Abstract
Let {L}_{\rho} be a uniformly convex modular function space with a strong Opial property. Let T:C\to C be an asymptotic pointwise nonexpansive mapping, where C is a ρa.e. compact convex subset of {L}_{\rho}. In this paper, we prove that the generalized Mann and Ishikawa processes converge almost everywhere to a fixed point of T. In addition, we prove that if C is compact in the strong sense, then both processes converge strongly to a fixed point.
MSC:47H09, 47H10.
1 Introduction
In 2008, Kirk and Xu [21] studied the existence of fixed points of asymptotic pointwise nonexpansive mappings T:C\to C, i.e.,
where {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}(x)\le 1, for all x,y\in C. Their main result (Theorem 3.5) states that every asymptotic pointwise nonexpansive selfmapping of a nonempty, closed, bounded and convex subset C of a uniformly convex Banach space X has a fixed point. As pointed out by Kirk and Xu, asymptotic pointwise mappings seem to be a natural generalization of nonexpansive mappings. The conditions on {\alpha}_{n} can be for instance expressed in terms of the derivatives of iterations of T for differentiable T. In 2009 these results were generalized by Hussain and Khamsi to metric spaces, [9].
In 2011, Khamsi and Kozlowski [18] extended their result proving the existence of fixed points of asymptotic pointwise ρnonexpansive mappings acting in modular function 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 pointwise ρnonexpansive mapping. This paper aims at filling this gap.
Let us recall that modular function spaces are natural generalization of both function and sequence variants of many important, from applications perspective, spaces like Lebesgue, Orlicz, MusielakOrlicz, Lorentz, OrliczLorentz, CalderonLozanovskii spaces and many others, see the book by Kozlowski [24] 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., [3–5, 8, 13, 14, 17–20, 24] 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 numerical 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., [1, 2, 6, 7, 9, 12, 16, 30–36, 38–42] and the works referred there. Kozlowski proved convergence to fixed point of some iterative algorithms of asymptotic pointwise nonexpansive mappings in Banach spaces [25] and the existence of common fixed points of semigroups of pointwise Lipschitzian mappings in Banach spaces [26]. Recently, weak and strong convergence of such processes to common fixed points of semigroups of mappings in Banach spaces has been demonstrated by Kozlowski and Sims [28].
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 applications 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 2 provides necessary preliminary material on modular function spaces.

(b)
Section 3 introduces the asymptotic pointwise nonexpansive mappings and related notions.

(c)
Section 4 deals with the Demiclosedness Principle which provides a critical stepping stone for proving almost everywhere convergence theorems.

(d)
Section 5 utilizes the Demiclosedness Principle to prove the almost everywhere convergence theorem for generalized Mann process.

(e)
Section 6 establishes the almost everywhere convergence theorem for generalized Ishikawa process.

(f)
Section 7 provides the strong convergence theorem for both generalized Mann and Ishikawa processes for the case of a strongly compact set C.
2 Preliminaries
Let Ω be a nonempty set and Σ be a nontrivial σalgebra of subsets of Ω. Let \mathcal{P} be a δring of subsets of Ω such that E\cap A\in \mathcal{P} for any E\in \mathcal{P} and A\in \mathrm{\Sigma}. Let us assume that there exists an increasing sequence of sets {K}_{n}\in \mathcal{P} such that \mathrm{\Omega}=\bigcup {K}_{n}. By ℰ we denote the linear space of all simple functions with supports from \mathcal{P}. By {\mathcal{M}}_{\mathrm{\infty}} we will denote the space of all extended measurable functions, i.e., all functions f:\mathrm{\Omega}\to [\mathrm{\infty},\mathrm{\infty}] such that there exists a sequence \{{g}_{n}\}\subset \mathcal{E}, {g}_{n}\le f and {g}_{n}(\omega )\to f(\omega ) for all \omega \in \mathrm{\Omega}. By {1}_{A} we denote the characteristic function of the set A.
Definition 2.1 Let \rho :{\mathcal{M}}_{\mathrm{\infty}}\to [0,\mathrm{\infty}] be a nontrivial, convex and even function. We say that ρ is a regular convex function pseudomodular if:

(i)
\rho (0)=0;

(ii)
ρ is monotone, i.e., f(\omega )\le g(\omega ) for all \omega \in \mathrm{\Omega} implies \rho (f)\le \rho (g), where f,g\in {\mathcal{M}}_{\mathrm{\infty}};

(iii)
ρ is orthogonally subadditive, i.e., \rho (f{1}_{A\cup B})\le \rho (f{1}_{A})+\rho (f{1}_{B}) for any A,B\in \mathrm{\Sigma} such that A\cap B\ne \mathrm{\varnothing}, f\in {\mathcal{M}}_{\mathrm{\infty}};

(iv)
ρ has the Fatou property, i.e., {f}_{n}(\omega )\uparrow f(\omega ) for all \omega \in \mathrm{\Omega} implies \rho ({f}_{n})\uparrow \rho (f), where f\in {\mathcal{M}}_{\mathrm{\infty}};

(v)
ρ is order continuous in ℰ, i.e., {g}_{n}\in \mathcal{E} and {g}_{n}(\omega )\downarrow 0 implies \rho ({g}_{n})\downarrow 0.
Similarly, as in the case of measure spaces, we say that a set A\in \mathrm{\Sigma} is ρnull if \rho (g{1}_{A})=0 for every g\in \mathcal{E}. We say that a property holds ρalmost everywhere if the exceptional set is ρnull. As usual, we identify any pair of measurable sets whose symmetric difference is ρnull as well as any pair of measurable functions differing only on a ρnull set. With this in mind we define
where each f\in \mathcal{M}(\mathrm{\Omega},\mathrm{\Sigma},\mathcal{P},\rho ) is actually an equivalence class of functions equal ρa.e. rather than an individual function. Where no confusion exists we will write ℳ instead of \mathcal{M}(\mathrm{\Omega},\mathrm{\Sigma},\mathcal{P},\rho ).
Definition 2.2 Let ρ be a regular function pseudomodular.

