R E S E A R C H
Open Access
Fixed point iteration processes for asymptotic
pointwise nonexpansive mapping in modular
function spaces
Buthinah A Bin Dehaish
1*and WM Kozlowski
2*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
LetLρbe a uniformly convex modular function space with a strong Opial property. LetT:C→Cbe an asymptotic pointwise nonexpansive mapping, whereCis a
ρ
-a.e. compact convex subset ofLρ. In this paper, we prove that the generalized Mann and Ishikawa processes converge almost everywhere to a fixed point ofT. In addition, we prove that ifCis compact in the strong sense, then both processes converge strongly to a fixed point.MSC: Primary 47H09; Secondary 47H10
Keywords: fixed point; nonexpansive mapping; fixed point iteration process; Mann process; Ishikawa process; modular function space; Orlicz space; Opial property; uniform convexity
1 Introduction
In , Kirk and Xu [] studied the existence of fixed points of asymptotic pointwise nonexpansive mappingsT:C→C,i.e.,
Tn(x) –Tn(y)≤αn(x)x–y,
wherelim supn→∞αn(x)≤, for allx,y∈C. Their main result (Theorem .) states that every asymptotic pointwise nonexpansive self-mapping of a nonempty, closed, bounded and convex subsetCof a uniformly convex Banach spaceXhas a fixed point. As pointed out by Kirk and Xu, asymptotic pointwise mappings seem to be a natural generalization of nonexpansive mappings. The conditions onαncan be for instance expressed in terms of the derivatives of iterations ofT for differentiableT. In these results were gener-alized by Hussain and Khamsi to metric spaces, [].
In , Khamsi and Kozlowski [] extended their result proving the existence of fixed points of asymptotic pointwiseρ-nonexpansive mappings acting in modular func-tion spaces. The proof of this important theorem is of the existential nature and does not describe any algorithm for constructing a fixed point of an asymptotic pointwiseρ -nonexpansive mapping. This paper aims at filling this gap.
Let us recall that modular function spaces are natural generalization of both func-tion and sequence variants of many important, from applicafunc-tions perspective, spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces
and many others, see the book by Kozlowski [] for an extensive list of examples and special cases. There exists an extensive literature on the topic of the fixed point theory in modular function spaces, see,e.g., [–, , , , –, ] and the papers referenced there.
It is well known that the fixed point construction iteration processes for generalized nonexpansive mappings have been successfully used to develop efficient and powerful nu-merical methods for solving various nonlinear equations and variational problems, often of great importance for applications in various areas of pure and applied science. There exists an extensive literature on the subject of iterative fixed point construction processes for asymptotically nonexpansive mappings in Hilbert, Banach and metric spaces, see,e.g., [, , , , , , , –, –] and the works referred there. Kozlowski proved conver-gence to fixed point of some iterative algorithms of asymptotic pointwise nonexpansive mappings in Banach spaces [] and the existence of common fixed points of semigroups of pointwise Lipschitzian mappings in Banach spaces []. Recently, weak and strong con-vergence of such processes to common fixed points of semigroups of mappings in Banach spaces has been demonstrated by Kozlowski and Sims [].
We would like to emphasize that all convergence theorems proved in this paper define constructive algorithms that can be actually implemented. When dealing with specific ap-plications of these theorems, one should take into consideration how additional properties of the mappings, sets and modulars involved can influence the actual implementation of the algorithms defined in this paper.
The paper is organized as follows:
(a) Section provides necessary preliminary material on modular function spaces. (b) Section introduces the asymptotic pointwise nonexpansive mappings and related
notions.
(c) Section deals with the Demiclosedness Principle which provides a critical stepping stone for proving almost everywhere convergence theorems.
(d) Section utilizes the Demiclosedness Principle to prove the almost everywhere convergence theorem for generalized Mann process.
(e) Section establishes the almost everywhere convergence theorem for generalized Ishikawa process.
(f ) Section provides the strong convergence theorem for both generalized Mann and Ishikawa processes for the case of a strongly compact setC.
2 Preliminaries
Letbe a nonempty set and be a nontrivialσ-algebra of subsets of. LetP be aδ -ring of subsets ofsuch thatE∩A∈P for anyE∈P andA∈. 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:
(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 define
M(,,P,ρ) =f ∈M∞;f(ω)<∞ρ-a.e., (.)
where eachf ∈M(,,P,ρ) is actually an equivalence class of functions equalρ-a.e. rather than an individual function. Where no confusion exists we will writeMinstead of
M(,,P,ρ).
Definition . Letρbe a regular function pseudomodular.
() We say thatρis a regular convex function semimodular ifρ(αf) = for everyα> impliesf= ρ-a.e.;
() We say thatρis 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.
Let us denoteρ(f,E) =ρ(fE) forf ∈M,E∈. It is easy to prove thatρ(f,E) is a func-tion pseudomodular in the sense of Def... in [] (more precisely, it is a funcfunc-tion pseu-domodular with the Fatou property). Therefore, we can use all results of the standard the-ory of modular function spaces as per the framework defined by Kozlowski in [–].
