R E S E A R C H
Open Access
Fixed point of
α
-ψ
-contractive type
mappings in uniform spaces
Muhammad Usman Ali
1, Tayyab Kamran
2and Erdal Karapınar
3,4**Correspondence:
[email protected] 3Department of Mathematics,
Atilim University, Incek, Ankara 06836, Turkey
4Nonlinear Analysis and Applied
Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, Saudi Arabia
Full list of author information is available at the end of the article
Abstract
In this paper, we shall investigate the existence and uniqueness of a fixed point of
α
-ψ
-contractive mappings in the context of uniform spaces. We shall also prove some common fixed point theorems by introducing the notion ofα
-admissible pairs. We shall construct some examples to support our novel results.MSC: 46T99; 47H10; 54H25
Keywords:
α
-ψ
-contractive;α
-admissible maps1 Introduction
One of the interesting metric fixed point results was given by Sametet al.[] by intro-ducing the notions ofα-admissible andα-ψ-contractive type mappings. They reported results via these new notions, and they extended and unified most of the related existing metric fixed point results in the literature. In particular, the authors [] showed that fixed point results via cyclic contractions are consequences of their related results. Naturally, many authors have started to investigate the existence and uniqueness of a fixed point theorem via admissible mappings and variations of the concept ofα-ψ-contractive type mappings, for reference see [–]. The notion of cyclic contraction was introduced by Kirket al.[]. The main advantage of the cyclic contraction is that the given mapping does not need to be continuous. It has been appreciated by several authors; seee.g.[–] and related references therein.
In this paper, we shall consider the characterization of the notions ofα-ψ-contractive andα-admissible mappings in the context uniform spaces. Further, we shall prove some fixed point theorems by using these concepts. We shall also useα-admissible pairs to in-vestigate the existence and uniqueness of a common fixed point in the setting of uniform spaces. We shall also establish some examples to illustrate the main results.
For the sake of completeness, we shall recollect some basic definitions and fundamental results. LetXbe a nonempty set. A nonempty family,ϑ, of subsets ofX×Xis called a uniform structure ofXit satisfies the following properties:
(i) ifGis inϑ, thenGcontains the diagonal{(x,x)|x∈X};
(ii) ifGis inϑandHis a subset ofX×Xwhich containsG, thenHis inϑ; (iii) ifGandHare inϑ, thenG∩His inϑ;
(iv) ifGis inϑ, then there existsHinϑ, such that, whenever(x,y)and(y,z)are inH, then(x,z)is inG;
(v) ifGis inϑ, then{(y,x)|(x,y)∈G}is also inϑ.
The pair (X,ϑ) is called a uniform space and the element ofϑis called entourage or neigh-borhood or surrounding. The pair (X,ϑ) is called a quasiuniform space (seee.g.[, ]) if property (v) is omitted.
Let={(x,x)|x∈X}be the diagonal of a nonempty setX. ForV,W∈X×X, we shall use the following setting in the sequel:
V◦W=(x,y)|there existz∈X: (x,z)∈Wand (z,y)∈V
and
V–=(x,y)|(y,x)∈V.
For a subsetV∈ϑ, a pair of pointsxandyare said to beV-close if (x,y)∈Vand (y,x)∈V. Moreover, a sequence{xn}inXis called a Cauchy sequence forϑ, if, for anyV∈ϑ, there existsN≥ such thatxnandxmareV-close forn,m≥N. For (X,ϑ), there is a unique topologyτ(ϑ) onXgenerated byV(x) ={y∈X|(x,y)∈V}whereV∈ϑ.
A sequence{xn} inXis convergent toxforϑ, denoted bylimn→∞xn=x, if, for any V∈ϑ, there existsn∈Nsuch thatxn∈V(x) for everyn≥n. A uniform space (X,ϑ) is
called Hausdorff if the intersection of all theV∈ϑis equal toofX, that is, if (x,y)∈V for allV∈ϑimpliesx=y. IfV=V–then we shall say that a subsetV∈ϑis symmetrical. Throughout the paper, we shall assume that eachV∈ϑis symmetrical. For more details, seee.g.[, –].
Now, we shall recall the notions ofA-distance andE-distance.
