract

In this paper, we introduce the concept of (α,

ψ)-rational type contractive mappings

and provide suﬃcient conditions for the existence and uniqueness of a ﬁxed point for such class of generalized nonlinear contractive mappings in the setting of

generalized metric spaces. We also deduce several interesting corollaries.

MSC: 47H10; 54H25

Keywords: ﬁxed point;

α-admissible; contraction mappings; metric space

1 Introduction and preliminaries

Fixed point theory has gained very large impetus due to its wide range of applications in various fields such as engineering, economics, computer science, and many others. It is well known that the contractive conditions are very indispensable in the study of fixed point theory, and Banach’s fixed point theorem [] for contraction mappings is one of the pivotal result in analysis. This theorem has been extended and generalized by various authors (see,e.g., [–]) in various abstract spaces, one of which is generalized metric space.

As pointed out in [], the topology of a generalized metric space has some disadvan-tages:

(T) A generalized metric does not need to be continuous.

(T) A convergent sequence in generalized metric spaces does not need to be Cauchy. (T) A generalized metric space does not need to be Hausdorﬀ, and hence the

uniqueness of the limits cannot be guaranteed.

In this paper, we introduce the concept of (α,ψ)-rational type contractive mappings and provide suﬃcient conditions for the existence and uniqueness of ﬁxed points for such class of generalized nonlinear contractive mappings in the framework of generalized met-ric spaces by caring the problems (T)-(T) mentioned above. We also deduce several interesting corollaries. The proved results generalize and extend various well-known re-sults in the literature. The techniques used in this paper have been studied and improved by various authors (see [–, ] and references cited therein).

To start with, we give some notations and introduce some deﬁnitions which will be used in the sequel.

Deﬁnition .[] LetXbe a nonempty set andd:X×X→[,∞) satisfy the following conditions, for allx,y∈Xand all distinctu,v∈Xeach of which is diﬀerent fromxandy:

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(GMS) d(x,y) = if and only ifx=y, (GMS) d(x,y) =d(y,x),

(GMS) d(x,y)≤d(x,u) +d(u,v) +d(v,y).

Then the mapdis called a generalized metric and abbreviated as GM. Here, the pair (X,d) is called a generalized metric space and abbreviated as GMS.

In the above deﬁnition, ifdsatisﬁes only (GMS) and (GMS), then it is called a semi-metric (see,e.g., []).

A sequence{xn}in a GMS (X,d) is GMS convergent to a limitxif and only ifd(xn,x)→ asn→ ∞.

A sequence{xn}in a GMS (X,d) is GMS Cauchy if and only if for every>  there exists a positive integerN() such thatd(xn,xm) <, for alln>m>N().

A GMS (X,d) is called complete if every GMS Cauchy sequence inXis GMS convergent. A mapping T : (X,d)→(X,d) is continuous if for any sequence{xn} inX such that d(xn,x)→ asn→ ∞, we haved(Txn,Tx)→ asn→ ∞.

The following assumption was suggested by Wilson [] to replace the triangle inequality with the weakened condition.

(W) For each pair of (distinct) pointsu,v, there is a numberru,v> such that for every z∈X,ru,v<d(u,z) +d(z,v).

Proposition .[] In a semimetric space,the assumption(W)is equivalent to the as-sertion that the limits are unique.

Proposition . [] Suppose that {xn} is a Cauchy sequence in a GMS (X,d) with

lim_n_→∞d(xn,u) = ,where u∈X.Thenlimn→∞d(xn,z) =d(u,z),for all z∈X.In particu-lar,the sequence{xn}does not converge to z if z=u.

Deﬁnition . LetXbe a nonempty set,T:X→Xandα:X×X→[,∞) be two map-pings. We say thatTis anα-admissible mapping ifα(x,y)≥ impliesα(Tx,Ty)≥, for all x,y∈X.

Deﬁnition . Let (X,d) be a GMS andα:X×X→[,∞).Xis calledα-regular GMS if, for a sequence{xn}inXsuch thatxn→xandα(xn,xn+)≥, there exists a subsequence

{xnk}of{xn}such thatα(xnk,x)≥ for allk∈N.