(1)
We say that ρ is a regular convex function semimodular if \rho (\alpha f)=0 for every \alpha >0 implies f=0 ρa.e.;

(2)
We say that ρ is a regular convex function modular if \rho (f)=0 implies f=0 ρa.e.;
The class of all nonzero regular convex function modulars defined on Ω will be denoted by ℜ.
Let us denote \rho (f,E)=\rho (f{1}_{E}) for f\in \mathcal{M}, E\in \mathrm{\Sigma}. It is easy to prove that \rho (f,E) is a function pseudomodular in the sense of Def.2.1.1 in [24] (more precisely, it is a function pseudomodular with the Fatou property). Therefore, we can use all results of the standard theory of modular function spaces as per the framework defined by Kozlowski in [22–24].
Remark 2.1 We limit ourselves to convex function modulars in this paper. However, omitting convexity in Definition 2.1 or replacing it by sconvexity would lead to the definition of nonconvex or sconvex regular function pseudomodulars, semimodulars and modulars as in [24].
Let ρ be a convex function modular.

(a)
A modular function space is the vector space {L}_{\rho}(\mathrm{\Omega},\mathrm{\Sigma}), or briefly {L}_{\rho}, defined by
{L}_{\rho}=\{f\in \mathcal{M};\rho (\lambda f)\to 0\text{as}\lambda \to 0\}. 
(b)
The following formula defines a norm in {L}_{\rho} (frequently called Luxemurg norm):
{\parallel f\parallel}_{\rho}=inf\{\alpha >0;\rho (f/\alpha )\le 1\}.
In the following theorem, we recall some of the properties of modular spaces that will be used later on in this paper.
Let\rho \in \mathrm{\Re}.

(1)
{L}_{\rho}, {\parallel f\parallel}_{\rho} is complete and the norm {\parallel \cdot \parallel}_{\rho} is monotone w.r.t. the natural order in ℳ.

(2)
{\parallel {f}_{n}\parallel}_{\rho}\to 0 if and only if \rho (\alpha {f}_{n})\to 0 for every \alpha >0.

(3)
If \rho (\alpha {f}_{n})\to 0 for an \alpha >0 then there exists a subsequence \{{g}_{n}\} of \{{f}_{n}\} such that {g}_{n}\to 0 ρa.e.

(4)
If \{{f}_{n}\} converges uniformly to f on a set E\in \mathcal{P} then \rho (\alpha ({f}_{n}f),E)\to 0 for every \alpha >0.

(5)
Let {f}_{n}\to f ρa.e. There exists a nondecreasing sequence of sets {H}_{k}\in \mathcal{P} such that {H}_{k}\uparrow \mathrm{\Omega} and \{{f}_{n}\} converges uniformly to f on every {H}_{k} (Egoroff theorem).

(6)
\rho (f)\le lim\hspace{0.17em}inf\rho ({f}_{n}) whenever {f}_{n}\to f ρa.e. (Note: this property is equivalent to the Fatou property.)

(7)
Defining {L}_{\rho}^{0}=\{f\in {L}_{\rho};\rho (f,\cdot )\mathit{\text{is order continuous}}\} and {E}_{\rho}=\{f\in {L}_{\rho};\lambda f\in {L}_{\rho}^{0}\mathit{\text{for every}}\lambda 0\} we have:

(a)
{L}_{\rho}\supset {L}_{\rho}^{0}\supset {E}_{\rho},

(b)
{E}_{\rho} has the Lebesgue property, i.e., \rho (\alpha f,{D}_{k})\to 0 for \alpha >0, f\in {E}_{\rho} and {D}_{k}\downarrow \mathrm{\varnothing}.

(c)
{E}_{\rho} is the closure of ℰ (in the sense of {\parallel \cdot \parallel}_{\rho}).
The following definition plays an important role in the theory of modular function spaces.
Definition 2.4 Let \rho \in \mathrm{\Re}. We say that ρ has the {\mathrm{\Delta}}_{2}property if
whenever {D}_{k}\downarrow \mathrm{\varnothing} and {sup}_{n}\rho ({f}_{n},{D}_{k})\to 0.
Theorem 2.2 Let\rho \in \mathrm{\Re}. The following conditions are equivalent:

(a)
ρ has {\mathrm{\Delta}}_{2},

(b)
{L}_{\rho}^{0} is a linear subspace of {L}_{\rho},

(c)
{L}_{\rho}={L}_{\rho}^{0}={E}_{\rho},

(d)
if \rho ({f}_{n})\to 0, then \rho (2{f}_{n})\to 0,

(e)
if \rho (\alpha {f}_{n})\to 0 for an \alpha >0, then {\parallel {f}_{n}\parallel}_{\rho}\to 0, 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 modular convergence. It is defined, among other important terms, in the following definition.
Definition 2.5 Let \rho \in \mathrm{\Re}.

(a)
We say that \{{f}_{n}\} is ρconvergent to f and write {f}_{n}\to f(\rho ) if and only if \rho ({f}_{n}f)\to 0.

(b)
A sequence \{{f}_{n}\} where {f}_{n}\in {L}_{\rho} is called ρCauchy if \rho ({f}_{n}{f}_{m})\to 0 as n,m\to \mathrm{\infty}.

(c)
A set B\subset {L}_{\rho} is called ρclosed if for any sequence of {f}_{n}\in B, the convergence {f}_{n}\to f(\rho ) implies that f belongs to B.

(d)
A set B\subset {L}_{\rho} is called ρbounded if sup\{\rho (fg);f\in B,g\in B\}<\mathrm{\infty}.

(e)
A set B\subset {L}_{\rho} is called strongly ρbounded if there exists \beta >1 such that {M}_{\beta}(B)=sup\{\rho (\beta (fg));f\in B,g\in B\}<\mathrm{\infty}.

