The theory of modular spaces was initiated by Nakano [34] in 1950 in connection with the theory of order spaces and redefined and generalized by Musielak and Orlicz [35] in 1959. Even though a metric is not defined, many problems in metric fixed point theory can be reformulated and solved in modular spaces (see, for instance, [36–39]). In particular, Dhompongsa et al. [40] have obtained some fixed point results for multivalued mappings in modular functions spaces.

Let us recall some basic concepts about modular function spaces (for more details the reader is referred to [41, 42]).

Let be a nonempty set and a nontrivial -algebra of subsets of . Let be a -ring of subsets of , 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 -finite measure space). By we denote the linear space of all simple functions with supports from . By we will denote the space of all measurable functions, that is, all functions such that there exist a sequence , and for all .

Let us recall that a set function is called a -subadditive measure if , for any and for any sequence of sets . By we denote the characteristic function of the set .

Definition 4.1.

A functional is called a function modular if:

(1) for any ;

(2) whenever for any , and

(3) is a -subadditive measure for every

(4) as decreases to for every , where

(5)if there exists such that , then for every

(6)for any , is order continuous on , that is, if and decreases to .

A -subadditive measure is said to be additive if whenever such that and .

The definition of is then extended to by

Definition 4.2.

A set is said to be -null if for every . A property is said to hold -almost everywhere (-a.e.) if the set is -null. For example, we will say frequently -a.e.

Note that a countable union of -null sets is still -null. In the sequel we will identify sets and whose symmetric difference is -null, similarly we will identify measurable functions which differ only on a -null set.

Under the above conditions, we define the function by . We know from [41] that satisfies the following properties:

(i) if and only if -a.e.

(ii) for every scalar with and .

(iii) if , and .

In addition, if the following property is satisfied

(iii) if , and ,

we say that is a convex modular.

A function modular is called -finite if there exists an increasing sequence of sets such that and .

The modular defines a corresponding modular space , which is given by

In general the modular is not subadditive and therefore does not behave as a norm or a distance. But one can associate to a modular an -norm. In fact, when is convex, the formula

defines a norm which is frequently called the Luxemburg norm. The formula

defines a different norm which is called the Amemiya norm. Moreover, and are equivalent norms. We can also consider the space

Definition 4.3.

A function modular is said to satisfy the -condition if

It is known that the -condition is equivalent to .

Definition 4.4.

A function modular is said to satisfy the -type condition if there exists such that for any we have *.*

In general, the -type condition and -condition are not equivalent, even though it is obvious that the -type condition implies the -condition.

Definition 4.5.

Let be a modular space.

(1)The sequence is said to be -convergent to if as .

(2)The sequence is said to be -a.e. convergent to if the set is -null.

(3)A subset of is called -a.e. closed if the -a.e. limit of a -a.e. convergent sequence of always belongs to .

(4)A subset of is called -a.e. compact if every sequence in has a -a.e. convergent subsequence in .

(5)A subset of is called -bounded if

We know by [41] that under the -condition the norm convergence and modular convergence are equivalent, which implies that the norm and modular convergence are also the same when we deal with the -type condition. In the sequel we will assume that the modular function is convex and satisfies the -type condition. Hence, the -convergence defines a topology which is identical to the norm topology.

In the same way as the Hausdorff distance defined on the family of bounded closed subsets of a metric space, we can define the analogue to the Hausdorff distance for modular function spaces. We will speak of -Hausdorff distance even though it is not a metric.

Definition 4.6.

Let be a nonempty subset of . We will denote by the family of nonempty -closed subsets of and by the family of nonempty -compact subsets of . Let be the -Hausdorff distance on , that is,

where is the -distance between and . A multivalued mapping is said to be a -contraction if there exists a constant such that

If it is valid when , then is called -nonexpansive.

A function is called a fixed point for a multivalued mapping if .

Dhompongsa et al. [40] stated the Banach Contraction Principle for multivalued mappings in modular function spaces.

Theorem 4.7 (see [40, Theorem ]).

Let be a convex function modular satisfying the -type condition, a nonempty -bounded -closed subset of , and a -contraction mapping, that is, there exists a constant such that

Then has a fixed point.

By using that result, they proved the existence of fixed points for multivalued -nonexpansive mappings.

Theorem 4.8 (see [40, Theorem ]).