Remark . We limit ourselves to convex function modulars in this paper. However, omit-ting convexity in Definition . or replacing it by s-convexity would lead to the definition of nonconvex or s-convex regular function pseudomodulars, semimodulars and modulars as in [].
Definition .[–] Letρbe a convex function modular.
(a) A modular function space is the vector spaceLρ(,), or brieflyLρ, defined by
Lρ=f∈M;ρ(λf)→asλ→.
(b) The following formula defines a norm inLρ(frequently called Luxemurg norm):
fρ=infα> ;ρ(f/α)≤.
Theorem .[–] Letρ∈ .
() Lρ,fρis complete and the norm · ρis monotone w.r.t. the natural order inM. () fnρ→if and only ifρ(αfn)→for everyα> .
() Ifρ(αfn)→for anα> then there exists a subsequence{gn}of{fn}such that gn→ρ-a.e.
() If{fn}converges uniformly tof on a setE∈Pthenρ(α(fn–f),E)→for everyα> .
() Letfn→f ρ-a.e. There exists a nondecreasing sequence of setsHk∈Psuch that Hk↑and{fn}converges uniformly tof on everyHk(Egoroff theorem).
() ρ(f)≤lim infρ(fn)wheneverfn→f ρ-a.e. (Note: this property is equivalent to the
Fatou property.)
() DefiningLρ={f ∈Lρ;ρ(f,·)is order continuous}and Eρ={f ∈Lρ;λf ∈L
ρfor everyλ> }we have:
(a) Lρ⊃L ρ⊃Eρ,
(b) Eρhas the Lebesgue property, i.e.,ρ(αf,Dk)→forα> ,f ∈EρandDk↓ ∅.
(c) Eρis the closure ofE(in the sense of · ρ).
The following definition plays an important role in the theory of modular function spaces.
Definition . Letρ∈ . We say thatρhas the-property if
sup n
ρ(fn,Dk)→
wheneverDk↓ ∅andsupnρ(fn,Dk)→.
Theorem . Letρ∈ . The following conditions are equivalent:
(a) ρhas,
(b) L
ρis a linear subspace ofLρ,
(c) Lρ=Lρ=Eρ,
(d) ifρ(fn)→, thenρ(fn)→,
(e) ifρ(αfn)→for anα> , thenfnρ→, i.e., the modular convergence is
equivalent to the norm convergence.
We will also use another type of convergence which is situated between norm and mod-ular convergence. It is defined, among other important terms, in the following definition.
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∈Lρis 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}<∞.
(f ) A setB⊂Lρis calledρ-compact if for any{fn}inCthere exists a subsequence{fnk}
(g) A setC⊂Lρis calledρ-a.e. closed if for any{fn}inCwhichρ-a.e. converges to somef, then we must havef ∈C.
(h) A setC⊂Lρis calledρ-a.e. compact if for any{fn}inC, there exists a subsequence {fnk}whichρ-a.e. converges to somef ∈C.
(i) Letf ∈LρandC⊂Lρ. Theρ-distance betweenf andCis defined as
dρ(f,C) =infρ(f–g);g∈C.
Let us note thatρ-convergence does not necessarily implyρ-Cauchy condition. Also, fn→f does not imply in generalλfn→λf,λ> . Using Theorem ., it is not difficult to prove the following:
Proposition . Letρ∈ .
(i) Lρisρ-complete,
(ii) ρ-ballsBρ(x,r) ={y∈Lρ;ρ(x–y)≤r}areρ-closed andρ-a.e. closed.
Let us compare different types of compactness introduced in Definition ..
Proposition . Letρ∈ . The following relationships hold for sets C⊂Lρ:
(i) IfCisρ-compact, thenCisρ-a.e. compact. (ii) IfCis · ρ-compact, thenCisρ-compact.
(iii) Ifρsatisfies, then · ρ-compactness andρ-compactness are equivalent inLρ.
Proof
(i) follows from Theorem . part (). (ii) follows from Theorem . part ().
(iii) follows from (.) and from Theorem . part (e).
3 Asymptotic pointwise nonexpansive mappings
Let us recall the modular definitions of asymptotic pointwise nonexpansive mappings and associated notions, [].
Definition . Letρ∈ and letC⊂Lρbe nonempty andρ-closed. A mappingT:C→ Cis called an asymptotic pointwise mapping if there exists a sequence of mappingsαn: C→[,∞) such that
ρTn(f) –Tn(g)≤α
n(f)ρ(f –g) for anyf,g∈Lρ.
(i) Ifαn(f) = for everyf ∈Lρand everyn∈N, thenT is calledρ-nonexpansive or shortly nonexpansive.
(ii) If{αn}converges pointwise toα:C→[, ), thenT is called asymptotic pointwise contraction.
(iii) Iflim supn→∞αn(f)≤for anyf ∈Lρ, thenT is called asymptotic pointwise nonexpansive.
