On Fixed Point Theorems for Multivalued Contractions

Abstract

Three concepts of multivalued contractions in complete metric spaces are introduced, and the conditions guaranteeing the existence of fixed points for the multivalued contractions are established. The results obtained in this paper extend genuinely a few fixed point theorems due to Ćirić (2009) Feng and Liu (2006) and Klim and Wardowski (2007). Five examples are given to explain our results.

1. Introduction and Preliminaries

Let be a metric space, and let , , and denote the families of all nonempty closed, all nonempty closed and bounded, and all nonempty compact subsets of , respectively. For any and , let and

(1.1)

Such a mapping is called a generalized Hausdorff metric ininduced by.

Throughout this paper, we assume that ,, and denote the sets of all real numbers, nonnegative real numbers, and positive integers, respectively.

The existence of fixed points for various multivalued contractive mappings had been studied by many authors under different conditions. For details, we refer the reader to [1–7] and the references therein. In 1969, Nadler Jr [7] extended the famous Banach Contraction Principle from single-valued mapping to multivalued mapping and proved the following fixed point theorem for the multivalued contraction.

Theorem 1.1 (see [7]).

Let be a complete metric space, and let be a mapping from into . Assume that there exists such that

(1.2)

Then, has a fixed point.

In 1989, Mizoguchi and Takahashi [6] generalized the Nadler fixed point theorem and got a fixed point theorem for the multivalued contraction as follows.

Theorem 1.2 (see [6]).

Let be a complete metric space, and let be a mapping from into . Assume that there exists a map such that

(1.3)

Then, has a fixed point.

In 2006, Feng and Liu [3] obtained a new extension of the Nadler fixed point theorem and proved the following fixed point theorem.

Theorem 1.3 (see [3]).

Let be a complete metric space, and let be a multivalued mapping from into . If there exist constants , such that for any there is satisfying

(1.4)

then has a fixed point in provided a function , is lower semicontinuous.

In 2007, Klim and Wardowski [5] improved the result of Feng and Liu and proved the following results.

Theorem 1.4 (see [5]).

Let be a complete metric space, and let be a multivalued mapping from into . Assume that

(a)the mapping , defined by , , is lower semi-continuous;

(b)there exist and satisfying

(1.5)

and for any there is satisfying

(1.6)

Then, has a fixed point in .

Theorem 1.5 (see [5]).

Let be a complete metric space, and let be a mapping from into . Assume that

(a)the mapping , defined by , , is lower semi-continuous;

(b)there exists a function satisfying

(1.7)

and for any there is satisfying

(1.8)

Then, has a fixed point in .

In 2008 and 2009, Ćirić [1, 2] introduced new multivalued nonlinear contractions and established a few nice fixed point theorems for the multivalued nonlinear contractions, one of which is as follows.

Theorem 1.6 (see [2]).

Let be a complete metric space, and let be a mapping from into . Assume that

(a)the mapping , defined by , , is lower semi-continuous;

(b)there exists a function , , satisfying

(1.9)

and for any there is satisfying

(1.10)

Then has a fixed point in .

The aim of this paper is both to introduce three new multivalued contractions in complete metric spaces and to prove the existence of fixed points for the multivalued contractions under weaker conditions than the ones in [2, 3, 5]. Five nontrivial examples are given to show that the results presented in this paper generalize substantially and unify the corresponding fixed point theorems of Ćirić [2], Feng and Liu [3], and Klim and Wardowski [5] and are different from those results of Mizoguchi and Takahashi [6] and Nadler Jr [7].

Next we recall and introduce the following result in [4] and some notions and terminologies.

Lemma 1.7 (see [4]).

Let be a complete metric space and . Then, for each and there exists an element such that

(1.11)

In the rest of this paper, for a multivalued mapping , we put

(1.12)

where and , for all . The function is said to be -orbitally lower semicontinuous at if is an arbitrary orbit of with impling that .

2. Main Results

In this section, we establish three fixed point theorems for three new multivalued contractions in complete metric spaces.

Theorem 2.1.

Let be a multivalued mapping from a complete metric space into such that

(2.1)

where and satisfy that

(2.2)

Then,

(a1) for each there exists an orbit of and such that ;

(a2) is a fixed point of in if and only if the function is -orbitally lower semi-continuous at .

