R E S E A R C H
Open Access
On the set of common fixed points of
semigroups of nonlinear mappings in
modular function spaces
Saud M Alsulami
1*and Walter M Kozlowski
2*Correspondence:
1Department of Mathematics, King
Abdulaziz University, P.O. Box 138381, Jeddah, 21323, Saudi Arabia Full list of author information is available at the end of the article
Abstract
We prove that the set of all common fixed points for a continuous nonexpansive semigroup of nonlinear mappings acting in modular function spaces can be represented as an intersection of fixed points sets of two nonexpansive mappings. This representation is then used to prove convergence of several iterative methods for construction of common fixed points of semigroups of nonlinear mappings. We also demonstrate an example how the results of this paper can be applied for constructing a stationary point of a process defined by the Urysohn integral operator. MSC: Primary 47H09; secondary 46B20; 47H10; 47H20; 47E30; 47J25
Keywords: fixed point; common fixed point; nonexpansive mapping; semigroup of nonlinear mappings; modular function space; Orlicz space; convex modular; fixed point iteration process; Mann process; Ishikawa process
1 Introduction
The purpose of this paper is to prove that the set of all common fixed points for a contin-uous nonexpansive semigroup of nonlinear mappings acting in modular function spaces can be represented as an intersection of fixed point sets of two nonexpansive mappings, where nonexpansiveness is understood in the modular sense. Modular function spaces are natural generalizations of both function and sequence variants of many important, from applications perspective, spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces and many others; see [, ] for an extensive list of examples and special cases.
The fixed point theory in modular function spaces originated in the seminal paper by Khamsi, Kozlowski and Reich []. In that paper, the authors showed that there exist mappings which areρ-nonexpansive but are not norm-nonexpansive. They demonstrated that for a mappingTto be norm nonexpansive in a modular function spaceLρ, a stronger
thanρ-nonexpansiveness assumption is needed:ρ(λ(T(x) –T(x)))≤ρ(λ(x–y)) for any λ≥. 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. It is worthwhile to mention that from the perspective of applications, modular type conditions are typically more easily verified than their metric or norm counterparts. For earlier and recent results of fixed point theory in modular function spaces, refer,e.g., to [, –].
Let us recall that a family {Tt}t≥of mappings forms a semigroup ifT(x) =x,Ts+t=
Ts(Tt(x)). Such a situation is quite typical in mathematics and applications. For instance, in the theory of dynamical systems, the modular function spaceLρwould define the state
space and the mapping (t,x)→Tt(x) would represent the evolution function of a dynami-cal system. The question about the existence of common fixed points, and about the struc-ture 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 time
t, 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 infinite dimensional. Therefore, it is natural to apply these results not only to deterministic dynamical systems but also to stochastic dynamical systems.
An existence of common fixed points ofρ-nonexpansive semigroups was demonstrated in []. However, a structure of the set of common fixed points can bea priorivery complicated and therefore it can be difficult to apply any methods of construction of such common fixed points, which is of a major importance for applications. In the current pa-per, we show that in the case of a continuous nonexpansive semigroup, the set of its com-mon fixed points can be actually represented by an intersection of fixed point sets of just two suitably chosen, nonexpansive mappings. The idea of such representation is known in Banach spaces; see,e.g., the paper by Suzuki [] and references therein. How-ever, the case of ρ-nonexpansive mappings acting in modular function spaces have not been investigated prior to the current paper. It is worthwhile to mention that we use only convexity of the function modularρas it does not need to have any triangle inequality of homogeneity properties. This shows the strength of the convexity assumptions because convexity ofρsuffices to prove both the existences and the representation of a set of com-mon fixed points.
We use this representation to show how the Mann and Ishikawa type iterative methods can be used for the construction of common fixed points of continuous nonexpansive semigroups. The idea of using such processes in this context can be traced back to the seminal s-s papers by Mann [], Krasnosel’skii [], Ishikawa [], Reich [, ], and others. See also an extensive body of work from the s and s [–], and more recent research from the current century [, –] and the works referred there. We also show an example how the results of this paper can be applied for constructing a stationary point of an Urysohn process.
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 be a nontrivialσ-algebra of subsets of. LetP be a δ-ring of subsets ofsuch thatE∩A∈P for anyE∈P andA∈. Let us assume that there exists an increasing sequence of setsKn∈Psuch that=
Kn. ByEwe denote the linear space of all simple functions with supports fromP. ByM∞we denote the space
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 is ρ-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 func-tion modular ifρ(f) = impliesf = ρ-a.e. The class of all nonzero regular convex func-tion 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→}.
