On α-ψ-Meir-Keeler contractive mappings

Karapınar et al. Fixed Point Theory and Applications 2013, 2013:94http://www.fixedpointtheoryandapplications.com/content/2013/1/94

RESEARCH Open Access

On α-ψ-Meir-Keeler contractive mappingsErdal Karapınar1, Poom Kumam2* and Peyman Salimi3

*Correspondence:[email protected];[email protected] of Mathematics,Faculty of Science, King Mongkut’sUniversity of Technology Thonburi,Bangkok, 10140, ThailandFull list of author information isavailable at the end of the article

AbstractIn this paper, we introduce the notion of α-ψ -Meir-Keeler contractive mappings via atriangular α-admissible mapping. We discuss the existence and uniqueness of a fixedpoint of such a mapping in the setting of complete metric spaces. We state a numberof examples to illustrate our results.MSC: 46N40; 47H10; 54H25; 46T99

Keywords: Meir-Keeler contractive mappings; triangular α-admissible mappings;fixed points

1 Introduction and preliminariesFixed-point theory is one of the most intriguing research fields in nonlinear analysis. Thenumber of authors have published papers and have increased continuously in the lastdecades. Themain reason for this involvement can be observed easily: Application poten-tial. Fixed point theory has an application in many disciplines such as chemistry, physics,biology, computer science and many branches of mathematics. Banach contraction map-ping principle or Banach fixed-point theorem is the most celebrated and pioneer result inthis direction: In a complete metric space, each contraction mapping has a unique fixedpoint. Following Banach [], many authors give various generalizations of this principle invarious space (see e.g. [–]). One of the interesting results was given by Samet et al. []by defining α-ψ-contractive mappings via admissible mappings, see also [].In this paper, we introduce an α-ψ-Meir-Keeler contractive mapping in the setting of

completemetric spaces via a triangularα-admissiblemapping.Weprove the existence anduniqueness of a fixed point of such a mapping. We also consider a number of examples toillustrate our results.

Definition Let f : X → X and α : X × X → (–∞, +∞). We say that f is a triangularα-admissible mapping if(T) α(x, y)≥ implies α(fx, fy) ≥ , x, y ∈ X ,(T)

{α(x, z) ≥ ,α(z, y) ≥ , implies α(x, y)≥ , x, y, z ∈ X .

Example Let X =R, fx = √x and α(x, y) = ex–y then f is a triangular α-admissible map-ping. Indeed, if α(x, y) = ex–y ≥ then x ≥ y which implies fx ≥ fy. That is, α(fx, fy) =efx–fy ≥ . Also, if

{α(x, z) ≥ ,α(z, y) ≥ then

{x – z ≥ ,z – y ≥ . That is, x – y≥ and so α(x, y) = ex–y ≥ .

Example Let X = R, fx = ex and α(x, y) = √x – y + . Hence, f is a triangular α-admissible mapping. Again, if α(x, y) = √x – y + ≥ then x ≥ y which implies fx ≥ fy.That is, α(fx, fy)≥ .

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Moreover, if{

α(x, z) ≥ ;α(z, y) ≥ , then x – y≥ , and hence, α(x, y)≥ .

Example Let X = [,+∞), fx = x + ln(x + ) and

α(x, y) = x

+ x–

y

y + + .

Then f is a triangular α-admissible mapping. In fact, if

α(x, y) = x

+ x–

y

y + + ≥

then x ≥ y. Hence, fx ≥ fy. That is, α(fx, fy) ≥ . Also,

α(x, z) + α(z, y) = x

+ x–

z

z + + +

z

+ z–

y

y + +

=x

+ x–

y

y + + ≤

(x

+ x–

y

y + +

)= α(x, y).

Thus, α(x, z) + α(z, y) ≤ α(x, y). Now, if{

α(x, z) ≥ ;α(z, y) ≥ , then α(x, y)≥ .

Example Let X = R, fx = x + √x and α(x, y) = x – y + . Then f is a triangular α-admissible mapping.

Example Let X = [,+∞), fx = x + ex and

α(x, y) =

⎧⎨⎩, if x, y ∈ [, ],, otherwise.