Definition .[, ] Let (X,ϑ) be a uniform space. A functionp:X×X→[,∞) is said to be anA-distance if, for anyV ∈ϑ, there existsδ> such that ifp(z,x)≤δand p(z,y)≤δfor somez∈X, then (x,y)∈V.
Definition .[, ] Let (X,ϑ) be a uniform space. A functionp:X×X→[,∞) is said to be anE-distance if
(i) pis anA-distance,
(ii) p(x,y)≤p(x,z) +p(z,y),∀x,y,z∈X.
Example .[, ] Let (X,ϑ) be a uniform space and letdbe a metric onX. It is evident that (X,ϑd) is a uniform space whereϑdis a set of all subsets ofX×Xcontaining a ‘band’ U ={(x,y)∈X|d(x,y) <}for some> . Moreover, ifϑ⊆ϑd, thendis anE-distance on (X,ϑ).
Lemma .[, ] Let(X,ϑ)be a Hausdorff uniform space and p be an A-distance on X. Let{xn}and{yn}be sequences in X and{αn},{βn}be sequences in[,∞)converging to. Then,for x,y,z∈X,the following results hold:
(a) Ifp(xn,y)≤αnandp(xn,z)≤βnfor alln∈N,theny=z.In particular,ifp(x,y) = andp(x,z) = ,theny=z.
(b) Ifp(xn,yn)≤αnandp(xn,z)≤βnfor alln∈N,then{yn}converges toz.
(c) Ifp(xn,xm)≤αnfor alln,m∈Nwithm>n,then{xn}is a Cauchy sequence in(X,ϑ).
Definition .[, ] Let (X,ϑ) be a uniform space andpbe anA-distance onX.
(i) XisS-complete if, for everyp-Cauchy sequence{xn}, there existsxinXwith
limn→∞p(xn,x) = .
(ii) Xisp-Cauchy complete if, for everyp-Cauchy sequence{xn}, there existsxinX withlimn→∞xn=xwith respect toτ(ϑ).
(iii) T:X→Xisp-continuous iflimn→∞p(xn,x) = implieslimn→∞p(T(xn),T(x)) = .
Remark . Let (X,ϑ) be a Hausdorff uniform space which isS-complete. If a sequence
{xn}be ap-Cauchy sequence, then we havelimn→∞p(xn,x) = . Regarding Lemma .(b), we derivelimn→∞xn=xwith respect to the topologyτ(ϑ), and henceS-completeness impliesp-Cauchy completeness.
Definition .[] LetXbe a nonempty set,ma positive integer andT:X→Xa map-ping.X=mi=Aiis said to be a cyclic representation ofXwith respect toTif
(i) Ai,i= , , . . . ,mare nonempty sets; (ii) T(A)⊂A, . . . ,T(Am–)⊂Am,T(Am)⊂A.
2 Main results
Letbe the family of functionsψ: [,∞)→[,∞) satisfying the following conditions:
() ψis nondecreasing; ()
+∞
n=ψn(t) <∞for allt> , whereψnis thenth iterate ofψ.
These functions are known in the literature as (c)-comparison functions. It is easily proved that ifψis a (c)-comparison function, thenψ(t) <tfor anyt> .
Definition .[] LetT:X→Xandα:X×X→[,∞). We shall say thatTisα -admis-sible if, for allx,y∈X, we have
α(x,y)≥ ⇒ α(Tx,Ty)≥.
We shall characterize the notion ofα-ψ-contractive mapping, introduced by Sametet al.[], in the context of uniform space as follows.
Definition . Let (X,ϑ) be a uniform space such thatpis anE-distance onXandT : X→Xbe a given mapping. We shall say thatT is anα-ψ-contractive mapping if there exist two functionsα:X×X→[,∞) andψ∈such that
α(x,y)p(Tx,Ty)≤ψp(x,y), for allx,y∈X. (.)
Theorem . Let (X,ϑ) be a S-complete Hausdorff uniform space such that p be an E-distance on X.Let T:X→X be anα-ψ-contractive mapping satisfying the following conditions:
(i) Tisα-admissible;
(ii) there existsx∈Xsuch thatα(x,Tx)≥andα(Tx,x)≥; (iii) Tisp-continuous.