Throughout the paper,F(T) denotes the set of ﬁxed points of the mappingT.

2 Main results

The contraction mappings considered in this paper are constructed via auxiliary functions deﬁned below. Letbe a family of functionsψ: [,∞)→[,∞) satisfying the following properties:

(i) ψis upper semi-continuous, strictly increasing; (ii) {ψn_(t)_}

n∈Nconverges toasn→ ∞, for allt> ;

(iii) ψ(t) <t, for everyt> .

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such that for allx,y∈Xthe following condition holds:

α(x,y)d(Tx,Ty)≤ψM(x,y), (.) where

M(x,y) =max

d(x,y),d(x,Tx),d(y,Ty),d(x,Tx)d(y,Ty)  +d(x,y) ,

d(x,Tx)d(y,Ty)  +d(Tx,Ty)

.

Next, we state and prove an existence and uniqueness theorem for ﬁxed point of (α,ψ )-rational type-I contractive mappings.

Theorem . Let(X,d)be a complete GMS,T:X→X be a self mapping andα:X×X→ [,∞)a given function.Suppose that the following conditions are satisﬁed:

(i) Tis anα-admissible mapping;

(ii) Tis an(α,ψ)-rational type-I contractive mapping;

(iii) there existsx∈Xsuch thatα(x,Tx)≥,α(x,Tx)≥;

(iv) eitherTis continuous,orXisα-regular.

Then T has a ﬁxed point x∗∈X and{Tnx}converges to x∗.Further,if for all x,y∈F(T),

we haveα(x,y)≥then T has a unique ﬁxed point in X.

Proof Letx∈Xsatisﬁesα(x,Tx)≥ andα(x,Tx)≥. We construct the sequence

{xn}inXasxn=Tnx=Txn–, forn∈N. It is obvious that ifxn=xn+, for somen∈N, thenxnis a ﬁxed point ofT. Consequently, we suppose thatxn= xn+for alln∈N.

SinceTisα-admissible,α(x,Tx) =α(x,x)≥ ⇒α(Tx,Tx) =α(x,x)≥ ⇒and

thus,α(Tx,Tx) =α(x,x)≥ . . . , and hence by induction, we getα(xn,xn+)≥ for all

n≥.

By similar arguments, since α(x,Tx) ≥ , we have α(x,x) = α(x,Tx) ≥ ,

α(Tx,Tx) =α(x,x)≥. By induction, we get α(xn,xn+)≥ for alln≥. Consider

(.) withx=xnandy=xn+. Clearly, we have

d(xn+,xn+) =d(Txn,Txn+)

≤α(xn,xn+)d(Txn,Txn+)

≤ψM(xn,xn+)

,

where

M(xn,xn+) =max

d(xn,xn+),d(xn,Txn),d(xn+,Txn+),

d(xn,Txn)d(xn+,Txn+)

 +d(xn,xn+)

,d(xn,Txn)d(xn+,Txn+)  +d(Txn,Txn+)

=max

d(xn,xn+),d(xn,xn+),d(xn+,xn+),

d(xn,xn+)d(xn+,xn+)

 +d(xn,xn+)

,d(xn,xn+)d(xn+,xn+)  +d(xn+,xn+)

=maxd(xn,xn+),d(xn+,xn+)

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sinced(xn,xn+)d(xn+,xn+)

+d(xn,xn+) ≤d(xn+,xn+) and

d(xn,xn+)d(xn+,xn+)

+d(xn+,xn+) ≤d(xn,xn+). If for somen, we haveM(xn,xn+) =d(xn+,xn+), then

d(xn+,xn+)≤ψ

M(xn,xn+)

=ψd(xn+,xn+)

<d(xn+,xn+), (.)

which is impossible. Hence,M(xn,xn+) =d(xn,xn+), for alln∈N,

d(xn+,xn+)≤ψ

M(xn,xn+)

=ψd(xn,xn+)

. (.)

