R E S E A R C H
Open Access
Common fixed points for pointwise
Lipschitzian semigroups in modular function
spaces
Buthinah A Bin Dehaish
1*, Mohamed A Khamsi
2,3and Wojciech M Kozlowski
4*Correspondence:
[email protected] 1Department of Mathematics, King Abdulaziz University, P.O. Box 53909, Jeddah, 21593, Saudi Arabia Full list of author information is available at the end of the article
Abstract
LetCbe a
ρ
-bounded,ρ
-closed, convex subset of a modular function spaceLρ. We investigate the existence of common fixed points for asymptotic pointwise nonexpansive semigroups of nonlinear mappingsTt:C→C,i.e.a family such thatT0(f) =f,Ts+t(f) =Ts◦Tt(f) and
ρ(T
(f) –T(g))
≤α
t(f)ρ
(f–g),where lim supt→∞
α
t(f)≤1 for everyf∈C. In particular, we prove that ifLρis uniformly convex, then the common fixed point is nonemptyρ
-closed and convex.MSC: Primary 47H09; secondary 46B20; 47H10; 47E10
Keywords: fixed point; modular function space; nonexpansive mapping; Orlicz space; pointwise Lipschitzian mapping; pointwise nonexpansive mapping; semigroup; uniform convexity
1 Introduction
The purpose of this paper is to prove the existence of common fixed points for semi-groups of nonlinear mappings acting in modular function spaces which are natural gen-eralizations of both function and sequence variants of many important, from applica-tions perspective, spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces and many others, see the book by Kozlowski [] for an ex-tensive list of examples and special cases. Earlier studies of fixed point theory in modu-lar function spaces can be found in [–], see also []. Recently, Khamsi and Kozlowski presented a series of fixed point results for pointwise contractions, asymptotic pointwise contractions, pointwise nonexpansive and asymptotic pointwise nonexpansive mappings acting in modular functions spaces [, ] (all these should be considered in the modular sense, not in the sense of the corresponding norms). These results are also new and of a big interest, even in a much simpler context of ‘plain’ modular contractions and nonexpansive mappings,i.e., without any pointwise and asymptotic complications.
In many cases, modular type conditions are much more natural as modular type as-sumptions can be more easily verified than their metric or norm counterparts. Further-more, there are also important results that can be proved only using the apparatus of modular function spaces. Khamsiet al.demonstrated in [] that a mappingT is
nonexpansive in a modular function spaceLρif and only if
ρλT(f) –T(g)≤ρλ(f –g) for anyλ≥.
They also gave an example of a mapping which is ρ-nonexpansive, but it is not norm-nonexpansive. From this perspective, the fixed point theory in modular function spaces should be considered as complementary to the fixed point theory in normed spaces and in metric spaces.
Let us recall that a family {Tt}t≥ of mappings forms a semigroup if T(x) =x and
Ts+t=Ts◦Tt, see Definition . below for details. Such a situation is quite typical in
math-ematics and applications. For instance, in the theory of dynamical systems, the modular function spaceLρwould define the state space and the mapping (t,f)→Tt(f) would
rep-resent the evolution function of a dynamical system. The question about the existence of common fixed points and about the structure of the set of common fixed points, can be interpreted as a question whether there exist points that are fixed during the state space transformationTtat any given point of timet, and if yes, what the structure of a set of such
points may look like. In the setting of this paper, the state space may be an infinite dimen-sional vector space. Therefore, it is natural to apply these results not only to deterministic dynamical systems but also to stochastic dynamical systems.