Proof By hypothesis (ii) of the theorem we havex∈Xsuch thatα(x,Tx)≥. Define the
sequence{xn}inXbyxn+=Txnfor alln∈N∪ {}. Ifxn=xn+for somen, thenu=xn
is a fixed point ofT. So, we can assume thatxn=xn+for alln. SinceTisα-admissible, we
have
α(x,x) =α(x,Tx)≥ ⇒ α(Tx,Tx) =α(x,x)≥.
Inductively, we have
α(xn,xn+)≥, for alln∈N∪ {}. (.)
From (.) and (.), it follows that, for alln∈N, we have
p(xn+,xn) =p(Txn,Txn–)≤α(xn,xn–)p(Txn,Txn–)≤ψ
p(xn,xn–)
. (.)
Iteratively, we derive
p(xn,xn+)≤ψn
p(x,x)
, for alln∈N.
Sincepis anE-distance, form>n, we have
p(xn,xm)≤p(xn,xn+) +· · ·+p(xm–,xm)
≤ψnp(x,x)
+ψn+p(x,x)
+· · ·+ψm–p(x,x)
. (.)
To show that{xn}is ap-Cauchy sequence, consider
Sn= n
k=
ψkp(x,x)
.
Thus from (.) we have
p(xn,xm)≤Sm––Sn–. (.)
Sinceψ∈, there existsS∈[,∞) such thatlimn→∞Sn=S. Thus by (.) we have
lim
n,m→∞p(xn,xm) = . (.)
Sincepis not symmetrical, by repeating the same argument we have
lim
n,m→∞p(xm,xn) = . (.)
Hence the sequence{xn}is ap-Cauchy in theS-complete spaceX. Thus, there existsu∈ Xsuch thatlimn→∞p(xn,u) = , which implieslimn→∞xn=u. SinceT isp-continuous, we havelimn→∞p(Txn,Tu) = , which implies thatlimn→∞(xn+,Tu) = . Hence we have
limn→∞p(xn,u) = andlimn→∞(xn,Tu) = . Thus by Lemma .(a) we haveu=Tu.
Theorem . Let (X,ϑ) be a S-complete Hausdorff uniform space such that p is an E-distance on X.Let T:X→X be anα-ψ-contractive mapping satisfying the following conditions:
(i) Tisα-admissible;
(ii) there existsx∈Xsuch thatα(x,Tx)≥andα(Tx,x)≥;
(iii) for any sequence{xn}inXwithxn→xasn→ ∞andα(xn,xn+)≥for each
n∈N∪ {},thenα(xn,x)≥for eachn∈N∪ {}. Then T has a fixed point u∈X.
Proof By following the proof of Theorem ., we know that{xn} is ap-Cauchy in the S-complete spaceX. Thus, there existsu∈Xsuch thatlimn→∞p(xn,u) = , which implies
limn→∞xn=u. By using (.) and assumption (iii), we get
p(xn,Tu)≤p(xn,xn+) +p(xn+,Tu)
≤p(xn,xn+) +α(xn,u)p(Txn,Tu) ≤p(xn,xn+) +ψ
p(xn,u)
.
Lettingn→ ∞in above inequality, we shall havelimn→∞p(xn,Tu) = . Hence we have
limn→∞p(xn,u) = andlimn→∞p(xn,Tu) = . Thus by Lemma .(a) we haveu=Tu.
Example . LetX={n :n∈N} ∪ {}be endowed with the usual metricd. Defineϑ=
{U|> }. It is easy to see that (X,ϑ) is a uniform space. DefineT:X→Xby
Tx= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ifx= ,
n+ ifx= n:n> , ifx= ,
(.)
andα:X×X→[,∞) by
α(x,y) = ⎧ ⎨ ⎩
ifx,y∈X–{},
otherwise, (.)
andψ(t) =t
for allt≥. One can easily see thatTisα-ψ-contractive andα-admissible
mapping. Also forx= we haveα(x,Tx) =α(Tx,x) = . Moreover, for any sequence {xn}inXwithxn→xasn→ ∞andα(xn–,xn) = for eachn∈Nwe haveα(xn,x) = for eachn∈N. Therefore by Theorem .,Thas a fixed point.