(f)
A set B\subset {L}_{\rho} is called ρcompact if for any \{{f}_{n}\} in C there exists a subsequence \{{f}_{{n}_{k}}\} and an f\in C such that \rho ({f}_{{n}_{k}}f)\to 0.

(g)
A set C\subset {L}_{\rho} is called ρa.e. closed if for any \{{f}_{n}\} in C which ρa.e. converges to some f, then we must have f\in C.

(h)
A set C\subset {L}_{\rho} is called ρa.e. compact if for any \{{f}_{n}\} in C, there exists a subsequence \{{f}_{{n}_{k}}\} which ρa.e. converges to some f\in C.

(i)
Let f\in {L}_{\rho} and C\subset {L}_{\rho}. The ρdistance between f and C is defined as
{d}_{\rho}(f,C)=inf\{\rho (fg);g\in C\}.
Let us note that ρconvergence does not necessarily imply ρCauchy condition. Also, {f}_{n}\to f does not imply in general \lambda {f}_{n}\to \lambda f, \lambda >1. Using Theorem 2.1, it is not difficult to prove the following:
Proposition 2.1 Let\rho \in \mathrm{\Re}.

(i)
{L}_{\rho} is ρcomplete,

(ii)
ρballs {B}_{\rho}(x,r)=\{y\in {L}_{\rho};\rho (xy)\le r\} are ρclosed and ρa.e. closed.
Let us compare different types of compactness introduced in Definition 2.5.
Proposition 2.2 Let\rho \in \mathrm{\Re}. The following relationships hold for setsC\subset {L}_{\rho}:

(i)
If C is ρcompact, then C is ρa.e. compact.

(ii)
If C is {\parallel \cdot \parallel}_{\rho}compact, then C is ρcompact.

(iii)
If ρ satisfies {\mathrm{\Delta}}_{2}, then {\parallel \cdot \parallel}_{\rho}compactness and ρcompactness are equivalent in {L}_{\rho}.
Proof

(i)
follows from Theorem 2.1 part (3).

(ii)
follows from Theorem 2.1 part (2).

(iii)
follows from (2.2) and from Theorem 2.2 part (e).
□
3 Asymptotic pointwise nonexpansive mappings
Let us recall the modular definitions of asymptotic pointwise nonexpansive mappings and associated notions, [18].
Definition 3.1 Let \rho \in \mathrm{\Re} and let C\subset {L}_{\rho} be nonempty and ρclosed. A mapping T:C\to C is called an asymptotic pointwise mapping if there exists a sequence of mappings {\alpha}_{n}:C\to [0,\mathrm{\infty}) such that

(i)
If {\alpha}_{n}(f)=1 for every f\in {L}_{\rho} and every n\in \mathbb{N}, then T is called ρnonexpansive or shortly nonexpansive.

(ii)
If \{{\alpha}_{n}\} converges pointwise to \alpha :C\to [0,1), then T is called asymptotic pointwise contraction.

(iii)
If {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}(f)\le 1 for any f\in {L}_{\rho}, then T is called asymptotic pointwise nonexpansive.

