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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 902692, 14 pages
http://dx.doi.org/10.1155/2013/902692
Research Article

Caristi Type Coincidence Point Theorem in Topological Spaces

1School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China
2School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Received 6 June 2013; Revised 7 August 2013; Accepted 13 August 2013

Academic Editor: Wei-Shih Du

Copyright © 2013 Jiang Zhu 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

A generalized Caristi type coincidence point theorem and its equivalences in the setting of topological spaces by using a kind of nonmetric type function are obtained. These results are used to establish variational principle and its equivalences in -complete spaces, bornological vector space, seven kinds of completed quasi-semimetric spaces equipped with -functions, uniform spaces with -distance, generating spaces of quasimetric family, and fuzzy metric spaces.

1. Introduction

Caristi’s fixed-point theorem [1, 2] and its equivalences, Ekeland variational principle [3, 4], and Takahashi minimization theorem are forceful tools in nonlinear analysis, control theory, and global analysis; see, for example, [35]. In the last two decades, Caristi’s fixed-point theorem and Ekeland variational principle have been generalized and extended in several directions. About these, one can refer to, for example, [132] and the references therein. In particular, in [25], a very general Ekeland variational principle and Caristi’s fixed-point theorem are presented, which give a unified approach to three classes of Ekeland type variational principle: in the first class, the underlying space is a sequentially complete uniform space (or equivalently, a sequentially complete F-type topological space), and the perturbation involves a family of topology generating pseudometrics (or quasimetrics); in the second class, the underlying space is a locally complete locally convex space (resp., a locally complete locally -convex space), and the perturbation involves a family of topology generating seminorms (resp., topology generating -homogeneous -pseudonorms) or involves a single Minkowski functional; in the third class, the underlying space is a complete metric space, and the perturbation involves a -distance or a -function. On the other hand, Banach fixed-point theorem has been extended to large class of nonmetric spaces which included -complete topological spaces, symmetric spaces, and quasimetric spaces (see, e.g., [3335]). But to our knowledge, neither Ekeland's variational principle nor any of its equivalents have been established in such -complete topological spaces.

Motivated by the aforementioned works, we attempt to give a unified approach to the previous works. A generalized Caristi type coincidence point theorem in the setting of topological spaces by using a kind of nonmetric type function is proved. As an application of this Caristi's coincidence point theorem, an Ekeland type variational principle and its equivalences in the setting of topological spaces are obtained. Also, these results present Caristi type coincidence point theorem, variational principle, and its equivalences in -complete topological spaces. Moreover, these results are used to establish variational principle and its equivalences in bornological vector space, seven kinds of completed quasi-semimetric spaces equipped with -functions, uniform spaces with -distance, generating spaces of quasimetric family, and fuzzy metric spaces. The results of this paper uniformly extend and generalize the corresponding results appeared in the literature [14, 613, 15, 25, 26, 28, 30, 32].

2. Caristi Type Coincidence Point Theorem

The primary goal of this section is to establish two equivalent generalized Caristi type coincidence point theorems in the setting of topological spaces by using a kind of nonmetric type function. As an application of these Caristi’s coincidence point theorems, equivalent generalized Caristi type common fixed point theorem, Caristi type fixed point theorem for set valued, Caristi type fixed point theorem for single-valued map, Ekeland type variational principles and its equivalences in the setting of topological spaces are obtained. To establish our main results, we need the following definitions.

Definition 1 (see [15]). Let be a topological space. An extended real-valued function is said to be sequentially lower monotone if for every sequence converging to and satisfying we have , for each .

Definition 2. Let be a topological space and a function. A proper function (i.e., is not identically to ) is said to be sequentially lower monotone with respect to (in short, sequentially lower monotone with respect to ) if for any sequence in satisfying , and for each , we have for each .

Definition 3. Let be a topological space and a function.(1) is said to be -complete [33, 34] if any sequence with implies that the sequence is convergent to some .(2) is said to be sequentially lower complete with respect to , if any sequence with implies that the sequence is convergent to some , and for any .(3)Let be a proper function. The topological space is said to be sequentially lower complete with respect to and if any sequence in satisfying and for all is convergent to some , and (2) holds for any .

Remark 4. It is clearly that if is sequentially lower complete w.r.t. , then for any proper function , is sequentially lower complete w.r.t. and .

Now, we can prove the following Caristi type coincidence point theorem in the setting of topological spaces.