The existence of common fixed points for families of contractions and nonexpansive mappings in the Banach spaces have been investigated since the early s, see,e.g., Belluce and Kirk [, ], Browder [], Bruck [], DeMarr [], Lim []. The asymptotic approach for finding common fixed points of semigroups of Lipschitzian (but not point-wise Lipschitzian) mappings has also been investigated for some time, see,e.g., Tan and Xu []. It is worthwhile mentioning the recent studies on the special case, when the pa-rameter set for the semigroup is equal to{, , , , . . .}andTn=Tn, thenth iterate of an
asymptotic pointwise nonexpansive mapping,i.e.,T:C→Csuch that there exists a se-quence of functionsαn:C→[,∞) with
Tn(f) –Tn(g)≤αn(f)f–g for anyf,g∈C
andlim supn→∞αn(f) = for any f ∈C. Kirk and Xu [] proved the existence of fixed
points for asymptotic pointwise contractions and asymptotic pointwise nonexpansive mappings in the Banach spaces, while Hussain and Khamsi extended this result to metric spaces [] and Khamsi and Kozlowski to modular function spaces [, ]. Kozlowski in [] and [] proved convergence to fixed points of some iterative algorithms, applied to asymptotic pointwise nonexpansive mappings in the Banach spaces, and the existence of common fixed points of semigroups of asymptotic pointwise nonexpansive semigroups in the Banach spaces []. Convergence of generalized Mann and Ishikawa algorithms to common points of such semigroups in Banach spaces was established in [] and []. In the context of modular function spaces, convergence to fixed points of some iterative algorithms, applied to asymptotic pointwise nonexpansive mappings, was proven by Bin Dehaish and Kozlowski in [].
mappings in modular functions spaces, to the pointwise asymptotic semigroups. How-ever, methods used in the current paper are substantially different due to the asymptotic behavior of semigroups in question. It is worth noting that existence of semigroups of nonexpansive mappings in modular function spaces was discussed by Khamsi [] in the context of Musielak-Orlicz spaces and discussed applications to differential equations.
2 Preliminaries
Let us introduce basic notions related to modular function spaces and related notation, which will be used in this paper. For further details, we refer the reader to preliminary sections of the recent articles [, , ] or to the survey article [], see also [, , ] for the standard framework of modular function spaces.
Letbe a nonempty set, and letbe a nontrivialσ-algebra of subsets of. LetPbe aδ-ring of subsets ofsuch thatE∩A∈Pfor anyE∈PandA∈. Let us assume that there exists an increasing sequence of setsKn∈P such that=
Kn. ByE we denote
the linear space of all simple functions with supports fromP. ByM∞we will denote the
space of all extended measurable functions,i.e., all functionsf :→[–∞,∞] such that there exists a sequence{gn} ⊂E,|gn| ≤ |f|andgn(ω)→f(ω) for allω∈. By Awe denote
the characteristic function of the setA.
Definition .[] Letρ:M∞→[,∞] be a nontrivial, convex and even function. We
say thatρis a regular convex function pseudomodular if:
(i) ρ() = ;
(ii) ρis monotone,i.e.,|f(ω)| ≤ |g(ω)|for allω∈impliesρ(f)≤ρ(g), where f,g∈M∞;
(iii) ρis orthogonally subadditive,i.e.,ρ(fA∪B)≤ρ(fA) +ρ(fB)for anyA,B∈such thatA∩B=∅,f ∈M;
(iv) ρhas the Fatou property,i.e.,|fn(ω)| ↑ |f(ω)|for allω∈impliesρ(fn)↑ρ(f), wheref ∈M∞;
(v) ρis order continuous inE,i.e.,gn∈Eand|gn(ω)| ↓impliesρ(gn)↓.
Similarly, as in the case of measure spaces, we say that a setA∈isρ-null ifρ(gA) =
for everyg∈E. We say that a property holdsρ-almost everywhere if the exceptional set isρ-null. As usual, we identify any pair of measurable sets, whose symmetric difference isρ-null, as well as any pair of measurable functions, differing only on aρ-null set. With this in mind, we defineM={f ∈M∞;|f(ω)|<∞ρ-a.e.}, where each element is actually
an equivalence class of functions equalρ-a.e. rather than an individual function.
