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Approximating fixed points of multivalued ρ-nonexpansive mappings in modular function spaces
Fixed Point Theory and Applications volume 2014, Article number: 34 (2014)
The existence of fixed points of single-valued mappings in modular function spaces has been studied by many authors. The approximation of fixed points in such spaces via convergence of an iterative process for single-valued mappings has also been attempted very recently by Dehaish and Kozlowski (Fixed Point Theory Appl. 2012:118, 2012). In this paper, we initiate the study of approximating fixed points by the convergence of a Mann iterative process applied on multivalued ρ-nonexpansive mappings in modular function spaces. Our results also generalize the corresponding results of (Dehaish and Kozlowski in Fixed Point Theory Appl. 2012:118, 2012) to the case of multivalued mappings.
MSC:47H09, 47H10, 54C60.
1 Introduction and preliminaries
The theory of modular spaces was initiated by Nakano  in connection with the theory of ordered spaces, which was further generalized by Musielak and Orlicz . The fixed point theory for nonlinear mappings is an important subject of nonlinear functional analysis and is widely applied to nonlinear integral equations and differential equations. The study of this theory in the context of modular function spaces was initiated by Khamsi et al.  (see also [4–8]). Kumam  obtained some fixed point theorems for nonexpansive mappings in arbitrary modular spaces. Kozlowski  has contributed a lot towards the study of modular function spaces both on his own and with his collaborators. Of course, most of the work done on fixed points in these spaces was of existential nature. No results were obtained for the approximation of fixed points in modular function spaces until recently Dehaish and Kozlowski  tried to fill this gap using a Mann iterative process for asymptotically pointwise nonexpansive mappings.
All above work has been done for single-valued mappings. On the other hand, the study of fixed points for multivalued contractions and nonexpansive mappings using the Hausdorff metric was initiated by Markin  (see also ). Later, an interesting and rich fixed point theory for such maps was developed which has applications in control theory, convex optimization, differential inclusion, and economics (see  and references cited therein). Moreover, the existence of fixed points for multivalued nonexpansive mappings in uniformly convex Banach spaces was proved by Lim . The theory of multivalued nonexpansive mappings is harder than the corresponding theory of single-valued nonexpansive mappings. Different iterative processes have been used to approximate the fixed points of multivalued nonexpansive mappings in Banach spaces.
Dhompongsa et al.  have proved that every ρ-contraction has a fixed point where ρ is a convex function modular satisfying the so-called -type condition, C is a nonempty ρ-bounded ρ-closed subset of and a family of ρ-closed subsets of C. By using this result, they asserted the existence of fixed points for multivalued ρ-nonexpansive mappings. Again their results are existential in nature. See also Kutbi and Latif .
In this paper, we approximate fixed points of ρ-nonexpansive multivalued mappings in modular function spaces using a Mann iterative process. We make the first ever effort to fill the gap between the existence and the approximation of fixed points of ρ-nonexpansive multivalued mappings in modular function spaces. In a way, the corresponding results of Dehaish and Kozlowski  are also generalized to the case of multivalued mappings.
Some basic facts and notation needed in this paper are recalled as follows.
Let Ω be a nonempty set and Σ 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 (for instance, can be the class of sets of finite measure in a σ-finite measure space). By , we denote the characteristic function of the set A in Ω. 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 .
Definition 1 Let be a nontrivial, convex and even function. We say that ρ is a regular convex function pseudomodular if
ρ is monotone, i.e., for any implies , where ;
ρ is orthogonally subadditive, i.e., for any such that , ;
ρ has Fatou property, i.e., for all implies , where ;
ρ is order continuous in ℰ, i.e., , and implies .
A set is said to be ρ-null if for every . A property is said to hold ρ-almost everywhere (ρ-a.e.) if the 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 is actually an equivalence class of functions equal ρ-a.e. rather than an individual function. Where no confusion exists we will write ℳ instead of .
Definition 2 Let ρ be a regular function pseudomodular. We say that ρ is a regular convex function modular if implies ρ-a.e.
It is known (see ) that ρ satisfies the following properties:
for every scalar α with and .
if , and .
ρ is called a convex modular if, in addition, the following property is satisfied:
(3′) if , and .
Definition 3 The convex function modular ρ defines the modular function space as
Generally, the modular ρ is not subadditive and therefore does not behave as a norm or a distance. However, the modular space can be equipped with an F-norm defined by
In the case ρ is convex modular,
defines a norm on the modular space , and it is called the Luxemburg norm.
The following uniform convexity type properties of ρ can be found in .
Definition 4 Let ρ be a nonzero regular convex function modular defined on Ω. Let , , . Define
and if .
As a conventional notation, .
Definition 5 A nonzero regular convex function modular ρ is said to satisfy if for every , , . Note that for every , for small enough. ρ is said to satisfy if for every , , there exists depending only upon s and ε such that for any .
Definition 6 Let be a modular space. The sequence is called:
ρ-convergent to if as ;
ρ-Cauchy, if as n and .