Denotingan(x) =max(αn(x), ), we note that without loss of generality we can assume that T is asymptotically pointwise nonexpansive if
ρTn(f) –Tn(g)≤an(f)ρ(f –g) for allf,g∈C,n∈N, (.) lim
n→∞an(f) = ,an(f)≥ for allf ∈C, andn∈N. (.)
Definebn(f) =an(f) – . In view of (.), we have
lim
n→∞bn(f) = . (.)
The above notation will be consistently used throughout this paper.
ByT(C) we will denote the class of all asymptotic pointwise nonexpansive mappings T:C→C.
In this paper, we will impose some restrictions on the behavior ofanandbn. This type of assumptions is typical for controlling the convergence of iterative processes for asymp-totically nonexpansive mappings, see,e.g., [].
Definition . DefineTr(C) as a class of allT∈T(C) such that
∞
n=
bn(x) <∞ for everyx∈C, (.)
anis a bounded function for everyn≥. (.)
We recall the following concepts related to the modular uniform convexity introduced in []:
Definition . Letρ∈ . We define the following uniform convexity type properties of the function modularρ: Lett∈(, ),r> ,ε> . Define
D(r,ε) =
(f,g);f,g∈Lρ,ρ(f)≤r,ρ(g)≤r,ρ(f–g)≥εr. Let
δt(r,ε) =inf
– rρ
tf+ ( –t)g; (f,g)∈D(r,ε)
, ifD(r,ε)=∅,
andδ(r,ε) = ifD(r,ε) =∅. We will use the following notational convention:δ=δ
.
Definition . We say thatρsatisfies (UC) if for everyr> ,ε> ,δ(r,ε) > . Note that for everyr> ,D(r,ε)=∅, forε> small enough. We say thatρsatisfies (UUC) if for everys≥,ε> there existsη(s,ε) > depending only onsandεsuch that
δ(r,ε) >η(s,ε) > for anyr>s.
Lemma . Letρ∈ be(UUC)and let t∈(, ). Then for every s> ,ε> there exists ηt(s,ε) > depending only on s andεsuch that
δt(r,ε) >ηt(s,ε) > for any r>s.
The notion of bounded away sequences of real numbers will be used extensively throughout this paper.
Definition . A sequence{tn} ⊂(, ) is called bounded away from if there exists < a< such thattn≥afor everyn∈N. Similarly,{tn} ⊂(, ) is called bounded away from if there exists <b< such thattn≤bfor everyn∈N.
We will need the following generalization of Lemma . from [] and being a modular equivalent of a norm property in uniformly convex Banach spaces, see,e.g., [].
Lemma . Letρ∈ be(UUC)and let{tk} ⊂(, )be bounded away fromand. If there exists R> such that
lim sup
n→∞ ρ(fn)≤R, lim supn→∞ ρ(gn)≤R, (.)
lim n→∞ρ
tnfn+ ( –tn)gn
=R, (.)
then
lim
n→∞ρ(fn–gn) = .
Proof Assume to the contrary that this is not the case and fix an arbitraryγ > . Passing to a subsequence if necessary, we may assume that there exists anε> such that
ρ(fn)≤R+γ, ρ(gn)≤R+γ, (.)
while
ρ(fn–gn)≥(R+γ)ε. (.)
Since{tn}is bounded away from and there exist <a<b< such thata≤tn≤bfor all naturaln. Passing to a subsequence if necessary, we can assume thattn→t∈[a,b]. For everyt∈[, ] andf,g∈D(R+γ,ε), let us defineλf,g(t) =ρ(tf + ( –t)g). Observe that the functionλf,g: [, ]→[,R+γ] is a convex function. Hence that the function
λ(t) =supλf,g(t) :f,g∈D(R+γ,ε)
(.)
is also convex on [, ], and consequently, it is a continuous function on [a,b]. Noting that
δt(R+γ,ε) = –
we conclude thatδt(R+γ,ε) is a continuous function oft∈[a,b]. Hence
lim n→∞δ
tn
(R+γ,ε) =δ t
(R+γ,ε). (.)
By (.) and (.)
δtn
(R+γ,ε)≤ – R+γρ
tnfn+ ( –tn)gn
. (.)
By (.) the left-hand side of (.) tends toδt
(R+γ,ε) asn→ ∞while the right-hand side tends toR+γγ in view of (.). Hence
δt
(R+γ,ε)≤
γ
R+γ. (.)
By (UUC) and by Lemma ., there existsηt
(R,ε) > satisfying
<ηt
(R,ε)≤δ t
(R+γ,ε). (.)
Combining (.) with (.) we get
<ηt
(R,ε)≤
γ
R+γ. (.)
Lettingγ→ we get a contradiction which completes the proof.
Let us introduce a notion of aρ-type, a powerful technical tool which will be used in the proofs of our fixed point results.
Definition . LetK⊂Lρbe convex andρ-bounded. A functionτ:K→[,∞] is called aρ-type (or shortly a type) if there exists a sequence{yn}of elements ofKsuch that for anyz∈Kthere holds
τ(z) =lim sup
n→∞ ρ(yn–z).