From the property (iii) ofψ, we conclude that

d(xn+,xn+) <d(xn,xn+), (.)

for everyn∈N. Combining (.) and (.), we deduced(xn+,xn+)≤ψn(d(x,x)), for all

n∈N. Using the property (ii) ofψ, it is clear that

lim

n→∞d(xn+,xn+) = . (.)

Consider now (.) withx=xn–andy=xn+. We have

d(xn,xn+) =d(Txn–,Txn+)

≤α(xn–,xn+)d(Txn–,Txn+)

≤ψM(xn–,xn+)

, (.)

where

M(xn–,xn+) =max

d(xn–,xn+),d(xn–,Txn–),d(xn+,Txn+),

d(xn–,Txn–)d(xn+,Txn+)

 +d(xn–,xn+)

,d(xn–,Txn–)d(xn+,Txn+)  +d(Txn–,Txn+)

=max

d(xn–,xn+),d(xn–,xn),d(xn+,xn+),

d(xn–,xn)d(xn+,xn+)

 +d(xn–,xn+)

,d(xn–,xn)d(xn+,xn+)  +d(xn,xn+)

. (.)

From (.) we haved(xn+,xn+) <d(xn–,xn). Deﬁnean=d(xn,xn+) andbn=d(xn,xn+).

Then

M(xn–,xn+) =max

an–,bn–,

bn–bn+

 +an–

,bn–bn+  +an

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IfM(xn–,xn+) =bn–, orb_+n–_a_nbn_–+orbn_+–_abn_n+then takinglim supasn→ ∞in (.) and using

(.) and upper semi-continuity ofψwe get

≤lim sup

n→∞

an≤lim sup n→∞

ψM(xn–,xn+)

=ψ

lim sup

n→∞

M(xn–,xn+) =ψ() = ,

and hence,

lim

n→∞an=nlim→∞d(xn,xn+) = .

IfM(xn–,xn+) =an–, then (.) implies

an≤ψ(an–) <an–,

due to the property (iii) ofψ. In other words, the sequence{an}is positive monotone decreasing, and hence, it converges to somet≥. Assume thatt> . Now, by (.), we get

t=lim sup

n→∞ an=lim supn→∞ ψ(an–) =ψ

lim sup

n→∞ an– =ψ(t) <t,

which is a contradiction. Therefore,

lim

n→∞an=nlim→∞d(xn,xn+) = . (.)

Now, we shall prove thatxn=xm, for alln=m. Assume on the contrary thatxn=xm, for somem,n∈Nwithn=m. Sinced(xp,xp+) > , for eachp∈N, without loss of generality,

we may assume thatm>n+ . Substitute againx=xn=xmandy=xn+=xm+in (.),

which yields

d(xn,xn+) =d(xn,Txn) =d(xm,Txm) =d(Txm–,Txm)

≤α(xm–,xm)d(Txm–,Txm)≤ψ

M(xm–,xm)

, (.)

where

M(xm–,xm) =max

d(xm–,xm),d(xm–,Txm–),d(xm,Txm),

d(xm–,Txm–)d(xm,Txm)  +d(xm–,xm)

,d(xm–,Txm–)d(xm,Txm)  +d(Txm–,Txm)

=max

d(xm–,xm),d(xm–,xm),d(xm,xm+),

d(xm–,xm)d(xm,xm+)

 +d(xm–,xm)

,d(xm–,xm)d(xm,xm+)  +d(xm,xm+)

=maxd(xm–,xm),d(xm,xm+)

. (.)

IfM(xm–,xm) =d(xm–,xm), then (.) implies

d(xn,xn+)≤ψ

d(xm–,xm)

≤ψm–nd(xn,xn+)

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If on the other handM(xm–,xm) =d(xm,xm+), then from (.) we have

d(xn,xn+)≤ψ

d(xm,xm+)

≤ψm–n+d(xn,xn+)

. (.)

Using the property (iii) ofψ, the two inequalities (.) and (.) imply

d(xn,xn+) <d(xn,xn+),

which is impossible.

Now, we shall prove that{xn}is a Cauchy sequence, that is,limn→∞d(xn,xn+k) = , for all k∈N. We have already proved the cases fork=  andk=  in (.) and (.), respectively. Take arbitraryk≥. We discuss two cases.