(iv)
If {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}(f)\le k for any f\in {L}_{\rho} with 0<k<1, then T is called strongly asymptotic pointwise contraction.
Denoting {a}_{n}(x)=max({\alpha}_{n}(x),1), we note that without loss of generality we can assume that T is asymptotically pointwise nonexpansive if
Define {b}_{n}(f)={a}_{n}(f)1. In view of (3.2), we have
The above notation will be consistently used throughout this paper.
By \mathcal{T}(C) we will denote the class of all asymptotic pointwise nonexpansive mappings T:C\to C.
In this paper, we will impose some restrictions on the behavior of {a}_{n} and {b}_{n}. This type of assumptions is typical for controlling the convergence of iterative processes for asymptotically nonexpansive mappings, see, e.g., [25].
Definition 3.2 Define {\mathcal{T}}_{r}(C) as a class of all T\in \mathcal{T}(C) such that
We recall the following concepts related to the modular uniform convexity introduced in [18]:
Definition 3.3 Let \rho \in \mathrm{\Re}. We define the following uniform convexity type properties of the function modular ρ: Let t\in (0,1), r>0, \epsilon >0. Define
Let
and {\delta}_{1}(r,\epsilon )=1 if {D}_{1}(r,\epsilon )=\mathrm{\varnothing}. We will use the following notational convention: {\delta}_{1}={\delta}_{1}^{\frac{1}{2}}.
Definition 3.4 We say that ρ satisfies (UC1) if for every r>0, \epsilon >0, {\delta}_{1}(r,\epsilon )>0. Note that for every r>0, {D}_{1}(r,\epsilon )\ne \mathrm{\varnothing}, for \epsilon >0 small enough. We say that ρ satisfies (UUC1) if for every s\ge 0, \epsilon >0 there exists {\eta}_{1}(s,\epsilon )>0 depending only on s and ε such that
We will need the following result whose proof is elementary. Note that for t=\frac{1}{2}, this result follows directly from Definition 3.4.
Lemma 3.1 Let\rho \in \mathrm{\Re}be(UUC1)and lett\in (0,1). Then for everys>0, \epsilon >0there exists{\eta}_{1}^{t}(s,\epsilon )>0depending only on s and ε such that
The notion of bounded away sequences of real numbers will be used extensively throughout this paper.
Definition 3.5 A sequence \{{t}_{n}\}\subset (0,1) is called bounded away from 0 if there exists 0<a<1 such that {t}_{n}\ge a for every n\in \mathbb{N}. Similarly, \{{t}_{n}\}\subset (0,1) is called bounded away from 1 if there exists 0<b<1 such that {t}_{n}\le b for every n\in \mathbb{N}.
We will need the following generalization of Lemma 4.1 from [18] and being a modular equivalent of a norm property in uniformly convex Banach spaces, see, e.g., [36].
Lemma 3.2 Let\rho \in \mathrm{\Re}be(UUC1)and let\{{t}_{k}\}\subset (0,1)be bounded away from 0 and 1. If there existsR>0such that
then
Proof Assume to the contrary that this is not the case and fix an arbitrary \gamma >0. Passing to a subsequence if necessary, we may assume that there exists an \epsilon >0 such that
while
Since \{{t}_{n}\} is bounded away from 0 and 1 there exist 0<a<b<1 such that a\le {t}_{n}\le b for all natural n. Passing to a subsequence if necessary, we can assume that {t}_{n}\to {t}_{0}\in [a,b]. For every t\in [0,1] and f,g\in {D}_{1}(R+\gamma ,\epsilon ), let us define {\lambda}_{f,g}(t)=\rho (tf+(1t)g). Observe that the function {\lambda}_{f,g}:[0,1]\to [0,R+\gamma ] is a convex function. Hence that the function
is also convex on [0,1], and consequently, it is a continuous function on [a,b]. Noting that
we conclude that {\delta}_{1}^{t}(R+\gamma ,\epsilon ) is a continuous function of t\in [a,b]. Hence
By (3.8) and (3.9)
By (3.12) the lefthand side of (3.13) tends to {\delta}_{1}^{{t}_{0}}(R+\gamma ,\epsilon ) as n\to \mathrm{\infty} while the righthand side tends to \frac{\gamma}{R+\gamma} in view of (3.7). Hence
By (UUC1) and by Lemma 3.1, there exists {\eta}_{1}^{{t}_{0}}(R,\epsilon )>0 satisfying
Combining (3.14) with (3.15) we get
Letting \gamma \to 0 we get a contradiction which completes the proof. □
Let us introduce a notion of a ρtype, a powerful technical tool which will be used in the proofs of our fixed point results.
Definition 3.6 Let K\subset {L}_{\rho} be convex and ρbounded. A function \tau :K\to [0,\mathrm{\infty}] is called a ρtype (or shortly a type) if there exists a sequence \{{y}_{n}\} of elements of K such that for any z\in K there holds
Note that τ is convex provided ρ is 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 3.3 below. First, let us recall the definition of the Opial property and the strong Opial property in modular function spaces, [15, 17].
Definition 3.7 We say that {L}_{\rho} satisfies the ρa.e. Opial property if for every \{{f}_{n}\}\in {L}_{\rho} which is ρa.e. convergent to 0 such that there exists a \beta >1 for which
the following inequality holds for any g\in {E}_{\rho} not equal to 0
Definition 3.8 We say that {L}_{\rho} satisfies the ρa.e. strong Opial property if for every \{{f}_{n}\}\in {L}_{\rho} which is ρa.e. convergent to 0 such that there exists a \beta >1 for which
the following equality holds for any g\in {E}_{\rho}
Remark 3.1 Note that the ρa.e. Strong Opial property implies ρa.e. Opial property [15].
Remark 3.2 Also, note that, by virtue of Theorem 2.1 in [15], every convex, orthogonally additive function modular ρ has the ρa.e. strong Opial property. Let us recall that ρ is called orthogonally additive if \rho (f,A\cup B)=\rho (f,A)+\rho (f,B) whenever A\cap B=\mathrm{\varnothing}. Therefore, all Orlicz and MusielakOrlicz 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 {L}^{p} spaces for 1\le p\ne 2 while the modular strong Opial property holds in {L}^{p} for all p\ge 1.
Lemma 3.3[27]
Let\rho \in \mathrm{\Re}. Assume that{L}_{\rho}has the ρa.e. strong Opial property. LetC\subset {E}_{\rho}be a nonempty, strongly ρbounded and ρa.e. compact convex set. Then any ρtype 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 3.1[18]
Assume\rho \in \mathrm{\Re}is(UUC1). Let C be a ρclosed ρbounded convex nonempty subset. Then anyT:C\to Casymptotically pointwise nonexpansive has a fixed point. Moreover, the set of all fixed pointsFix(T)is ρclosed.
4 Demiclosedness Principle
The following modular version of the Demiclosedness Principle will be used in the proof of our convergence Theorem 5.1. Our proof the Demiclosedness Principle uses the parallelogram inequality valid in the modular spaces with the (UUC1) property (see Lemma 4.2 in [18]). We start with a technical result which will be used in the proof of Theorem 4.1.
Lemma 4.1 Let\rho \in \mathrm{\Re}. LetC\subset {L}_{\rho}be a convex set, and letT\in {\mathcal{T}}_{r}(C). If\{{x}_{k}\}is a ρapproximate fixed point sequence for T, that is, \rho (T({x}_{k}){x}_{k})\to 0ask\to \mathrm{\infty}, then for every fixedm\in \mathbb{N}there holds
ask\to \mathrm{\infty}.
Proof It follows from 3.5 that there exists a finite constant M>0 such that
Using the convexity of ρ and the ρnonexpansiveness of T, we get
as k\to \mathrm{\infty}. □
Corollary 4.1 If, under the hypothesis of Lemma 4.1, ρ satisfies additionally the{\mathrm{\Delta}}_{2}condition, then\rho ({T}^{m}({x}_{k}){x}_{k})\to 0ask\to \mathrm{\infty}.
The version of the Demiclosedness Principle used in this paper (Theorem 4.1) requires the uniform continuity of the function modular ρ in the sense of the following definition (see, e.g., [17]).
Definition 4.1 We say that \rho \in \mathrm{\Re} is uniformly continuous if to every \epsilon >0 and L>0, there exists \delta >0 such that
provided \rho (h)<\delta and \rho (g)\le 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 [37] or in sequence Orlicz spaces [11] the uniform continuity of the Orlicz modular is equivalent to the {\mathrm{\Delta}}_{2}type condition.
Theorem 4.1 Demiclosedness Principle. Let\rho \in \mathrm{\Re}. Assume that:

(1)
ρ is (UCC1),

(2)
ρ has strong Opial property,

(3)
ρ has {\mathrm{\Delta}}_{2} property and is uniformly continuous.
LetC\subset {L}_{\rho}be a nonempty, convex, strongly ρbounded and ρclosed, and letT\in {\mathcal{T}}_{r}(C). Let\{{x}_{n}\}\subset C, andx\in C. If{x}_{n}\to xρa.e. and\rho (T({x}_{n}){x}_{n})\to 0, thenx\in F(T).
Proof Let us recall that by definition of uniform continuity of ρ to every \epsilon >0 and L>0, there exists \delta >0 such that
provided \rho (h)<\delta and \rho (g)\le L. Fix any m\in \mathbb{N}. Noting that \rho ({x}_{n}x)\le M<\mathrm{\infty} due to the strong ρboundedness of C and that \rho ({T}^{m}({x}_{n}){x}_{n})\to 0 by Corollary (4.1), it follows from (4.5) with g={x}_{n}x and h={T}^{m}({x}_{n}){x}_{n} that
as n\to \mathrm{\infty}. Hence
Define the ρtype φ by
By (4.7) we get
Hence, for every y\in C there holds
Using (4.10) with y=x and by passing with m to infinity, we conclude that
Since ρ satisfies the strong Opial property, it also satisfies the Opial property. Since {x}_{n}\to x\phantom{\rule{0.25em}{0ex}}\rhoa.e., it follows via the Opial property that for any y\ne x
which implies that
Combining (4.11) with (4.13), we have
that is,
We claim that
Assume to the contrary that (4.16) does not hold, that is,
By {\mathrm{\Delta}}_{2}, it follows from (4.17) that \rho (\frac{{T}^{m}(x)x}{2}) does not tend to zero. By passing to a subsequence if necessary, we can assume that there exists 0<t<M such that
for m\in \mathbb{N}, which implies that
for every m,n\in \mathbb{N}. Hence,
for every m,n\in \mathbb{N}. Applying the modular parallelogram inequality valid in (UCC1) modular function spaces, see Lemma 4.2 in [18],
where \rho (z)\le r, \rho (y)\le r and max\{\rho (z),\rho (y)\}\ge s for 0<s<r, with r=M, s=\frac{t}{4}, z={x}_{n}x, y={T}^{m}(x), we get
Note that by (4.13)
Combining (4.22) with (4.23), we obtain
which implies
Letting m\to \mathrm{\infty} and applying (4.15), we get
Using the properties of Ψ, we conclude that \rho (x{T}^{m}(x)) tends to zero itself, which contradicts our assumption (4.17). Hence, \rho (x{T}^{m}(x))\to 0 as m\to \mathrm{\infty}. Clearly, then \rho (x{T}^{m+1}(x))\to 0 as m\to \mathrm{\infty}, that is, {T}^{m+1}(x)\to x(\rho ) while {T}^{m+1}(x)\to T(x)(\rho ) by ρcontinuity of T. By the uniqueness of the ρlimit, we obtain T(x)=x, that is, x\in F(T). □
5 Convergence of generalized Mann iteration process
The following elementary, easy to prove, lemma will be used in this paper.
Lemma 5.1[2]
Suppose\{{r}_{k}\}is a bounded sequence of real numbers and\{{d}_{k,n}\}is a doublyindex sequence of real numbers which satisfy
for eachk,n\ge 1. Then\{{r}_{k}\}converges to anr\in \mathbb{R}.
Following Mann [29], let us start with the definition of the generalized Mann iteration process.
Definition 5.1 Let T\in {\mathcal{T}}_{r}(C) and let \{{n}_{k}\} be an increasing sequence of natural numbers. Let \{{t}_{k}\}\subset (0,1) be bounded away from 0 and 1. The generalized Mann iteration process generated by the mapping T, the sequence \{{t}_{k}\}, and the sequence \{{n}_{k}\} denoted by gM(T,\{{t}_{k}\},\{{n}_{k}\}) is defined by the following iterative formula:
Definition 5.2 We say that a generalized Mann iteration process gM(T,\{{t}_{k}\},\{{n}_{k}\}) is well defined if
Remark 5.1 Observe that by the definition of asymptotic pointwise nonexpansiveness, {lim}_{k\to \mathrm{\infty}}{a}_{k}(x)=1 for every x\in C. Hence we can always select a subsequence \{{a}_{{n}_{k}}\} such that (5.2) holds. In other words, by a suitable choice of \{{n}_{k}\}, we can always make gM(T,\{{t}_{k}\},\{{n}_{k}\}) well defined.
The following result provides an important technique which will be used in this paper.
