R E V I E W
Open Access
Fixed point results for generalized mappings
Farhan Golkarmanesh
1*, Abdullah E Al-Mazrooei
2, Vahid Parvaneh
3and Abdul Latif
2*Correspondence:
1Department of Mathematics,
Sanandaj Branch, Islamic Azad University, Sanandaj, Iran Full list of author information is available at the end of the article
Abstract
In this paper, first we establish fixed point results for weak asymptotic pointwise contraction type mappings in metric spaces. Then we study the existence of fixed points for weak asymptotic pointwise nonexpansive type mappings inCAT(0) spaces. Our results improve and extend some corresponding known results in the literature.
MSC: 47H09; 47H10; 54H25
Keywords: asymptotic center; asymptotic pointwise contraction type; convexity structure;T-stable; weak asymptotic pointwise nonexpansive type;CAT(0) space
1 Introduction
The notion of asymptotic pointwise contraction was introduced by Kirk [] as follows. Let (M,d) be a metric space. A mappingT:M→Mis called an asymptotic pointwise contraction if there exists a functionα:M→[, ) such that, for each integern≥,
dTnx,Tny≤αn(x)d(x,y) for eachx,y∈M,
whereαn→αpointwise onM.
Moreover, Kirk and Xu [] proved that ifC be a weakly compact convex subset of a Banach spaceEandT:C→Can asymptotic pointwise contraction, thenThas a unique fixed pointv∈Cand for eachx∈Cthe sequence of Picard iterates{Tnx}converges in
norm tov.
Rakotch [] proved that if Mbe a complete metric space and f :M→M satisfies
d(f(x),f(y))≤α(d(x,y))d(x,y), for allx,y∈M, whereα: [,∞)→[, ) is monotonically decreasing, thenf has a unique fixed pointzand{fn(x)}converges toz, for eachx∈M.
Boyd and Wong [] proved that ifMbe a complete metric space andf :M→M sat-isfiesd(f(x),f(y))≤ψ(d(x,y))d(x,y), for allx,y∈M, whereψ: [,∞)→[,∞) is upper semicontinuous from the right and satisfies ≤ψ(t) <tfort> , thenf has a unique fixed pointzand{fn(x)}converges toz, for eachx∈M.
Using the diameter of an orbit, Walter [] obtained a result that may be stated as follows: Let (M,d) be a complete metric space and letT:M→Mbe a mapping with bounded orbits. If there exists a continuous, increasing functionϕ:R+→R+for whichϕ(r) <rfor
everyr> and
d(Tx,Ty)≤ϕdiamOT(x,y)
for everyx,y∈M,
whereOT(x,y) ={Tnx} ∪ {Tny}, thenThas a unique fixed pointx. Moreover,{Tnx}
con-verges tox, for eachx∈M.
In [], the authors proved coincidence results for asymptotic pointwise nonexpansive mappings. Kirk [], proved that ifMbe a complete metric space andT:M→Msatisfies
d(Tnx,Tny)≤φ
n(d(x,y)), for allx,y∈M, whereφn: [,∞)→[,∞),φn→φuniformly
on the range ofdandφis continuous withφ(s) <sfor alls> , thenT has a unique fixed pointzand{Tn(x)}converges toz, for eachx∈M.
References [, , –] present simple and elegant proofs of fixed point results for point-wise contraction, asymptotic pointpoint-wise contraction, and asymptotic nonexpansive map-pings.
Very recently, Saeidi [] introduced the concept of (weak) asymptotic pointwise con-traction type mappings.
Let (M,d) be a metric space. A mappingT:M→Mis said to be of asymptotic pointwise contraction type (resp. of weak asymptotic pointwise contraction type) ifTNis continuous
for some integerN≥ and there exists a functionα:M→[, ) such that, for eachxinM,
lim sup
n→∞
sup
y∈M
dTnx,Tny–αn(x)d(x,y)
≤ (.)
resp. lim inf
n→∞ supy∈M
dTnx,Tny–αn(x)d(x,y)
≤
, (.)
whereαn→αpointwise onM. Taking
rn(x) =sup y∈M
dTnx,Tny–αn(x)d(x,y)
∈R+∪ {∞}
it can easily be seen from (.) (resp. (.)) that
lim
n→∞rn(x) = (.)
resp. lim inf
n→∞ rn(x)≤
(.)
for allx∈Mand
dTnx,Tny≤αn(x)d(x,y) +rn(x). (.)