Proof.

Put

(2.3)

It follows from (2.1) that for each there exists satisfying

(2.4)

which together with (2.3) yield that

(2.5)

Continuing this process, we choose easily an orbit of satisfying

(2.6)

which imply that

(2.7)

Now, we claim that

(2.8)

Notice that the ranges of , (2.2), and (2.3) ensure that

(2.9)

Using (2.7) and (2.9), we conclude that is a nonnegative and nonincreasing sequence, which means that

(2.10)

for some . Suppose that . Taking limits superior as in (2.7) and using (2.2), (2.3), (2.9), and (2.10), we get that

(2.11)

which is a contradiction. Thus, ; that is, (2.8) holds.

Next, we show that is a Cauchy sequence. Let

(2.12)

It follows from (2.2), (2.3), (2.9), and (2.12) that

(2.13)

Let and . In light of (2.12) and (2.13), we deduce that there exists some such that

(2.14)

which together with (2.6) and (2.7) yield that

(2.15)

which imply that

(2.16)

which give that

(2.17)

which implies that is a Cauchy sequence because . It follows from completeness of that there is some such that .

Suppose that is -orbitally lower semi-continuous in . It follows from (2.8) that

(2.18)

which means that because is closed.

Suppose that is a fixed point of in . For any orbit of with and , we have

(2.19)

that is, is -orbitally lower semi-continuous in . This completes the proof.

Theorem 2.2.

Let be a multivalued mapping from a complete metric space into such that

(2.20)

where satisfies that

(2.21)

Then,

(a1) for each there exists an orbit of and such that ;

(a2) is a fixed point of in if and only if the function is -orbitally lower semi-continuous at .

Proof.

It follows from Lemma 1.7 that for each there exists with

(2.22)

which together with (2.20) and (2.21) ensures that (2.1) and (2.2) hold with and . Thus, Theorem 2.2 follows from Theorem 2.1. This completes the proof.

Theorem 2.3.

Let be a multivalued mapping from a complete metric space into such that

(2.23)

where and satisfy that

(2.24)

and one of and is nondecreasing. Then,

(a1) for each there exists an orbit of and such that ;

(a2) is a fixed point of in if and only if the function is -orbitally lower semi-continuous at .

Proof.

Put

(2.25)

It follows from the ranges of , (2.24), and (2.25) that

(2.26)

As in the proof of Theorem 2.1, we select an orbit of satisfying

(2.27)

which imply that

(2.28)

(2.29)

Using (2.26) and (2.28), we conclude easily that is a nonnegative and nonincreasing sequence. Consequently there exists some satisfying

(2.30)

Now we claim that

(2.31)

Suppose to the contrary, that is, there exists some satisfying . Note that one of and is nondecreasing. In view of (2.25), (2.26), and (2.29), we get that

(2.32)

which is a contradiction. Thus, (2.31) holds. Therefore, there exists some such that

(2.33)

Next, we show that . Suppose that . Taking limits superior as in (2.28) and using (2.24), (2.25), (2.30), and (2.33), we get that

(2.34)

which is impossible. Thus, . Let

(2.35)

It follows from (2.24), (2.25), (2.33), and (2.35) that

(2.36)

Let and . By means of (2.35) and (2.36), we infer that there exists some such that

(2.37)

which together with (2.27) and (2.28) yield that

(2.38)

The rest of the proof is similar to that of Theorem 2.1 and is omitted. This completes the proof.

3. Remarks and Examples

In this section, we construct five examples to illustrate the superiority and applications of the results presented in this paper.

Remark 3.1.

In case and for all , where and are constants in with , then Theorem 2.1 reduces to a result, which is a generalization of Theorem 1.3. The following example reveals that Theorem 2.1 extends both essentially Theorem 1.3 and is different from Theorems 1.1 and 1.2.

Example 3.2.

Let be endowed with the Euclidean metric , and let be defined by

(3.1)

It is easy to see that

(3.2)

is -orbitally lower semi-continuous in and . Define and by

(3.3)

Obviously,

(3.4)

For , we have

(3.5)

For , we infer that

(3.6)

For , there exists satisfying

(3.7)

For , there exists satisfying

(3.8)

That is, the conditions of Theorem 2.1 are fulfilled. It follows from Theorem 2.1 that has a fixed point in . However, we cannot invoke each of Theorems 1.1–1.3 to show that the mapping has a fixed point in .