Case . Suppose thatk= m+ , wherem≥. Using the quadrilateral inequality (GMS), we have

d(xn,xn+k) =d(xn,xn+m+)≤d(xn,xn+) +d(xn+,xn+) +· · ·+d(xn+m,xn+m+)

≤

n+m

p=n

ψpd(x,x)

≤

+∞

p=n

ψpd(x,x)

→ asn→ ∞. (.)

Case . Suppose thatk= m, wherem≥. Again, by applying the quadrilateral inequal-ity, we have

d(xn,xn+k) =d(xn,xn+m)≤d(xn,xn+) +d(xn+,xn+) +· · ·+d(xn+m–,xn+m)

≤d(xn,xn+) +

n+m–

p=n+

ψpd(x,x)

≤d(xn,xn+) + +∞

p=n

ψpd(x,x)

→ asn→ ∞, (.)

sincelimn→∞=  because of (.). In both of the above cases, we havelimn→∞d(xn,xn+k) = , for allk≥. Hence we conclude that{xn}is a Cauchy sequence in (X,d). Since (X,d) is complete, there existsx∗∈Xsuch that

lim

n→∞d

xn,x∗

= . (.)

We will show next that the limitx∗of the sequence{xn}is a ﬁxed point ofT. First, we suppose thatT is continuous. Then from (.) we have

lim

n→∞d

Txn,Tx∗

= lim

n→∞d

xn+,Tx∗

= .

Due to Proposition ., we conclude that

x∗=Tx∗,

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Now, we suppose thatXisα-regular. Then there exists a subsequence{xnk}of{xn}such thatα(xnk–,x

∗₎_≥_{ for all}_k_∈_{N. Now, from inequality (.) with}_x₌_x

nk andy=x ∗_{, we}

obtain

dxnk+,Tx

∗₌_d_Tx nk,Tx

∗

≤αxnk,x ∗_d_Tx

nk,Tx ∗

≤ψMxnk,x

∗_, _(.)

where

Mxnk,x

∗₌_max_d_x nk,x

∗_,_d(x

nk,Txnk),d

x∗,Tx∗,

d(xnk,Txnk)d(x ∗_,_Tx∗₎

 +d(xnk,x∗)

,d(xnk,Txnk)d(x ∗_,_Tx∗₎

 +d(Txnk,Tx∗)

=max

dxnk,x ∗_,_d(x

nk,xnk+),d

x∗,Tx∗,

d(xnk,xnk+)d(x∗,Tx∗)  +d(xnk,x∗)

,d(xnk,xnk+)d(x∗,Tx∗)  +d(xnk+,Tx∗)

. (.)

Lettingk→ ∞in (.), we obtain M(xnk,x

∗_{) =}_d(x∗_,_Tx∗_{). Therefore, upon taking the}

limit ask→ ∞, in inequality (.), we haved(x∗,Tx∗)≤ψ(d(x∗,Tx∗)) <d(x∗,Tx∗), which impliesx∗=Tx∗, that is,x∗is a ﬁxed point ofT.

Finally, suppose thatx∗andy∗are two ﬁxed points ofTsuch thatx∗=y∗. Then by the hypothesis,α(x∗,y∗)≥. Hence, from (.) withx=x∗andy=y∗we have

dx∗,y∗=dTx∗,Ty∗

≤αx∗,y∗dTx∗,Ty∗

≤ψMx∗,y∗,

where

Mx∗,y∗=max

dx∗,y∗,dx∗,Tx∗,dy∗,Ty∗,

d(x∗,Tx∗)d(y∗,Ty∗)  +d(x∗,y∗) ,

d(x∗,Tx∗)d(y∗,Ty∗)  +d(Tx∗,Ty∗)

=dx∗,y∗. (.)