It is easy to see that the class of asymptotic pointwise contraction type mappings con-tains the class of an asymptotic pointwise contraction mappings, but the converse is not true []. Furthermore, ifCis a nonempty weakly compact subset of a Banach spaceE
andT :C→Ca mapping of weak asymptotic pointwise contraction type, thenT has a unique fixed pointv∈Cand, for eachx∈C, the sequence of Picard iterates{Tnx}
con-verges in norm tov(see []). Golkarmanesh and Saeidi [] obtained some results for this mappings in modular spaces.
In this paper, motivated by [, –], we combine the above results and obtain fixed point results for classes of mappings that extend the notions of asymptotic contraction and asymptotic pointwise contraction introduced by Kirk [] and Saeidi [].
2 Preliminaries
A(M), the class of the admissable subsets ofM, defines a convexity structure on any metric spaceM. Recall that a subset ofMis admissable if it is a nonempty intersection of closed balls.
At this point we introduce some notation which will be used throughout the remainder of this work. For a subsetAof a metric spaceM, set:
rx(A) =sup{d(x,y) :y∈A}; R(A) =inf{rx(A) :x∈A};
diam(A) =sup{d(x,y) :x,y∈A}; CA(A) ={x∈A:rx(A) =R(A)};
cov(A) ={B:Bis a closed ball andB⊇A}.
diam(A) is called the diameter ofA,R(A) is called the Chebyshev radius ofA,CA(A) is
called the Chebyshev center ofAandcov(A) is called the cover ofA.
Definition ([]) LetFbe a convexity structure onM.
(i) We will say thatFis compact if any family(Aα)α∈of elements ofF, has a
nonempty intersection providedα∈FAα =∅for any finite subsetF⊂.
(ii) We will say thatFis normal if for anyA∈F, not reduced to one point, we have
R(A) <diam(A).
(iii) We will say thatFis uniformly normal if there existsc∈(, )such that, for any
A∈F, not reduced to one point, we haveR(A)≤c(diam(A)). It is easy to check that
c≥/.
LetMbe a metric space andF a convexity structure. We will say that a function:
M→MisF-convex if{x:(x)≤r} ∈F for anyr≥. Also we define a type to be a function:M→[,∞] such that
(u) =lim sup
n→∞ d(xn,u),
where (xn) is a bounded sequence inM. Types are very useful in the study of the geometry
of Banach spaces and the existence of fixed point of mappings. We will say that a convexity structureFonMisT-stable if types areF-convex. We have the following lemma.
Lemma ([]) Let M be a metric space andFa compact convexity structure on M which is T -stable.Then,for any type,there exists x∈M such that
(x) =inf
(x) :x∈M.
Hussain and Khamsi [] and Nicolae [] proved the following results in metric spaces.
Theorem ([]) Let M be a bounded metric space.Assume that there exists a convexity structureFwhich is compact and T -stable.T:M→M be an asymptotic pointwise con-traction.Then T has a unique fixed point x.Moreover,the orbit{Tnx}converges to xfor
each x∈M.
Theorem ([]) Let M be a bounded metric space,T:M→M and suppose that there exists a convexity structureFwhich is compact and T -stable.Assume that
dTnx,Tny≤αn(x)rx
OT(y)
whereαn:M→Rfor each n∈Nand the sequence{αn}n∈Nconverges pointwise to a
func-tionα:M→[, ).Then T has a unique fixed point x.Moreover,the orbit{Tnx}converges
to x,for each x∈M.
