Journal of Applied Mathematics

Volume 2011, Article ID 985797, 22 pages

http://dx.doi.org/10.1155/2011/985797

## The Existence of Cone Critical Point and Common Fixed Point with Applications

Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 824, Taiwan

Received 6 May 2011; Accepted 15 August 2011

Academic Editor: Ya Ping Fang

Copyright © 2011 Wei-Shih Du. 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

We first establish some new critical point theorems for nonlinear dynamical systems in cone metric spaces or usual metric spaces, and then we present some applications to generalizations of Dancš-Hegedüs-Medvegyev's principle and the existence theorem related with Ekeland's variational principle, Caristi's common fixed point theorem for multivalued maps, Takahashi's nonconvex minimization theorem, and common fuzzy fixed point theorem. We also obtain some fixed point theorems for weakly contractive maps in the setting of cone metric spaces and focus our research on the equivalence between scalar versions and vectorial versions of some results of fixed point and others.

#### 1. Introduction

In 1983, Dancš et al. [1] proved the following interesting existence theorem of critical point (or stationary point) for a nonlinear dynamical system.

*Dancš-Hegedüs-Medvegyev's Principle [1]*

Let be a complete metric space. Let be a multivalued map with nonempty values. Suppose that the following conditions are satisfied:(i)for each , we have , and is closed;(ii), with implies ;(iii)for each and each , we have . Then there exists such that .

Dancš-Hegedüs-Medvegyev's Principle has been popularly investigated and applied in various fields of applied mathematical analysis and nonlinear analysis, see, for example, [2, 3] and references therein. It is well known that the celebrated Ekeland's variational principle can be deduced by the detour of using Dancš-Hegedüs-Medvegyev's principle, and it is equivalent to the Caristi's fixed point theorem, to the Takahashi's nonconvex minimization theorem, to the drop theorem, and to the petal theorem. Many generalizations in various different directions of these results in metric (or quasi-metric) spaces and more general in topological vector spaces have been studied by several authors in the past; for detail, one can refer to [2–12].

Let be a topological vector space (t.v.s. for short) with its zero vector . A nonempty subset of is called *a convex cone* if and for . A convex cone is said to be *pointed* if . For a given proper, pointed, and convex cone in , we can define a partial ordering with respect to by will stand for and , while will stand for , where denotes the interior of .

In the following, unless otherwise specified, we always assume that is a locally convex Hausdorff t.v.s. with its zero vector , a proper, closed, convex, and pointed cone in with , and a partial ordering with respect to . Denote by and the set of real numbers and the set of positive integers, respectively.

Fixed point theory in -metric and -normed spaces was studied and developed by Perov [13], Kvedaras et al. [14], Perov and Kibenko [15], Mukhamadiev and Stetsenko [16], Vandergraft [17], Zabrejko [18], and references therein. In 2007, Huang and Zhang [19] reintroduced such spaces under the name of cone metric spaces and investigated fixed point theorems in such spaces in the same work. Since then, the cone metric fixed point theory is prompted to study by many authors; for detail, see [20–29] and references therein.

Very recently, in order to improve and extend the concept of cone metric space in the sense of Huang and Zhang, Du [23] first introduced the concepts of *TVS-cone metric* and *TVS-cone metric *space as follows.

*Definition 1.1 (see [23]). * Let be a nonempty set. A vector-valued function is said to be a *TVS*-*cone metric*, if the following conditions hold: for all , and if and only if ; for all , ; for all . The pair is then called a *TVS*-*cone metric space*.

*Definition 1.2 (see [23]). *Let be a *TVS-cone metric* space, , and, let be a sequence in . (i) is said to *TVS-cone converge to * if, for every with , there exists a natural number such that for all . We denote this by *cone*- or as and call the *TVS-cone limit* of .(ii) is said to be a *TVS-cone Cauchy sequence* if, for every with , there is a natural number such that for all , .(iii) is said to be *TVS*-*cone complete* if every *TVS-*cone Cauchy sequence in is *TVS-cone* convergent. In [23], the author proved the following important results.

Theorem 1.3 (see [23]). *Let be a TVS-cone metric spaces. Then defined by is a metric, where the nonlinear scalarization function is defined by
*

*Example 1.4. *Let , , , , and . Define by
Then is a *TVS-cone* complete metric space. It is easy to verify that
so is a metric on , and is a complete metric space.