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) A setB⊂Lρis calledρ-closed if for any sequence offn∈B, the convergence
fn→f(ρ)implies thatf belongs toB.
(d) A setB⊂Lρis calledρ-bounded ifsup{ρ(f –g) :f∈B,g∈B}<∞. (e) A setB⊂Lρis called stronglyρ-bounded if there existsβ> such that
Mβ(B) =sup{ρ(β(f –g)) :f ∈B,g∈B}<∞.
Sinceρfails in general the triangle identity, many of the known properties of limit may not extend toρ-convergence. For example,ρ-convergence does not necessarily imply the ρ-Cauchy condition. However, it is important to remember that theρ-limit is unique when it exists. The following proposition brings together a few facts that will be often used in the proofs of our results.
Proposition . Letρ∈ .
(i) Lρisρ-complete.
(ii) ρ-ballsBρ(x,r) ={y∈Lρ:ρ(x–y)≤r}areρ-closed andρ-a.e.closed.
(iii) Ifρ(αfn)→for anα> ,then there exists a subsequence{gn}of{fn}such that
(iv) ρ(f)≤lim infρ(fn)wheneverfn→f ρ-a.e. (Note:this property is equivalent to the Fatou property.)
We will also need the definition of the -property of a function modular; see,e.g., [, ].
Definition . Letρ∈ . We say thatρhas the-property if
sup n
ρ(fn,Dk)→
wheneverDk↓ ∅andsupnρ(fn,Dk)→.
The modular equivalents of uniform convexity were introduced in [].
Definition . Letρ∈ . We define the following uniform convexity type properties 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)=∅,
andδ(r,ε) = ifD(r,ε) =∅. We say thatρsatisfies (UC) if for everyr> ,ε> ,
δ(r,ε) > . Note that for everyr> ,D(r,ε)=∅forε> small enough. (ii) We say thatρsatisfies (UUC) if for everys≥,ε> there exists
η(s,ε) >
depending onsandεsuch that
δ(r,ε) >η(s,ε) > forr>s.
Let us also introduce the modular definition of strict convexity following [].
Definition . We say thatρis strictly convex (SC) ifρ(f) =ρ(g) and
ρ λf + ( –λ)g=λρ(f) + ( –λ)ρ(g)
imply thatf =g, whereλ∈(, ) andf,g∈Lρ.
Proposition . By Proposition.from[]it follows that ifρis(UUC),then it is also
(SC).
that a functionω: (,∞)×X×X→[,∞] is called a convex modular metric on a setX
if: (i)x=yif and only ifωλ(x,y) = for allλ> ; (ii)ωλ(x,y) =ωλ(y,x) for allλ> ,x∈X, y∈X; (iii)ωλ+μ(x,y)≤λ+λμωλ(x,z) +λμ+μωμ(z,y). In this context, givenx∈X, the modular metric space aroundx, denoted byXω(x), is defined as
Xω(x) =
x∈X:ωλ(x,x)→ asλ→ ∞
.
Furthermore,Xω(x) can be endowed with a metric given by
dω∗(x,y) =inf
λ> :ωλ(x,y)≤
.
Given a modular function spaceLρ, whereρis a convex function modular, it is not difficult
to demonstrate that the formula
ωλ(f,g) =ρ
f –g
λ
defines a modular metric onLρ. Moreover, we have
f=gρ=d∗ω(f,g)
for anyf,g∈Lρ.
Let us also introduce modular definitions of Lipschitzian and nonexpansive mappings and associated definitions of semigroups of nonlinear mappings acting within a modular function space.
Definition .[] Letρ∈ and let C⊂Lρ be nonempty andρ-closed. A mapping T:C→Cis calledρ-Lipschitzian if there exists a constantL> such that
ρ T(f) –T(g)≤Lρ(f–g) for anyf,g∈Lρ.
Tis called aρ-nonexpansive mapping ifL= .
For any mappingT, byF(T) we denote the set of all fixed points ofT. The following theorem is an immediate consequence of Theorem . in [].
Theorem . Assume that ρ ∈ is(UUC). Let C be a ρ-closed, ρ-bounded convex nonempty subset.Then anyρ-nonexpansive mapping T:C→C has a fixed point. More-over,the set of all fixed pointsFix(T)isρ-closed and convex.