Lemma 5.2 Let\rho \in \mathrm{\Re}be(UUC1). LetC\subset {L}_{\rho}be a ρclosed, ρbounded and convex set. LetT\in {\mathcal{T}}_{r}(C)and let\{{n}_{k}\}\subset \mathbb{N}. Assume that a sequence\{{t}_{k}\}\subset (0,1)is bounded away from 0 and 1. Let w be a fixed point of T andgM(T,\{{t}_{k}\},\{{n}_{k}\})be a generalized Mann process. Then there existsr\in \mathbb{R}such that
Proof Let w\in F(T). Since
it follows that for every n\in \mathbb{N},
Denote {r}_{p}=\rho ({x}_{p}w) for every p\in \mathbb{N} and {d}_{k,n}={diam}_{\rho}(C){\sum}_{i=k}^{k+n1}{b}_{{n}_{i}}(w). Observe that since T\in {\mathcal{T}}_{r}(C), it follows that {lim\hspace{0.17em}sup}_{k\to \mathrm{\infty}}{lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{d}_{k,n}=0. By Lemma 5.1, there exists an r\in \mathbb{R} such that {lim}_{k\to \mathrm{\infty}}\rho ({x}_{k}w)=r as claimed. □
The next result will be essential for proving the convergence theorems for iterative process.
Lemma 5.3 Let\rho \in \mathrm{\Re}be(UUC1). LetC\subset {L}_{\rho}be a ρclosed, ρbounded and convex set, andT\in {\mathcal{T}}_{r}(C). Assume that a sequence\{{t}_{k}\}\subset (0,1)is bounded away from 0 and 1. Let\{{n}_{k}\}\subset \mathbb{N}andgM(T,\{{t}_{k}\},\{{n}_{k}\})be a generalized Mann iteration process. Then
and
Proof By Theorem 3.1, T has at least one fixed point w\in C. In view of Lemma 5.2, there exists r\in \mathbb{R} such that
Note that
and that
Set {f}_{k}={T}^{{n}_{k}}({x}_{k})w, {g}_{k}={x}_{k}w, and note that {lim\hspace{0.17em}sup}_{k\to \mathrm{\infty}}\rho ({g}_{k})\le r by (5.7), and {lim\hspace{0.17em}sup}_{k\to \mathrm{\infty}}\rho ({f}_{k})\le r by (5.8). Observe also that
Hence, it follows from Lemma 3.2 that
which by the construction of the sequence \{{x}_{k}\} is equivalent to
as claimed. □
In the next lemma, we prove that under suitable assumption the sequence \{{x}_{k}\} 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 5.3 A strictly increasing sequence \{{n}_{i}\}\subset \mathbb{N} is called quasiperiodic if the sequence \{{n}_{i+1}{n}_{i}\} is bounded, or equivalently, if there exists a number p\in \mathbb{N} such that any block of p consecutive natural numbers must contain a term of the sequence \{{n}_{i}\}. The smallest of such numbers p will be called a quasiperiod of \{{n}_{i}\}.
Lemma 5.4 Let\rho \in \mathrm{\Re}be(UUC1)satisfying{\mathrm{\Delta}}_{2}. LetC\subset {L}_{\rho}be a ρclosed, ρbounded and convex set, andT\in {\mathcal{T}}_{r}(C). Let\{{t}_{k}\}\subset (0,1)be bounded away from 0 and 1. Let\{{n}_{k}\}\subset \mathbb{N}be such that the generalized Mann processgM(T,\{{t}_{k}\},\{{n}_{k}\})is well defined. If, in addition, the set of indices\mathcal{J}=\{j;{n}_{j+1}=1+{n}_{j}\}is quasiperiodic, then\{{x}_{k}\}is an approximate fixed point sequence, i.e.,
Proof Let p\in \mathbb{N} be a quasiperiod of \mathcal{J}. Observe that it is enough to prove that \rho (T({x}_{k}){x}_{k})\to 0 as k\to \mathrm{\infty} through \mathcal{J}. Indeed, let us fix \epsilon >0. From \rho (T({x}_{k}){x}_{k})\to 0 as k\to \mathrm{\infty} through \mathcal{J} it follows that
for sufficiently large k. By the quasiperiodicity of \mathcal{J}, to every positive integer k, there exists {j}_{k}\in \mathcal{J} such that k{j}_{k}\le p. Assume that kp\le {j}_{k}\le k (the proof for the other case is identical). Since T is ρLipschitzian with the constant M=sup\{{a}_{1}(x);x\in C\}, there exist a 0<\delta <\frac{\epsilon}{3} such that
Note that by (5.6) and by {\mathrm{\Delta}}_{2}, \rho (p({x}_{k+1}{x}_{k}))<\frac{\delta}{p} for k sufficiently large. This implies that
and therefore,
which demonstrates that
as k\to \mathrm{\infty}. By {\mathrm{\Delta}}_{2} again, we get \rho (T({x}_{k}){x}_{k})\to \mathrm{\infty}.
To prove that \rho (T({x}_{k}){x}_{k})\to 0 as k\to \mathrm{\infty} through \mathcal{J}, observe that, since {n}_{k+1}={n}_{k}+1 for such k, there holds
which tends to zero in view of (5.5), (5.6) and (5.2). □
The next theorem is the main result of this section.
Theorem 5.1 Let\rho \in \mathrm{\Re}. Assume that:

(1)
ρ is (UCC1),

(2)
ρ has Strong Opial Property,

(3)
ρ has {\mathrm{\Delta}}_{2} property and is uniformly continuous.
LetC\subset {L}_{\rho}be nonempty, ρa.e. compact, convex, strongly ρbounded and ρclosed, and letT\in {\mathcal{T}}_{r}(C). Assume that a sequence\{{t}_{k}\}\subset (0,1)is bounded away from 0 and 1. Let\{{n}_{k}\}\subset \mathbb{N}andgM(T,\{{t}_{k}\},\{{n}_{k}\})be a welldefined generalized Mann iteration process. Assume, in addition, that the set of indices\mathcal{J}=\{j;{n}_{j+1}=1+{n}_{j}\}is quasiperiodic. Then there existsx\in F(T)such that{x}_{n}\to xρa.e.
Proof Observe that by Theorem 4.1 in [18], the set of fixed points F(T) is nonempty, convex and ρclosed. Note also that by Lemma 3.1 in [27], it follows from the strong Opial property of ρ that any ρtype attains its minimum in C. By Lemma 5.4, the sequence \{{x}_{k}\} is an approximate fixed point sequence, that is,
as k\to \mathrm{\infty}. Consider y,z\in C, two ρa.e. cluster points of \{{x}_{k}\}. There exits then \{{y}_{k}\}, \{{z}_{k}\} subsequences of \{{x}_{k}\} such that {y}_{k}\to yρa.e. and {z}_{k}\to zρa.e. By Theorem 4.1, y\in F(T) and z\in F(T). By Lemma 5.2, there exist {r}_{y},{r}_{z}\in \mathbb{R} such that
We claim that y=z. Assume to the contrary that y\ne z. Then, by the strong Opial property, we have
The contradiction implies that y=z. Therefore, \{{x}_{k}\} has at most one ρa.e. cluster point. Since, C is ρa.e. compact it follows that the sequence \{{x}_{k}\} has exactly one ρa.e. cluster point, which means that \rho ({x}_{k})\to xρa.e. Using Theorem 4.1 again, we get x\in F(T) as claimed. □
Remark 5.2 It is easy to see that we can always construct a sequence \{{n}_{k}\} with the quasiperiodic properties specified in the assumptions of Theorem 5.1. When constructing concrete implementations of this algorithm, the difficulty will be to ensure that the constructed sequence \{{n}_{k}\} is not “too sparse” in the sense that the generalized Mann process gM(T,\{{t}_{k}\},\{{n}_{k}\}) remains well defined. The similar quasiperiodic type assumptions are common in the asymptotic fixed point theory, see, e.g., [2, 25, 28].
6 Convergence of generalized Ishikawa iteration process
The twostep Ishikawa iteration process is a generalization of the onestep Mann process. The Ishikawa iteration process, [10], provides more flexibility in defining the algorithm parameters, which is important from the numerical implementation perspective.
Definition 6.1 Let T\in {\mathcal{T}}_{r}(C) and let \{{n}_{k}\} be an increasing sequence of natural numbers. Let \{{t}_{k}\}\subset (0,1) be bounded away from 0 and 1, and \{{s}_{k}\}\subset (0,1) be bounded away from 1. The generalized Ishikawa iteration process generated by the mapping T, the sequences \{{t}_{k}\}, \{{s}_{k}\}, and the sequence \{{n}_{k}\} denoted by gI(T,\{{t}_{k}\},\{{s}_{k}\},\{{n}_{k}\}) is defined by the following iterative formula:
Definition 6.2 We say that a generalized Ishikawa iteration process gI(T,\{{t}_{k}\},\{{s}_{k}\},\{{n}_{k}\}) is well defined if
Remark 6.1 Observe that, by the definition of asymptotic pointwise nonexpansiveness, {lim}_{k\to \mathrm{\infty}}{a}_{k}(x)=1 for every x\in C. Hence we can always select a subsequence \{{a}_{{n}_{k}}\} such that (6.2) holds. In other words, by a suitable choice of \{{n}_{k}\}, we can always make gI(T,\{{t}_{k}\},\{{s}_{k}\},\{{n}_{k}\}) well defined.
Lemma 6.1 Let\rho \in \mathrm{\Re}be(UUC1). LetC\subset {L}_{\rho}be a ρclosed, ρbounded and convex set. LetT\in {\mathcal{T}}_{r}(C)and let\{{n}_{k}\}\subset \mathbb{N}. Let\{{t}_{k}\}\subset (0,1)be bounded away from 0 and 1, and\{{s}_{k}\}\subset (0,1)be bounded away from 1. Letw\in F(T)andgI(T,\{{t}_{k}\},\{{s}_{k}\},\{{n}_{k}\})be a generalized Ishikawa process. There exists then anr\in \mathbb{R}such that{lim}_{k\to \mathrm{\infty}}\rho ({x}_{k}w)=r.
Proof Define {T}_{k}:C\to C by
It is easy to see that {x}_{k+1}={T}_{k}({x}_{k}) and that F(T)\subset F({T}_{k}). Moreover, a straight calculation shows that each {T}_{k} satisfies
where
and
Note that {A}_{k}(x)\ge 1, which follows directly from the fact that {a}_{{n}_{k}}(x)\ge 1 and from (6.5). Using (6.5) and the fact that {M}_{k}(w)=w, we have
Fix any M>1. Since {lim}_{k\to \mathrm{\infty}}{a}_{{n}_{k}}(w)=1, it follows that there exists a {k}_{0}\ge 1 such that for k>{k}_{0}, {a}_{{n}_{k}}(w)\le M. Therefore, using the same argument as in the proof of Lemma 5.2, we deduce that for k>{k}_{0} and n>1
Arguing like in the proof of Lemma 5.2, we conclude that there exists an r\in \mathbb{R} such that {lim}_{k\to \mathrm{\infty}}\rho ({x}_{k}w)=r. □
Lemma 6.2 Let\rho \in \mathrm{\Re}be(UUC1). LetC\subset {L}_{\rho}be a ρclosed, ρbounded and convex set. LetT\in {\mathcal{T}}_{r}(C)and let\{{n}_{k}\}\subset \mathbb{N}. Let\{{t}_{k}\}\subset (0,1)be bounded away from 0 and 1, and\{{s}_{k}\}\subset (0,1)be bounded away from 1. LetgI(T,\{{t}_{k}\},\{{s}_{k}\},\{{n}_{k}\})be a generalized Ishikawa process. Define
Then
or equivalently
Proof By Theorem 3.1, F(T)\ne \mathrm{\varnothing}. Let us fix w\in F(T). By Lemma 6.1, {lim}_{k\to \mathrm{\infty}}\rho ({x}_{k}w) exists. Let us denote it by r. Since w\in F(T), T\in {\mathcal{T}}_{r}(C), and {lim}_{k\to \mathrm{\infty}}\rho ({x}_{k}w)=r by Lemma 6.1, we have the following:
Note that
Applying Lemma 3.2 with {u}_{k}={T}^{{n}_{k}}({y}_{k})w and {v}_{k}={x}_{k}w, we obtain the desired equality {lim}_{k\to \mathrm{\infty}}\rho ({T}^{{n}_{k}}({y}_{k}){x}_{k})=0, while (6.11) follows from (6.10) via the construction formulas for {x}_{k+1} and {y}_{k}. □
Lemma 6.3 Let\rho \in \mathrm{\Re}be(UUC1)satisfying{\mathrm{\Delta}}_{2}. LetC\subset {L}_{\rho}be a ρclosed, ρbounded and convex set. LetT\in {\mathcal{T}}_{r}(C)and let\{{n}_{k}\}\subset \mathbb{N}. Let\{{t}_{k}\}\subset (0,1)be bounded away from 0 and 1, and\{{s}_{k}\}\subset (0,1)be bounded away from 1. LetgI(T,\{{t}_{k}\},\{{s}_{k}\},\{{n}_{k}\})be a welldefined generalized Ishikawa process. Then
Proof Let {y}_{k}={s}_{k}{T}^{{n}_{k}}({x}_{k})+(1{s}_{k}){x}_{k}. Hence
Since \{{s}_{k}\}\subset (0,1) is bounded away from 1, there exists 0<s<1 such that {s}_{k}\le s for every k\ge 1. Hence,
The righthand side of this inequality tends to zero because \rho ({T}^{{n}_{k}}({x}_{k}){y}_{k})\to 0 by Lemma 6.2 and ρ satisfies {\mathrm{\Delta}}_{2}. □
Lemma 6.4 Let\rho \in \mathrm{\Re}be(UUC1)satisfying{\mathrm{\Delta}}_{2}. LetC\subset {L}_{\rho}be a ρclosed, ρbounded and convex set, andT\in {\mathcal{T}}_{r}(C). Let\{{t}_{k}\}\subset (0,1)be bounded away from 0 and 1 and\{{s}_{k}\}\subset (0,1)be bounded away from 1. Let\{{n}_{k}\}\subset \mathbb{N}be such that the generalized Ishikawa processgI(T,\{{t}_{k}\},\{{s}_{k}\},\{{n}_{k}\})is well defined. If, in addition, the set\mathcal{J}=\{j;{n}_{j+1}=1+{n}_{j}\}is quasiperiodic, then\{{x}_{k}\}is an approximate fixed point sequence, i.e.,
Proof The proof is analogous to that of Lemma 5.4 with (6.11) used instead of (5.6) and (6.14) replacing (5.5). □
Theorem 6.1 Let\rho \in \mathrm{\Re}. Assume that