3 Fixed point results for asymptotic pointwise contractive type mappings in metric spaces
In this section, we generalize the results obtained by Hussain and Khamsi [] and Nicolae [] for the wider class of weak asymptotic pointwise contraction type mappings.
Theorem Let(M,d)be a bounded metric space.Assume that there exists a convexity
structureFwhich is compact and T -stable.Let T:M→M be a weak asymptotic pointwise contraction type mapping.Then T has a unique fixed point x.Moreover,the orbit{Tnx}
converges to xfor each x∈M.
Proof Fixx∈Mand define a functionf by
f(u) =lim sup
n→∞ d
Tnx,u, u∈M.
SinceFis compact andT-stable, there existsx∈Msuch that
f(x) =inf
f(x);x∈M.
Let us show thatf(x) = . Indeed, for anym≥ we have
fTmx
=lim sup
n→∞ d
Tnx,Tmx
=lim sup
n→∞ d
Tm+nx,Tmx
=lim sup
n→∞ d
TmTnx,Tmx
≤lim sup
n→∞ αm(x)d
Tnx,x
+rm(x)
=αm(x)f(x) +rm(x),
which implies
f(x) =inf
f(x);x∈C≤fTmx
≤αm(x)f(x) +rm(x). (.)
Now, by (.) and (.), we obtain
f(x)≤lim inf
m→∞ αm(x)f(x) +rm(x)
=α(x)f(x),
which forcesf(x) = asα(x) < . Hence,d(Tnx,x)→ asn→ ∞. From this and the
continuity ofTN, for someN≥, it follows that
TNx=TN
lim
n→∞T
nx= lim n→∞T
n+Nx=x
namely,xis a fixed point ofTN. Now, repeating the above proof forxinstead ofx, we
deduce that{Tnx
}is convergent to an element ofM. ButTkNx=xfor allk≥. Hence,
Tnx
→x. We show thatTx=x. For this purpose, consider an arbitrary> . Then
there exists ak> such thatd(Tnx,x) <for alln>k. So, choosing a natural number
k>k/N, we obtain
d(Tx,x) =d
TTkNx
,x
=dTkN+x,x
<.
Since the choice of> is arbitrary, we getTx=x.
It is easy to verify thatThas only one fixed point. Indeed, ifa,b∈Mare two fixed points ofT, then we have
d(a,b) =dTna,Tnb≤αn(a)d(a,b) +rn(a).
Takinglim infin the above inequality, it follows thatd(a,b)≤α(a)d(a,b). Sinceα(a) < ,
we immediately geta=b.
In the following, we present an example in a bounded metric space which shows that a mapping of asymptotic pointwise contraction type is not necessary an asymptotic point-wise contraction.
Example LetM=ni=Ii(Ii= [, ]), equipped with the Euclidean norm. ThenMis a
bounded metric space. For each (x,x, . . . ,xn)∈M, define T(x,x, . . . ,xn) =
f(x),f(x), . . . ,f(xn),
,
wheref : [, ]→[, ] is some discontinuous function withf() = . We deduce thatTis discontinuous, and then it would not be an asymptotic pointwise contraction. But we see thatTnx= for allx∈M, and soTis of asymptotic pointwise contraction type.
Theorem Let(M,d)be a bounded metric space,T:M→M and suppose there exists a convexity structureFwhich is compact and T-stable and TNis continuous for some integer N≥.Assume
lim inf
n→∞ supy∈M
dTnx,Tny–αn(x)rx
OT(y)
≤ for every x,y∈M,
whereαn:M→Rfor each n∈N and the sequence{αn}converges pointwise to a function α:M→[, ).Then T has a unique fixed point z.Moreover,the orbit{Tnx}converges to z for each x∈M.
Proof Taking
γn(x) =sup y∈M
dTnx,Tny–αn(x)rx
OT(y)
∈R,
it can easily be seen that
lim inf
for allx∈M, and
dTnx,Tny≤αn(x)rx
OT(y)
+γn(x). (.)
Fixx∈Mand define a functionf by
f(u) =lim sup
n→∞
dTnx,u, u∈M.