Definition .[] We say that a regular function pseudomodularρis a regular convex
function modular ifρ(f) = impliesf = ρ-a.e. The class of all nonzero regular convex function modulars, defined onwill be denoted by.
Definition .[, , ] Letρbe a convex function modular. A modular function space
is the vector spaceLρ={f ∈M;ρ(λf)→ asλ→}. In the vector spaceLρ, the following
formula
fρ=inf
α> ;ρ
f α
≤
The following notions will be used throughout the paper.
Definition .[] Letρ∈ .
(a) We say that{fn}isρ-convergent tof and writefn→f(ρ)if and only ifρ(fn–f)→. (b) A sequence{fn}, wherefn∈Lρ, is calledρ-Cauchy ifρ(fn–fm)→asn,m→ ∞. (c) We say thatLρisρ-complete if and only if anyρ-Cauchy sequence inLρis
ρ-convergent.
(d) A setB⊂Lρis calledρ-closed if for any sequence offn∈B, the convergence fn→f(ρ)implies thatf belongs toB.
(e) A setB⊂Lρis calledρ-bounded ifsup{ρ(f–g);f ∈B,g∈B}<∞.
Sinceρfails in general the triangle identity, many of the known properties of limit may not extend to theρ-convergence. For example, theρ-convergence does not necessarily imply theρ-Cauchy condition. However, it is important to remember that theρ-limit is unique when it exists. The following proposition brings together few facts that will be often used in the proofs of our results.
Proposition .[] Letρ∈ .
(i) Lρisρ-complete.
(ii) ρ-ballsBρ(f,r) ={g∈Lρ;ρ(f –g)≤r}areρ-closed.
(iii) Ifρ(αfn)→for anα> ,then there exists a subsequence{gn}of{fn}such that gn→ρ-a.e.
(iv) ρ(f)≤lim infn→∞ρ(fn),wheneverfn→f ρ-a.e. (Note:this property is equivalent to
the Fatou property.)
Let us recall the definition of an asymptotic pointwise nonexpansive mapping acting in a modular function space.
Definition .[] Letρ∈ , and letC⊂Lρbe nonempty andρ-closed. A mappingT :
C→Cis called
(i) a pointwise Lipschitzian mapping, if there existsα:C→[,∞)such that ρT(f) –T(g)≤α(f)ρ(f –g) for anyf,g∈C;
(ii) an asymptotic pointwise nonexpansive, if there exists a sequence of mappings αn:C→[,∞)such that
ρTn(f) –Tn(g)≤αn(f)ρ(f–g) for anyf,g∈C andlim supn→∞αn(f)≤for anyf ∈Lρ.
A pointf ∈Cis called a fixed point ofT, wheneverT(f) =f. The set of fixed points ofT
will be denoted byF(T).
This definition is now extended to a one-parameter family of mappings.
Definition . A one-parameter familyF={Tt:t≥}of mappings fromCinto itself is
(i) T(f) =f forf ∈C;
(ii) Tt+s(f) =Tt(Ts(f))forf ∈Candt,s∈[,∞);
(iii) for eacht≥,Ttis an asymptotic pointwise nonexpansive mapping,i.e., there exists a functionαt:C→[,∞)such that
ρTt(f) –Tt(g)
≤αt(f)ρ(f –g) for allf,g∈C (.)
such thatlim supt→∞αt(f)≤for everyf∈C;
(iv) for eachf ∈C, the mappingt→Tt(f)isρ-continuous.
For eacht≥, letF(Tt) denote the set of its fixed points. Define then the set of all common
fixed points set for mappings fromFas the following intersection
F(F) =
t≥
F(Tt).
Note that without loss of generality, we may assumeαt(f)≥ for anyt≥ andf ∈C,
andlim supt→∞αt(f) =limt→∞αt(f) = .
3 Existence of common fixed points
The conceptρ-type is a powerful technical tool, which is used in the proofs of many fixed point results. The definition of aρ-type is based on a given sequence. In this work, we generalize this definition to be adapted to one-parameter family of mappings.