Definition .[] A one-parameter familyF={Tt :t≥} of mappings fromCinto itself is said to be aρ-Lipschitzian (resp.ρ-nonexpansive) semigroup onCifF satisfies the following conditions:
(i) T(x) =xforx∈C;
(ii) Tt+s(x) =Tt(Ts(x))forx∈Candt,s≥;
Definition . A semigroupF={Tt :t≥}is called strongly continuous if for every
z∈C, the following function
z(t) =ρ Tt(z) –z
(.)
is continuous at everyt∈[,∞).
Definition . A semigroupF ={Tt :t≥} is called continuous if for everyz∈C, the mapping t−→Tt(z) isρ-continuous at everyt∈[,∞), i.e.,ρ(Ttn(z) –Tt(z))→ astn→t.
ByF(F) we denote the set of common fixed points of the semigroupF.
Let us finish this section with the existence theorem for semigroups of nonexpansive mappings acting in modular function spaces.
Theorem .[] Assume thatρ∈ is(UUC).Let C be aρ-closedρ-bounded convex nonempty subset.LetFbe a nonexpansive semigroup on C.Then the set F(F)of common fixed points is nonempty,ρ-closed and convex.
3 Representation theorems
Let us start with the following result which relates to Bruck’s theorem in Banach spaces, see [].
Theorem . Letρ∈ be a strictly convex function modular.Let C⊂Lρand let T and S be twoρ-nonexpansive mappings from C into X with a common fixed point.Then,for eachλ∈(, ),a mapping U:C→X defined by U(x) =λS(x) + ( –λ)T(x)for x∈X is
ρ-nonexpansive and F(U) =F(S)∩F(T).
Proof A straightforward calculation shows that the mappingU isρ-nonexpansive. It is also clear thatF(S)∩F(T)⊂F(U). Therefore, to complete the proof, we need only to prove the converse inclusion.
To this end, let us fixx∈F(U) andw∈F(S)∩F(T). Let us calculate:
ρ(x–w) =ρ λS(x) + ( –λ)T(x) –w
≤λρ S(x) –w+ ( –λ)ρ T(x) –w
=λρ S(x) –S(w)+ ( –λ)ρ T(x) –T(w)
≤λρ(x–w) + ( –λ)ρ(x–w). (.)
In particular, (.) yields the following
ρ(x–w) =ρ S(x) –w=ρ T(x) –w. (.)
Indeed, let us assume to the contrary that
Combining (.) with (.), we have
ρ(x–w)≤λρ S(x) –w+ ( –λ)ρ T(x) –w
<λρ(x–w) + ( –λ)ρ(x–w) =ρ(x–w), (.)
which is impossible. Since the same reasoning can be applied assuming thatρ(T(x) –w) < ρ(x–w), we conclude that the claim (.) holds. Setf =S(x) –wandg=T(x) –wand observe that (.) implies thatρ(f) =ρ(g). Straight calculation shows that
ρ λf+ ( –λ)g=ρ U(x) –w. (.)
On the other hand, it follows from (.) and from the assumptionx∈F(U) that
λρ(f) + ( –λ)ρ(g) =λρ S(x) –w+ ( –λ)ρ T(x) –w
=λρ(x–w) + ( –λ)ρ(x–w) =ρ(x–w) =ρ U(x) –w. (.)
Comparing (.) to (.), we obtain immediately
ρ λf+ ( –λ)g=λρ(f) + ( –λ)ρ(g), (.)
which by the strict convexity ofρimplies thatf =g, and consequently thatS(x) =T(x). Compute
x=U(x) =λS(x) + ( –λ)T(x) =λS(x) + ( –λ)S(x) =S(x), (.)
and hence x∈F(S). Similarly, we can prove thatx∈F(T). Hence, x∈F(S)∩F(T) as
claimed.
Theorem . Letρ∈ and letF={Tt:t≥}be a continuous semigroup of mappings
on a subset C of Lρ.Let{αn}be a sequence of nonnegative numbers converging toα∈[,∞) such thatαn=αfor all n∈N.Then the following representation of the set of all common fixed points ofFholds
F(F) =
∞
n=
F(Tαn). (.)
Proof We only need to prove that∞n=F(Tαn)⊂F(F) as the other direction is trivial. Let z∈Cbe such thatTαn(z) =zfor everyn∈N.
Observe first that if{tn}is a sequence of nonnegative real numbers such thatTtn(z) =z
andtn→t, wheret∈[,∞), thenTt(z) =z. Indeed,
ρ
Tt(z) –z
≤
ρ Tt(z) –Ttn(z)
+
ρ Ttn(z) –z
=
ρ Tt(z) –Ttn(z)
→, (.)