In the sequel, we shall investigate the uniqueness of a fixed point. For this purpose, we shall introduce the following condition.
(H) For allx,y∈Fix(T), there existsz∈Xsuch thatα(z,x)≥andα(z,y)≥.
Here,Fix(T) denotes the set of fixed points ofT.
The following theorem guarantees the uniqueness of a fixed point.
Proof Suppose, on the contrary, thatv∈Xis another fixed point ofT. From (H), there existsz∈Xsuch that
α(z,u)≥ and α(z,v)≥. (.)
Owing to the fact thatTisα-admissible, from (.), we have
αTnz,u≥ and αTnz,v≥, for alln∈N∪ {}. (.)
We define the sequence{zn}inXbyzn+=Tzn=Tnzfor alln∈N∪ {}andz=z. From
(.) and (.), we have
p(zn+,u) =p(Tzn,Tu)≤α(zn,u)p(Tzn,Tu)≤ψ
p(zn,u)
, (.)
for alln∈N∪ {}. This implies that
p(zn,u)≤ψn
p(z,u)
, for alln∈N.
Lettingn→ ∞in the above inequality, we obtain
lim
n→∞p(zn,u) = . (.)
Similarly,
lim
n→∞p(zn,v) = . (.)
From (.) and (.) together with Lemma .(a), it follows that u=v. Thus we have proved thatuis the unique fixed point ofT.
Definition .[] A pair of two self-mappingsT,S:X→Xis said to beα-admissible, if, for anyx,y∈Xwithα(x,y)≥, we haveα(Tx,Sy)≥ andα(Sx,Ty)≥.
Definition . Let (X,ϑ) be a uniform space. A pair of two self-mappingsT,S:X→Xis said to be anα-ψ-contractive pair if
α(x,y)maxp(Tx,Sy),p(Sx,Ty)≤ψp(x,y), (.)
for anyx,y∈X, whereψ∈.
Theorem . Let (X,ϑ) be a S-complete Hausdorff uniform space such that p is an E-distance on X.Suppose that the pair of T,S:X→X is anα-ψ-contractive pair satis-fying the following conditions:
(i) (T,S)isα-admissible;
(ii) there existsx∈Xsuch thatα(x,Tx)≥andα(Tx,x)≥;
(iii) for any sequence{xn}inXwithxn→xasn→ ∞andα(xn,xn+)≥for each
Proof By hypothesis (ii) of the theorem, we havex∈X such that α(x,Tx)≥ and
α(Tx,x)≥. Since (T,S) is anα-admissible pair, we can construct a sequence such that
Txn=xn+, Sxn+=xn+ and
α(xn,xn+)≥, α(xn+,xn)≥, for alln∈N∪ {}.
From (.) for alln∈N∪ {}, we have
p(xn+,xn+) =p(Txn,Sxn+)
≤α(xn,xn+)max
p(Txn,Sxn+),p(Sxn,Txn+)
≤ψp(xn,xn+)
.
Hence, we conclude that
p(xn+,xn+)≤ψ
p(xn,xn+)
. (.)
Similarly, we find that
p(xn+,xn+) =p(Sxn+,Txn+)
≤α(xn+,xn+)max
p(Txn+,Sxn+),p(Sxn+,Txn+)
≤ψp(xn+,xn+)
.
Hence, we derive
p(xn+,xn+)≤ψ
p(xn+,xn+)
. (.)
Thus from (.) and (.), and by induction, we get
p(xn,xn+)≤ψn
p(x,x)
, for alln∈N. (.)
We shall show that{xn}is ap-Cauchy sequence, Sincepis anE-distance, form>n, we have
p(xn,xm)≤p(xn,xn+) +· · ·+p(xm–,xm)
≤ψnp(x,x)
+ψn+p(x,x)
+· · ·+ψm–p(x,x)
. (.)
Now, we shall consider
Sn= n
k=
ψkp(x,x)
.
Thus, from (.) we have
Sinceψ∈, there existsS∈[,∞) such thatlimn→∞Sn=S. Thus, by (.) we have
lim
n,m→∞p(xn,xm) = . (.)