Definition . LetK⊂Lρbe convex andρ-bounded.
() A functionτ:K→[,∞]is called aρ-type (or shortly a type) if there exists a one-parameter family{ht}t≥of elements ofKsuch that for anyf∈Kthere holds
τ(f) = inf
M>
sup
t≥M
ρ(ht–f)
.
() Letτ be a type. A sequence{gn}is called a minimizing sequence ofτ if
lim
n→∞τ(gn) =inf
τ(f) :f∈K. Note thatτ is convex, providedρis convex.
Let us recall the modular equivalents of uniform convexity introduced in [].
Definition . Letρ∈ . We define the following uniform convexity (UC) type
proper-ties of the function modularρ:
(i) Letr> ,ε> . Define
D(r,ε) =(f,g) :f,g∈Lρ,ρ(f)≤r,ρ(g)≤r,ρ(f –g)≥εr
.
Let
δ(r,ε) =inf
–
rρ
f+g
: (f,g)∈D(r,ε) ifD(r,ε)=∅,
(ii) We say thatρsatisfies (UUC) if there existsη(s,ε) > , for everys≥, andε>
such that
δ(r,ε) >η(s,ε) > forr>s.
The following technical lemma is very useful throughout this paper (see [] for its proof ).
Lemma . Letρ∈ be(UUC).Let R> .Assume that{fn}and{gn}are in Lρsuch that
lim sup
n→∞ ρ(fn)≤R; lim supn→∞ ρ(gn)≤R and nlim→∞ρ
fn+gn
=R.
Then we must havelimn→∞ρ(fn–gn) = .
The following property plays in the theory of modular function space a role similar to the reflexivity in the Banach spaces, see,e.g., [].
Definition . We say thatLρ has property (R) if and only if every nonincreasing
se-quence{Cn}of nonempty,ρ-bounded,ρ-closed and convex subsets ofLρhas a nonempty
intersection.
Similarly as in the Banach space case, the modular uniform convexity implies the prop-erty (R).
Theorem .[] Letρ∈ be(UUC),then Lρhas a property(R).
The next lemma is the generalization of the minimizing sequence property for types defined by the sequences in Lemma . in [] to the one-parameter semigroup case.
Lemma . Assumeρ∈ is(UUC).Let C be a nonempty,ρ-bounded,ρ-closed and con-vex subset of Lρ.Letτ be a type defined by a one-parameter family{ht}t≥in C.
(i) Ifτ(f) =τ(f) =inff∈Cτ(f),thenf=f.
(ii) Any minimizing sequence{fn}ofτ isρ-convergent.Moreover,theρ-limit of{fn}is
independent of the minimizing sequence.
Proof First, let us prove (i). Letf,f∈Csuch thatτ(f) =τ(f) =inff∈Cτ(f). Let us consider
two cases.
Case :inff∈Cτ(f) = . Since
ρ
f–f
=ρ
f–ht+ht–f
≤ρ(f–ht) +ρ(ht–f)
for anyt≥, we get
ρ
f–f
≤sup
t≥M
ρ(f–ht) +sup t≥M
for anyM> . Since
τ(f) = inf
M>
sup
t≥M
ρ(f –ht)
= lim
M→∞tsup≥M
ρ(f –ht)
for anyf ∈C, we get
ρ
f–f
≤τ(f) +τ(f) = ,
which impliesf=fas claimed.
Case :inff∈Cτ(f) > . Assume to the contrary thatf=f. Set
R=inf
f∈Cτ(f) and ε=
ρ(f–f)
R .
Letν∈(,R). Thenρ(f–f) = Rε≥(R+ν)ε. Using the definition ofτ, we deduce that
there existsMν> such that
sup
t≥Mν
ρ(f–ht)≤τ(f) +ν=R+ν and sup
t≥Mν
ρ(f–ht)≤τ(f) +ν=R+ν.