SinceFis compact andT-stable, there existsz∈Msuch that
f(z) =inff(x) :x∈M.
Let us prove thatf(z) = . Indeed, for anym≥ we have
fTmz=lim sup
n→∞
dTnx,Tmz
=lim sup
n→∞
dTm+nx,Tmz
=lim sup
n→∞
dTmTnx,Tmz
≤lim sup
n→∞
αm(z)rz
OT
Tnx+γm(z)
=αm(z)lim sup n→∞
rz
OT
Tnx+γm(z),
which implies
f(z) =inff(x);x∈C≤fTmz≤αm(z)lim sup n→∞
rz
OT
Tnx+γm(z). (.)
By (.) we havelim infn→∞γm(z)≤, thus, for the subsequence{γmk(z)}of{γm(z)}, we have
lim
k→∞γmk(z)≤. (.)
Now, by (.) and (.) we obtain
f(z)≤lim inf
k→∞
αmk(z)lim sup
n→∞ rz
OT
Tnx+γmk(z)
=α(z)f(z),
which forcesf(z) = asα(z) < . Hence,d(Tnx,z)→ asn→ ∞. From this and the
con-tinuity ofTN, for someN≥, it follows that
TNz=TN
lim
n→∞T
nx= lim n→∞T
n+Nx=z;
namely,zis a fixed point ofTN. Now, repeating the above proof forzinstead ofx, we
de-duce thatTnzis convergent to a member ofM. ButTkNz=zfor allk≥. Hence,Tnz→z.
k> such thatd(Tnz,z) <for alln>k. So, by choosing a natural numberk>k/N, we
obtain
d(Tz,z) =dTTkNx
,z=dTkN+z,z<. Since> is arbitrary, we getTz=z.
Assume thatT has two fixed pointsa,b∈M, then, for eachn∈N,
d(a,b) =dTna,Tnb≤αn(a)ra
OT(b)
+γn(a) =αn(a)d(a,b) +γn(a).
Taking thelim infin the above inequality, it follows that
d(a,b)≤α(a)d(a,b).
Sinceα(a) < , we immediately geta=b.
4 Fixed point results for weak asymptotic pointwise nonexpansive type mappings in metric spaces
In this section we introduce weak asymptotic pointwise nonexpansive type mappings in metric spaces and we extend the results found of [].
Definition Let (M,d) be a metric space. A mappingT:M→Mis said to be of asymp-totic pointwise nonexpansive type (resp. weak asympasymp-totic pointwise nonexpansive type) ifTN is continuous for some integerN≥ and there exists a sequenceα
n:M→[, +∞)
such that
lim sup
n→∞ supy∈M
dTnx,Tny–αn(x)d(x,y)
≤ (.)
resp. lim inf
n→∞ supy∈M
dTnx,Tny–αn(x)d(x,y)
≤. (.)
wherelim supn→∞αn(x)≤. Taking
rn(x) =sup y∈M
dTnx,Tny–αn(x)d(x,y)
∈R+∪ {∞}
it can easily be seen from (.) (resp. (.)) that
lim
n→∞rn(x) = (.)
resp. lim inf
n→∞ rn(x)≤
(.)
for allx∈Mand
dTnx,Tny≤αn(x)d(x,y) +rn(x). (.)
taken to be the infimum of the lengh of all rectifiable paths joining them. In this case,dis said to be a length metric (otherwise, known an inner metric or intrinsic metric). In the case that there is no rectifiable path joins two points of the space, the distance between them is said to be∞.
A geodesic path joiningx∈Mtoy∈M(or, more briefly, a geodesic fromxtoy) is a map
cfrom a closed interval [,l]⊂RtoMsuch thatc() =x,c(l) =y, andd(c(t),c(t)) =|t–t|
for allt,t∈[,l]. In particular,cis an isometry andd(x,y) =l. The imageαofcis called a geodesic (or metric) segment joiningxandy. (M,d) is said to be a geodesic space if every two points of (M,d) are joined by a geodesic.Mis said to be uniquely geodesic if there is exactly one geodesic joiningxandyfor eachx,y∈M, which we will denote by [x,y], called the segment joiningxtoy.