The above observation implies in particular thatTα(z) =z. Let us defineβn=|αn–α|> , wheren∈N. From assumptions it follows that eachβnis a positive real number and that βn→. Note that
max{αn,α}=min{αn,α}+βn. (.)
Hence, denotingmn=min{αn,α}andMn=max{αn,α}, we have
Tβn(z) =Tβn◦Tmn(z) =Tβn+mn(z) =TMn(z) =z. (.)
Fix anyt> . By Lemma in [], there exists a sequence{kn}inN∪ {}such that
t=
∞
i=
kiβi. (.)
Denoting
sn= n
i=
kiβi, (.)
we obtain for eachn∈Nwithsn>
Tsn(z) =T
kn βn◦T
kn–
βn–◦ · · · ◦T
k β ◦T
k
β(z) =z. (.)
SinceT(z) =z, it follows thatTsn(z) =zfor everyn∈N. Becausesn→tandFis a con-tinuous semigroup, we conclude, as previously observed, thatTt(z) =zwhich concludes
the proof of the theorem.
The following technical result about real numbers (Lemma in []) will be used in the proof of our next representation theorem.
Lemma .[] Letαandβbe positive real numbers satisfyingα/β∈/Q.Define sequences
{αn}in(,∞)and{kn}inNas follows:
(i) α=max{α,β};
(ii) α=min{α,β};
(iii) kn= [αn/αn+]for alln∈N;
(iv) αn+=αn–knαn+for alln∈N.
Then the following hold:
(a) <αn+<αnfor alln∈N;
(b) kn∈Nfor alln∈N;
(c) αn/αn+∈/Qfor alln∈N;
(d) {αn}converges to.
Theorem . Letρ∈ and letF={Tt:t≥}be a continuous semigroup of mappings
on a subset C of Lρ.Letα> andβ> be two real numbers such thatα/β∈/Q.Then
Proof We only need to prove that
F(Tα)∩F(Tβ)⊂F(F) (.)
as the converse inclusion is obvious. To this end, let us fixz∈Cbe such thatz∈F(Tα)∩ F(Tβ). Let{αn}in (,∞) and{kn}inNbe two sequences defined as in Lemma in [].
We will show that
Tαn(z) =Tαn+(z) =z (.)
for everyn∈N,
Tα(z) =z and Tα(z) =z. (.)
Thus, equation (.) holds forn= .
Now suppose thatTαn(z) =Tαn+(z) =z. Then we have
Tαn+(z) =Tαn+◦T
kn
αn+(z) =Tαn++knαn+(z) =Tαn(z) =Tαn+(z) =z. (.)
Hence, by induction,Tαn(z) =zfor everyn∈Nand consequently,
F(Tα)∩F(Tβ)⊂
∞
n=
F(Tαn). (.)
Since, by construction,{αn}is a sequence of positive numbers converging to ∈[,∞), it follows from Theorem . that
F(F) =
∞
n=
F(Tαn). (.)
Combining (.) with (.), we obtain the desired inclusion (.) which completes the
proof.
The next result is an immediate consequence of the fact that the uniform convexity (UUC) implies the strict convexity (SC) ofρ(Proposition .), and of Theorems ., . and ..
Theorem . Letρ∈ be(UUC),and letF={Tt:t≥}be a continuous semigroup of ρ-nonexpansive mappings on aρ-closed,ρ-bounded,convex,nonempty subset of Lρ.Let
α> andβ> be two real numbers such thatα/β∈/Q.Fix an arbitraryλ∈(, ).Then
F(F) =F λTα+ ( –λ)Tβ
. (.)
4 Convergence of Mann iteration processes
We concluded the previous section with Theorem . which says that, under suit-able assumptions, the set of all common fixed points of a continuous semigroup of ρ-nonexpansive mappings is nonempty and can be represented as the set of all fixed points of just oneρ-nonexpansive mapping. In this section we demonstrate how this result can be applied to the construction of such a common fixed point. This idea can be summarized as follows: using the results of the previous sections, we can reduce a problem of constructing a common fixed point for a semigroup of mappings to a problem of constructing a fixed point for just oneρ-nonexpansive mapping. There exist well-known algorithms for solv-ing the latter problem ussolv-ing generalized Mann and Ishikawa iteration processes, see []. In the current section, we prove the convergence of the Mann iterative process to a common fixed point of a continuous semigroup. Let us start with the definition of the Mann process, see [].