Hence, f is a triangular α-admissible mapping.

Lemma Let f be a triangular α-admissible mapping. Assume that there exists x ∈ Xsuch that α(x, fx) ≥ . Define sequence {xn} by xn = f nx. Then

α(xm,xn) ≥ for all m,n ∈N with m < n.

Proof Since there exist x ∈ X such that α(x, fx) ≥ then from (T), we deduce thatα(x,x) = α(fx, f x)≥ . By continuing this process, we get

α(xn,xn+) ≥ for all n ∈ N∪ . (.)

Suppose thatm < n. Since{

α(xm ,xm+)≥ ,α(xm+,xm+)≥ , then from (T) we have α(xm,xm+) ≥ .

Again, since{

α(xm ,xm+) ≥ ,α(xm+,xm+) ≥ , then we deduce α(xm,xm+) ≥ .

By continuing this process, we get α(xm,xn) ≥ . �

Denotewith� the family of nondecreasing functionsψ : [, +∞) → [, +∞) continuousin t = such that• ψ(t) = if and only if t = ,• ψ(t + s) ≤ ψ(t) +ψ(s).



2 Main resultsDefinition Let (X,d) be a metric space and ψ ∈ � . Suppose that f : X → X is a trian-gular α-admissible mapping satisfying the following condition: for each ε > there existsδ > such that

ε ≤ ψ(d(x, y)

)< ε + δ implies α(x, y)ψ

(d(fx, fy)

)< ε (.)

for all x, y ∈ X. Then f is called an α-ψ-Meir-Keeler contractive mapping.

Remark Let f be an α-ψ-Meir-Keeler contractive mapping. Then

α(x, y)ψ(d(fx, fy)

)< ψ

(d(x, y)

)

for all x, y ∈ X when x = y. Also, if x = y then d(fx, fy) = . i.e.,

α(x, y)ψ(d(fx, fy)

) ≤ ψ(d(x, y)

)

for all x, y ∈ X.

Theorem Let (X,d) be a complete metric space. Suppose that f is a continuous α-ψ-Meir-Keeler contractive mapping and that there exists x ∈ X such that α(x, fx) ≥ , thenf has a fixed point.

Proof Let x ∈ X and define a sequence {xn} by xn = f nx for all n ∈ N. If xn = xn+ forsome n ∈ N∪ {}, then obviously f has a fixed point. Hence, we suppose that

xn = xn+ (.)

for all n ∈ N∪{}.We have d(xn,xn+) > for all n ∈N∪{}. Nowdefine sn = ψ(d(xn,xn+)).By Remark , we deduce that for all n ∈ N∪ {},

α(xn,xn+)ψ(d(xn+,xn+)

)= α(xn,xn+)ψ

(d(fxn, fxn+)

)< ψ

(d(xn,xn+)

).

Then by applying Lemma

ψ(d(xn+,xn+)

)< ψ

(d(xn,xn+)

).

Hence, the sequence {sn} is decreasing in R+ and so it is convergent to s ∈ R+. We willshow that s = . Suppose, to the contrary, that s > . Hence, we have

< s < ψ(d(xn,xn+)

)for all n ∈N∪ {}. (.)

Let ε = s > . Then by hypothesis, there exists a δ(ε) > such that (.) holds. On the otherhand, by the definition of ε, there exists n ∈N such that

ε < sn = ψ(d(xn ,xn+)

)< ε + δ.



Now by (.), we have

sn+ = ψ(d(xn+,xn+)

) ≤ α(xn ,xn+)ψ(d(xn+,xn+)

)= α(xn ,xn+)ψ

(d(fxn , fxn+)

)< ε = s

which is a contradiction. Hence, s = , that is, limn→+∞ sn = . Now, by the continuity of ψin t = , we have limn→+∞ d(xn,xn+) = . For given ε > , by the hypothesis, there exists aδ = δ(ε) > such that (.) holds. Without loss of generality, we assume δ < ε. Since s = ,then there exists N ∈ N such that

sn– = ψ(d(xn–,xn)

)< δ, for all n ≥ N . (.)