Sinceρis (UUC), there existsη(R,ε) > such that
δ(R+ν,ε)≥η(R,ε)
for anyν∈(,R). So for anyt≥Mν, we have
ρ
f+f
–ht
≤(R+ν) –δ(R+ν,ε)≤(R+ν) –η(R,ε).
Hence
τ
f+f
≤ sup
t≥Mν
ρ
f+f
–ht
≤(R+ν) –η(R,ε).
SinceCis convex, we get
R≤τ
f+f
≤(R+ν) –η(R,ε).
If we letν→, we will get
R≤R –η(R,ε),
which is impossible, sinceR> andη(R,ε) > . Therefore, we must havef=f.
Next, we prove (ii). DenoteR=infg∈Cτ(g). For anyn≥, let us set
where convρ(A) is the intersection of allρ-closed convex subset of C, which contains
A⊂C. SinceCis itselfρ-closed and convex, we getKn⊂Cfor anyn≥. Property (R) will
then implyKn=∅. Let us fix then arbitraryf ∈
Kn,g∈Candε> . By definition of τ(g), there existsMε> such thatsupt≥Mερ(g–ht)≤τ(g) +ε. Letn≥Mε. Then for any
t≥n, we haveρ(g–ht)≤τ(g) +ε,i.e.,ht∈Bρ(g,τ(g) +ε). SinceBρ(g,τ(g) +ε) isρ-closed
and convex, we getKn⊂Bρ(g,τ(g) +ε). Hencef ∈Bρ(g,τ(g) +ε),i.e.,
ρ(g–f)≤τ(g) +ε. (.) Sinceεwas taken arbitrarily greater than , we getρ(g–f)≤τ(g) for anyg∈C. Let{fn}
be a minimizing sequence forτ. IfR= , then, since{fn}is a minimizing sequence, we get
limn→∞τ(fn) =R= . Using (.), we can see thatρ(fn–f)≤τ(fn) for anyn≥. Hence
{fn}isρ-convergent tof. Since selection off was independent of{fn}, it follows that any
minimizing sequence isρ-convergent tof ifR= . We can assume, therefore, thatR> . For anyn≥, let us set
dn=sup i,j≥n
ρ(fi–fj). (.)
We claim that{fn} isρ-Cauchy. Assume to the contrary that this is not the case. Since
the sequence{dn}is decreasing and{fn}is notρ-Cauchy, we get d:=infn≥dn> . Set ε=dR > . Let us fix arbitraryν∈(,R). Sincelimn→∞τ(fn) =R, there existsn≥ such
that for anyn≥n, we have
τ(fn)≤R+ ν
. (.)
Letn≥n. By (.), there existsin,jn≥ such that
ρ(fin–fjn) >dn– d
≥
d
= Rε> (R+ν)ε.
Using the definition ofτand (.), we deduce the existence ofM> such that
sup
t≥M
ρ(fin–ht)≤τ(fin) + ν
≤R+ν and
sup
t≥M
ρ(fjn–ht)≤τ(fjn) + ν
≤R+ν. Hence
ρ
fin+fjn
–ht
≤(R+ν) –δ(R+ν,ε)
for anyt≥M. Sinceρis (UUC), there existsη(R,ε) > such thatδ(R+ν,ε)≥η(R,ε).
Hence
ρ
fin+fjn
–ht
for anyt≥M. Hence
R≤τ
fin+fjn
≤sup
t≥M ρ
fjn+fjn
–ht
≤(R+ν) –η(R,ε)<R.
Using the definition ofR, we get
R≤(R+ν) –η(R,ε)
for anyν∈(,R). If we letν→, we getR≤R( –η(R,ε)). This contradiction implies that
{fn}isρ-Cauchy. SinceLρisρ-complete, we deduce that{fn}isρ-convergent as claimed.
In order to finish the proof of (ii), let us show that the ρ-limit of {fn} is
indepen-dent of the minimizing sequence. Indeed, let{gn}be another minimizing sequence ofτ.