Note thatτis convex providedρis convex. A typical method of proof for the fixed point theorems in Banach and metric spaces is to construct a fixed point by finding an element on which a specific type function attains its minimum. To be able to proceed with this method, one has to know that such an element indeed exists. This will be the subject of Lemma . below. First, let us recall 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 exists aβ> for which
sup n
ρ(βfn)
<∞, (.)
the following inequality holds for anyg∈Eρnot equal to
lim inf
Definition . We say thatLρsatisfies theρ-a.e. strong Opial property if for every{fn} ∈ Lρwhich isρ-a.e. convergent to such that there exists aβ> 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 impliesρ-a.e. Opial property [].
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≥.
Lemma .[] Letρ∈ . Assume that Lρhas theρ-a.e. strong Opial property. Let C⊂ Eρbe a nonempty, stronglyρ-bounded andρ-a.e. compact convex set. Then anyρ-type defined in C attains its minimum in C.
Let us finish this section with the fundamental fixed point existence theorem which will be used in many places in the current paper.
Theorem . [] Assume ρ ∈ is (UUC). Let C be a ρ-closed ρ-bounded convex nonempty subset. Then any T:C→C asymptotically pointwise nonexpansive has a fixed point. Moreover, the set of all fixed pointsFix(T)isρ-closed.
4 Demiclosedness Principle
The following modular version of the Demiclosedness Principle will be used in the proof of our convergence Theorem .. Our proof the Demiclosedness Principle uses the paral-lelogram inequality valid in the modular spaces with the (UUC) property (see Lemma . in []). We start with a technical result which will be used in the proof of Theorem ..
Lemma . Let ρ∈ . Let C⊂Lρ be a convex set, and let T∈Tr(C). If {xk} is a ρ -approximate fixed point sequence for T, that is,ρ(T(xk) –xk)→as k→ ∞, then for every fixed m∈Nthere holds
ρ
Tm(x k) –xk m
→, (.)
Proof It follows from . that there exists a finite constantM> such that
m–
j=
supaj(x);x∈C
≤M. (.)
Using the convexity ofρand theρ-nonexpansiveness ofT, we get
ρ
Tm(xk) –xk m
=ρ
m
m–
j=
Tj+(xk) –Tj(xk)
≤
m m–
j=
ρTj+(xk) –Tj(xk)
≤ρT(xk) –xk
m– j=
aj(xn) +
≤
m(M+ )ρ
T(xk) –xk
→, (.)
ask→ ∞.
Corollary . If, under the hypothesis of Lemma .,ρsatisfies additionally the con-dition, thenρ(Tm(x
k) –xk)→as k→ ∞.
The version of the Demiclosedness Principle used in this paper (Theorem .) 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 to everyε> andL> , there existsδ> such that
ρ(g) –ρ(g+h)≤ε, (.)
providedρ(h) <δandρ(g)≤L.
Let us mention that the uniform continuity holds for a large class of function modulars. For instance, it can be proved that in Orlicz spaces over a finite atomless measure [] or in sequence Orlicz spaces [] the uniform continuity of the Orlicz modular is equivalent to the-type condition.
Theorem . Demiclosedness Principle. Letρ∈ . Assume that:
() ρis(UCC),
() ρhas strong Opial property,
() ρhasproperty and is uniformly continuous.
Let C⊂Lρ be a nonempty, convex, stronglyρ-bounded andρ-closed, and let T∈Tr(C). Let{xn} ⊂C, and x∈C. If xn→xρ-a.e. andρ(T(xn) –xn)→, then x∈F(T).
Proof Let us recall that by definition of uniform continuity ofρto everyε> andL> , there existsδ> such that
providedρ(h) <δandρ(g)≤L. Fix anym∈N. Noting thatρ(xn–x)≤M<∞due to the strongρ-boundedness ofCand thatρ(Tm(x
n) –xn)→ by Corollary (.), it follows from (.) withg=xn–xandh=Tm(xn) –xnthat
ρ(xn–x) –ρ
xn–x+Tm(xn) –xn→, (.) asn→ ∞. Hence
lim sup
n→∞ ρ(xn–x) =lim supn→∞ ρ
Tm(xn) –x
. (.)
Define theρ-typeϕby
ϕ(x) =lim sup
n→∞ ρ(xn–x). (.)
By (.) we get
ϕ(x) =lim sup
n→∞ ρ
Tm(xn) –x
. (.)
Hence, for everyy∈Cthere holds
ϕTm(y)=lim sup
n→∞ ρ
Tm(xn) –Tm(y)
≤am(y)lim sup
n→∞ ρ(xn–y) =am(y)ϕ(y). (.)
Using (.) withy=xand by passing withmto infinity, we conclude that
lim sup
m→∞ ϕ
Tm(x)≤ϕ(x). (.)