We will prove that for any fixed k ≥ N,

ψ(d(xk ,xk+l)

) ≤ ε, for all l ∈N, (.)

holds. Note that (.), by (.), holds for l = . Suppose the condition (.) is satisfied forsomem ∈N. For l =m + , by (.), we get

ψ(d(xk–,xk+m)

) ≤ ψ(d(xk–,xk) + d(xk ,xk+m)

)≤ ψ

(d(xk–,xk)

)+ψ

(d(xk ,xk+m)

)< ε + δ. (.)

If ψ(d(xk–,xk+m)) ≥ ε, then by (.) we get

ψ(d(xk ,xk+m+)

) ≤ α(xk–,xk+m)ψ(d(xk ,xk+m+)

)= α(xk ,xk+m+)ψ

(d(fxk–, fxk+m)

)< ε

and hence (.) holds.If ψ(d(xk–,xk+m)) < ε, by Remark , we get

ψ(d(xk ,xk+m+)

) ≤ α(xk–,xk+m)ψ(d(xk ,xk+m+)

) ≤ ψ(d(xk–,xk+m)

)< ε.

Consequently, (.) holds for l =m+ . Hence, ψ(d(xk ,xk+l)) ≤ ε for all k ≥ N and l ≥ ,which means

d(xn,xm) < ε, for allm ≥ n≥ N. (.)

Hence {xn} is a Cauchy sequence. Since (X,d) is complete, there exists z ∈ X such thatxn → z as n→ ∞. Now, since, f is continuous then

fz = f(

limn→∞xn

)= lim

n→∞xn+ = z,

that is, f has a fixed point. �



Theorem Let (X,d) be a complete metric space and let f be a α-ψ-Meir-Keeler con-tractive mapping. If the following conditions hold:

(i) there exists x ∈ X such that α(x, fx) ≥ ,(ii) if {xn} is a sequence in X such that α(xn,xn+) ≥ for all n and xn → x as n→ +∞,

then α(xn,x)≥ for all n.Then f has a fixed point.

Proof Following the proof of the Theorem , we say that α(xn,xn+) ≥ for all n ∈ N∪{}and that there exist z ∈ X such that xn → z as n → +∞. Hence, from (ii) α(xn, z) ≥ . ByRemark , we have

ψ(d(fz, z)

) ≤ ψ(d(fz, fxn) + d(fxn, z)

)≤ ψ

(d(fz, fxn)

)+ψ

(d(fxn, z)

)≤ α(xn, z)ψ

(d(fz, fxn)

)+ψ

(d(fxn, z)

)≤ ψ

(d(z,xn)

)+ψ

(d(xn+, z)

).

By taking limit as n → +∞, in the above inequality, we get ψ(d(fz, z)) ≤ , that is,d(fz, z) = . Hence, fz = z. �

Next, we give some examples to validate our main result.

Example Let X = [,∞) and d(x, y) = |x – y| be a metric on X. Define f : X → X by

fx =

⎧⎨⎩

x , if x ∈ [, ],x, if x ∈ (,∞),

and α(x, y) =

⎧⎨⎩, if x, y ∈ [, ],–, otherwise

and ψ(t) = t. Clearly, (X,d) is a complete metric space. We show that f is a triangular

α-admissible mapping. Let x, y ∈ X, if α(x, y) ≥ then x, y ∈ [, ]. On the other hand, forall x, y ∈ [, ] we have fx ≤ and fy≤ . It follows that α(fx, fy) ≥ . Also, if α(x, z)≥ andα(z, y) ≥ then x, y, z ∈ [, ] and hence, α(x, y)≥ . Thus, the assertion holds by the samearguments. Notice that α(, f ) ≥ .Now, if {xn} is a sequence in X such that α(xn,xn+) ≥ for all n ∈ N ∪ {} and xn → x

as n → +∞, then {xn} ⊂ [, ] and hence x ∈ [, ]. This implies that α(xn,x) ≥ for alln ∈N∪ {}. Let x, y ∈ [, ]. Without loss of generality, take x ≤ y. Then

ψ(d(fx, fy)

)=y

–x

,

ψ(d(x, y)

)=y–x.