Theorem 1.5 (see [23]). *Let be a TVS-cone metric space, let , and let be a sequence in . Then the following statements hold. *(a)*If TVS-cone converges to , then as .*(b)*If is a TVS-cone Cauchy sequence in , then is a Cauchy sequence (in usual sense) in .*

The paper is organized as follows. In Section 2, we first establish some new critical point theorems for nonlinear dynamical systems in cone metric spaces or usual metric spaces, and then we present some applications to generalizations of Dancš-Hegedüs-Medvegyev's principle and the existence theorem related with Ekeland's variational principle, Caristi's common fixed point theorem for multivalued maps, Takahashi's nonconvex minimization theorem, and the common fuzzy fixed point theorem. Section 3 is dedicated to the study of fixed point theorems for weakly contractive maps in the setting of cone metric spaces. In Section 4, we focus our research on the equivalence between scalar versions and vectorial vesions of some results of fixed point and others.

#### 2. Critical Point Theorems in Cone Metric Spaces

Let be a nonempty set. A fuzzy set in is a function of into . Let be the family of all fuzzy sets in . A fuzzy map on is a map from into . This enables us to regard each fuzzy map as a two-variable function of into . Let be a fuzzy map on . An element of is said to be a fuzzy fixed point of if (see, e.g., [4, 5, 30–32]). Let be a multivalued map. A point is called to be *a critical point* (or *stationary point*) [1–3, 7, 32] of if .

Recall that the nonlinear scalarization function is defined by

Lemma 2.1 (see [6, 23, 29, 33]). *For each and , the following statements are satisfied: *(i)*,*(ii)*,*(iii)*,*(iv)*,*(v)* is positively homogeneous and continuous on ,*(vi)*if (i.e., ), then ,*(vii)* for all ,. *

*Remark 2.2. *Notice that the reverse statement of (vi) in Lemma 2.1 (i.e., or ) does not hold in general. For example, let , let , and let . Then is a proper, closed, convex, and pointed cone in with and . For , it is easy to see that , and . By applying (iii) and (iv) of Lemma 2.1, we have while .

*Definition 2.3. *Let be a nonempty subset of a *TVS-cone metric* space . (i)The *TVS-cone* closure of , denoted , is defined by
Obviously,.(ii) is said to be *TVS*-*cone closed* if .(iii) is said to be *TVS-cone open* if the complement of A is *TVS-cone* closed.

If , and , then is a metric in usual sense, and the closure of is denoted by .

Theorem 2.4. *Let be a TVS-cone metric space and let
**
Then is a topology on induced by . *

*Proof. *Clearly, and are *TVS-cone* closed in . Thus, and are *TVS-cone* open in , and hence , . Let , . Then and are *TVS-cone* closed in . We claim that . Let . Then such that as . Without loss of generality, we may assume that there exists a subsequence of . Since as , we have . So , and; hence, is *TVS-cone* closed in . From
we see that is *TVS-cone* open in and .

Let be any index set, and let . We show that . For each , set . Thus, is *TVS-cone* closed in for all . Let . Then such that as . For each , since and , . Hence, . So , and then is *TVS*-*cone* closed in . Since
is *TVS*-*cone* open in , and .

Therefore, by above, we prove that is a topology on .

The following result is simple, but it is very useful in this paper.

Lemma 2.5. *Let be a t.v.s., a convex cone with in , and let , , . Then the following statements hold. *(i)*.*(ii)*If and , then .*(iii)*If and , then .*(iv)*If and , then .*(v)*If and , then .*(vi)*If and , then .*(vii)*If and , then .*(viii)*If and , then . *

*Proof. *The conclusion (i) follows from the facts that the set is open in , and is a convex cone. By the transitivity of partial ordering , we have the conclusion (ii). To see (iii), since and , it follows from (i) that
which means that . The proofs of conclusions (iv)–(viii) are similar to (iii).

*Definition 2.6. *Let be a *TVS-cone* metric space. A nonempty subset of is said to be *TVS-cone compact* if every sequence in has a *TVS-cone* convergent subsequence whose *TVS-cone* limit is an element of .

If is *TVS-cone* compact, then we say that is a *TVS-cone* compact metric space.