Sinceρsatisfies the strong Opial property, it also satisfies the Opial property. Sincexn→ xρ-a.e., it follows via the Opial property that for anyy=x
ϕ(x) =lim sup
n→∞ ρ(xn–x) <lim supn→∞ ρ(xn–y) =ϕ(y), (.)
which implies that
ϕ(x) =infϕ(y) :y∈C. (.) Combining (.) with (.), we have
ϕ(x)≤lim sup
m→∞
ϕTm(x)≤ϕ(x), (.)
that is,
lim sup
m→∞
ϕTm(x)=ϕ(x). (.)
We claim that
lim m→∞ρ
Assume to the contrary that (.) does not hold, that is,
ρTm(x) –xdoes not tend to zero. (.)
By, it follows from (.) thatρ(T
m(x)–x
) does not tend to zero. By passing to a subse-quence if necessary, we can assume that there exists <t<Msuch that
ρ
Tm(x) –x
>t> , (.)
form∈N, which implies that
ρ(xn–x) +ρ
xn–Tm(x)
> t
, (.)
for everym,n∈N. Hence,
maxρ(xn–x),ρ
xn–Tm(x)
≥ t
(.)
for everym,n∈N. Applying the modular parallelogram inequality valid in (UCC) mod-ular function spaces, see Lemma . in [],
ρ
z+y
≤
ρ (z) +
ρ (y) –
r,s, rρ(z–y)
, (.)
whereρ(z)≤r,ρ(y)≤randmax{ρ(z),ρ(y)} ≥sfor <s<r, withr=M,s=t,z=xn–x, y=Tm(x), we get
ρ
xn–
x+Tm(x)
≤
ρ (x
n–x) + ρ
x
n–Tm(x)
– M, t , Mρ
x–Tm(x). (.)
Note that by (.)
ϕ(x)≤ϕ
x+Tm(x)
=lim sup n→∞
ρ
xn–
x+Tm(x)
. (.)
Combining (.) with (.), we obtain
ϕ(x)≤ lim supn→∞
ρ(xn–x)
+
lim supn→∞ ρ
x
n–Tm(x)
– M, t , Mρ
x–Tm(x), (.)
which implies ≤ M, t , Mρ
x–Tm(x)≤ ϕ
Tm(x)– ϕ
Lettingm→ ∞and applying (.), we get
≤lim sup
m→∞
M, t ,
Mρ
x–Tm(x)
≤
lim supm→∞ ϕ
Tm(x)– ϕ
(x)≤. (.)
Using the properties of , we conclude that ρ(x–Tm(x)) tends to zero itself, which contradicts our assumption (.). Hence, ρ(x–Tm(x))→ asm→ ∞. Clearly, then
ρ(x–Tm+(x))→ asm→ ∞, that is,Tm+(x)→x(ρ) whileTm+(x)→T(x)(ρ) byρ -continuity ofT. By the uniqueness of theρ-limit, we obtainT(x) =x, that is,x∈F(T).
5 Convergence of generalized Mann iteration process
The following elementary, easy to prove, lemma will be used in this paper.
Lemma .[] Suppose{rk}is a bounded sequence of real numbers and{dk,n}is a doubly-index sequence of real numbers which satisfy
lim sup k→∞
lim sup n→∞
dk,n≤and rk+n≤rk+dk,n
for each k,n≥. Then{rk}converges to an r∈R.
Following Mann [], let us start with the definition of the generalized Mann iteration process.
Definition . LetT∈Tr(C) and let{nk}be an increasing sequence of natural numbers. Let {tk} ⊂(, ) be bounded away from and . The generalized Mann iteration pro-cess generated by the mappingT, the sequence{tk}, and the sequence{nk}denoted by gM(T,{tk},{nk}) is defined by the following iterative formula:
xk+=tkTnk(xk) + ( –tk)xk, wherex∈Cis chosen arbitrarily. (.)
Definition . We say that a generalized Mann iteration processgM(T,{tk},{nk}) is well defined if
lim sup
k→∞ ank(xk) = . (.)
Remark . Observe that by the definition of asymptotic pointwise nonexpansiveness, limk→∞ak(x) = for every x ∈C. Hence we can always select a subsequence {ank}
such that (.) holds. In other words, by a suitable choice of{nk}, we can always make gM(T,{tk},{nk}) well defined.
The following result provides an important technique which will be used in this paper.
and. Let w be a fixed point of T and gM(T,{tk},{nk})be a generalized Mann process. Then there exists r∈Rsuch that
lim
k→∞ρ(xk–w) =r. (.)
Proof Letw∈F(T). Since
ρ(xk+–w)≤tkρ
Tnk(x
k) –w
+ ( –tk)ρ(xk–w)
=tkρ
Tnk(x
k) –Tnk(w)
+ ( –tk)ρ(xk–w)
≤tk
+bnk(w)
ρ(xk–w) + ( –tk)ρ(xk–w)
=tkbnk(w)ρ(xk–w) +ρ(xk–w)
≤bnk(w)diamρ(C) +ρ(xk–w),
it follows that for everyn∈N,
ρ(xk+n–w)≤ρ(xk–w) +diamρ(C) k+n–
i=k
bni(w). (.)