Clearly, by taking δ = ε the condition (.) holds. Otherwise, α(x, y) = –. Hence, for givenε > we have α(x, y)ψ(d(fx, fy))≤ < ε. Hence, conditions of Theorem holds and f hasa fixed point. But, if x, y ∈ (,∞) and

ε ≤ d(x, y) < ε + δ,



where ε > and δ > . Then

d(fx, fy) = |x – y| = d(x, y) ≥ d(x, y)≥ ε.

That is, the Meir-Keeler theorem cannot applied for this example.


fx =

⎧⎪⎪⎨⎪⎪⎩

( – x), if x ∈ [, ],√–x

+sin(x) , if x ∈ (, ],√x–x+ , if x ∈ (,∞)

and ψ(t) = t,

α(x, y) =

⎧⎨⎩, if x, y, z ∈ [, ],–, otherwise.

Let x, y ∈ [, ]. Without loss of generality, take x≤ y. Then

ψ(d(fx, fy)

)=(y – x),

ψ(d(x, y)

)=(y – x).

Clearly, by taking δ = ε the condition (.) holds. Otherwise, α(x, y) = –. Hence, for givenε > we have α(x, y)ψ(d(fx, fy))≤ < ε. Hence, conditions of Theorem holds and f hasa fixed point.

Denote with �st the family of strictly nondecreasing functions ψst : [, +∞) → [, +∞)continuous in t = such that• ψst(t) = if and only if t = ,• ψst(t + s)≤ ψst(t) +ψst(s).

Definition Let (X,d) be a metric space and ψst ∈ �st . Suppose that f : X → X is atriangular α-admissible mapping satisfying the following condition: for each ε > thereexists δ > such that

ε ≤ ψst(M(x, y)

)< ε + δ implies α(x, y)ψst

(d(fx, fy)

)< ε (.)

for all x, y ∈ X where

M(x, y) = max

{d(x, y),d(fx,x),d(fy, y),

[d(fx, y) + d(x, fy)

]}.

Then f is called generalized an α-ψst-Meir-Keeler contractive mapping.



Remark Let f be a generalized α-ψst-Meir-Keeler contractive mapping. Then

α(x, y)ψst(d(fx, fy)

)< ψst

(M(x, y)

)

for all x, y ∈ X when M(x, y) > . Also, if M(x, y) = then x = y which implies ψ(d(fx,fy)) = , i.e.,

α(x, y)ψst(d(fx, fy)

) ≤ ψst(M(x, y)

)

for all x, y ∈ X.

Proposition Let (X,d) be a metric space and f : X → X a generalized α-ψst-Meir-Keeler contractive mapping, if there exists x ∈ X such that α(x, fx) ≥ . Thenlimn→∞ d(f n+x, f nx) = .

Proof Define a sequence {xn} by xn = f nx for all n ∈N. If xn = xn+ for some n ∈N∪{},then obviously f has a fixed point. Hence, we suppose that

xn = xn+ (.)

for all n ∈ N ∪ {}. Then we have M(xn+,xn) > for every n ≥ . Then by Lemma andRemark , we have

ψst(d(xn+,xn+)

) ≤ α(xn,xn+)ψst(d(xn+,xn+)

)= α(xn,xn+)ψst

(d(fxn, fxn+)

)< ψst

(M(xn,xn+)

)

= ψst

(max

{d(xn,xn+),d(fxn,xn),d(fxn+,xn+),

[d(fxn,xn+) + d(xn, fxn+)

]})

≤ ψst(max

{d(xn,xn+),d(xn+,xn+)

}).

Now, since ψst is strictly nondecreasing then, we get

d(xn+,xn+) < max{d(xn+,xn),d(xn+,xn+)

}.

