- Open Access
Common fixed points for pointwise Lipschitzian semigroups in modular function spaces
© Bin Dehaish et al.; licensee Springer. 2013
- Received: 28 February 2013
- Accepted: 24 July 2013
- Published: 12 August 2013
Let C be a ρ-bounded, ρ-closed, convex subset of a modular function space . We investigate the existence of common fixed points for asymptotic pointwise nonexpansive semigroups of nonlinear mappings , i.e. a family such that , and
where for every . In particular, we prove that if is uniformly convex, then the common fixed point is nonempty ρ-closed and convex.
MSC:47H09, 46B20, 47H10, 47E10.
- fixed point
- modular function space
- nonexpansive mapping
- Orlicz space
- pointwise Lipschitzian mapping
- pointwise nonexpansive mapping
- uniform convexity
The purpose of this paper is to prove the existence of common fixed points for semigroups of nonlinear mappings acting in modular function spaces which 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 the book by Kozlowski  for an extensive list of examples and special cases. Earlier studies of fixed point theory in modular function spaces can be found in [2–4], see also . Recently, Khamsi and Kozlowski presented a series of fixed point results for pointwise contractions, asymptotic pointwise contractions, pointwise nonexpansive and asymptotic pointwise nonexpansive mappings acting in modular functions spaces [6, 7] (all these should be considered in the modular sense, not in the sense of the corresponding norms). These results are also new and of a big interest, even in a much simpler context of ‘plain’ modular contractions and nonexpansive mappings, i.e., without any pointwise and asymptotic complications.
They also gave an example of a mapping which is ρ-nonexpansive, but it is not norm-nonexpansive. From this perspective, the fixed point theory in modular function spaces should be considered as complementary to the fixed point theory in normed spaces and in metric spaces.
Let us recall that a family of mappings forms a semigroup if and , see Definition 2.6 below for details. Such a situation is quite typical in mathematics and applications. For instance, in the theory of dynamical systems, the modular function space would define the state space and the mapping would represent the evolution function of a dynamical system. The question about the existence of common fixed points and about the structure of the set of common fixed points, can be interpreted as a question whether there exist points that are fixed during the state space transformation at 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 an infinite dimensional vector space. Therefore, it is natural to apply these results not only to deterministic dynamical systems but also to stochastic dynamical systems.
and for any . Kirk and Xu  proved the existence of fixed points for asymptotic pointwise contractions and asymptotic pointwise nonexpansive mappings in the Banach spaces, while Hussain and Khamsi extended this result to metric spaces  and Khamsi and Kozlowski to modular function spaces [6, 7]. Kozlowski in  and  proved convergence to fixed points of some iterative algorithms, applied to asymptotic pointwise nonexpansive mappings in the Banach spaces, and the existence of common fixed points of semigroups of asymptotic pointwise nonexpansive semigroups in the Banach spaces . Convergence of generalized Mann and Ishikawa algorithms to common points of such semigroups in Banach spaces was established in  and . In the context of modular function spaces, convergence to fixed points of some iterative algorithms, applied to asymptotic pointwise nonexpansive mappings, was proven by Bin Dehaish and Kozlowski in .
In this paper, we extend the definition of asymptotic pointwise nonexpansive mappings to semigroups of mappings and prove some common fixed point results in the context of modular function spaces. Therefore, our results generalize the results of Kozlowski , who proved the existence of common fixed points for semigroups of nonexpansive mappings in modular functions spaces, to the pointwise asymptotic semigroups. However, methods used in the current paper are substantially different due to the asymptotic behavior of semigroups in question. It is worth noting that existence of semigroups of nonexpansive mappings in modular function spaces was discussed by Khamsi  in the context of Musielak-Orlicz spaces and discussed applications to differential equations.
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 [6, 7, 22] or to the survey article , see also [1, 25, 26] for the standard framework of modular function spaces.
Let Ω be a nonempty set, and let Σ be a nontrivial σ-algebra of subsets of Ω. Let be a δ-ring of subsets of Ω such that for any and . Let us assume that there exists an increasing sequence of sets such that . By ℰ we denote the linear space of all simple functions with supports from . By we will denote the space of all extended measurable functions, i.e., all functions such that there exists a sequence , and for all . By we denote the characteristic function of the set A.
Definition 2.1 
ρ is monotone, i.e., for all implies , where ;
ρ is orthogonally subadditive, i.e., for any such that , ;
ρ has the Fatou property, i.e., for all implies , where ;
ρ is order continuous in ℰ, i.e., and implies .