Denoterp=ρ(xp–w) for everyp∈Nanddk,n=diamρ(C)
k+n–
i=k bni(w). Observe that
sinceT∈Tr(C), it follows thatlim supk→∞lim supn→∞dk,n= . By Lemma ., there exists anr∈Rsuch thatlimk→∞ρ(xk–w) =ras claimed.
The next result will be essential for proving the convergence theorems for iterative pro-cess.
Lemma . Letρ∈ be(UUC). Let C⊂Lρbe aρ-closed,ρ-bounded and convex set, and T∈Tr(C). Assume that a sequence{tk} ⊂(, )is bounded away fromand. Let {nk} ⊂Nand gM(T,{tk},{nk})be a generalized Mann iteration process. Then
lim k→∞ρ
Tnk(x
k) –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→∞
ρTnk(x
k) –w
=lim sup k→∞
ρTnk(x
k) –Tnk(w)
≤lim sup
and that
lim k→∞ρ
tk
Tnk(x
k) –w
+ ( –tk)(xk–w)
= lim
k→∞ρ(xk+–w) =r. (.)
Set fk =Tnk(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
tkfk+ ( –tk)gk
= lim k→∞ρ
tkTnk(xk) + ( –tk)xk–w
= lim
k→∞ρ(xk+–w) =r. (.)
Hence, it follows from Lemma . that
lim k→∞ρ
Tnk(x
k) –xk
= lim
k→∞ρ(fk–gk) = , (.)
which by the construction of the sequence{xk}is equivalent to
lim
k→∞ρ(xk+–xk) = , (.)
as claimed.
In the next lemma, we prove that under suitable assumption the sequence{xk}becomes an approximate fixed point sequence, which will provide an important step in the proof of the generalized Mann iteration process convergence. First, we need to recall the following notions.
Definition . A strictly increasing sequence{ni} ⊂Nis called quasi-periodic if the se-quence{ni+–ni} is bounded, or equivalently, if there exists a numberp∈Nsuch that any block ofpconsecutive natural numbers must contain a term of the sequence{ni}. The smallest of such numberspwill be called a quasi-period of{ni}.
Lemma . Letρ∈ be(UUC)satisfying. Let C⊂Lρbe aρ-closed,ρ-bounded and convex set, and T∈Tr(C). Let{tk} ⊂(, )be bounded away fromand. Let{nk} ⊂Nbe such that the generalized Mann process gM(T,{tk},{nk})is well defined. If, in addition, the set of indicesJ ={j;nj+= +nj}is quasi-periodic, then{xk}is an approximate fixed point sequence, i.e.,
lim k→∞ρ
T(xk) –xk
= . (.)
Proof Letp∈Nbe a quasi-period ofJ. Observe that it is enough to prove thatρ(T(xk) – xk)→ ask→ ∞throughJ. Indeed, let us fixε> . Fromρ(T(xk) –xk)→ ask→ ∞ throughJ it follows that
ρT(xk) –xk
for sufficiently large k. By the quasi-periodicity ofJ, to every positive integerk, there existsjk∈J such that|k–jk| ≤p. Assume thatk–p≤jk≤k(the proof for the other case is identical). SinceTisρ-Lipschitzian with the constantM=sup{a(x);x∈C}, there exist a <δ<ε
such that
ρT(x) –T(y)<ε ifρ(x–y) <δ. (.) Note that by (.) and by,ρ(p(xk+–xk)) <δp forksufficiently large. This implies that
ρ(xk–xjk)≤
p
ρp(xk–xk–)
+· · ·+ρp(xjk+–xjk)
≤pδ
p=δ, (.)
and therefore,
ρ
xk–T(xk)
≤
ρ(xk–xjk) +
ρ
xjk–T(xjk)
+ ρ
T(xjk) –T(xk)
≤δ+ε +
ε
<ε, (.)
which demonstrates that
ρ
xk–T(xk)
→ (.)
ask→ ∞. Byagain, we getρ(T(xk) –xk)→ ∞.
To prove thatρ(T(xk) –xk)→ ask→ ∞throughJ, observe that, sincenk+=nk+ for suchk, there holds
ρ
xk–T(xk)
≤
ρ(xk–xk+) + ρ
xk+–Tnk+(xk+)
+ ρ
Tnk+(x
k+) –Tnk+(xk)
+ ρ
TTnk(x
k) –T(xk)
≤
ρ(xk–xk+) + ρ
xk+–Tnk+(xk+)
+
ank+(xk+)ρ(xk–xk+) + Mρ
Tnk(x
k) –xk
(.)
which tends to zero in view of (.), (.) and (.).
The next theorem is the main result of this section.
Theorem . Letρ∈ . Assume that:
() ρis(UCC),
() ρhas Strong Opial Property,
() ρhasproperty and is uniformly continuous.
Proof Observe that by Theorem . in [], the set of fixed pointsF(T) is nonempty, con-vex andρ-closed. Note also that by Lemma . in [], it follows from the strong Opial property ofρthat anyρ-type attains its minimum inC. By Lemma ., the sequence{xk} is an approximate fixed point sequence, that is,
ρT(xk) –xk
→ (.)
ask→ ∞. Considery,z∈C, twoρ-a.e. cluster points of{xk}. There exits 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=limk→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. Since,Cisρ-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.