Sincepis not symmetrical, by repeating the same argument we have
lim
n,m→∞p(xm,xn) = . (.)
Hence the sequence{xn}isp-Cauchy in theS-complete spaceX. Thus, there existsu∈X such thatlimn→∞p(xn,u) = , which implieslimn→∞Txn=limn→∞Sxn+=u. By using
(.) and assumption (iii), we get
p(xn,Tu)≤p(xn,xn+) +p(xn+,Tu)
=p(xn,xn+) +p(Sxn+,Tu)
≤p(xn,xn+) +α(xn+,u)max
p(Txn+,Su),p(Sxn+,Tu)
≤p(xn,xn+) +ψ
p(xn+,u)
. (.)
Lettingn→ ∞in (.), we havep(xn,Tu) = . Hence we havelimn→∞p(xn,u) = and
limn→∞p(xn,Tu) = . Thus by Lemma .(a) we haveu=Tu. Analogously, one can derive
u=Su. Thereforeu=Tu=Su.
Remark . Note that Theorem . is valid if one replaces condition (ii) with
(ii) there existsx∈Xsuch thatα(x,Sx)≥andα(Sx,x)≥.
We shall get the following result by lettingS=Tin Theorem ..
Corollary . Let (X,ϑ) be a S-complete Hausdorff uniform space such that p is an E-distance on X.Suppose that a mapping T:X→X is satisfying the condition
α(x,y)maxp(Tx,y),p(x,Ty)≤ψp(x,y),
for any x,y∈X,whereψ∈.Also suppose that the following conditions are satisfied:
(i) Tisα-admissible;
(ii) there existsx∈Xsuch thatα(x,Tx)≥andα(Tx,x)≥;
(iii) for any sequence{xn}inXwithxn→xasn→ ∞andα(xn,xn+)≥for each
n∈N∪ {},thenα(xn,x)≥for eachn∈N∪ {}. Then T has a fixed point.
Example . Let (X,d) is a dislocated metric space whereX={
n :n∈N} ∪ {} and d(x,y) =max{x,y}. Defineϑ={U|> }, whereU={(x,y)∈X:d(x,y) <d(x,x) +}. It is
easy to see that (X,ϑ) is a uniform space. DefineT:X→Xby
Tx= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ifx= ,
n+ ifx= n:n> ,
ifx= ,
andS:X→Xby
Sx= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ifx= ,
n ifx= n:n> , ifx= ,
(.)
andα:X×X→[,∞) by
α(x,y) = ⎧ ⎨ ⎩
ifx,y∈X–{},
otherwise,
(.)
and ψ(t) = t
for all t≥. One can easily see that (T,S) is an α-ψ-contractive and
α-admissible pair. Also forx= we haveα(x,Tx) =α(Tx,x) = . Moreover, for any
sequence{xn}inXwithxn→xasn→ ∞andα(xn,xn+)≥ for eachn∈N∪ {}, we
haveα(xn,x)≥ for eachn∈N. Therefore by Theorem .,TandShave a common fixed point.
To investigate the uniqueness of a common fixed point, we shall introduce the following condition.
(I) For eachx,y∈CFix(T,S), we haveα(x,y)≥, whereCFix(T,S)is the set of all common fixed points ofTandS.
Theorem . Adding the condition(I)to the hypothesis of Theorem.,we obtain the uniqueness of the common fixed point of T and S.
Proof On the contrary suppose thatu,v∈Xare two distinct common fixed points ofT andS. From (I) and (.) we have
p(u,v)≤α(u,v)maxp(Tu,Sv),p(Su,Tv)≤ψp(u,v)<p(u,v),
which is impossible forp(u,v) > . Consequently, we havep(u,v) = . Analogously, one can show thatp(u,u) = . Thus we haveu=v, which is a contradiction to our assumption. HenceTandShave a unique common fixed point.
3 Consequences
3.1 Standard contractions on uniform space
Taking in Theorem .,α(x,y) = for allx,y∈X, we shall obtain immediately the following fixed point theorems.
