Journal of Applied Mathematics

Volume 2012 (2012), Article ID 786061, 13 pages

doi:10.1155/2012/786061

Research Article

Some Fixed Point Theorems for Nonlinear Set-Valued Contractive Mappings

Zeqing Liu,¹ Zhihua Wu,¹ Shin Min Kang,² and Sunhong Lee²

¹Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China

²Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 10 April 2012; Accepted 9 May 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Zeqing Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Four fixed point theorems for nonlinear set-valued contractive mappings in complete metric spaces are proved. The results presented in this paper are extensions of a few well-known fixed point theorems. Two examples are also provided to illustrate our results.

1. Introduction and Preliminaries

The existence of fixed points for various set-valued contractive mappings had been researched by many authors under different conditions, see, for example, [1–9] and the references cited therein. In 1969, Nadler [7] proved a well-known fixed point theorem for the set-valued contraction mapping  (1.1) below.

Theorem 1.1 (see [7]). Let be a complete metric space and be a set-valued mapping such that

where is a constant. Then has a fixed point.

In 1972, Reich [8] extended Nadler's result and established an interesting fixed point theorem for the set-valued contraction mapping (1.2) below.

Theorem 1.2 (see [8]). Let be a complete metric space and satisfy that

where

Then has a fixed point.

In [8] Reich posed the question whether Theorem 1.2 is also true for the set-valued contractive mapping with (1.2). The affirmative answer under the hypothesis of , for  all was given by Mizoguchi and Takahashi in [6]. They deduced the following fixed point theorem which is a generalization of the Nadler fixed point theorem.

Theorem 1.3 (see [6]). Let be a complete metric space and satisfy (1.2), where

Then has a fixed point.

Remark 1.4. It is clear that the mappings in Theorems 1.1–1.3 are continuous on .

Remark 1.5. Each of Theorems 1.2 and 1.3 ensures that has a fixed point , which together with (1.2) implies that , that is, is defined at . Thus the domain of in each of (1.3) and (1.4) should be but not .

The aim of this paper is to present four fixed point theorems for some nonlinear set-valued contractive mappings. Our results extend, improve, and unify the corresponding results in [6–8]. Two nontrivial examples are given to show that our results are genuine generalizations or different from these results in [6–8].

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

where

(a)is nondecreasing on ;(b), for  all ;(c) is subadditive in , that is,

(d). Clearly (a)–(d) imply that(e) is strictly inverse on , that is, if there exist satisfying , then .

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

Such a mapping is called a generalized Hausdorff metric induced b in . It is well known that is a metric on . Let be a set-valued mapping, and be defined by

A sequence is said to be an orbit o if it satisfies that and for each . The function is said to be if for each orbit of with , we have that .

2. Main Results

The following lemmas play important roles in this paper.

Lemma 2.1. Let be a metric space and . Then for each and there exists satisfying .

Proof. Suppose that there exist and such that

which yields that

which is a contradiction. This completes the proof.

Lemma 2.2. Let be a metric space, and . Then for each and there exists such that

Proof. Let and . Now we consider two possible cases as follows.

Case  1. Suppose that . It follows from (b) and (d) that . Since is a closed subset of , it follows that . Put . Clearly (2.3) holds.

Case  2. Suppose that . Note that (b) and (d) mean that

Choose and . Lemma 2.1 ensures that there exists satisfying , which together with (a) and (c) gives that

That is, (2.3) holds. This completes the proof.

Now we prove four fixed point theorems for the nonlinear set-valued contractive mappings (2.6), (2.25), (2.26), and (2.36) below in complete metric spaces.

Theorem 2.3. Let be a complete metric space and satisfy that

where and

Then for each , there exists an orbit of and such that . Furthermore, is fixed point of if and only if the function defined by (1.8) is orbitally lower semicontinuous at .

Proof. Let be any initial point and choose . It follows from (2.6), (2.7) and Lemma 2.2 that for there exists satisfying

and for there exists satisfying

Repeating the above argument we obtain a sequence such that for and for , there exists satisfying

Suppose that there exists some satisfying . It follows from (a), (b), and (2.10) that for all . It is clear the conclusion of Theorem 2.3 holds.

Suppose that for any . It follows that for each . Note that (b), (2.7), and (2.10) give that is a positive and decreasing sequence. It follows from (e) that is decreasing. Therefore, there exist constants and satisfying

Notice that (2.7) implies that there exists a constant satisfying

Taking upper limits in (2.10) and by (2.11) and (2.12) we get that

which implies that .

Next we assert that . Since is a decreasing sequence, it follows from (a) and (2.11) that

that is, , which together with (b) and (d) yields that .