A geodesic metric spaceΔ(x,x,x) in a geodesic metric space (M,d) consists of three
points inM (the vertices ofΔ) and a geodesic segment between each pair of vertices (the edges ofΔ). A comparison triangleΔ(x,x,x) in (M,d) is a triangleΔ¯(x,x,x) :=
Δ(¯x,x¯,x¯) in Mk such that dR(¯xi,x¯j) =d(xi,xj) fori,j∈ {, , }. Ifk> it is further
assumed that perimeter ofΔ(x,x,x) is less than Dk, whereDkdenotes the diameter
ofM
k. Such a triangle always exists.
A geodesic metric space is said to be aCAT(k) space if all geodesic triangles of appro-priate size satisfy the followingCAT(k) comparison axiom.
CAT(k): LetΔbe a geodesic triangle inMandΔ¯⊂Mkbe a comparison triangle forΔ. ThenΔis said to satisfy theCAT(k) inequality if for allx,y∈Δand all comparison points ¯
x,y¯∈ ¯Δ,
d(x,y)≤dx¯,y¯.
CompleteCAT() spaces are often called Hadamard spaces. These spaces are of partic-ular relevance to this study.
Finally, we observe that ifx,y,yare points of aCAT() space and ifyis the midpoint
of the segment [y,y], which we denote by y⊕y, then theCAT() inequality implies
d
x,y⊕y
≤ d(x,y)
+
d(x,y)
–
d(y,y)
.
LetMbe a complete CAT() space andx,y∈M, then for any α∈[, ] there exists a unique pointαx⊕( –α)y∈[x,y] such that
dz,αx⊕( –α)y≤αd(z,x) + ( –α)d(z,y), (.)
for anyz∈M.
LetMbe a completeCAT() space. A subsetC⊂Mis convex if for anyx,y∈Cwe have [x,y]⊂C. Any type function achieves its infimum,i.e., for any bounded sequence{xn}in
aCAT() spaceM, there existsω∈Msuch thatf(ω) =inf{f(x) :x∈M}, where
f(x) =lim sup
n→∞ d(xn,x).
Proof Fixx∈Cand define a functionf by
f(u) =lim sup
n→∞ d
Tnx,u, u∈C.
Letx∈Cbe such thatf(x) =inf{f(x);x∈M}=f. According to the proof of Theorem
we havef(Tnx
)≤αn(x)f+rn(x), for anyn≥. TheCAT() inequality implies
d
Tn(x),T
mx
⊕Thx
≤ d
Tnx,Tmx
+ d
Tnx,Thx
– d
Tmx,Thx
.
If we letngo to infinity, we get
f≤f
Tmx
⊕Thx
≤ f
Tmx
+ f
Thx
– d
Tmx,Thx
,
which implies
dTmx,Thx
≤fαm(x) + αh(x) –
+ rn(x) + rh(x)
+ f
αm(x)rm(x) +αh(x)rh(x)
.
SinceTis of weak asymptotic pointwise nonexpansive type, we get
lim sup
m,h→∞
dTmx,Thx
≤,
which implies{Tnx}is a Cauchy sequence. Letv=limn→∞Tnx. By the proof of
Theo-rem Tv=vandFix(T) is closed. In order to prove thatFix(T) is convex, it is enough to provex⊕y∈Fix(T), wheneverx,y∈Fix(T). Letz=x⊕y. TheCAT() inequality implies
dTnz,z≤
d
x,Tnz+ d
y,Tnz– d(x,y)
for anyn≥. Since
dx,Tnz=dTnx,Tnz≤αn(z)d(z,x) +rn(z)
=
αn(z) d(x,y)
+rn(z)
and
dy,Tnz=dTny,Tnz≤αn(z)d(z,y) +rn(z)
=
αn(z) d(x,y)
+rn(z)
we get
dTnz,z≤
αn(z) – d(x,y)+rn(z) + αn(z)rn(z)d(x,y)
Before we state the next and final result of this work, we need the following notation:
{xn}z if and only if f(z) =inf x∈Cf(x),
whereCis a closed convex subset which contains the bounded sequence{xn}andf(x) =
lim supn→∞d(xn,x).