Definition . Letρ∈ ,C⊂Lρ, and letTbe aρ-nonexpansive self-mapping onC. Let
σ ∈(, ). The Mann iteration process generated by the mappingT and the constantσ, denoted byM(T,σ), is defined by the following iterative formula:
xk+=σT(xk) + ( –σ)xk, wherex∈Cis chosen arbitrarily. (.)
We will need the following technical results.
Lemma .[, ] Letρ∈ be(UUC)and letσ∈(, ).If there exists R> such that
lim sup n→∞
ρ(fn)≤R, lim sup n→∞
ρ(gn)≤R, (.)
lim
n→∞ρ σfn+ ( –σ)gn
=R, (.)
then
lim
n→∞ρ(fn–gn) = .
Lemma . Letρ∈ be(UUC),C⊂Lρ be aρ-closed,ρ-bounded and convex set.Let T:C→C beρ-nonexpansive,and letσ∈(, ).Denote by{xk}a sequence of elements of C generated by a Mann process M(T,σ).Assume that w is a fixed point of T.Then there exists r∈Rsuch that
lim
k→∞ρ(xk–w) =r. (.)
Proof Since
ρ(xk+–w)≤σρ T(xk) –w
+ ( –σ)ρ(xk–w)
=σρ T(xk) –T(w)
+ ( –σ)ρ(xk–w)
≤σρ(xk–w) + ( –σ)ρ(xk–w)
it follows that{xk}is a nonincreasing sequence of nonnegative numbers hence it is
con-vergent to a numberr∈R.
Lemma . Letρ∈ be(UUC),C⊂Lρ be aρ-closed,ρ-bounded and convex set.Let T:C→C beρ-nonexpansive,and letσ∈(, ).Denote by{xk}a sequence of elements of C generated by a Mann process M(T,σ).Then
lim
k→∞ρ T(xk) –xk
= (.)
and
lim
k→∞ρ(xk+–xk) = . (.)
Proof By Theorem .,Thas at least one fixed pointw∈C. In view of Lemma ., there existsr∈Rsuch that
lim
k→∞ρ(xk–w) =r. (.)
Note that
lim sup k→∞
ρ T(xk) –w=lim sup k→∞
ρ T(xk) –T(w)≤lim sup k→∞
ρ(xk–w)≤r, (.)
and that
lim
k→∞ρ σ T(xk) –w
+ ( –σ)(xk–w)
= lim
k→∞ρ(xk+–w) =r. (.)
Set fk = T(xk) – w, gk =xk – w, and note that lim supk→∞ρ(gk) ≤ r by (.), and lim supk→∞ρ(fk)≤rby (.). Observe also that
lim
k→∞ρ σfk+ ( –σ)gk
= lim
k→∞ρ σT(xk) + ( –σ)xk–w
= lim
k→∞ρ(xk+–w) =r. (.)
Hence, it follows from Lemma . that
lim
k→∞ρ T(xk) –xk
= lim
k→∞ρ(fk–gk) = , (.)
which by the construction of the sequence{xk}is equivalent to
lim
k→∞ρ(xk+–xk) = , (.)
as claimed.
Let us recall now the definition of the Opial property and the strong Opial property in modular function spaces [, ].
Definition . We say thatLρ satisfies theρ-a.e. Opial property if for every{fn} ∈Lρ
which isρ-a.e. convergent to such that there existsβ> for which
sup n
ρ(βfn)<∞, (.)
the following inequality holds for anyg∈Eρnot equal to
lim inf
n→∞ ρ(fn)≤lim infn→∞ ρ(fn+g). (.)
Definition . We say thatLρsatisfies theρ-a.e. strong Opial property if for every{fn} ∈ Lρwhich isρ-a.e. convergent to such that there existsβ> for which
sup n
ρ(βfn)<∞, (.)
the following equality holds for anyg∈Eρ
lim inf
n→∞ ρ(fn+g) =lim infn→∞ ρ(fn) +ρ(g). (.)
Remark . Note that theρ-a.e. strong Opial property implies theρ-a.e. Opial prop-erty [].
Remark . Also note that, by virtue of Theorem . in [], every convex, orthogonally additive function modularρhas theρ-a.e. strong Opial property. Let us recall thatρis called orthogonally additive ifρ(f,A∪B) =ρ(f,A) +ρ(f,B) wheneverA∩B=∅. Therefore, all Orlicz and Musielak-Orlicz spaces must have the strong Opial property.