Theorem 5. Let be a topological space, a function, a proper, bounded from below, sequentially lower monotone function with respect to , and a nondecreasing function. Assume that is sequentially lower complete with respect to and . Let be a nonempty subset of , a surjective function, an index set, and, for each , a multivalued map. Then the following conclusions hold and are equivalent.(1)Suppose that for each with , there exists , such thatThen for any , there exists a coincidence point of and ; that is, , such that (2)Suppose that for each with , there exists an and , such that (3) holds. Then for any , there exists a coincidence point of and ; that is, , such that (4) holds.

Proof. (1) We take an , since ; without loss of generality, we can assume that . Since is a surjective function, there exists , such that . If , then the conclusion holds. Otherwise, by the supposition, there exists , such thatThus, Hence, Obviously, for any , . Thus, we can take such that Assume that has been taken, and for . If , then the conclusion holds. Otherwise, Note that for any , we have . Thus, we can take such that It remains to consider the case that there is an infinite sequence which satisfies (10) and (11). From (10), we know that is a decreasing sequence. Since is nondecreasing, it follows from (11) that Since is sequentially lower complete w.r.t. and , there exists such that and for any . Assume that , . We claim that the conclusion holds for . Since is sequentially lower monotone, we have That is (4) holds. If , then there exists , such that It follows from that there exists a subsequence such that From this and (14) we know that there exists an such that for all , From (13) we have that is, , . From (10) we have By letting , we get that Combing with (13) we have which contradicts (14). Thus the conclusion of Theorem 5(1) holds.
(2) It is clear that Theorem 5(1) Theorem 5(2). Now, we prove that Theorem 5(2) Theorem 5(1). Assume that the conditions of Theorem 5(1) are satisfied; then, for each , if , there exists , such that (3) holds. Then we get that and . For each , we define by It is clear that if and only if . Also, satisfies the condition of Theorem 5(2). Then by the conclusion of Theorem 5(2), there exists a coincidence point of and ; that is, , such that (4) holds. Therefore, there exists a coincidence point of and ; that is, , such that (4) holds. That is Theorem 5(1) holds. The proof is completed.

Remark 6. If for any with implies that , then the conclusion (1) of Theorem 5 can be rewritten as: for each with there exists , such that

In particular, if and is the identity map in Theorem 5, then we obtain the following generalized Caristi type common fixed point theorem, Caristi type fixed point theorem for set-valued, and single-valued map.

Theorem 7. Let be a topological space, a function, a proper, bounded from below, sequentially lower monotone function with respect to , and a nondecreasing function. Assume that is sequentially lower complete w.r.t. and . Then the following conclusions hold and are all equivalent to Theorem 5.(1)Let be an index set, and, for each , let be a multivalued map.Suppose further that for each with , there exists , such that Then for any , there exists a common fixed point of ; that is, , such that (2)Let be a multivalued map.Suppose further that for each with , there exists , such that (23) holds. Then for any , has a fixed point , such that (24) holds.(3)Let be a map.Suppose further that for each with , there exists , such that (23) holds. Then for any , has a fixed point , such that (24) holds.

Proof. It is clear that the following implications hold: Theorem 5(1) Theorem 7(1) Theorem 7(2) Theorem 7(3).
Now, we prove that Theorem 7(3) Theorem 5(1). Assume that the conditions of Theorem 5 hold. It is similar to the proof of Lemma 2.1 in [36] that, by using the axiom of choice, we can prove that there exists a subset such that and is one-to-one. Define a map by where such that either or Then satisfies the condition of Theorem 7(3); thus for any , has a fixed point , such that holds. Since , there exists , such that . Then by the definition of , we get that . That is, the conclusion of Theorem 5 holds. The proof is completed.

The following corollary is an extension of the results in [19, 20]. In Corollary 8, we remove the condition that is nondecreasing, which is used in [19, 20].

Corollary 8. Let be a completed metric space. Suppose that satisfies and that is lower semicontinuous on , and there exist and two real numbers , such that and one of the following conditions is satisfied:(i), is nonnegative on , and there exist and such that (ii), is nonnegative on , and there exist and such that Then each Caristi type mapping (i.e., satisfying ) has a fixed point .

Proof. Case (i). It follows from and (28) that is a bounded from below, lower semicontinuous function on . Let Then, as in the proof of the Theorem 1 in [20] that is a complete metric space, , and Thus we have, by (29), Define a function by . Let be a sequence in , such that This implies that is a Cauchy sequence in . Since is a complete metric space, is a convergent sequence in . If , then for any , . Thus, is sequentially lower complete w.r.t. . Clearly, all conditions of Theorem 7 are satisfied. Therefore, has a fixed point in .
Case (ii). Let Then we have Thus, the conclusion can be deduced by Case (i). The proof is completed.