Corollary . Let (X,ϑ) be a S-complete Hausdorff uniform space such that p is an E-distance on X and T:X→X be a given mapping.Suppose that there exists a function ψ∈such that
p(Tx,Ty)≤ψp(x,y),
By substitutingψ(t) =kt, wherek∈[, ), in Corollary ., we shall get the following.
Corollary . Let (X,ϑ) be a S-complete Hausdorff uniform space such that p is an E-distance on X.Suppose that T:X→X is a given mapping satisfying
p(Tx,Ty)≤kp(x,y),
for all x,y∈X.Then T has a unique fixed point.
Taking in Theorem .α(x,y) = for allx,y∈X, we shall obtain immediately the fol-lowing common fixed point theorem.
Corollary . Let (X,ϑ) be a S-complete Hausdorff uniform space such that p is an
E-distance on X and T,S:X→X are given mappings.Suppose that there exists a func-tionψ∈such that
maxp(Tx,Sy),p(Sx,Ty)≤ψp(x,y),
for all x,y∈X.Then T and S have a unique common fixed point.
Corollary . Let (X,ϑ) be a S-complete Hausdorff uniform space such that p is an E-distance on X and T:X→X be a given mapping.Suppose that there exists a function ψ∈such that
maxp(Tx,y),p(x,Ty)≤ψp(x,y),
for all x,y∈X.Then T has a unique fixed point.
3.2 Cyclic contraction on uniform space
Corollary . Let (X,ϑ) be a S-complete Hausdorff uniform space such that p is an E-distance on X and A, A are nonempty closed subsets of X with respect to the
topo-logical space(X,τ(ϑ)).Let T:Y→Y be a mapping,where Y=i=Ai.Suppose that the following conditions hold:
(i) T(A)⊆AandT(A)⊆A;
(ii) there exists a functionψ∈such that
p(Tx,Ty)≤ψp(x,y), for all(x,y)∈A×A.
Then T has a unique fixed point that belongs to A∩A.
Proof SinceAandAare closed subsets ofX, (Y,d) is anS-complete Hausdorff uniform
space. Define the mappingα:Y×Y→[,∞) by
α(x,y) = ⎧ ⎨ ⎩
if (x,y)∈(A×A)∪(A×A),
From (ii) and the definition ofα, we can write
α(x,y)p(Tx,Ty)≤ψp(x,y),
for allx,y∈Y. ThusTis anα-ψ-contractive mapping.
Let (x,y)∈Y×Y such thatα(x,y)≥. If (x,y)∈A×A, from (i), (Tx,Ty)∈A×A,
which implies thatα(Tx,Ty)≥. If (x,y)∈A×A, from (i), (Tx,Ty)∈A×A, which
implies thatα(Tx,Ty)≥. Thus in all cases, we haveα(Tx,Ty)≥. This implies thatTis α-admissible.
Also, from (i), for anya∈A, we have (a,Ta)∈A×A, which implies thatα(a,Ta)≥.
Now, let{xn}be a sequence inXsuch thatα(xn,xn+)≥ for allnandxn→x∈Xas n→ ∞. This implies from the definition ofαthat
(xn,xn+)∈(A×A)∪(A×A), for alln.
Since (A×A)∪(A×A) is a closed subset ofXwith respect to the topological space
(X,τ(ϑ)), we get
(x,x)∈(A×A)∪(A×A),
which implies that x∈A ∩A. Thus we can easily get from the definition of α that
α(xn,x)≥ for alln.
Finally, letx,y∈Fix(T). From (i), this implies thatx,y∈A∩A. So, for anyz∈Y, we
haveα(z,x)≥ andα(z,y)≥. Thus condition (H) is satisfied.
Now, all the hypotheses of Theorem . are satisfied, and we deduce thatThas a unique fixed point that belongs toA∩A(from (i)).
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this article.
Authors’ contributions
All authors have equally contributed in writing this article. All authors have read and approved the final manuscript.
Author details
1Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, H-12, Islamabad, Pakistan. 2Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan.3Department of Mathematics, Atilim University, Incek, Ankara 06836, Turkey.4Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, Saudi Arabia.
Acknowledgements
The authors thank the anonymous referees for their remarkable comments, suggestions, and ideas that helped to improve this paper.
Received: 1 April 2014 Accepted: 13 June 2014 Published:22 Jul 2014
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