Theorem 2.7. *Let be a nonempty subset of a TVS-cone metric space . Then the following statements hold. *(a)*If is a closed set in the metric space , then is TVS-cone closed in and , where .*(b)*If is TVS-cone compact, then it is TVS-cone closed.*(c)*If is TVS-cone closed and is TVS-cone complete, then is also TVS -cone complete.*(d)*If is TVS-cone compact, then is TVS-cone complete.*(e)*If is TVS-cone compact, then is (sequentially) compact in the metric space .*

*Proof. *Applying Theorem 1.3, is a metric on . Let be a closed set in the metric space . By Theorem 1.5, we have
which implies that is *TVS-cone* closed in and . Hence, the conclusion (a) holds.

Next, assume that is *TVS-cone* compact in . Let . Then there exists such that as . By the *TVS*-*cone* compactness of , there exist and such that as . Applying Theorem 1.5, as and as . By the uniqueness of limit, , and; hence, . So is *TVS-cone* closed in , and (b) is proved.

To see (c), let be a *TVS-cone* Cauchy sequence in . Since is *TVS*-*cone* complete, there exists such that as . Hence, , which show that is *TVS-cone* complete.

Let us verify (d). Given with , and let be a *TVS-cone* Cauchy sequence in . Then there exists such that for all , . Since is *TVS-cone* compact, there exists a subsequence of , and such that as . For , there exists such that for all . Let ,. For any , since
by (iii) of Lemma 2.5, we have . So is *TVS*-*cone* convergent to . Therefore, is *TVS*-*cone* complete.

The conclusion (e) is obvious. The proof is completed.

Let be a subset of a *TVS-cone* metric space . We denote

It is obvious that in implies .

Now, we first introduce the concepts of fitting nest.

*Definition 2.8. *A sequence of subsets of a *TVS-cone* metric space is said to be a *fitting nest *if it satisfies the following properties: (FN1) for each ,(FN2)for any with , there exists such that for all .

*Remark 2.9. *(a) It is easy to observe that if , , and , then is a metric, and Assumption (FN2) is equivalent to if Assumption (FN1) holds, where is the diameter of . Indeed, “ (FN2)” is obvious. Conversely, if (FN2) holds, then, by (FN1), for any , there exists such that for all . This show that .

(b) Let be a metric space. Then a sequence in is a fitting nest for each and .

The following intersection theorem in *TVS-cone metric* spaces is one of the main results of this paper.

Theorem 2.10. *Let be a fitting nest in a TVS-cone metric space . Then the following statements hold. *(a)*.*(b)*If is TVS-cone complete and is TVS-cone closed in for all , then contains precisely one point. *

*Proof. *(a) Let be given. Then . By (FN2) and (iv) of Lemma 2.1, there exists such that
for all , which implies . By (FN1), we obtain
Hence, .

(b) Given with . By (FN2), there exists such that for all . For each , choose . Then, for , with ; since from (FN1), we have
Hence, is a *TVS-cone* Cauchy sequence in . By the *TVS-cone* completeness of , there exists , such that *TVS-cone* converges to . For any , from the *TVS-cone *closedness of and as , we have
So , and hence . Finally, we claim . For each , applying (a), we have
Hence, or, equivalency, , which gives the required result (b).

Theorem 2.11. *Let be a TVS-cone complete metric space. Then there exists a nonempty proper subset of , such that contains infinite points of , and is a complete metric space, where .*

*Proof. *Let be a fitting nest in . Following the same argument as in the proof of (b), we can obtain a sequence satisfying (1),(2) is a *TVS*-*cone* Cauchy sequence in ,(3)*TVS-cone *converges to some point in .

Applying Theorem 1.5 with (2) and (3), we know that is a Cauchy sequence in , and or as . Let . Therefore, is a complete metric space.

The celebrated Cantor intersection theorem [2] in metric spaces can be proved by Theorem 2.10 and Remark 2.9.

Corollary 2.12 (Cantor). *Let be a metric space, and let be a sequence of closed subsets of satisfying for each and . Then contains precisely one point.*

The following existence theorems relate with critical point and common fuzzy fixed point for a nonlinear dynamical system in *TVS-cone* complete metric spaces or complete metric spaces.