In Theorem 9, by using Theorem 5, we present a generalized Ekeland type variational principle, maximal element theorem for a family of multivalued maps, equilibrium theorem, and a generalized Takahashi minimization theorem in topological spaces and prove the equivalence among these results.

Theorem 9. Let be a topological space, a function, a proper, bounded from below, sequentially lower monotone function with respect to , and a nondecreasing function. Assume that is sequentially lower complete with respect to and . Then for any , the following conclusions hold, and they are equivalent to Theorem 5.(I)(Ekeland type variational principle in topological spaces) There exists , such that and (II) (Maximal element for a family of multivalued maps in topological spaces) Let be any index set, and, for each , let be a multivalued map. Assume that for each with , there exists , such that (23) holds. Then there exists , such that and for each .(III)(Equilibrium theorem in topological spaces) Let be a proper, bounded from below, sequentially lower monotone function in the first argument. Suppose that there exists such that for each with there exists such that If is sequentially lower complete w.r.t. and , then there exists , such that and .(IV)(Generalized Takahashi minimization theorem in topological spaces) Suppose that for any with , there exists such that (23) holds. Then there exists , such that and .

Proof. “Theorem 5(1) Theorem 9(II).” If the conclusion of (II) does not hold, then for any , there exists , such that . By the hypotheses of (II), there exists , such that (23) holds; thus, . Let , (the identical map of ), Then the conditions of Theorem 5(1) are satisfied for , and . Thus, from Theorem 5(1) there exists such that . This is a contradiction with the definition of . Therefore, there exists with such that for any .
“(II) (III)” Let From this we know that if , then . By the hypotheses of (III) there exists , such that By using (II) for , there exists , such that and for any ; that is , for any .
“(III) (IV)” Let . If then . Fix ; then if and only if . By the hypothesis of (IV), there exists , such that either or Define by . Then ; thus, the hypotheses of (III) are satisfied for and . It follows from (III) that there exists , such that and , for any . This implies that and , that is, .
“(IV) (I)” If (I) does not hold, then for any with , there exists , such that This implies that condition of (IV) holds on . Then, by (IV) there exists , such that and . This is a contradiction with (46). Thus, (I) holds.
“(I) Theorem 5(1)” From (I), there exists , such that and (37) holds. Since is a surjective mapping, there exists a , such that . We claim that . If , by the hypotheses of Theorem 5(1), there exists , such that (3) holds. This is a contradiction with (37). Thus, and . That is, Theorem 5(1) holds.
The proof is completed.

Remark 10. Theorem 5–Theorem 9 also present Caristi type coincidence point theorem, Ekeland type variational principle, and their equivalences in -complete topological spaces. Moreover, from [34] we know that -complete topological spaces include -complete symmetric (semimetric) spaces and complete quasimetric spaces.

3. Applications to Some Non-Metric Spaces

In this section, we show that our results in section two can be used with many nonmetric spaces. The reader may refer to the references [6, 13, 15, 25, 28, 37] for the notions and symbols in this section.

In [6], the authors introduce the concept of -function in quasimetric spaces which generalizes the notion of the -function and -distance, and they also prove an Ekeland variational principle as well as its equivalences in such spaces.

For the convenience of the reader we present the main concept of quasimetric space in the following (refer to [38]).

Let be a nonempty set. A real valued function is said to be a quasi-semimetric on if the following conditions are satisfied:  (QM1) and for all ; (QM2) for all .

If further (QM3) implies for all ,then is said to be a quasimetric on . A nonempty set together with a quasimetric (or quasi-semimetric ) is called a quasimetric space (or quasi-semimetric space), and it is denoted by . If is a quasi-semimetric space, for and , we define the balls in by the formula—the open ball, and —the closed ball.

The topology of a quasi-semimetric can be defined starting from the family of neighborhoods of an arbitrary point : such that such that .

The convergence of a sequence to with respect to can be characterized by .

Definition 11. Let be a quasi-semimetric space. A sequence in is said to be(i)left -Cauchy if for each there is a point in and an integer such that for all ;(ii)right -Cauchy if for each there is a point in and an integer such that for all ;(iii)-Cauchy if for each there is an integer such that for all ;(iv)right -Cauchy if for each there is an integer such that for all ;(v)left -Cauchy if for each there is an integer such that for all ;(vi)weakly left (right) -Cauchy if for each there is an integer such that () for all ;(vii)corresponding to the seven definitions of Cauchy sequence in a quasi-semimetric space, we have seven notions of completeness: is said to be left (right) -, [weakly] left (right) -, or -sequentially complete if every left (right) -, [weakly] left (right) -, or - (resp.) Cauchy sequence in converges to some point in (with respect to the topology induced on by ).