Note that the Opial property in the norm sense does not necessarily hold for several classical Banach function spaces. For instance, the norm Opial property does not hold for
Lpspaces for ≤p= , while the modular strong Opial property holds inLpfor allp≥. The version of the demiclosedness principle we use in this paper requires the uniform continuity of the function modularρin the sense of the following definition (see,e.g., []).
Definition . We say thatρ∈ is uniformly continuous if for everyε> andL> ,
there existsδ> such that
ρ(g) –ρ(g+h)≤ε, (.)
providedρ(h) <δandρ(g)≤L.
Theorem .(Demiclosedness principle []) Letρ∈ .Assume that () ρis(UUC),
() ρhas the strong Opial property,
() ρhas theproperty and is uniformly continuous.
Let C⊂Lρbe nonempty,convex,stronglyρ-bounded andρ-closed,T:C→C beρ -non-expansive,and x∈C.If xn→xρ-a.e.andρ(T(xn) –xn)→,then x∈F(T).
We are now ready to prove the following version of the Mann process convergence the-orem for a singleρ-nonexpansive mapping.
Theorem . Letρ∈ .Assume that () ρis(UUC),
() ρhas the strong Opial property,
() ρhas theproperty and is uniformly continuous.
Let C⊂Lρ be nonempty,ρ-a.e.compact,convex,strongly ρ-bounded andρ-closed.Let T :C→C beρ-nonexpansive andσ ∈(, ).Denote by{xk}a sequence of elements of C generated by a Mann process M(T,σ).Then there exists x∈F(T)such that xn→xρ-a.e.
Proof Observe that by Theorem . the set of fixed pointsF(T) is nonempty, convex and ρ-closed. By Lemma . the sequence{xk}is an approximate fixed point sequence, that is,
ρ T(xk) –xk
→ (.)
ask→ ∞. Considery,z∈C, twoρ-a.e. cluster points of{xk}. There exist then{yk},{zk}
subsequences of{xk}such thatyk→yρ-a.e., andzk→zρ-a.e. By Theorem .,y∈F(T) andz∈F(T). By Lemma ., there existry,rz∈Rsuch that
ry= lim
k→∞ρ(xk–y), rz=klim→∞ρ(xk–z). (.)
We claim thaty=z. Assume to the contrary thaty=z. Then, by the strong Opial property, we have
ry=lim inf
k→∞ ρ(yk–y) <lim infk→∞ ρ(yk–z) =lim inf
k→∞ ρ(zk–z) <lim infk→∞ ρ(zk–y) =ry. (.) The contradiction implies thaty=z. Therefore,{xk}has at most oneρ-a.e. cluster point. SinceCisρ-a.e. compact, it follows that the sequence{xk}has exactly oneρ-a.e. cluster point, which means thatρ(xk)→xρ-a.e. Using Theorem . again, we getx∈F(T) as
claimed.
Let us combine now Theorem . with Theorem . to demonstrate the convergence of an iterative algorithm to a common fixed point of a semigroup of nonlinear mappings in modular function spaces.
() ρhas the strong Opial property,
() ρhas theproperty and is uniformly continuous.
Let C⊂Lρ be nonempty,ρ-a.e.compact,convex,strongly ρ-bounded andρ-closed.Let F ={Tt :t≥}be a continuous semigroup ofρ-nonexpansive mappings on C.Assume
thatα> andβ> are two real numbers such thatα/β∈/Q.Fixλ,κ∈(, )such that
κ+λ< .Define a sequence{xn}in C by x∈C and
xn+=κTα(xn) +λTβ(xn) + ( –κ–λ)xn (.)
for natural n≥.Then{xn}ρ-a.e.converges to a common fixed point of the semigroupF.
Proof Define a mappingSby
S= κ κ+λTα+
λ
κ+λTβ, (.)
and observe thatS:C→Cisρ-nonexpansive. Fix anyx∈Cand let{xn}be generated by the Mann processM(S,σ) whereσ=κ+λ
xn+=σS(xn) + ( –σ)xn, (.)
which is exactly the sequence defined by (.). By Theorem . there existsx∈F(S) such thatxn→xρ-a.e. By Theorem .F(F) =F(S), hencex∈F(F). The proof is complete.
5 Convergence of Ishikawa iteration processes
The Ishikawa iteration process [] is a two-step process generalization of the Mann pro-cess. From the numerical point of view, the Ishikawa iteration process provides more flex-ibility in defining the algorithm parameters, and hence providing a better control over the speed of convergence of the algorithm.