Theorem 2.13. *Let be a TVS-cone complete metric space, let a map, and let be a multivalued map with nonempty values. Let be any index set. For each , let be a fuzzy map on . Suppose that the following conditions are satisfied. *(H1)*For each , is TVS-cone closed in .*(H2)* with implies and .*(H3)*For any with for each , it satisfies the following:**For any with , there exists such that for all .*(H4)*For any , there exists such that . ** Then there exists such that*(a)* for all .*(b)*. *

*Proof. *Let be given. Define a sequence by and for . Hence, for all from (H2). For each , let . By (H2) and (H3), we know that is a fitting nest in . By (H1), is *TVS-cone* closed in for all . Applying Theorem 2.10, there exists such that . Since for all , by (H2), we obtain
which implies . For each , by (H4), . By (H2) again, we have . Therefore, . The proof is completed.

*Remark 2.14. *Let , let , and let , then is a metric, and Assumption (H3) in Theorem 2.13 is equivalent to

for any with for each , we have .

The following critical point theorems are immediate from Theorem 2.13.

Theorem 2.15. *Let be a complete metric space, let be a map and let be a multivalued map with nonempty values. Let be any index set. For each , let be a fuzzy map on . Suppose that the following conditions are satisfied. **For each , is closed.** with implies and .**For any with for each , we have .**For any , there exists such that . ** Then there exists such that *(a)* for all .*(b)*.*

Corollary 2.16. *Let be a TVS- cone complete metric space, and let be a multivalued map with nonempty values. Suppose that the following conditions are satisfied. *(i)*For each , is TVS-cone closed in .*(ii)* with implies .*(iii)*For any with , for each , it satisfies the following:**For any with , there exists such that for all .**Then there exists such that .*

*Proof. *Let be a fuzzy map on defined by for all , , and let be an identity map. Therefore, the conclusion follows from Theorem 2.13.

Corollary 2.17. *Let be a complete metric space, and let be a multivalued map with nonempty values. Suppose that the following conditions are satisfied. *(i)*For each , is closed.*(ii)* with implies .*(iii)*For any with , for each , we have . **
Then there exists such that .*

Theorem 2.18. *Let be a complete metric space, and let be a multivalued map with nonempty values such that with implies . Then the following statements holds. *(1)*If a sequence in satisfies for each and , then .*(2)*If has the following property :
then there exists a sequence in satisfying for each and .*

*Proof. *(1) Let in with for each , and let . For any , by our hypothesis, , and; hence,
So for each . Since , we have
That is, .

(2) Suppose that has the property . Define a function by
We first note that for some . Indeed, on the contrary, assume that for all . Let be given. Set . Since , , and then there exists such that . Since , there exists such that . Continuing in the process, we can obtain a sequence , such that, for each ,

,. By , we have . On the other hand, by , we also obtain , a contradiction. Therefore, there exists such that . Let . Since
we have , and there exists such that
Since , we have and . So there exists such that
Continuing in this way, we can construct a sequence in satisfying, for each , ,. From , we have . By , it follows that .

*Remark 2.19. *In general, under the same assumptions of Theorem 2.18, does not always imply . For example, let with the metric . Then is a complete metric space. For each , define by . Thus, , with implies . Choose in with ; for all , we have while .

The following result is also a generalized Dancš-Hegedüs-Medvegyev's principle with common fuzzy fixed point. Notice that we do not assume for all .

Theorem 2.20. *Let be a complete metric space. Let be a multivalued map with nonempty values. Let be any index set. For each , let be a fuzzy map on . Suppose that the following conditions are satisfied. *(D1)*For each , is closed.*(D2)*, with implies .*(D3)*( Property ) for any with for each , we have .*(D4)

*For any , there exists such that .*

*Then there exists such that*(a)

*for all ,*(b)

*.*

*Proof. *By conclusion (2) of Theorem 2.18, there exists a sequence in satisfying for each and . For each , let . By (D2) and , we see that is a fitting nest in . Applying Theorem 2.10, there exists such that . Since for all , by (D2) again, we obtain
which implies . For each , by (D4), . The proof is completed.

The following existence theorem relate with common fixed point for multivalued maps and critical point for a nonlinear dynamical system in *TVS-cone* complete metric spaces.

Theorem 2.21. *Let , , and be the same as in Theorem 2.13. Assume that conditions (H1), (H2), and (H3) in Theorem 2.13 hold. Let be any index set. For each , let be a multivalued map with nonempty values. Suppose that, for each , there exists . Then there exists such that *(a)* is a common fixed point for the family (i.e., for all ),*(b)*. *

*Proof. *For each , define a fuzzy map on by
where is the characteristic function for an arbitrary set . Note that for . Then, for any , there exists such that . So (H4) in Theorem 2.13 holds. Therefore, the result follows from Theorem 2.13.