Remark 12. The implications between the seven notions of Cauchyness (refer to [38]) are as follows: -Cauchy left and right -Cauchy, left (right) -Cauchy weakly left (right) -Cauchy left (right) -Cauchy.

Definition 13 (see [6]). Let be a quasi-semimetric space. A function is called a -function on if the following conditions are satisfied: (Q1) for all , ; (Q2) if , is a sequence in such that it converges to a point (with respect to the quasi-semimetric) and for some , then ; (Q3) for any , there exists such that and imply .

Lemma 14. Let be a quasi-semimetric space with one of seven completeness defined in Definition 11(vii). If is a -function on , then is sequentially lower complete w.r.t. .

Proof. Assume that is a sequence in and . Let , then we have and for any , By (Q3), for any , there exists such that and imply . For the , there exists , such that for any . It follows from (47) that for any , we have Thus (Q3) implies that . That is, is a -Cauchy sequence. Therefore, by Remark 12, is any one of seven Cauchy sequences in Definition 11. Thus, converges to some . Equation (47) and (Q2) imply that . This shows that . For any , it follows from (Q1) that Therefore Thus, is sequentially lower complete w.r.t. . The proof is completed.

From Lemma 14 and Theorems 5, 7, and 9, we can get the following Ekeland type variational principle and its equivalences in quasi-semimetric spaces equipped with -functions, which also generalize the results in [6, 12].

Theorem 15. Let be a complete quasi-semimetric space with one of seven completeness defined in Definition 11(vii) and a -function on . Let be a proper, bounded from below, sequentially lower monotone function and a nondecreasing function. Let be a nonempty subset of , a surjective function, an index set and for each , a multivalued map. Then the following conclusions hold and are equivalent.(1)Suppose that for each with , there exists , such thatThen for any , there exists a coincidence point of and ; that is, , such that (2)Suppose that for each with , there exists an and , such that (51) holds. Then for any , there exists a coincidence point of and ; that is, , such that (52) holds.(3)Let be an index set, and for each , let be a multivalued map.Suppose further that for each with , there exists , such that Then for any , there exists a common fixed point of ; that is, , such that (4) (Ekeland type variational principle in quasi-semimetric spaces) For any , there exists , such that and Moreover, the rest of corresponding equivalent principles in Theorem 9 hold.

In particularly, if is a complete quasi-metric space, then from Theorem 15, we have the following results.

Theorem 16. Let be a complete quasimetric space with one of seven completeness defined in Definition 11(vii) and a -function on . Let be a proper, bounded from below, sequentially lower monotone function and a nondecreasing function. Let be a nonempty subset of , a surjective function, an index set and for each , a multivalued map. Then the following conclusions hold and are equivalent. (1)Suppose that for each with , there exists , such that Then for any , there exists a coincidence point of and ; that is, , such that (2)Suppose that for each with , there exists an and , such that (56) holds. Then for any , there exists a coincidence point of and ; that is, , such that (57) holds.(3)Let be an index set, and for each , let be a multivalued map. Suppose further that for each with , there exists , such that Then for any , there exists a common fixed point of ; that is, , such that (4) (Ekeland type variational principle in quasimetric spaces) For any , there exists , such that and

Proof. The equivalence of the conclusions (1)–(4) is clear. We only prove (4). It follows from (4) of Theorem 15 that for any , there exists , such that and If satisfies (60), then conclusion (4) is proved. Otherwise, there exists , such that If and satisfy (62), then and . By using (Q3) in Definition 13, we get that . It follows from (QM1) that , and then (QM3) implies that . That is, there is only one point which satisfies (62). Let Then . Since , we can imply that . This shows that for any , ; that is, satisfies (60). The proof is completed.

Definition 17 (see, e.g., [37]). Let be a real vector space; a collection of subsets of is called a vector bornology on , if it satisfies the following conditions: (B1) implies that ; (B2) and imply that ; (B3) implies that ; (B4) implies that  (B5) for any bounded interval , implies that In view of (B5), if , so is its balanced hull which is defined by .

Definition 18. The ordered pair is called a bornological vector space (in short: BVS), and every element of is called a bounded subset (with respect to ).