Hence, the case where

max{d(xn+,xn),d(xn+,xn+)

}= d(xn+,xn+)

is not possible. Therefore, we deduce that

d(xn+,xn+) < d(xn+,xn) (.)

for all n. That is, {d(xn+,xn)}∞n= is a decreasing sequence in R+ and it converges to ε ∈R+,that is,

limn→∞ψst

(d(xn+,xn)

)= lim

n→∞ψst(M(xn+,xn)

)= ψst(ε). (.)



Notice that ε = inf{p(xn,xn+) : n ∈ N}. Let us prove that ε = . Suppose, to the contrary,that ε > . Then ψ(ε) > . Regarding (.) together with the assumption that f is a gener-alized α-ψst-Meir-Keeler contractive mapping, for ψst(ε), there exists δ > and a naturalnumberm such that

ψst(ε) ≤ ψst(M(xm,xm+)

)< ψst(ε) + δ

implies that

ψst(d(xm+,xm+)

) ≤ α(xm,xm+)ψst(d(xm+,xm+)

)= α(xm,xm+)ψst

(d(fxm, fxm+)

)<ψst(ε).

Now, since ψst is strictly nondecreasing then we get

d(xm+,xm+) < ε

which is a contradiction since ε = inf{p(xn,xn+) : n ∈N}. Then ε = and so limn→∞ d(xn+,xn) = . �

Theorem Let (X,d) be a complete metric space and f : X → X a orbitally contin-uous generalized α-ψst-Meir-Keeler contractive mapping, if there exist x ∈ X such thatα(x, fx) ≥ . Then, f has a fixed point.

Proof Define xn+ = f n+x for all n ≥ . We want to prove that limm,n→∞ d(xn,xm) = . Ifthis is not, then there exist ε > and a subsequence {xn(i)} of {xn} such that

d(xn(i),xn(i+)) > ε. (.)

For this ε > , there exists δ > such that ε ≤ ψst(M(x, y)) < ε + δ implies thatα(x, y)ψst(d(fx, fy)) < ε. Put r = min{ε, δ} and sn = d(xn,xn+) for all n ≥ . From Proposi-tion , there exists n such that

sn = d(xn,xn+) <r

(.)

for all n ≥ n. Let n(i) > n. We get n(i) ≤ n(i + ) – . If d(xn(i),xn(i+)–) ≤ ε + r , then

d(xn(i),xn(i+)) ≤ d(xn(i),xn(i+)–) + d(xn(i+)–,xn(i+))

≤ d(xn(i),xn(i+)–) + d(xn(i+)–,xn(i+))

< ε +r+ sn(i+)– < ε +

r

< ε

which contradicts the assumption (.). Therefore, there are values of k such that n(i) ≤k ≤ n(i + ) and d(xn(i),xk) > ε + r

. Now if d(xn(i),xn(i)+)≥ ε + r , then

sn(i) = d(xn(i),xn(i)+)≥ ε +r> r + r

>r



which is a contradiction with (.). Hence, there are values of k with n(i) ≤ k ≤ n(i + )such that d(xn(i),xk) < ε + r

. Choose smallest integer k with k ≥ n(i) such that d(xn(i),xk) ≥ε + r

. Thus, d(xn(i),xk–) < ε + r and so

d(xn(i),xk) ≤ d(xn(i),xk–) + d(xk–,xk)

≤ d(xn(i),xk–) + d(xk–,xk) < ε +r+r= ε +

r.

Now, we can choose a natural number k satisfying n(i) ≤ k ≤ n(i + ) such that

ε +r

≤ d(xn(i),xk) < ε +r. (.)

Therefore, we obtain

d(xn(i),xk) < ε +r

< ε + r, (.)

d(xn(i),xn(i)+) = dn(i) <r< ε + r, (.)

and

d(xk ,xk+) = dk <r< ε + r. (.)

Thus, we have

[d(xn(i),xk+) + d(xn(i)+,xk)

]

≤ [d(xn(i),xk) + d(xk ,xk+) + d(xn(i)+,xn(i)) + d(xn(i),xk)

]

≤ [d(xn(i),xk) + d(xk ,xk+) + d(xn(i)+,xn(i)) + d(xn(i),xk)

]

= d(xn(i),xk) +[sk + sn(i)]

< ε +r

+

[r+r

]= ε + r. (.)

