Abstract

We give some initial properties of a subset of modular metric spaces and introduce some fixed-point theorems for multivalued mappings under the setting of contraction type. An appropriate example is as well provided. The stability of fixed points in our main theorems is also studied.

1. Introduction and Preliminaries

The field of metric fixed-point theory has been widely investigated since 1922, when Banach [1] had proved his contraction principle. We are going to recall this well-known theorem before we continue over on.

A self-mapping on a metric space is called a contraction if there exists such that for all . The contraction principle simply stated that, if is complete, such a mapping has a unique fixed point.

One of the most influenced generalizations of Banach’s theorem is traced to Nadler [2]. In 1969, via Hausdorff’s concept of a distance between two arbitrary sets, Nadler proved the contraction principle for multivalued mappings in complete metric spaces. Also, some authors extended Nadler’s principle and established fixed-point theorems for multivalued mappings in metric spaces and other spaces (see [3–9]). One of the most interesting studies are the extensions of such principle in modular spaces and modular function spaces (see [10–12] and references therein).

Lately, in 2010, Chistyakov [13] introduced the notion of a modular metric space which is a new generalization of a metric space. We will give a short revisit to modular and modular metric spaces as follows.

Definition 1.1. Let be a linear space over with as its zero element. A functional is said to be a modular on if for any , the following conditions hold:(i) if and only if ,(ii),(iii) whenever and .The linear subspace is called a modular space.

Definition 1.2 (see [13]). Let be a nonempty set. A function is said to be a metric modular on if satisfying, for all , the following conditions hold:(i) for all if and only if ,(ii) for all ,(iii) for all .Suppose , the set is called a modular metric space generated by and induced by . If its generator does not play any role in the situation, we will write instead of .
Observe that a metric modular on is nonincreasing with respect to . We can simply show this assertion by using the condition (iii) itself. For any and , we have For each and , we set and . Consequently, from (1.2), we have .
If, for any , a metric modular on possesses a finite value and for all , then is a metric on .

Recently, Mongkolkeha et al. [14] have introduced some notions and established some fixed-point results in modular metric spaces. We now state some notions and results in [14] in the following.

Definition 1.3 (see [14]). Let be a modular metric space. (i)The sequence in is said to be convergent if there exists such that , as for all .(ii)The sequence in is said to be a Cauchy sequence if , as for all .(iii) is said to be complete if every Cauchy sequence in converges.(iv)A subset of is said to be closed if the limit of a convergent sequence of always belongs to .(v)A subset of is said to be bounded if, for all , = .Along this paper, we will use the following alternative notions of convergence and Cauchyness, which are equivalent to the notions given above.
Let be a modular metric space and be a sequence in .(i)A point is called a limit of if for each , there exists such that for every with . A sequence that has a limit is said to be convergent (or converges to ) and will be written as .(ii)A sequence in is said to be a Cauchy sequence if, for each , there exists such that for every with .Moreover, we observe that the limit of any sequence in is unique.

Definition 1.4 (see [14]). Let be a modular metric space. A self-mapping on is said to be a contraction if there exists such that for all and .

Theorem 1.5 (see [14]). Let be a complete modular metric space and a contraction on . Then, the sequence converges to the unique fixed point of in for any initial .

The purpose of this paper is to study some properties of a subset of modular metric spaces, establish and extend some fixed-point theorems of Mongkolkeha et al. [14] to multivalued mappings in modular metric spaces.

2. Some Properties of a Subset of Modular Metric Spaces

In this section, we study some properties of a subset of modular metric spaces, some of which will take advantages in the proof of our main theorems. Throughout this paper, let denotes the set of all nonempty closed bounded subsets of and denotes the set of all nonempty closed subsets of .

Let be a non-empty subset of a modular metric space . For , we denotes .

For , define and the Hausdorff metric modular . Notice that is not symmetric.