The previous proof will show that{gn}is also ρ-convergent. In order to prove that the ρ-limits of{fn}and{gn}are equal, let us show thatlimn→∞ρ(fn–gn) = . Assume not,i.e.,
limn→∞ρ(fn–gn)= . Without loss of generality, we may assume that there existsd>
such thatρ(fn–gn)≥dfor anyn≥. Setε=dR> . Letν∈(,R). Sincelimn→∞τ(fn) =
limn→∞τ(gn) =R, there existsn≥ such that for anyn≥, we haveτ(fn)≤R+ν and τ(gn)≤R+ν. Fixn≥n. Then
ρ(fn–gn)≥d= Rε> (R+ν)ε.
Using the definition ofτ, we deduce the existence ofM> such that
sup
t≥M
ρ(fn–ht)≤τ(fn) + ν
≤R+ν and
sup
t≥M
ρ(gn–ht)≤τ(gn) + ν
≤R+ν. Hence
ρ
fn+gn
–ht
≤(R+ν) –δ(R+ν,ε)
for anyt≥M. Sinceρis (UUC), there existsη(R,ε) > such thatδ(R+ν,ε)≥η(R,ε) for anyν> . Hence
ρ
fn+gn
–ht
≤(R+ν) –η(R,ε)
for anyt≥M. So
τ
fn+gn
≤sup
t≥M ρ
fn+gn
–ht
≤(R+ν) –η(R,ε).
Using the definition ofR, we get
for anyν∈(,R). If we letν→, we get R≤R( –η(R,ε)). This contradiction implies
limn→∞ρ(fn–gn) = . The Fatou property will finally imply that
ρ(f–g)≤lim inf
n→∞ ρ(fn–gn),
wheref is theρ-limit of{fn}andgis theρ-limit of{gn}. Henceρ(f–g) = ,i.e.,f =g.
Using Lemma ., we are ready to prove our common fixed point result for asymptotic pointwise nonexpansive semigroups.
Theorem . Assumeρ∈ is(UUC).Let C be aρ-closed,ρ-bounded convex nonempty
subset.LetF={Tt:t≥}be an asymptotic pointwise nonexpansive semigroup on C.Then F has a common fixed point,and the set F(F)of common fixed points isρ-closed and convex.
Proof Let us fixf ∈Cand define the function
τ(g) = inf
M>
sup
t≥M
ρTt(f) –g
.
SinceCisρ-bounded, we haveτ(g)≤diamρ(C) < +∞for anyg∈C. Henceτ=inf{τ(g) :
g∈C}exists and is finite. For anyn≥, there existsgn∈C, such that
τ≤τ(gn) <τ+
n.
Therefore,limn→∞τ(zn) =τ,i.e.,{gn}is a minimizing sequence forτ. By Lemma ., there
existsg∈Csuch that{gn}ρ-converges tog. Let us now prove thatg∈F(F). Note that
ρTs+t(f) –Ts(h)
≤αs(h)ρ
Tt(f) –h
fors,t> andh∈C. Using the definition ofτ, we get
τTs(h)
≤ sup
t+s≥M
ρTs+t(f) –Ts(h)
≤αs(h) sup t≥M–s
ρTt(f) –h
for anyM>s, which implies
τTs(h)
≤αs(h)τ(h). (.)
Sincelims→∞αs(g) = , there existss> such that for anys≥s, we haveαs(g) < + .