Now, the inequalities (.)-(.) imply thatM(xn(i),xk) < ε + r ≤ ε + δ and so ψst(M(xn(i),xk)) < ψst(ε + δ) ≤ ψst(ε) + ψst(δ) the fact that f is a generalized α-ψst-Meir-Keeler con-tractive mapping yields that,

d(xn(i)+,xk+)≤ α(xn(i),xk)ψst(d(xn(i)+,xk+)

)<ψst(ε).

Then d(xn(i)+,xk+) < ε. We deduce,

d(f n(i)x, f kx

) ≤ d(f n(i)x, f n(i)+x

)+ d

(f n(i)+x, f kx

)≤ d

(f n(i)x, f n(i)+x

)+ d

(f n(i)+x, f kx

)≤ d

(f n(i)x, f n(i)+x

)+ d

(f n(i)+x, f k+x

)+ d

(f k+x, f kx

).



Hence, from this with (.), (.) and (.), we obtain

d(xn(i)+,xk+) ≥ d(xn(i),xk) – d(xn(i),xn(i)+) – d(xk ,xk+)

> ε +r–r–r= ε,

which is a contradiction. We obtained that limm,n→∞ d(xn,xm) = and so {xn = f nx} isa Cauchy sequence. Since, X is complete, then there exists z ∈ X such that f nx → z asn→ ∞. Now, since f is orbitally continuous, then z = fz. �


fx =

⎧⎨⎩

x , if x ∈ [, ], + lnx, if x ∈ (,∞)

and ψst(t) = t,

α(x, y) =

⎧⎨⎩, if x, y ∈ [, ],, otherwise.

Clearly, f is a triangular α-admissible mapping and orbitally continuous. Let x, y ∈ [, ].Without loss of generality, take x≤ y. Then

ψst(d(fx, fy)

)=y

–x

,

ψst(M(x, y)

)= ψst

(max

{y – x,x –

x

, y –

y

})= max

{y–x,x–x

,y–y

}.

Clearly, by taking δ = ε the condition (.) holds. Otherwise, α(x, y) = . Hence, for givenε > , we have = α(x, y)ψst(d(fx, fy)) < ε. Hence, condition of Theorem is held and fhas a fixed point.

Theorem Assume that all the hypotheses of Theorem ( and ) hold. Adding thefollowing conditions:(iii) for all x = y ∈ X there exists v ∈ X such that α(x, v)≥ and α(v, y)≥ ,

we obtain the uniqueness of the fixed point of f .

Proof Suppose that z and z* are two fixed points of f such that z = z*. Then α(z, v)≥ andα(v, z*) ≥ . Hence, from (T), we have α(z, z*) ≥ . Now, by Remark , we get

d(z, z*

)= d

(fz, fz*

) ≤ α(z, z*

)d(fz, fz*

)< d

(z, z*

)

which is a contradiction and so z = z*. Similarly, for Theorem , we can observe that f hasa unique fixed point. �

We can obtain the following corollaries intermediately.



Corollary Let (X,d) be a completemetric space and f : X → X is self-mapping. Supposethat for each ε > , there exists δ > such that

ε ≤ ψ(d(x, y)

)< ε + δ implies ψ

(d(fx, fy)

)<

ε

L,

where ψ ∈ � and L ≥ . Then f has a unique fixed points.

Corollary Let (X,d) be a complete metric space and f : X → X a orbitally continuousself-mapping. Suppose that for each ε > there exists δ > such that

ε ≤ ψst(M(x, y)

)< ε + δ implies ψst

(d(fx, fy)

)<

ε

L,

where ψst ∈ �st , L ≥ and

M(x, y) = max

{d(x, y),d(fx,x),d(fy, y),

[d(fx, y) + d(x, fy)

]}.

Then f has unique fixed points.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Author details1Department of Mathematics, Atilim University, Incek, Ankara, 06836, Turkey. 2Department of Mathematics, Faculty ofScience, King Mongkut’s University of Technology Thonburi, Bangkok, 10140, Thailand. 3Department of Mathematics,Astara Branch, Islamic Azad University, Astara, Iran.