Remark . It is easy to see that we can always construct a sequence{nk}with the quasi-periodic properties specified in the assumptions of Theorem .. When constructing crete implementations of this algorithm, the difficulty will be to ensure that the con-structed sequence{nk}is not “too sparse” in the sense that the generalized Mann process gM(T,{tk},{nk}) remains well defined. The similar quasi-periodic type assumptions are common in the asymptotic fixed point theory, see,e.g., [, , ].
6 Convergence of generalized Ishikawa iteration process
The two-step Ishikawa iteration process is a generalization of the one-step Mann process. The Ishikawa iteration process, [], provides more flexibility in defining the algorithm parameters, which is important from the numerical implementation perspective.
Definition . LetT∈Tr(C) and let{nk}be an increasing sequence of natural numbers. Let{tk} ⊂(, ) be bounded away from and , and{sk} ⊂(, ) be bounded away from . The generalized Ishikawa iteration process generated by the mappingT, the sequences {tk},{sk}, and the sequence{nk}denoted bygI(T,{tk},{sk},{nk}) is defined by the following iterative formula:
xk+=tkTnk
skTnk(xk) + ( –sk)xk
+ ( –tk)xk,
Definition . We say that a generalized Ishikawa iteration processgI(T,{tk},{sk},{nk}) is well defined if
lim sup
k→∞ ank(xk) = . (.)
Remark . Observe that, by the definition of asymptotic pointwise nonexpansiveness, limk→∞ak(x) = for every x ∈C. Hence we can always select a subsequence {ank}
such that (.) holds. In other words, by a suitable choice of{nk}, we can always make gI(T,{tk},{sk},{nk}) well defined.
Lemma . Letρ∈ be(UUC). Let C⊂Lρbe aρ-closed,ρ-bounded and convex set. Let T∈Tr(C)and let{nk} ⊂N. Let{tk} ⊂(, )be bounded away fromand, and{sk} ⊂(, ) be bounded away from. Let w∈F(T)and gI(T,{tk},{sk},{nk})be a generalized Ishikawa process. There exists then an r∈Rsuch thatlimk→∞ρ(xk–w) =r.
Proof DefineTk:C→Cby
Tk(x) =tkTnk
skTnk(x) + ( –sk)x
+ ( –tk)x, x∈C. (.)
It is easy to see thatxk+=Tk(xk) and thatF(T)⊂F(Tk). Moreover, a straight calculation shows that eachTksatisfies
ρTk(x) –Tk(y)
≤Ak(x)ρ(x–y), (.)
where
Ak(x) = +tkank
Mk(x)
+skank(x) –sk
–tk, (.)
and
Mk(x) =skTnk(x) + ( –sk)x. (.)
Note thatAk(x)≥, which follows directly from the fact thatank(x)≥ and from (.).
Using (.) and the fact thatMk(w) =w, we have
Bk(w) =Ak(w) – =tk
+skank(w)
ank(w) –
≤ +ank(w)
bnk(w). (.)
Fix anyM> . Sincelimk→∞ank(w) = , it follows that there exists ak≥ such that for
k>k,ank(w)≤M. Therefore, using the same argument as in the proof of Lemma ., we
deduce that fork>kandn>
ρ(xk+n–w)≤ρ(xk–w) +diamρ(C) k+n–
i=k Bi(w)
≤ρ(xk–w) +diamρ(C)( +M) k+n–
i=k
Arguing like in the proof of Lemma ., we conclude that there exists anr∈Rsuch that
limk→∞ρ(xk–w) =r.
Lemma . Letρ∈ be(UUC). Let C⊂Lρbe aρ-closed,ρ-bounded and convex set. Let T∈Tr(C)and let{nk} ⊂N. Let{tk} ⊂(, )be bounded away fromand, and{sk} ⊂(, ) be bounded away from. Let gI(T,{tk},{sk},{nk})be a generalized Ishikawa process. Define
yk=skTnk(xk) + ( –sk)xk. (.)
Then
lim k→∞ρ
Tnk(y
k) –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) =rby Lemma ., we have the following:
lim sup k→∞
ρTnk(y
k) –w
=lim sup
k→∞ ρ
Tnk(y
k) –Tnk(w)
≤lim sup k→∞
ank(w)ρ(yk–w) =lim sup
k→∞
ank(w)ρ
skTnk(xk) + ( –sk)xk–w
≤lim sup k→∞
skank(w)ρ
Tnk(x
k) –w
+ ( –sk)ank(w)ρ(xk–w)
≤lim sup k→∞
skank(w)ρ(xk–w) + ( –sk)ank(w)ρ(xk–w)
≤r. (.)
Note that
lim k→∞ρ
tk
Tnk(y
k) –w
+ ( –tk)(xk–w)
= lim k→∞ρ
tkTnk(yk) + ( –tk)xk–w
= lim
k→∞ρ(xk+–w) =r. (.)