(1)
ρ is (UCC1),

(2)
ρ has Strong Opial Property,

(3)
ρ has {\mathrm{\Delta}}_{2} property and is uniformly continuous.
LetC\subset {L}_{\rho}be nonempty, ρa.e. compact, convex, strongly ρbounded and ρclosed, and letT\in {\mathcal{T}}_{r}(C). LetT\in {\mathcal{T}}_{r}(C). Let\{{t}_{k}\}\subset (0,1)be bounded away from 0 and 1, and\{{s}_{k}\}\subset (0,1)be bounded away from 1. Let\{{n}_{k}\}be such that the generalized Ishikawa processgI(T,\{{t}_{k}\},\{{s}_{k}\},\{{n}_{k}\})is well defined. If, in addition, the set\mathcal{J}=\{j;{n}_{j+1}=1+{n}_{j}\}is quasiperiodic, then\{{x}_{k}\}generated bygI(T,\{{t}_{k}\},\{{s}_{k}\},\{{n}_{k}\})converges ρa.e. to a fixed pointx\in F(T).
Proof The proof is analogous to that of Theorem 5.1 with Lemma 5.4 replaced by Lemma 6.4, and Lemma 5.2 replaced by Lemma 6.1. □
7 Strong convergence
It is interesting that, provided C is ρcompact, both generalized Mann and Ishikawa processes converge strongly to a fixed point of T even without assuming the Opial property.
Theorem 7.1 Let\rho \in \mathrm{\Re}satisfy conditions(UUC1)and{\mathrm{\Delta}}_{2}. LetC\subset {L}_{\rho}be a ρcompact, ρbounded and convex set, and letT\in {\mathcal{T}}_{r}(C). Let\{{t}_{k}\}\subset (0,1)be bounded away from 0 and 1, and\{{s}_{k}\}\subset (0,1)be bounded away from 1. Let\{{n}_{k}\}be such that the generalized Mann processgM(T,\{{t}_{k}\},\{{n}_{k}\}) (resp. Ishikawa processgI(T,\{{t}_{k}\},\{{s}_{k}\},\{{n}_{k}\})) is well defined. Then there exists a fixed pointx\in F(T)such that then\{{x}_{k}\}generated bygM(T,\{{t}_{k}\},\{{n}_{k}\}) (resp. gI(T,\{{t}_{k}\},\{{s}_{k}\},\{{n}_{k}\})) converges strongly to a fixed point of T, that is
Proof By the ρcompactness of C, we can select a subsequence \{{x}_{{p}_{k}}\} of \{{x}_{k}\} such that there exists x\in C with
Note that
which tends to zero by Lemma 5.3 (resp. Lemma 6.4) and by (7.2). By {\mathrm{\Delta}}_{2} it follows from (7.3) that
Observe that by the convexity of ρ and by ρnonexpansiveness of T, we have
which tends to zero by (7.4) and by Lemma 5.3 (resp. Lemma 6.4). Hence \rho (T(x)x)=0 which implies that x\in F(T). Applying Lemma 5.2 (resp. Lemma 6.1), we conclude that {lim}_{k\to \mathrm{\infty}}\rho ({x}_{k}x) exists. By (7.4) this limit must be equal to zero which implies that
□
Remark 7.1 Observe that in view of the {\mathrm{\Delta}}_{2} assumption, the ρcompactness of the set C assumed in Theorem 7.1 is equivalent to the compactness in the sense of the norm defined by ρ.
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Acknowledgements
The authors would like to thank MA Khamsi for his valuable suggestions to improve the presentation of the paper.
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Dehaish, B.A.B., Kozlowski, W. Fixed point iteration processes for asymptotic pointwise nonexpansive mapping in modular function spaces. Fixed Point Theory Appl 2012, 118 (2012). https://doi.org/10.1186/168718122012118
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DOI: https://doi.org/10.1186/168718122012118
Keywords
 fixed point
 nonexpansive mapping
 fixed point iteration process
 Mann process
 Ishikawa process
 modular function space
 Orlicz space
 Opial property
 uniform convexity