*Remark 2.22. *(a) Theorems 2.20 and 2.21 all generalize and improve the primitive Dancš-Hegedüs-Medvegyev's principle.

(b) Corollary 2.17 is a special case of Theorem 2.20 or Theorem 2.21.

The following result is a special case of [32, Theorem 4.1], but it can also be proved by applying Theorem 2.20 (please follow a similar argument as in the proof of [32, Theorem 4.1]).

Theorem 2.23. *Let be a complete metric space, let be a proper l.s.c. and bounded from below function, and let be a nondecreasing function. Let be any index set. For each , let be a fuzzy map on . Suppose that, for each , there exists such that and . Then, for each with , there exists such that *(a)*,*(b)* for all with ,*(c)* for all . **Moreover, if further assume that**
(H) for any with , there exists with such thatthen .*

By using Theorem 2.23, one can immediately obtain the following existence theorem related to generalized Ekeland's variational principle, generalized Takahashi's nonconvex minimization theorem, and generalized Caristi's common fixed point theorem for multivalued maps.

Theorem 2.24. *Let , , and be be the same as in Theorem 2.23. Let be any index set. For each , let be a multivalued map with nonempty values such that, for each , there exists such that . Then, for each with , there exists such that *(a)*.*(b)* for all with .*(c)* is a common fixed point for the family . **Moreover, if further assume that**
(H) for any with , there exists with such that , then .*

#### 3. Fixed Point Theorems in Cone Metric Spaces

In this section, motivated by the recent results of Abbas and Rhoades [21], we will present some generalizations of those in *TVS-cone* complete metric spaces.

Theorem 3.1. *Let be a TVS-cone complete metric space. Suppose that are two self-maps of satisfying
**
for all , , where , , and . Then the following statements hold:*(a)*There exists a nonempty proper subset of , such that contains infinite points of and is a complete metric space, where .*(b)* and have a unique common fixed point in (in fact, the unique common fixed point of and belongs to ). Moreover, for each , the mixed iterative sequence , defined by and for , TVS-cone converges to the common fixed point.*(c)*Any fixed point of is a fixed point of , and conversely.*

*Proof. *Since is a locally convex Hausdorff’s t.v.s. with its zero vector , let denote the topology of , and let be the base at consisting of all absolutely convex neighborhood of . Let
Then is a family of seminorms on . For each , let
and let
Then is a base at , and the topology generated by is the weakest topology for such that all seminorms in are continuous and . Moreover, given any neighborhood of , there exists such that (see, e.g., [34, Theorem 12.4 in II.12, Page 113] or the proofs of [28, Theorem 3.1] and [29, Theorem 2.1]).

Let be given. First, from our hypothesis, we have
If is a fixed point of , then, by using (3.1),
implies that
or
Since and is pointed, we have . So , and; hence, ; that is, is a common fixed point of and . Otherwise, if , we will define the mixed iterative sequence by and for . Then . We claim that is a *TVS-cone* Cauchy’s sequence in . Let .

By (3.5), we know . For each , we have
which implies that
Similarly, we also obtain
Hence, for each ,
Therefore, for with ,
Given with (i.e., , there exists a neighborhood of such that . Therefore, there exists with such that , where
for some , and . Let , and let . If , since each is a seminorm, we have and
for all and all . If , since , , and hence there exists such that for all . So, for each and any , we obtain
Therefore, for any , ( for all , and hence . So we obtain
or
for all . For , with , by (3.13), (3.18), and Lemma 2.5, it follows that
Hence, is a *TVS-cone* Cauchy sequence in . By the *TVS-cone* completeness of , there exists , such that *TVS-cone* converges to . On the other hand, applying Theorem 1.5, is a Cauchy sequence in , and or as . Let . Then is a complete metric space, and the conclusion (a) holds.

To see (b), it suffices to show that is the unique common fixed point of and . By (v) and (vi) of Lemma 2.1, the assumption (3.1) implies
For any , by (3.20),
which implies that
Since is a metric and as , the right-hand side of (3.22) approaches zero as . Hence, or . Also, since