Hence, we getd(x∗,y∗)≤ψ(d(x∗,y∗)) <d(x∗,y∗), which is possible only ifd(x∗,y∗) = , that is,x∗=y∗. HenceThas a unique ﬁxed point. Deﬁnition . Let (X,d) be a generalized metric space andα:X×X→R+_{. A mapping}

T:X→Xis said to be (α,ψ)-rational type-II contractive mapping if there exists aψ∈, such that, for allx,y∈X, the following condition holds:

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where

M(x,y) =max

d(x,y),d(x,Tx),d(y,Ty),

d(x,Tx)d(y,Ty)  +d(x,y) +d(x,Ty) +d(y,Tx),

d(x,Ty)d(x,y)

 +d(x,Tx) +d(y,Tx) +d(y,Ty)

.

For this class of mappings we state a similar existence and uniqueness theorem.

Theorem . Let(X,d)be a complete generalized metric space,T:X→X be a self map-ping,andα:X×X→R.Suppose that the following conditions are satisﬁed:

(i) Tis anα-admissible mapping;

(ii) Tis an(α,ψ)-rational type-II contractive mapping;

(iii) there existsx∈Xsuch thatα(x,Tx)≥andα(x,Tx)≥;

Then T has a ﬁxed point x∗∈X and{Tn_x

}converges to x∗.Further,if for all x,y∈F(T),

we haveα(x,y)≥,then T has a unique ﬁxed point in X.

Proof The proof can be done by following the lines of the proof of Theorem .. The following example illustrating Theorem . is inspired by [].

Example . LetXbe a finite set defined asX={, , , }. Defined:X×X→[,∞) as:

d(, ) =d(, ) =d(, ) =d(, ) = ,

d(, ) =d(, ) = ,

d(, ) =d(, ) =d(, ) =d(, ) = ,

d(, ) =d(, ) =d(, ) =d(, ) =d(, ) =d(, ) = .

The functiondis not a metric onX. Indeed, note that

 =d(, )≥d(, ) +d(, ) =  +  = ,

that is, the triangle inequality is not satisﬁed. However,dis a generalized metric onXand, moreover, (X,d) is a complete generalized metric space. DeﬁneT:X→Xas

T =T =T = , T = ,

α(x,y) asα(x,y) =  andψ(t) =_t. Then, forx= , ,  andy= , , , we have

α(x,y)d(Tx,Ty) = ≤ψM(x,y)= .

On the other hand, forx= , ,  andy=  we obtain

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and

M(x, ) =max

d(x, ),d(x,T),d(y,T),d(x,Tx)d(y,Ty)  +d(x, ) ,

d(x,Tx)d(y,Ty)  +d(Tx,T)

= ,

and hence

α(x, )d(Tx,T) = ≤  = .

Forx= ,y= , the contraction condition is obvious. Clearly,Tsatisﬁes the conditions of Theorem . and has a unique ﬁxed pointx= .

3 Some consequences

In this section we give some consequences of the main results presented above. Speciﬁ-cally, we apply our results to generalized metric spaces endowed with a partial order.

Deﬁnition . Let (X,) be a partially ordered set. A mappingT:X→Xis said to be nondecreasing with respect toif for everyx,y∈X xyimpliesTxTy.

Deﬁnition . Let (X,d,) be a partially ordered GMS.Xis called regular GMS if, when-ever{xn}is a sequence inXsuch thatxn→xandxnxn+, then there exists a subsequence

{xnk}of{xn}such thatxnkxfor allk∈N.

Theorem . Let(X,d,)be a partially ordered complete generalized metric space and T:X→X be a nondecreasing self mapping.Suppose that the following conditions are sat-isﬁed:

(i) There exists a functionψ∈for which

d(Tx,Ty)≤ψM(x,y), (.)

where

M(x,y) =max

d(x,y),d(x,Tx),d(y,Ty),d(x,Tx)d(y,Ty)  +d(x,y) ,

d(x,Tx)d(y,Ty)  +d(Tx,Ty)

for allx,y∈Xwithxy.

(ii) There existsx∈Xsuch thatxTxandxTx.

(iii) EitherTis continuous,orXis regular.

}converges to x∗.

Proof Deﬁne a mappingα:X×X→[,∞) as follows.

α(x,y) =

⎧ ⎨ ⎩

, ifxyoryx, , otherwise.