Theorem Let M be a complete CAT() metric space. Let C be a bounded closed
nonempty convex subset of M.Let T:C→C be a weak asymptotic pointwise nonexpansive type.Let{xn} ∈C be an approximate fixed point sequence,i.e.,limn→∞d(xn,Txn) = and
{xn}z.Then we have Tz=z.
Proof Since{xn}is an approximate fixed point sequence, we have
f(x) =lim sup
n→∞ d
Tmxn,x
,
for anym≥. Hencef(Tmx)≤α
m(x)f(x) +rm(x) (see the proof of Theorem ). In
partic-ular,limn→∞f(Tmz) =f(z). TheCAT() inequality implies
d
xn,
z⊕Tnz
≤ d(xn,z)
+
d
xn,Tmz
– d
z,Tmz,
for anym,n≥. Ifn→ ∞, we will get
f
z⊕Tnz
≤ f(z)
+
f
Tmz– d
z,Tmz,
for anym≥. The definition ofzimplies
f(z)≤ f(z)
+
f
Tmz– d
z,Tmz
for anym≥, or
dz,Tmz≤fTmz– f(z)≤αm(z)f(z) +rm(z)
– f(z).
Lettingm→ ∞, we will getlimn→∞d(z,Tmz) = . The rest of the proof is similar to the
one used for Theorem .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Sanandaj Branch, Islamic Azad University, Sanandaj, Iran.2Department of Mathematics,
King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia. 3Young Researchers and Elite Club, Kermanshah
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University. The authors, therefore, acknowledge with thanks the DSR for technical and financial support.
Received: 28 April 2014 Accepted: 8 October 2014 Published:22 Oct 2014 References
1. Kirk, WA: Asymptotic pointwise contraction, plenary lecture. In: Proceedings of the 8th International Conference on Fixed Point Theory and Its Applications, Chiang Mai University, Thailand, July 16-22 (2007)
2. Kirk, WA, Xu, H-K: Asymptotic pointwise contractions. Nonlinear Anal.69, 4706-4712 (2008) 3. Rakotch, E: A note on contractive mappings. Proc. Am. Math. Soc.13, 459-465 (1962) 4. Boyd, D, Wong, JSW: On nonlinear contraction. Proc. Am. Math. Soc.20, 458-464 (1969)
5. Walter, W: Remarks on a paper F. Browder about contraction. Nonlinear Anal. TMA5(1), 21-25 (1981)
6. Espínloa, R, Hussain, N: Common fixed points for multimaps metric spaces. Fixed Point Theory Appl.2010, Article ID 204981 (2010)
7. Kirk, WA: Fixed points asymptotic contractions. J. Math. Anal. Appl.277(2), 645-650 (2003)
8. Kirk, WA: Contraction mappings and extensions. In: Kirk, WA, Sims, B (eds.) A Handbook of Metric Fixed Point Theory, pp. 1-34. Kluwer Academic, Dordrecht (2001)
9. Hussain, N, Khamsi, MA: On asymptotic pointwise contractions in metric spaces. Nonlinear Anal.71, 4423-4429 (2009) 10. Nicolae, A: Generalized asymptotic pointwise contractions and nonexpansive mappings involving orbits. Fixed Point
Theory Appl.2010, Article ID 458265 (2010)
11. Saeidi, S: Mapping under asymptotic contractive type conditions. J. Nonlinear Convex Anal. (in press)
12. Golkarmanesh, F, Saeidi, S: Asymptotic pointwise contractive type in modular function spaces. Fixed Point Theory Appl.2013, Article ID 101 (2013)
10.1186/1687-1812-2014-217