Let be a convex function modular satisfying the -type condition, a nonempty -a.e. compact -bounded convex subset of , and a -nonexpansive mapping. Then has a fixed point.

They also applied the above theorem to obtain fixed point results in the Banach space (resp., ) for multivalued mappings whose domains are compact in the topology of the convergence locally in measure (resp., -topology).

Consider the space for a -finite measure with the usual norm. Let be a bounded closed convex subset of for and a multivalued nonexpansive mapping. Because of uniform convexity of , it is known that has a fixed point. For , can fail to have a fixed point even in the singlevalued case for a weakly compact convex set (see [43]). However, since is a modular space where for all , Theorem 4.8 implies the existence of a fixed point when we define mappings on a -a.e. compact -bounded convex subset of . Thus the following can be stated.

Corollary 4.9 (see [40, Corollary ]).

Let be as above, a nonempty bounded convex set which is compact for the topology of the convergence locally in measure, and a nonexpansive mapping. Then has a fixed point.

In the case of the space we also can obtain a bounded closed convex set and a nonexpansive mapping which is fixed point free. Indeed, consider the following easy and well-known example.

Let

Define a nonexpansive mapping by

then is a fixed point free map. However, if we consider where , for all , then -a.e. convergence and -convergence are identical on bounded subsets of (see [36]). This fact leads to the following corollary.

Corollary 4.10 (see [40, Corollary ]).

Let be a nonempty -compact convex subset of and a nonexpansive mapping. Then has a fixed point.

Next we will give a property of closed convex bounded subsets of more general than weak star compactness which implies the fixed point property for nonexpansive mappings.

Domínguez et al. introduced in [44] some compactness conditions concerning proximinal subsets called Property (P). Following this idea we will use the following similar notion for modular function spaces.

Definition 4.11.

Let be a nonempty -closed convex -bounded subset of . It is said that has Property () if for every which is the -a.e. limit of a sequence in , the set is a nonempty and -compact subset of , where

Using that notion and the following two lemmas, we obtain a new fixed point result for multivalued -nonexpansive mappings.

Lemma 4.12 (see [40, Lemma ]).

Let be a convex function modular satisfying the -type condition, , and a nonempty -compact subset of . Then there exists such that

Lemma 4.13 (see [37, Lemma ]).

Let be a function modular satisfying the -type condition, and be a sequence in such that and there exists such that . Then,

Theorem 4.14.

Let be a convex function modular satisfying the -type condition, a nonempty -closed -bounded convex subset of satisfying Property such that every sequence in has a -a.e. convergent subsequence in , and a -nonexpansive mapping. Then has a fixed point.

Proof.

Fix . For each , the -contraction is defined by

By Theorem 4.7, we can conclude that has a fixed point, say . It is easy to see that

By our assumptions, we can assume, by passing through a subsequence, that for some . By Lemma 4.12, for each there exists such that

Now we are going to show that for each . Taking any , from the -compactness of and Lemma 4.12, we can find such that

and we can assume, by passing through a subsequence, that for some . From above and using Lemma 4.13, it follows that

On the other hand, by Lemma 4.13 we also have

Thus, we deduce , which implies that and so .

Now we define the mapping by . From [45, Proposition ] we know that the mapping is upper semicontinuous. Since is a nonempty -compact convex set and the -topology is a norm-topology, we can apply the Kakutani-Bohnenblust-Karlin Theorem (see [14]) to obtain a fixed point for and hence for .

If we apply the previous theorem in the particular case of the space for a -finite measure with the usual norm, we obtain the following result, which can be also deduced from [44, Theorem ].

Corollary 4.15.

Let be as above, a nonempty closed bounded convex set which satisfies Property (P). Suppose, in addition, that every sequence in has a convergent locally in measure subsequence in . If is a nonexpansive mapping, then has a fixed point.

If we consider now the space , then the assumption of existence of a -convergent subsequence for every sequence in can be removed and we can state the following result.

Corollary 4.16.

Let be a nonempty closed bounded convex subset of which satisfies Property (P). If is a nonexpansive mapping, then has a fixed point.

Notice that in there exists a subset with Property (P) which is not -compact.

Example 4.17 (see [44, Example ]).

Let be a bounded sequence of nonnegative real numbers and let be the standard Schauder basis of . It is clear that the set , where , is never weakly star compact. Nevertheless, by using [46, Example ] it is easy to show that has Property (P) if and only if is nonempty and finite.