Similarly, as in the case of measure spaces, we say that a set is ρ-null if for every . We say that a property holds ρ-almost everywhere if the exceptional set is ρ-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 a ρ-null set. With this in mind, we define , where each element is actually an equivalence class of functions equal ρ-a.e. rather than an individual function.
Definition 2.2 
We say that a regular function pseudomodular ρ is a regular convex function modular if implies ρ-a.e. The class of all nonzero regular convex function modulars, defined on Ω will be denoted by ℜ.
defines a norm, frequently called Luxembourg norm.
The following notions will be used throughout the paper.
Definition 2.4 
We say that is ρ-convergent to f and write if and only if .
A sequence , where , is called ρ-Cauchy if as .
We say that is ρ-complete if and only if any ρ-Cauchy sequence in is ρ-convergent.
A set is called ρ-closed if for any sequence of , the convergence implies that f belongs to B.
A set is called ρ-bounded if .
Since ρ fails in general the triangle identity, many of the known properties of limit may not extend to the ρ-convergence. For example, the ρ-convergence does not necessarily imply the ρ-Cauchy condition. However, it is important to remember that the ρ-limit is unique when it exists. The following proposition brings together few facts that will be often used in the proofs of our results.
Proposition 2.1 
ρ-balls are ρ-closed.
If for an , then there exists a subsequence of such that ρ-a.e.
, whenever ρ-a.e. (Note: this property is equivalent to the Fatou property.)
Let us recall the definition of an asymptotic pointwise nonexpansive mapping acting in a modular function space.
Definition 2.5 
- (i)a pointwise Lipschitzian mapping, if there exists such that
- (ii)an asymptotic pointwise nonexpansive, if there exists a sequence of mappings such that
and for any .
A point is called a fixed point of T, whenever . The set of fixed points of T will be denoted by .
This definition is now extended to a one-parameter family of mappings.
for and ;
- (iii)for each , is an asymptotic pointwise nonexpansive mapping, i.e., there exists a function such that(2.1)
for each , the mapping is ρ-continuous.
Note that without loss of generality, we may assume for any and , and .
The concept ρ-type is a powerful technical tool, which is used in the proofs of many fixed point results. The definition of a ρ-type is based on a given sequence. In this work, we generalize this definition to be adapted to one-parameter family of mappings.
- (1)A function is called a ρ-type (or shortly a type) if there exists a one-parameter family of elements of K such that for any there holds
- (2)Let τ be a type. A sequence is called a minimizing sequence of τ if
Note that τ is convex, provided ρ is convex.
Let us recall the modular equivalents of uniform convexity introduced in .
- (i)Let , . DefineLet
and if . We say that ρ satisfies (UC) if for every , , . Note that for every , , for small enough.
- (ii)We say that ρ satisfies (UUC) if there exists , for every , and such that
The following technical lemma is very useful throughout this paper (see  for its proof).
Then we must have .
The following property plays in the theory of modular function space a role similar to the reflexivity in the Banach spaces, see, e.g., .
Definition 3.3 We say that has property (R) if and only if every nonincreasing sequence of nonempty, ρ-bounded, ρ-closed and convex subsets of has a nonempty intersection.
Similarly as in the Banach space case, the modular uniform convexity implies the property (R).
Theorem 3.1 
Let be (UUC), then has a property (R).
The next lemma is the generalization of the minimizing sequence property for types defined by the sequences in Lemma 4.3 in  to the one-parameter semigroup case.
If , then .
Any minimizing sequence of τ is ρ-convergent. Moreover, the ρ-limit of is independent of the minimizing sequence.
Proof First, let us prove (i). Let such that . Let us consider two cases.
which implies as claimed.
which is impossible, since and . Therefore, we must have .
for any . If we let , we get . This contradiction implies that is ρ-Cauchy. Since is ρ-complete, we deduce that is ρ-convergent as claimed.
where f is the ρ-limit of and g is the ρ-limit of . Hence , i.e., . □
Using Lemma 3.2, we are ready to prove our common fixed point result for asymptotic pointwise nonexpansive semigroups.
Theorem 3.2 Assume is (UUC). Let C be a ρ-closed, ρ-bounded convex nonempty subset. Let be an asymptotic pointwise nonexpansive semigroup on C. Then ℱ has a common fixed point, and the set of common fixed points is ρ-closed and convex.
by letting , we get for any , i.e., . □
This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. (247-006-D1433). The authors, therefore, acknowledge with thanks technical and financial support of DSR.
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