Consistent with , the ρ-distance from an to a set is given as follows:
Definition 7 A subset is called:
ρ-closed if the ρ-limit of a ρ-convergent sequence of D always belongs to D;
ρ-a.e. closed if the ρ-a.e. limit of a ρ-a.e. convergent sequence of D always belongs to D;
ρ-compact if every sequence in D has a ρ-convergent subsequence in D;
ρ-a.e. compact if every sequence in D has a ρ-a.e. convergent subsequence in D;
A set is called ρ-proximinal if for each there exists an element such that . We shall denote the family of nonempty ρ-bounded ρ-proximinal subsets of D by , the family of nonempty ρ-closed ρ-bounded subsets of D by and the family of ρ-compact subsets of D by . Let be the ρ-Hausdorff distance on , that is,
A multivalued mapping is said to be ρ-nonexpansive if
A sequence is called bounded away from 0 if there exists such that for every . Similarly, is called bounded away from 1 if there exists such that for every .
Lemma 1 (Lemma 3.2 )
Let ρ satisfy and let be bounded away from 0 and 1. If there exists such that
The above lemma is an analogue of a famous lemma due to Schu  in Banach spaces.
A function is called a fixed point of if . The set of all fixed points of T will be denoted by .
Lemma 2 Let be a multivalued mapping and
Then the following are equivalent:
, that is, .
, that is, for each .
, that is, . Further where denotes the set of fixed points of .
Proof . Since , so . Therefore, for any , implies that . Hence . That is, .
. Since , so by definition of we have . Thus by ρ-closedness of Tf. □
Definition 8 A multivalued mapping is said to satisfy condition (I) if there exists a nondecreasing function with , for all such that for all .
It is a multivalued version of condition (I) of Senter and Dotson  in the framework of modular function spaces.
2 Main results
We prove a key result giving a major support to our ρ-convergence result for approximating fixed points of multivalued ρ-nonexpansive mappings in modular function spaces using a Mann iterative process.
Theorem 1 Let ρ satisfy and D a nonempty ρ-closed, ρ-bounded and convex subset of . Let be a multivalued mapping such that is a ρ-nonexpansive mapping. Suppose that . Let be defined by the Mann iterative process:
where and is bounded away from both 0 and 1. Then
Proof Let . By Lemma 2, . Moreover, by the same lemma, . To prove that exists for all , consider
By convexity of ρ, we have
Hence exists for each .
We now prove that
As , it suffices to prove that
and so in view of (2.1), we have
from (2.1), (2.2), (2.3), and Lemma 1, we have
Now we are all set for our convergence result for approximating fixed points of multivalued ρ-nonexpansive mappings in modular function spaces using the Mann iterative process as follows.
Theorem 2 Let ρ satisfy and D a nonempty ρ-compact, ρ-bounded and convex subset of . Let be a multivalued mapping such that is ρ-nonexpansive mapping. Suppose that . Let be as defined in Theorem 1. Then ρ-converges to a fixed point of T.
Proof From ρ-compactness of D, there exists a subsequence of such that for some . To prove that q is a fixed point of T, let g be an arbitrary point in and f in . Note that
By Theorem 1, we have . This gives . Hence q is a fixed point of . Since the set of fixed points of is the same as that of T by Lemma 2, ρ-converges to a fixed point of T. □
Theorem 3 Let ρ satisfy and D a nonempty ρ-closed, ρ-bounded and convex subset of . Let be a multivalued mapping with and and satisfying condition (I) such that is ρ-nonexpansive mapping. Let be as defined in Theorem 1. Then ρ-converges to a fixed point of T.
Proof From Theorem 1, exists for all . If , there is nothing to prove. We assume . Again from Theorem 1, so that
Hence exists. We now prove that . By using condition (I) and Theorem 1, we have
Since l is a nondecreasing function and , it follows that .
Next, we show that is a ρ-Cauchy sequence in D. Let be arbitrarily chosen. Since , there exists a constant such that for all , we have
In particular, . There must exist a such that
Now for , we have
Hence is a ρ-Cauchy sequence in a ρ-closed subset D of , and so it must converge in D. Let . That q is a fixed point of T now follows from Theorem 2. □
We now give some examples. The first one shows the existence of a mapping satisfying the condition (I) whereas the second one shows the existence of a mapping satisfying all the conditions of Theorem 3.
Example 1 Let (the collection of all real valued measurable functions on ). Note that is a modular function space with respect to
Let . Obviously D is a nonempty closed and convex subset of . Define as
Define a continuous and nondecreasing function by . It is obvious that for all . Hence T satisfies the condition (I).
Example 2 The real number system ℝ is a space modulared by . Let . Obviously D is a nonempty closed and convex subset of ℝ. Define as
Define a continuous and nondecreasing function by . It is obvious that for all .
Note that when . Hence is nonexpansive. Moreover, by Lemma 2, for all . Thus defined by where ρ-converges to a fixed point of T.
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The first author owes a lot to Professor Wataru Takahashi from whom he started learning the very alphabets of Fixed Point Theory during his doctorate at Tokyo Institute of Technology, Tokyo, Japan. He is extremely indebted to Professor Takahashi and wishes him a long healthy active life. The authors are thankful to the anonymous referees for giving valuable comments.
The authors declare that they have no competing interests.
Both authors worked on the manuscript. Both read and approved the final manuscript.
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Khan, S.H., Abbas, M. Approximating fixed points of multivalued ρ-nonexpansive mappings in modular function spaces. Fixed Point Theory Appl 2014, 34 (2014) doi:10.1186/1687-1812-2014-34
- fixed point
- multivalued ρ-nonexpansive mapping
- iterative process
- modular function space