Proposition 2.1. Let be a modular metric space and . Then, the following properties hold.(i) for all .(ii) for all .(iii) for all .(iv) for all .

Proof. (i) By the definition of , we have, for all , that Since is closed in , we get for all . That is, for all .
(ii) It is obvious that for all and . Hence, .
(iii) Observe that, if , then
(iv) Let , , and . Then, which implies that Since is arbitrary, we have Similarly, since is arbitrary, we can deduce that

Proposition 2.2. Let be a modular metric space. Then, for all .

Proof. Suppose is arbitrary. For and , we have and . Hence, we get Similarly, we have Hence, we have

Proposition 2.3. Let be a modular metric space generated by . Then, is a modular metric space generated by and is induced by .

Proof. For , we have Since and , we have .
By the definition of and Proposition 2.1, it is clear that for all and for all if and only if .
Again, by Proposition 2.1, we have for all . Therefore, is a modular metric space generated by and is induced by .

Remark 2.4. Note that the metric modular depends on , so the completeness of implies the completeness of .

Now, we are arriving at the most important lemma used in our proof of main theorems.

Lemma 2.5. Let and . Then, for , there exists a point such that .

Proof. Let , be arbitrary. Since , we claim that is not a lower bound of the set . Therefore, there exists for which and hence .

3. Fixed-Point Theorems for Multivalued Mappings

In this section, we extend the result by Mongkolkeha et al. [14] under the multivalued setting and hereby obtain some corollaries. Beforehand, we will give the notion of a multivalued -contraction in modular metric spaces.

Definition 3.1. Let be a modular metric space. A multivalued mapping is said to be a multivalued -contraction if there exists such that for all and . In this case, the least number which satisfies the inequality (3.1) is said to be the contraction constant.

Remark 3.2. For a sequence in , it is obvious that, if and is a multivalued -contraction on , then .

Theorem 3.3. Let be a complete modular metric space and a multivalued -contraction on with contraction constant . Then, has a fixed point in .

Proof. Let be arbitrary and . By Lemma 2.5, there exists such that Similarly, by this procedure, we define a sequence in such that and for all . Hence, by the multivalued -contractivity, we have Thus, by induction, we deduce that Notice that and . Now, since for all . Without loss of generality, suppose and . Observe that, for arbitrary , for all for some , and hence is a Cauchy sequence. Then, the completeness of implies that for some . Consequently, the sequence converges to , that is, for all . Since , we have which implies that . Since is closed, it follows that .

Example 3.4. Let , defined by . Clearly, for any generator . Now, we define a multivalued mapping given by We have . Therefore, is a multivalued -contraction with contraction constant , and we have that 0 and 1 are fixed points of .

Remark 3.5. Note that our result does not assure the uniqueness of a fixed point, as illustrated in the above example.

We next present the local version of Theorem 3.3.

Theorem 3.6. Let be a complete modular metric space, and . Suppose there exists for which for all , and for all . Then, has a fixed point in .

Proof. To prove this theorem, we only need to show that is complete and . To show that is complete, suppose that is a Cauchy sequence in . Since is complete, for some for all . Since, for each , , we get As , we have . Therefore, is complete.
Now, we prove the latter. For any , let . Observe that, for all , This implies that for all . Applying Theorem 3.3 to complete the proof.

In the following theorem, we prove the existence of fixed points for a mapping introduced in 1969 by Kannan [15] in view of multivalued mappings in modular metric spaces.

Theorem 3.7. Let be a complete modular metric space and a multivalued mapping such that there exists such that for all and . Then, has a fixed point in .

Proof. Let the sequence be constructed as in the proof of Theorem 3.3, so we get, for all , for all . Observe that Further, set , we obtain As in the proof of Theorem 3.3, we conclude that is a Cauchy sequence. The completeness of implies that for some .
Now, we show that is a fixed point of . Observe that Again, we have that As , we have . Since is closed, we have . Therefore, is a fixed point of in .