Then the existence conditions of Theorem . hold and henceT has a ﬁxed point which is the limit of the sequence{Tn_x

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Theorem . Let(X,d,)be a partially ordered complete generalized metric space and T:X→X be a nondecreasing self mapping.Suppose that the following conditions are sat-isﬁed:

(i) There exist a functionψ∈for which

d(Tx,Ty)≤ψM(x,y), (.)

where

M(x,y) =max

d(x,Ty)d(x,y)

for allx,y∈Xwithxy;

(ii) there existsx∈Xsuch thatxTxandxTx;

(iii) eitherTis continuous,orXis regular.

Proof Employing again a mappingα:X×X→[,∞) deﬁned as

α(x,y) =

⎧ ⎨ ⎩

, ifxyoryx, , otherwise,

we observe that the existence conditions of Theorem . hold and hence,T has a ﬁxed point which is the limit of the sequence{Tn_x

}.

Several particular cases can also be deduced from the above results.

Corollary . Let(X,d)be a complete generalized metric space,T be a self mapping,T: X→X,andα:X×X→R.Suppose that the following conditions are satisﬁed:

(i) Tis anα-admissible mapping; (ii) Tsatisﬁes

d(Tx,Ty)≤kM(x,y), (.)

where

M(x,y) =max

d(x,Ty)d(x,y)

for somek∈[, );

(iii) there existsx∈Xsuch thatα(x,Tx)≥,α(x,Tx)≥;

Then T has a ﬁxed point x∗∈X and{Tnx}converges to x∗.Further,if,for all x,y∈F(T),

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Proof Defineψ(t) =kt. Clearly,ψ∈. By Theorem .,T has a unique fixed point. Corollary . Let(X,d,)be a partially ordered complete generalized metric space and T :X→X be a nondecreasing mapping.Suppose that the following conditions are satis-fied:

(i)

d(Tx,Ty)≤kM(x,y), (.)

where

M(x,y) =max

d(x,Ty)d(x,y)

for allx,y∈Xwithxyand somek∈[, );

(ii) there existsx∈Xsuch thatxTxandxTx;

(iii) eitherTis continuous,orXis regular.

Proof Deﬁneα:X×X→[,∞) as

α(x,y) =

⎧ ⎨ ⎩

, ifxyoryx, , otherwise.

Corollary . implies thatThas a ﬁxed point.

Competing interests

The authors declare that there is no conﬂict of interests regarding the publication of this article.

Authors’ contributions

All authors contributed equally and signiﬁcantly in writing this article. All authors read and approved the ﬁnal version of manuscript.

Author details

1_{Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Jeddah, Saudi Arabia.}2_{King Abdulaziz University,}

Jeddah, Saudi Arabia.3_{Department of Mathematics, Khalsa College of Engineering & Technology, Punjab Technical} University, Amritsar, 143001, India.4_{Centre Universitaire Polydisciplinaire Kelaa des Sraghna, Université Cadi Ayyad, B.P.} 263, Marrakech, Maroc.5_{Laboratoire de Mathématiques et de Dynamique de Populations, Université Cadi Ayyad,} Marrakech, Maroc. 6_{Department of Mathematics, Atılım University, Ankara, Turkey.}

Acknowledgements

The authors would like to thank referees for their useful comments and suggestions for the improvement of the paper.

Received: 21 January 2015 Accepted: 17 May 2015

References

1. Banach, S: Sur les opérations dans les ensembles abstraits et leur application aux équations integrales. Fundam. Math.3, 133-181 (1922)

2. Branciari, A: A ﬁxed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces. Publ. Math. (Debr.)57, 31-37 (2000)

3. Aydi, H, Karapýnar, E, Samet, B: Fixed points for generalized (α,ψ)-contractions on generalized metric spaces. J. Inequal. Appl.2014, 229 (2014)

4. Aydi, H, Karapınar, E, Lakzian, H: Fixed point results on the class of generalized metric spaces. Math. Sci.6, 46 (2012) 5. Bilgili, N, Karapınar, E: A note on ‘common ﬁxed points for (ψ,α,β)-weakly contractive mappings in generalized

(12)