Put . It follows from (2.12) that , which gives that . Notice that (2.11), (2.12), and ensure that there exist and satisfying

which implies that

Note that (2.10) and (2.16) mean that

Given . Since , it follows from (b) that there exists satisfying

which together with (2.17), (a), and (c) gives that

In view of (e) and (2.19), we deduce that , for all , which means that is a Cauchy sequence. Hence there exists such that by completeness of .

Suppose that is orbitally lower semicontinuous at . Since is an orbit of with , it follows that

Using (2.6) and (2.7), we infer that

which together with (e), (2.11), and implies that

that is, , which together with (2.20) yields that

which gives that , that is, .

Conversely, suppose that is a fixed point of . Let be an arbitrarily orbit of with . It is clear that

which implies that is orbitally lower semicontinuous at . This completes the proof.

Notice that for each . In light of Theorem 2.3, we have

Theorem 2.4. Let be a complete metric space and satisfy that

where and satisfies (2.7). Then for each , there exists an orbit of and such that . Furthermore, is fixed point of if and only if the function defined by (1.8) is orbitally lower semicontinuous at .

If in (2.6) is replaced by , one has

Theorem 2.5. Let be a complete metric space and satisfy that

where and satisfies (2.7). Then for each , there exists an orbit of and such that . Furthermore, is fixed point of if and only if the function defined by (1.8) is orbitally lower semicontinuous at .

Proof. Let be any initial point and choose . It follows from (2.7), (2.26), and Lemma 2.2 that for there exists such that

Repeating the above argument we obtain a sequence satisfying for each ,

Suppose that for some . It is easy to verify that for all and the conclusion of Theorem 2.5 holds.

Suppose that for each . It follows that and are positive sequences. Combining (2.7), (2.28), (2.29), (b) and (e), we infer that and are both positive and decreasing, so do and . It follows that there exist constants and satisfying

Notice that (2.7) implies that there exists a constant such that

Taking upper limits in (2.29) and by (2.30) and (2.31) we get that

which implies that , which together with (2.30) and (a) ensures that

that is, , which gives that by (b) and (d). It follows from (2.28), (2.30), and (2.31) that

which yields that . Notice that (2.30) and (a) guarantee that

which together with (b) and (d) yields that . The rest of the proof is similar to that of Theorem 2.3 and is omitted. This completes the proof.

The result below follows from Theorem 2.5.

Theorem 2.6. Let be a complete metric space and satisfy that

where and satisfies (2.7). Then for each , there exists an orbit of and such that . Furthermore, is fixed point of   if and only if the function defined by (1.8) is orbitally lower semicontinuous at .

3. Comparisons and Examples

Now we construct two examples to compare the results in Section 2 with the corresponding results in [6–8].

Remark 3.1. Theorems 2.3 and 2.4 extend Theorems 1.1–1.3, and Theorems 2.5 and 2.6 are different from Theorems 1.1–1.3, respectively, in the following ways:

(1)the ranges of the nonlinear set-valued contractive mappings in Theorems 2.3–2.6 are more general than the ranges and of the set-valued contraction mappings in Theorems 1.1–1.3, respectively;(2)the orbit lower semicontinuity at some of the functions in Theorems 2.3 and 2.4 is weaker than the continuity of the set-valued contraction mappings in in Theorems 1.1–1.3, respectively;(3)the set-valued contraction mappings (1.1) and (1.2) are special cases of the nonlinear set-valued contractive mapping (2.6) with because

Example 3.2 below shows that Theorems 2.3 and 2.4 extend substantively Theorems 1.1–1.3, respectively.

Example 3.2. Let and be the standard metric in . Let , and be defined by

respectively. It is clear that , satisfies (2.7) and

is orbitally lower semicontinuous in . In order to prove (2.6) holds, we consider two possible cases.

Case  1. Let and . It is clear that

Case  2. Let and . It follows that

that is, (2.6) holds. Therefore all assumptions of Theorems 2.3 and 2.4 are satisfied. It follows from each of Theorems 2.3 and 2.4 that has a fixed point in . However, we cannot invoke any one of Theorems 1.1–1.3 to show the existence of fixed points for the mapping in . Indeed, taking and , we get that

for any and

for any mapping with each of (1.3) and (1.4).

Next we construct an example to explain Theorems 2.5 and 2.6.

Example 3.3. Let and be the standard metric in . Define , and by

respectively. It is easy to see that (2.7) holds and

is orbitally lower semicontinuous in . In order to check (2.26), we have to consider two cases as follows.

Case  1. Let and . It is clear that

Case  2. Let and . It follows that

For , we have

For , we infer that

For , we get that

Hence (2.26) holds. Thus all assumptions of Theorems 2.5 and 2.6 are satisfied. It follows from each of Theorems 2.5 and 2.6 that has a fixed point in .

Taking and , we deduce that

for any , and

for any mapping with each of (1.3) and (1.4). That is, Theorems 1.1–1.3 are inapplicable in proving the existence of fixed points for the nonlinear set-valued contractive mapping .

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