Abstract
Let
MSC: 47H09, 47H10.
Keywords:
fixed point; nonexpansive mapping; fixed point iteration process; Mann process; Ishikawa process; modular function space; Orlicz space; Opial property; uniform convexity1 Introduction
In 2008, Kirk and Xu [21] studied the existence of fixed points of asymptotic pointwise nonexpansive mappings
where
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., [35,8,13,14,1720,24] 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., [1,2,6,7,9,12,16,3036,3842] 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
Definition 2.1 Let
(i)
(ii) ρ is monotone, i.e.,
(iii) ρ is orthogonally subadditive, i.e.,
(iv) ρ has the Fatou property, i.e.,
(v) ρ is order continuous in ℰ, i.e.,
Similarly, as in the case of measure spaces, we say that a set
where each
Definition 2.2 Let ρ be a regular function pseudomodular.
(1) We say that ρ is a regular convex function semimodular if
(2) We say that ρ is a regular convex function modular if
The class of all nonzero regular convex function modulars defined on Ω will be denoted by ℜ.
Let us denote
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
(b) The following formula defines a norm in
In the following theorem, we recall some of the properties of modular spaces that will be used later on in this paper.
Let
(1)
(2)
(3) If
(4) If
(5) Let
(6)
(7) Defining
(a)
(b)
(c)
The following definition plays an important role in the theory of modular function spaces.
Definition 2.4 Let
whenever
Theorem 2.2Let
(a) ρhas
(b)
(c)
(d) if
(e) if
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
(a) We say that
(b) A sequence
(c) A set
(d) A set
(e) A set
(f) A set
(g) A set
(h) A set
(i) Let
Let us note that ρconvergence does not necessarily imply ρCauchy condition. Also,
Proposition 2.1Let
(i)
(ii) ρballs
Let us compare different types of compactness introduced in Definition 2.5.
Proposition 2.2Let
(i) IfCisρcompact, thenCisρa.e. compact.
(ii) IfCis
(iii) Ifρsatisfies
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
(i) If
(ii) If
(iii) If
(iv) If
Denoting
Define
The above notation will be consistently used throughout this paper.
By
In this paper, we will impose some restrictions on the behavior of
Definition 3.2 Define
We recall the following concepts related to the modular uniform convexity introduced in [18]:
Definition 3.3 Let
Let
and
Definition 3.4 We say that ρ satisfies
We will need the following result whose proof is elementary. Note that for
Lemma 3.1Let
The notion of bounded away sequences of real numbers will be used extensively throughout this paper.
Definition 3.5 A sequence
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 3.2Let
then
Proof Assume to the contrary that this is not the case and fix an arbitrary
while
Since
is also convex on
we conclude that
By (3.8) and (3.9)
By (3.12) the lefthand side of (3.13) tends to
By
Combining (3.14) with (3.15) we get
Letting
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
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
the following inequality holds for any
Definition 3.8 We say that
the following equality holds for any
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
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
Lemma 3.3[27]
Let
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
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
Lemma 4.1Let
as
Proof It follows from 3.5 that there exists a finite constant
Using the convexity of ρ and the ρnonexpansiveness of T, we get
as
Corollary 4.1If, under the hypothesis of Lemma 4.1, ρsatisfies additionally the
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
provided
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
Theorem 4.1Demiclosedness Principle. Let
(1) ρis
(2) ρhas strong Opial property,
(3) ρhas
Let
Proof Let us recall that by definition of uniform continuity of ρ to every
provided
as
Define the ρtype φ by
By (4.7) we get
Hence, for every
Using (4.10) with
Since ρ satisfies the strong Opial property, it also satisfies the Opial property. Since
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
for
for every
for every
where
Note that by (4.13)
Combining (4.22) with (4.23), we obtain
which implies
Letting
Using the properties of Ψ, we conclude that
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
for each
Following Mann [29], let us start with the definition of the generalized Mann iteration process.
Definition 5.1 Let
Definition 5.2 We say that a generalized Mann iteration process
Remark 5.1 Observe that by the definition of asymptotic pointwise nonexpansiveness,
The following result provides an important technique which will be used in this paper.
Lemma 5.2Let
Proof Let
it follows that for every
Denote
The next result will be essential for proving the convergence theorems for iterative process.
Lemma 5.3Let
and
Proof By Theorem 3.1, T has at least one fixed point
Note that
and that
Set
Hence, it follows from Lemma 3.2 that
which by the construction of the sequence
as claimed. □
In the next lemma, we prove that under suitable assumption the sequence
Definition 5.3 A strictly increasing sequence
Lemma 5.4Let
Proof Let
for sufficiently large k. By the quasiperiodicity of
Note that by (5.6) and by
and therefore,
which demonstrates that
as
To prove that
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 5.1Let
(1) ρis
(2) ρhas Strong Opial Property,
(3) ρhas
Let
Proof Observe that by Theorem 4.1 in [18], the set of fixed points
as
We claim that
The contradiction implies that
Remark 5.2 It is easy to see that we can always construct a sequence
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
Definition 6.2 We say that a generalized Ishikawa iteration process
Remark 6.1 Observe that, by the definition of asymptotic pointwise nonexpansiveness,
Lemma 6.1Let
Proof Define
It is easy to see that
where
and
Note that
Fix any
Arguing like in the proof of Lemma 5.2, we conclude that there exists an
Lemma 6.2Let
Then
or equivalently
Proof By Theorem 3.1,
Note that
Applying Lemma 3.2 with
Lemma 6.3Let
Proof Let
Since
The righthand side of this inequality tends to zero because
Lemma 6.4Let
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 6.1Let
(1) ρis
(2) ρhas Strong Opial Property,
(3) ρhas
Let
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 7.1Let
Proof By the ρcompactness of C, we can select a subsequence
Note that
which tends to zero by Lemma 5.3 (resp. Lemma 6.4) and by (7.2). By
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
□
Remark 7.1 Observe that in view of the
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.
Acknowledgements
The authors would like to thank MA Khamsi for his valuable suggestions to improve the presentation of the paper.
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