6. Erhan, IM, Karapınar, E, Sekulic, T: Fixed points of (ψ,φ) contractions on rectangular metric spaces. Fixed Point Theory Appl.2012, 138 (2012)

7. Karapınar, E: Discussion on (α,ψ) contractions on generalized metric spaces. Abstr. Appl. Anal.2014, Article ID 962784 (2014)

8. Karapınar, E: A discussion on ‘α-ψ-Geraghty contraction type mappings’. Filomat28(4), 761-766 (2014) 9. Karapınar, E:α-ψ-Geraghty contraction type mappings and some related ﬁxed point results. Filomat28(1), 37-48

(2014)

10. Samet, B, Vetro, C, Vetro, P: Fixed point theorem forα-ψcontractive type mappings. Nonlinear Anal.75, 2154-2165 (2012)

11. Berzig, M, Chandok, S, Khan, MS: Generalized Krasnoselskii ﬁxed point theorem involving auxiliary functions in bimetric spaces and application to two-point boundary value problem. Appl. Math. Comput.248, 323-327 (2014) 12. Berzig, M, Rus, MD: Fixed point theorems forα-contractive mappings of Meir-Keeler type and applications. Nonlinear

Anal., Model. Control19(2), 178-198 (2014)

13. Chandok, S, Choudhury, BS, Metiya, N: Some ﬁxed point results in ordered metric spaces for rational type expressions with auxiliary functions. J. Egypt. Math. Soc. (2014). doi:10.1016/j.joems.2014.02.002

14. Amini-Harandi, A, Emami, H: A ﬁxed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary diﬀerential equations. Nonlinear Anal.72, 2238-2242 (2010)

15. Jachymski, J: Equivalent conditions for generalized contractions on (ordered) metric spaces. Nonlinear Anal.74, 768-774 (2011)

16. Mohammadi, B, Rezapour, S, Shahzad, N: Some results on ﬁxed points ofα-ψ-Ciric generalized multifunctions. Fixed Point Theory Appl.2013, 24 (2013)

17. Kutbi, MA, Chandok, S, Sintunavarat, W: Optimal solutions for nonlinear proximalCN-contraction mapping in metric

space. J. Inequal. Appl.2014, 193 (2014)

18. Wilson, WA: On semimetric spaces. Am. J. Math.53(2), 361-373 (1931)

19. Chandok, S, Narang, TD, Taoudi, MA: Some common ﬁxed point results in partially ordered metric spaces for generalized rational type contraction mappings. Vietnam J. Math.41(3), 323-331 (2013)

20. Kirk, WA, Shahzad, N: Generalized metrics and Caristi’s theorem. Fixed Point Theory Appl.2013, Article ID 129 (2013) 21. Kutbi, MA, Sintunavarat, W: Fixed point theorems for generalizedwα-contraction multivalued mappings in

α-complete metric spaces. Fixed Point Theory Appl.2014, 139 (2014)

22. Kumam, P, Sintunavarat, W: The existence of ﬁxed point theorems for partialq-set valued quasi-contractions in

b-metric spaces and related results. Fixed Point Theory Appl.2014, 226 (2014)

23. Latif, A, Roldan, A, Sintunavarat, W: On commonα-fuzzy ﬁxed points with applications. Fixed Point Theory Appl.

2014, 234 (2014)

24. Yamaod, O, Sintunavarat, W: Some ﬁxed point results for generalized contraction mappings with cyclic α-β-admissible mapping in multiplicative metric spaces. J. Inequal. Appl.2014, 448 (2014)

25. Kutbi, MA, Sintunavarat, W: On new ﬁxed point results forα,ψ,ξ-contractive multi-valued mappings onα-complete metric spaces and their consequences. Fixed Point Theory Appl.2015, 2 (2015)

26. Latif, A, Sintunavarat, W, Ninsri, A: Approximate ﬁxed point theorems for partial generalized convex contraction mappings inα-complete metric spaces. Taiwan. J. Math.19(1), 315-333 (2015)

27. La Rosa, V, Vetro, P: Common ﬁxed points forα,ψ,φ-contractions in generalized metric spaces. Nonlinear Anal., Model. Control19(1), 43-54 (2014)