Repeating this argument, one will finds>s+ such that for anys≥s, we haveαs(g) <
+. By induction, we will construct a sequence{sn}of positive numbers such thatsn+<
sn+n and for anys≥sn, we haveαs(gn) < +n. Let us fixt≥. Then inequality (.) will
imply
τTsn+t(gn)
≤αsn+t(gn)τ(gn)≤
+
n
for anyn≥. In particular, we get that{Tsn+t(gn)}is a minimizing sequence ofτ. Therefore,
Lemma . implies that{Tsn+t(gn)}ρ-converges togfor anyt≥. In particular, we have
{Tsn(gn)}ρ-converges tog. Since
ρTsn+t(gn) –Tt(g)
≤αt(g)ρ
Tsn(gn) –g
, we get{Tsn+t(gn)}ρ-converges toTt(g). Finally, using
ρ
Tt(g) –g
≤ρTt(g) –Tsn+t(gn)
+ρTsn+t(gn) –g
,
we getTt(g) =g. Sincetwas arbitrarily positive, we getg∈F(F),i.e.,F(F) is not empty.
Next, let us prove thatF(F) isρ-closed. Let{fn}inF(F)ρ-convergent tof. Since ρTs(fn) –Ts(f)
≤αs(f)ρ(fn–f)
for any n≥ and s> , we get {Ts(fn)} ρ-converges toTs(f). Since fn∈F(F), we get
{Ts(fn)}={fn}. In other words, {fn} ρ-converges toTs(f) and f. The uniqueness of the ρ-limit implies then thatTs(f) =f for anys≥,i.e.,f ∈F(F). Therefore,F(F) isρ-closed.
Let us finish the proof of Theorem . by showing thatF(F) is convex. It is sufficient to show that
h=f+g ∈F(F)
for anyf,g∈F(F). Without loss of generality, we will assume thatf =g. Lets> . We have
ρf –Ts(h)
=ρTs(f) –Ts(h)
≤αs(f)ρ(f –h)
and
ρg–Ts(h)
=ρTs(g) –Ts(h)
≤αs(g)ρ(g–h).
Sinceρ(f –h) =ρ(g–h) =ρ(f–g) and
ρ
f–g
≤
ρ
f–Ts(h)
+ ρ
g–Ts(h)
,
we conclude that
lim
s→∞ρ
f –Ts(h)
=lim
s→∞ρ
g–Ts(h)
=ρ
f–g
.
Similarly, we have
ρ
f –h+Ts(h)
≤
ρ(f–h) + ρ
f–Ts(h)
and
ρ
g–h+Ts(h)
≤
ρ(g–h) + ρ
g–Ts(h)
Since
ρ
f–g
≤
ρ
f–h+Ts(h)
+ ρ
g–h+Ts(h)
,
we conclude that
lim
s→∞ρ
f –h+Ts(h)
= lim
s→∞ρ
g–h+Ts(h)
=ρ
f–g
.
Therefore, we have
lim
s→∞ρ
f –Ts(h)
=lim
s→∞ρ
f–h+Ts(h)
=ρ(f–h).
Lemma ., applied toAt=f–Ts(h) andBt=Ts(h) –g, implies thatρ(At–Bt)→. Hence
lim
s→∞ρ
h–Ts(h)
= lim
s→∞ρ
At–Bt
≤ lim
s→∞ρ(At–Bt) = .
Clearly, we will getlims→∞ρ(h–Ts+t(h)) = , for anyt≥. Since
ρTt(h) –Ts+t(h)
≤αt(h)ρ
h–Ts(h)
,
we getlims→∞ρ(Tt(h) –Ts+t(h)) = . Finally, using the inequality
ρ
h–Tt(h)
≤
ρ
h–Ts+t(h)
+ ρ
Tt(h) –Ts+t(h)
,
by lettings→ ∞, we getTt(h) =hfor anyt≥,i.e.,h∈F(F).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors participated in the design of this work and performed equally. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, King Abdulaziz University, P.O. Box 53909, Jeddah, 21593, Saudi Arabia.2Department of
Mathematical Sciences, University of Texas at El Paso, El Paso, Texas, USA.3Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran, 31261, Saudi Arabia.4School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia.
Acknowledgements
This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. (247-006-D1433). The authors, therefore, acknowledge with thanks technical and financial support of DSR.
Received: 28 February 2013 Accepted: 24 July 2013 Published: 12 August 2013
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doi:10.1186/1687-1812-2013-214