Definition . Letρ∈ ,C⊂Lρand letTbe aρ-nonexpansive self-mapping onC. Let
σ,τ ∈(, ). The Ishikawa iteration process generated by the mappingTand the constants σ andτ, denoted byI(T,σ,τ), is defined by the following iterative formula:
xk+=σT τT(xk) + ( –τ)xk
+ ( –σ)xk, wherex∈Cis chosen arbitrarily. (.)
Lemma . Letρ∈ be(UUC).Let C⊂Lρbe aρ-closed,ρ-bounded and convex set.Let T:C→C beρ-nonexpansive,and letσ,τ ∈(, ).Denote by{xk}a sequence of elements of C generated by the Ishikawa process I(T,σ,τ).Assume that w is a fixed point of T.Then there exists r∈Rsuch thatlimk→∞ρ(xk–w) =r.
Proof Using a similar calculation to the one used in the proof of Lemma ., it is not difficult to prove that{xk}is a nonincreasing sequence of nonnegative numbers, hence it
is convergent to a numberr∈R.
of C generated by the Ishikawa process I(T,σ,τ).Define
yk=τT(xk) + ( –τ)xk. (.)
Then
lim
k→∞ρ T(yk) –xk
= (.)
or,equivalently,
lim
k→∞ρ(xk+–xk) = . (.)
Proof By Theorem .,F(T)=∅. Let us fixw∈F(T). By Lemma .,limk→∞ρ(xk–w) exists. Let us denote it by r. Sincew∈F(T), T ∈Tr(C) and limk→∞ρ(xk–w) =r, by Lemma . we have the following:
lim sup
k→∞ ρ T(yk) –w
=lim sup
k→∞ ρ T(yk) –T(w)
≤lim sup
k→∞ ρ(yk–w)
=lim sup k→∞
ρ τT(xk) + ( –τ)xk–w
≤lim sup
k→∞ τρ T(xk) –w
+ ( –τ)ρ(xk–w)
≤lim sup k→∞
τρ(xk–w) + ( –τ)ρ(xk–w)
≤r. (.)
Note that
lim
k→∞ρ σ T(yk) –w
+ ( –σ)(xk–w)
= lim
k→∞ρ σT(yk) + ( –σ)xk–w
= lim
k→∞ρ(xk+–w) =r. (.)
Applying Lemma . withuk=T(yk) –wandvk=xk–w, we obtain the desired equality limk→∞ρ(T(yk) –xk) = , while (.) follows from (.) via the construction formulas for
xk+andyk.
Remark . Please note that Lemma . and Lemma . are special cases of analogous but more general results obtained for asymptotic pointwise nonexpansive mappings, see Lemma . and Lemma . in []. Since the proofs for theρ-nonexpansive mappings are much simpler, the authors decided to include them in the current paper for the sake of clarity and completeness.
Lemma . Letρ∈ be(UUC)satisfying.Let C⊂Lρbe aρ-closed,ρ-bounded and convex set.Let T:C→C beρ-nonexpansive,and letσ,τ ∈(, ).Denote by{xk}a se-quence of elements of C generated by the Ishikawa process I(T,σ,τ).Then
lim
k→∞ρ T(xk) –xk
Proof Letyk=τT(xk) + ( –τ)xk. Hence
T(xk) –xk=
–τ T(xk) –yk
. (.)
Sinceτ∈(, ), there exists <s< such thatτ≤s. Hence,
ρ T(xk) –xk
=ρ
–τ T(xk) –yk
≤ρ
–s T(xk) –yk
. (.)
The right-hand side of this inequality tends to zero because ρ(T(xk) –yk) → by
Lemma . and becauseρsatisfies.
Using Lemma . instead of Lemma ., and Lemma . instead of Lemma ., and arguing in a similar way as in the proof of Theorem ., we can obtain the following con-vergence result for the Ishikawa process.
Theorem . Letρ∈ .Assume that () ρis(UUC),
() ρhas the strong Opial property,
() ρhas theproperty and is uniformly continuous.
Let C⊂Lρbe nonempty,ρ-a.e.compact,convex,stronglyρ-bounded andρ-closed.Let T: C→C beρ-nonexpansive andσ∈(, ),τ∈(, ).Denote by{xk}a sequence of elements of C generated by an Ishikawa process I(T,σ,τ).Then there exists x∈F(T)such that xn→x ρ-a.e.