Applying Lemma . withuk=Tnk(yk) –wandvk=xk–w, we obtain the desired equality limk→∞ρ(Tnk(yk) –xk) = , while (.) follows from (.) via the construction formulas
forxk+andyk.
Lemma . Letρ∈ be(UUC)satisfying. Let C⊂Lρ be aρ-closed, ρ-bounded and convex set. Let T∈Tr(C)and let{nk} ⊂N. Let{tk} ⊂(, )be bounded away from and, and{sk} ⊂(, )be bounded away from. Let gI(T,{tk},{sk},{nk})be a well-defined generalized Ishikawa process. Then
lim k→∞ρ
Tnk(x
k) –xk
Proof Letyk=skTnk(xk) + ( –sk)xk. Hence
Tnk(x
k) –xk= –sk
Tnk(x
k) –yk
. (.)
Since{sk} ⊂(, ) is bounded away from , there exists <s< such thatsk≤sfor every k≥. Hence,
ρTnk(x
k) –xk
=ρ
–sk
Tnk(x
k) –yk
≤ρ
–s
Tnk(x
k) –yk
. (.)
The right-hand side of this inequality tends to zero because ρ(Tnk(x
k) –yk)→ by
Lemma . andρsatisfies.
Lemma . Letρ∈ be(UUC)satisfying. Let C⊂Lρ be aρ-closed, ρ-bounded and convex set, and T∈Tr(C). Let{tk} ⊂(, )be bounded away fromandand{sk} ⊂ (, )be bounded away from. Let{nk} ⊂Nbe such that the generalized Ishikawa process gI(T,{tk},{sk},{nk})is well defined. If, in addition, the setJ ={j;nj+= +nj} is quasi-periodic, then{xk}is an approximate fixed point sequence, i.e.,
lim k→∞ρ
T(xk) –xk
= . (.)
Proof The proof is analogous to that of Lemma . with (.) used instead of (.) and
(.) replacing (.).
Theorem . Letρ∈ . Assume that
() ρis(UCC),
() ρhas Strong Opial Property,
() ρhasproperty and is uniformly continuous.
Let C⊂Lρ be nonempty,ρ-a.e. compact, convex, stronglyρ-bounded andρ-closed, and let T ∈ Tr(C). Let T ∈Tr(C). Let {tk} ⊂ (, ) be bounded away from and , and {sk} ⊂(, ) be bounded away from . Let {nk} be such that the generalized Ishikawa process gI(T,{tk},{sk},{nk})is well defined. If, in addition, the setJ ={j;nj+= +nj}is quasi-periodic, then{xk}generated by gI(T,{tk},{sk},{nk})convergesρ-a.e. to a fixed point x∈F(T).
Proof The proof is analogous to that of Theorem . with Lemma . replaced by Lemma ., and Lemma . replaced by Lemma ..
7 Strong convergence
It is interesting that, providedCisρ-compact, both generalized Mann and Ishikawa pro-cesses converge strongly to a fixed point ofT even without assuming the Opial property.
there exists a fixed point x∈F(T)such that then{xk}generated by gM(T,{tk},{nk})(resp. gI(T,{tk},{sk},{nk})) converges strongly to a fixed point of T , that is
lim
k→∞ρ(xk–x) = . (.)
Proof By theρ-compactness ofC, we can select a subsequence{xpk}of{xk}such that there
existsx∈Cwith
lim k→∞ρ
T(xpk) –x
= . (.)
Note that
ρ
xpk–x
≤
ρ
xpk–T(xpk)
+ ρ
T(xpk–x)
, (.)
which tends to zero by Lemma . (resp. Lemma .) and by (.). Byit follows from (.) that
ρ(xpk–x)→ ask→ ∞. (.)
Observe that by the convexity ofρand byρ-nonexpansiveness ofT, we have
ρ
T(x) –x
≤
ρ
T(x) –T(xpk)
+ ρ
T(xpk) –xpk
+
ρ(xpk–x)
≤
ρ(x–xpk) +
ρ
T(xpk) –xpk
+
ρ(xpk–x), (.)
which tends to zero by (.) and by Lemma . (resp. Lemma .). Henceρ(T(x) –x) = which implies thatx∈F(T). Applying Lemma . (resp. Lemma .), we conclude that limk→∞ρ(xk–x) exists. By (.) this limit must be equal to zero which implies that
lim
k→∞ρ(xk–x) = . (.)
Remark . Observe that in view of theassumption, theρ-compactness of the setC assumed in Theorem . is equivalent to the compactness in the sense of the norm defined byρ.
Competing interests
The authors declare that they have no competing interests.
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 53909, Jeddah 21593, Saudi Arabia.2School of
Acknowledgements
The authors would like to thank MA Khamsi for his valuable suggestions to improve the presentation of the paper.
Received: 4 April 2012 Accepted: 2 July 2012 Published: 20 July 2012 References
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doi:10.1186/1687-1812-2012-118
