Fixed Point Theorems of Set-Valued Mappings in Partially Ordered Hausdorff Topological Spaces

Abstract

In this work, several fixed point theorems of set-valued monotone mappings and set-valued Caristi-type mappings are proved in partially ordered Hausdorff topological spaces, which indeed extend and improve many recent results in the setting of metric spaces.

1. Introductions

In 1976, Caristi [1] proved Caristi’s fixed point theorem [1, 2], which has been the subject of intensive research in the past decades, and has found many applications in nonlinear analysis. Recall that this general fixed point theorem states that each mapping has a fixed point provided that is a complete metric space and there exists a lower semicontinuous and bounded below function such that for each . Kirk [2] gave an elegant proof of primitive Caristi’s result by investigating the existence of maximal elements of a partially ordered metric space , where is a partial order defined by Since then, Kirk’s method has been widely used in the generalizations of primitive Caristi’s result and the study of fixed point theorems of monotone mappings with respect to a partial order introduced by a functional and many satisfactory fixed point results have been obtained in metric spaces (see [3–10]).

The purpose of this paper is to generalize the results of [3–10] to general topological spaces. Under suitable assumptions, we proved several fixed point theorems of set-valued monotone mappings and set-valued Caristi-type mappings in partially ordered Hausdorff topological spaces, which indeed extend and improve many recent results in the setting of metric spaces.

2. Main Results

Let be a Hausdorff topological space and let be a partial order on . For each , let and let .

Let be a nonempty subset of and , where denote the family of all nonempty subset of . is increasing on , if for each with and each , there exists such that ; is quasi-increasing on , if for each with and each , there exists such that . has compact value on , if is compact for each . is a Caristi-type mapping, if for each , there exists such that .

Let be the inverse partial order of . It is clear that is increasing on with respect to if is quasi-increasing on with respect to .

In this paper, we make the following assumptions:for each totally ordered set , there exists a sequence such that for each with , there exists some such that ;for each totally ordered set , there exists a sequence such that for each with , there exists some such that ;for each increasing sequence , there exists a subsequence and some such that ;for each decreasing sequence , there exists a subsequence and some such that ;for each with for each , if there exists some such that and , then ;for each sequence with for some and each , if there exists such that , then ;there exists a functional such that

Theorem 1. Let be a Hausdorff topological space, let be a partial order on , and let . Assume that , , and are satisfied. If there exists such that , is increasing and has compact value on . Then has a maximal fixed point in ; that is, let be a fixed point of in such that , and then .

Proof. Let Clearly, is nonempty since . We divide the proof into four steps.
Step  1. We show that holds on . Let be an arbitrary increasing sequence. By , there exists a subsequence and some such that Let in (4), and then by we have ; that is, since for each . For arbitrary given , we have and hence by . Moreover the arbitrary property of forces the following: Since , there exists such that By (5), and the increasing property of on , there exists such that for each . This together with (6) implies that Since has compact value on , then is compact and so there exists a subsequence and such that Note that for each , and then by (4), (8), and , we have . This implies that and so since . Hence holds on .
Step  2. We show that each increasing sequence has an upper bound in . Since holds on by Step  1, there exists a subsequence and some such that Note that for arbitrary given , for all , and then by (9) and we have . Moreover the arbitrary property of forces for each ; that is, is an upper bound of .
Step  3. We show that each totally ordered set has an upper bound in , where is a directed set.
If there exists such that , then is an upper bound of and hence the proof is finished. Thus we may assume that for each . By , there is a sequence such that there exists such that Set Note that is totally ordered, and then is well defined and is an increasing sequence with for all . By step 2, has an upper bound in ; denote it by . Moreover by (10) and (11), which implies that is an upper bound of .
Step  4. We show that has a maximal fixed point in . By Zorn’s lemma, has a maximal element ; that is, for each with , we must have . Since , there exists such that . Moreover by the increasing property of on , there exists such that . This indicates that and hence . Finally the maximality of in implies that ; that is, is a maximal fixed point of in . The proof is complete.

Theorem 2. Let be a Hausdorff topological space, let be a partial order on , and let . Assume that , , and are satisfied. If there exists such that , is quasi-increasing and has compact value on . Then has a minimal fixed point in ; that is, let be a fixed point of in such that , and then .

Proof. Let be the inverse partial order of . It is clear that and are satisfied with respect to by and . Set and . Obviously, , , , and is increasing and has compact value on . Applying Theorem 1 on , we find that has a maximal fixed point corresponding to . Let be a fixed point of . If , then and hence by the maximality of corresponding to ; that is, is a minimal fixed point of in corresponding to . The proof is complete.

Example 3. Let with the usual metric for each and the usual order for each . It is easy to check that are satisfied.
Let be defined by Clearly, is an increasing mapping and has compact value on . Note that and ; then by Theorem 1, has a maximal fixed point .
Let be defined by Clearly, is a quasi-increasing mapping and has compact value on . Note that and ; then by Theorem 2, has a minimal fixed point .
The following theorem extends primitive Caristi’s result to Hausdorff topological spaces.

Theorem 4. Let be a Hausdorff topological space and let be a partial order on . Assume that , , and are satisfied. Then each set-valued Caristi-type mapping has fixed point in .

Proof. In analogy to Step  2 in the proof of Theorem 1, by and , we can prove that each increasing sequence has an upper bound in . Thus following Step  3 in the proof of Theorem 1, each totally ordered chain of has an upper bound by . Moreover by Zorn’s lemma, has a maximal element; denote it by . Note that there exists some such that ; then by the maximality of and hence is a fixed point of . The proof is complete.