Again, let us combine Theorem . with Theorem . to demonstrate the convergence of an Ishikawa-type, two-step iterative algorithm to a common fixed point of a semigroup of nonlinear mappings in modular function spaces. Please note that, as said before, typ-ical implementations of the Ishikawa algorithm are convergent at a faster pace than the corresponding Mann schema.
Theorem . Letρ∈ .Assume that () ρis(UUC),
() ρhas the strong Opial property,
() ρhas theproperty and is uniformly continuous.
Let C⊂Lρ be nonempty,ρ-a.e.compact,convex,strongly ρ-bounded andρ-closed.Let F ={Tt :t≥}be a continuous semigroup ofρ-nonexpansive mappings on C.Assume
thatα> andβ> are two real numbers such thatα/β∈/Q.Fixλ,κ∈(, )such that
κ+λ< .Define a sequence{xn}in C by x∈C and
xn+=κTα κTα(xn) +λTβ(xn) + ( –κ–λ)xn
+λTβ κTα(xn) +λTβ(xn) + ( –κ–λ)xn
+ ( –κ–λ)xn (.)
for natural n≥.Then{xn}ρ-a.e.converges to a common fixed point of the semigroupF.
Proof Similarly as in the Mann process case, let us define a mappingSby
S= κ κ+λTα+
λ
and note thatS:C→Cisρ-nonexpansive. Fix anyx∈Cand let{xn}be generated by the two-step Ishikawa processI(S,σ,τ), whereσ=κ+λandτ=κ+λ,
xn+=σS τS(xn) + ( –τ)xn
+ ( –σ)xn, (.)
which is exactly the sequence defined by (.). By Theorem . there existsx∈F(S) such that xn→x ρ-a.e. By Theorem .F(F) =F(S), hencex∈F(F), which completes the
proof.
6 Application to construction of a stationary point of the Urysohn process
In this section we provide an example how the results of the preceding sections can be utilized for constructing a stationary point of a process defined by the Urysohn operator
T(f)(x) =
k x,y,f(y)dy+f(x),
wherefis a fixed function andf : [, ]→Ris Lebesgue measurable. For the kernelk, we assume that
(a) k: [, ]×[, ]×R+→R+is Lebesgue measurable,
(b) k(x,y, ) = ,
(c) k(x,y,·)is continuous, convex and increasing to+∞, (d) k(x,y,t)dx> fort> andy∈(, ).
Assume in addition that for almost allt∈[, ] and for any two measurable functionsf,g, there holds
k t,u,k u,v,f(v)–k u,v,g(v)dv
du
≤
k t,u,f(u) –g(u)du.
Settingρ(f) ={k(x,y,|f(y)|)dy}dxand using Jensen’s inequality, it is easy to show that ρis a convex function modular on the space of measurable functions defined in [, ], and thatρ(T(f) –T(g))≤ρ(f –g), that is,Tis nonexpansive with respect toρ. Let us fixr> and setC={f ∈Eρ:ρ(f–f)≤r}. It is easy to see thatT:C→C. If we assume additionally that there exist a constantM> and a Lebesgue-integrable functionh: [, ]×[, ]→ [,∞) such that for everyu≥ andx,y∈[, ],
k(x,y, u)≤Mk(x,y,u) +h(x,y),
then the modularρhas the propertyin the sense of Definition .. It can be shown, see [], that, givenf ∈C, the following initial value problem
⎧ ⎨ ⎩
u() =f,
u(t) + (I–T)u(t) = (.)
has a solutionuf : [, +∞]→C. As proved by Khamsi in [], the formula
defines the semigroup ofρ-nonexpansive mappings. Note thatρin this example is orthog-onally additive and hence it has the strong Opial property, see []. Therefore, assuming ρis (UUC) and uniformly continuous, see [, , ] for several criteria, we can use our methods (Theorem . and Theorem .) to construct a common fixed point of the semi-group{St}which will be a stationary point of the Urysohn process defined by the evolution function (t,f)→uf(t)∈C.
Competing interests
The authors declare that they have no competing interest.
Authors’ contributions
Both authors equally participated in all stages of preparations of this article. Both authors read and approved the final manuscript.
Author details
1Department of Mathematics, King Abdulaziz University, P.O. Box 138381, Jeddah, 21323, Saudi Arabia.2School of
Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia.
Acknowledgements
This research was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (400/130/1433). The authors, therefore, acknowledge with thanks the DSR financial support. The authors would like to thank the referee for the valuable suggestions to improve the presentation of the paper.
Received: 18 July 2013 Accepted: 4 December 2013 Published:03 Jan 2014
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