AcknowledgementsThe authors would like to thank the referees for their valuable comments and suggestions. Also, the second author wouldlike to thank the Commission on Higher Education, the Thailand Research Fund and the King Mongkut’s University ofTechnology Thonburi (Grant No. MRG5580213) for financial support during the preparation of this manuscript. The thirdauthor is thankful for support of Astara Branch, Islamic Azad University, during this research.

Received: 23 December 2012 Accepted: 27 March 2013 Published: 12 April 2013

References1. Banach, S: Sur les opérations dans les ensembles abstraits et leur application aux equations itegrales. Fundam. Math.

3, 133-181 (1922)2. Berinde, V: General constructive fixed point theorems for Ciric-type almost contractions in metric spaces. Carpath. J.

Math. 24(2), 10-19 (2008)3. Boyd, DW, Wong, JSW: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458-464 (1969)4. Chatterjea, SK: Fixed point theorems. C. R. Acad. Bulgare Sci. 25, 727-730 (1972)5. Ciric, L, Samet, B, Aydi, H, Vetro, C: Common fixed points of generalized contractions on partial metric spaces and an

application. Appl. Math. Comput. 218, 2398-2406 (2011)6. Edelstein, M: On fixed points and periodic points under contraction mappings. J. Lond. Math. Soc. 37, 74-79 (1962)7. Hardy, GC, Rogers, T: A generalization of fixed point theorem of S. Reich. Can. Math. Bull. 16, 201-206 (1973)8. Holmes, RD: On fixed and periodic points under sets of mappings. Can. Math. Bull. 12, 813-822 (1969)9. Jaggi, DS: Some unique fixed point theorems. Indian J. Pure Appl. Math. 8(2), 223-230 (1977)

10. Kannan, R: Some results on fixed points-II. Am. Math. Mon. 76, 71-76 (1969)11. Karapinar, E: Fixed point theory for cyclic weak φ-contraction. Appl. Math. Lett. 24, 822-825 (2011)12. Karapınar, E: Weak φ-contraction on partial metric spaces. J. Comput. Anal. Appl. 14(2), 206-210 (2012)13. Karapinar, E, Erhan, IM: Fixed point theorems for operators on partial metric spaces. Appl. Math. Lett. 24, 1894-1899

(2011)14. Karapınar, E: Generalizations of Caristi Kirk’s theorem on partial metric spaces. Fixed Point Theory Appl. 2011, 4 (2011).

doi:10.1186/1687-1812-2011-415. Karapınar, E: Some nonunique fixed point theorems of Ciric type on cone metric spaces. Abstr. Appl. Anal. 2010,

Article ID 123094 (2010)16. Karapınar, E: Fixed point theorems in cone Banach spaces. Fixed Point Theory Appl. 2009, Article ID 609281 (2009)17. Karapınar, E, Yuksel, U: Some common fixed point theorems in partial metric spaces. J. Appl. Math. 2011, Article ID

263621 (2011)


http://dx.doi.org/10.1186/1687-1812-2011-4


18. Karapınar, E: A note on common fixed point theorems in partial metric spaces. Miskolc Math. Notes 12(2), 185-191(2011)

19. Kirk, WA, Srinavasan, PS, Veeramani, P: Fixed points for mapping satisfying cyclical contractive conditions. Fixed PointTheory Appl. 4, 79-89 (2003)

20. Reich, S: Some remarks concerning contraction mappings. Can. Math. Bull. 14, 121-124 (1971)21. Samet, B, Vetro, C, Vetro, P: Fixed point theorem for α-ψ contractive type mappings. Nonlinear Anal. 75, 2154-2165

(2012)22. Karapinar, E, Samet, B: Generalized (α-ψ ) contractive type mappings and related fixed point theorems with

applications. Abstr. Appl. Anal. 2012, Article ID 793486 (2012)

doi:10.1186/1687-1812-2013-94Cite this article as: Karapınar et al.: On α-ψ -Meir-Keeler contractive mappings. Fixed Point Theory and Applications 20132013:94.


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