The following lemma shows that the conditions and are not hard to be satisfied.

Lemma 5. Let be a nonempty set and let be a partial order on . If is satisfied and is bounded below (resp., bounded above), then (resp., ) is satisfied.

Proof. We only show is satisfied, and the proof of the other case is similar.
Let be a totally ordered set of , where is a directed set, and set . Note that is bounded below; then exists, and so there exists a subset of such that Let be an element such that ; then by . Suppose that for each . By , for each , and consequently, we have by (15). This is a contradiction, and so there exists some such that This shows that is satisfied. The proof is complete.

Remark 6. It follows from Lemma 5 that Theorems 1–4 are still valid while (resp., ) is replaced with provided that is bounded below (resp., bounded above).

3. Applications to Metric Spaces

In this section, we shall show that most of the fixed point results in the setting of metric spaces of [3–10] could be derived from Theorems 1–4.

Let be a metric space and let be a partial order on . We list the conditions used in [3, 6, 9, 10] as follows: is a continuous, nondecreasing, and subadditive (i.e., for each ) function with ;there exists a functional and a function with such that for each with , where is a partialorder on ;for each , the order interval is closed;for each , the order interval is closed; is nondecreasing.

Remark 7. (i) It is easy to check that , , and .
(ii) Let be a relation on defined by where and . It follows from [3, Lemma  4.1] that introduced by (17) is a partial order on provided that is satisfied. Clearly, the partial order on introduced by (17) is certainly a partial order such that is satisfied, and hence is satisfied.

Lemma 8. Let be a complete metric space and let be a partial order on . Assume that and are satisfied. Then (i)if is bounded below, then is satisfied;(ii)if is bounded above, then is satisfied.

Proof. (i) Let be an increasing sequence. It suffices to show that is a Cauchy sequence. If otherwise, there exists an increasing subsequence and such that Then by and , we have which implies that is a decreasing sequence of reals and hence convergent since is bounded below. Moreover by (19), we have Let , and by we have , which is a contradiction. And consequently, is a Cauchy sequence.
(ii) Let be a decreasing sequence. It suffices to show that is a Cauchy sequence. If otherwise, there exists a decreasing subsequence and such that (18) is satisfied. Then by and , we have which implies that is an increasing sequence of reals and hence convergent since is bounded above. Moreover by (21), we have Let , and by we have , which is a contradiction. And consequently, is a Cauchy sequence. The proof is complete.

By Lemma 8, Remark 6 and (i) of Remark 7, we have the following two corollaries.

Corollary 9. Let be a complete metric space, let be a partial order on , and let . Assume that are satisfied, is a bounded below (resp., bounded above) functional, there exists such that (resp., ), and is increasing (resp., quasi-increasing) and has compact value on (resp., ). Then has a maximal fixed point (resp., a minimal fixed point ).

Corollary 10 (see [6, Theorem  5]). Let be a complete metric space, and let be a partial order on . Assume that , , and are satisfied and is a bounded below functional. Then each set-valued Caristi-type mapping has fixed point in .

In analogy to the proof of [6, Lemma  1], we can prove the following lemma by (i) of Remark 7.

Lemma 11. Let be a complete metric space and let be a relation on introduced by (17). Assume that is satisfied. Then(i)if is lower semicontinous on , then is satisfied;(ii)if is upper semicontinuous on , then is satisfied;(iii)if is continuous, then is satisfied.

The following two corollaries directly follow from Corollaries 9 and 10, Lemma 11, and Remark 7.

Corollary 12. Let be a complete metric space, let be a relation on introduced by (17), and let . Assume that is satisfied, is a continuous and bounded below (resp., bounded above) functional, there exists such that (resp., ), and is increasing (resp., quasi-increasing) and has compact value on (resp., ). Then has a maximal fixed point (resp., a minimal fixed point ).

Corollary 13 (see [3, Theorem 4.2]). Let be a complete metric space and let . Assume that is satisfied, is a lower semicontinuous and bounded below functional, and for each , there exists such that . Then has fixed point in .

Remark 14. (i) Theorems  3 and  4 in [8] are special cases of Corollary 12 with .
(ii) Note that each single-valued mapping naturally has compact value on . Then (i) of both Theorems  1 and 2 in [10] immediately follows from Corollary 9, and Theorem  2 in [9] is directly derived from Corollary 12. Moreover, if , then Corollary 12 is reduced to Theorems  3 and 4 in [7].

Remark 15. It follows from Remark  2 in [6] that generalized Caristi’s fixed point theorems obtained by Feng and Liu [3], Khamsi [4], and Li [5] are equivalent to primitive Caristi’s result [1] and all the relating results in [3–5] could be obtained by Corollary 10; contrarily, Corollary 10 could not be derived from the results of [3–5]. Therefore, Theorem 4 indeed improves the results in [3–6].

Conflict of Interests

The authors declare that they there is no conflict of interests.

Authors’ Contribution

The authors have contributed in obtaining the new results presented in this paper. All authors read and approved the final paper.

Acknowledgments

The work was supported by Natural Science Foundation of China (11161022), Natural Science Foundation of Jiangxi Province (20114BAB211006, 20122BAB201015), Educational Department of Jiangxi Province (GJJ12280, GJJ13297), and Program for Excellent Youth Talents of JXUFE (201201).