Definition 19 (see, e.g., [28, 37]). Let be a bornological vector space. (i)A sequence in is said to be Mackey-convergent (or -convergent) to a point , denoted by , if there is a balanced and a sequence of positive real numbers such that and for any . Also, we say that is a bornological limit of .(ii)A sequence in is said to be Mackey-Cauchy (or -Cauchy) if there is a balanced and a double sequence of positive real numbers such that and for any .(iii) is said to be Mackey-closed (or -closed) if it contains all bornological limits of any sequences in .(iv) is said to be Mackey-complete (or -complete) if every -Cauchy sequence in will be -convergent to some element in .(v)A BVS is said to be separated if every -convergent sequence is -convergent to exactly one bornological limit.

Remark 20. From Lemma 2.13 in [28] we know that if is a -complete subset, then is -closed. On the other hand, if is -complete and is -closed, then is -complete. For the details about BVS, one can refer to [17, 28, 37].

The collection of all (complements of) -closed subsets of defines a topology on , and we called it bornological topology. Therefore, endowed this topology is a topological space (but, from Remark 2.4 in [28], we can see that it is rarely a vector topology with respect to the algebraic structure of ). In the following, we will assume that is separated; that is (v) in Definition 19 holds.

Let be a separated bornological vector space and a positively homogeneous subadditive function. By Lemma 4.4 in [28], for any nonzero if satisfies the following condition: (P1) the set is -complete and bounded.

Lemma 21. Let be defined by . If (P1) holds, then is sequentially lower complete with respect to .

Proof. Let be a sequence in and . Then for any , there exists a positive integer , such that . Since is subadditive, we get that for any , From this we have So we have . For any , let , and then we have , and . It follows from that where , is the (bounded) balanced hull of . That is, is -Cauchy. Since is -complete, is -convergent. Thus, is also -convergent. Assume that is -convergent to a point . If we set then , and hence whenever . It follows from (68) that for . Since is -closed by (P1), we have Consequently, , and hence . Since , we have That is, Thus, is sequentially lower complete with respect to . The proof is completed.

From Lemma 21 and Theorem 9, we can get the following Ekeland type variational principle in bornological vector space, which is also proved in [17, 28].

Theorem 22 (Ekeland type variational principle in bornological vector space). Let be a separated bornological vector space and a positively homogeneous subadditive function satisfying the condition (P1). Let be a proper, bounded from below, sequentially lower monotone function and a nondecreasing function. Then for any , there exists , such that and Moreover, the corresponding equivalent principles in Theorem 9 hold.

Definition 23 (see [25]). Let be a uniform space. An extended real-valued function is called a -distance on if the following conditions are satisfied: (q1) for any , ; (q2) every sequence with is a Cauchy sequence and in the case implies that in ; (q3) for , and imply .

Here means that for any , there exists such that for all .

Definition 24 (see [25]). Let be a uniform space and a -distance on . A proper function is said to be sequentially lower monotone with respect to (in short, sequentially lower monotone with respect to ) if for any sequence in satisfying , and , we have for each .

Definition 25 (see [25]). Let be a uniform space, a -distance on , and a proper function. is said to be sequentially complete with respect to if for any sequence in satisfying and for each , there exists such that .

Lemma 26. Let be a uniform space, a -distance on , and a sequentially lower monotone with respect to , proper function, bounded from below. If is sequentially lower complete with respect to , then is sequentially lower complete with respect to and .

Proof. Assume that is a sequence in with for each and . Let , then we have that , and for any , This shows that . By using Definition 23 and Definition 25, we get that there exists such that and converges to . Then for any , by we have that Thus, is sequentially lower complete w.r.t. and . The proof is completed.

From Theorems 5, 7, and 9, we can get the following Caristi type coincidence point theorem and Ekeland type variational principle in uniform space equipped with -distance. From Lemma 26, we also see that this is a transformation of the results appeared in [25].

Theorem 27. Let be a uniform space, a -distance on , and a sequentially lower monotone with respect to , proper function, bounded from below. Let be a nonempty subset of , a surjective function, an index set, and, for each , a multivalued map. If is sequentially complete with repect to and and is a nondecreasing function, then the following conclusions hold and are equivalent.(1)Suppose that for each with , there exists , such that Then for any , there exists a coincidence point of and ; that is, , such that (2)Suppose that for each with , there exists an and , such that (76) holds. Then for any , there exists a coincidence point of and ; that is, , such that (77) holds.(3)Let be an index set, and, for each , let