In fact, for any , we consider two possible cases as follows.

Case 1.

Let. Take and . Notice that and . It is clear that

(3.9)

Case 2.

Let. Put and . It follows that

(3.10)

Set and . Note that and . It is easy to verify that

(3.11)

That is, the conditions of Theorem 1.3 do not hold.

Put and . Clearly

(3.12)

for any with . That is, the conditions of Theorems 1.1 and 1.2 do not hold.

Remark 3.3.

If and for all , then Theorem 2.1 changes into a result, which is an extension of Theorem 1.6. The following example demonstrates that Theorem 2.1 generalizes substantially Theorem 1.6.

Example 3.4.

Let be endowed with the Euclidean metric . Let be defined by

(3.13)

It is easy to see that

(3.14)

is -orbitally lower semi-continuous in and . Define and by

(3.15)

Obviously,

(3.16)

For , there exists satisfying

(3.17)

For , there exists satisfying

(3.18)

That is, the conditions of Theorem 2.1 are fulfilled. It follows from Theorem 2.1 that has a fixed point in . However, Theorem 1.6 is inapplicable in ensuring the existence of fixed points for the mapping in because there does not exist and satisfying the assumptions of Theorem 1.6. In fact, for any and , we consider the following two possible cases.

Case 1.

Let  . Put . Note that . If , we see that

(3.19)

If , then we infer that

(3.20)

Case 2.

Let  . Put . Note that . Suppose that . Let . It is clear that

(3.21)

Take . Obviously,

(3.22)

Suppose that . As in the proof of Case 1 stated first, we conclude similarly the conclusion of Case 1 stated after then. Therefore, the assumptions of Theorem 1.6 do not hold.

Remark 3.5.

The following example is an application of Theorem 2.2.

Example 3.6.

Let be endowed with the Euclidean metric . Let be defined by

(3.23)

It is easy to see that

(3.24)

is -orbitally lower semi-continuous in and . Define by

(3.25)

Clearly,

(3.26)

For , we infer that

(3.27)

For , we conclude that

(3.28)

For , we obtain that

(3.29)

For , we get that

(3.30)

Therefore, all assumptions of Theorem 2.2 are satisfied. It follows from Theorem 2.2 that has a fixed point in .

Remark 3.7.

If and for all , then Theorem 2.3 comes down to a result, which extends Theorem 1.5. The following example shows that Theorem 2.3 is a genuine generalization of Theorem 1.5.

Example 3.8.

Let be endowed with the Euclidean metric . Define , , and by

(3.31)

respectively. Clearly, and

(3.32)

is continuous in . Note that for each there exists such that

(3.33)

It is easy to verify that the assumptions of Theorem 2.3 are satisfied. Consequently, Theorem 2.3 guarantees that has a fixed point in . But does not satisfy the conditions of Theorem 1.5 because is not compact for all .

Remark 3.9.

In case and for all , then Theorem 2.3 reduces to a result, which extends Theorem 1.4. The following example reveals that Theorem 2.3 generalizes properly Theorem 1.4.

Example 3.10.

Let and be as in example 3.1. Clearly,

(3.34)

is -orbitally lower semi-continuous in and . Define and by

(3.35)

It is easy to verify that is nondecreasing and

(3.36)

For , we have

(3.37)

For , we infer that

(3.38)

For , there exists satisfying

(3.39)

For , there exists , satisfying

(3.40)

That is, the conditions of Theorem 2.3 are fulfilled. It follows from Theorem 2.3 that has a fixed point in . However, we cannot use Theorem 1.4 to show that the mapping has a fixed point in since there does not exist and satisfying the assumptions in Theorem 1.4. In fact, for any and , we consider two possible cases as follows.

Case 1.

Let. Take and . Note that . It is clear that

(3.41)

Case 2.

Let. Put and . It follows that

(3.42)

Let and . Note that . It is easy to verify that

(3.43)

Therefore, the assumptions of Theorem 1.4 are not satisfied.