Critical Point Theorems for Nonlinear Dynamical Systems and Their Applications

Abstract

We present some new critical point theorems for nonlinear dynamical systems which are generalizations of Dancš-Hegedüs-Medvegyev's principle in uniform spaces and metric spaces by applying an abstract maximal element principle established by Lin and Du. We establish some generalizations of Ekeland's variational principle, Caristi's common fixed point theorem for multivalued maps, Takahashi's nonconvex minimization theorem, and common fuzzy fixed point theorem for -functions. Some applications to the existence theorems of nonconvex versions of variational inclusion and disclusion problems in metric spaces are also given.

1. Introduction

In 1983, Dancš et al. [1] proved the following existence theorem of critical point (or stationary point or strict fixed 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 that ,

(iii)for each and each , we have .

Then there exists such that .

The famous Dancš-Hegedüs-Medvegyev's Principle is an important tool in various fields of applied mathematical analysis and nonlinear analysis. A number of generalizations of these results have been investigated by several authors; for example, see [2, 3] and references therein.

In 1963, Bishop and Phelps [4] proved a fundamental theorem concerning the density of the set of support points of a closed convex subset of a Banach space by using a maximal element principle in certain partially ordered complete subsets of a normed linear space. Later, the famous Brézis-Browder's maximal element principle [5] was established and applied to deal with nonlinear problems. Many generalizations in various different directions of maximal element principle have been studied in the past; for example, see [2, 3, 6–10] and references therein. However, few literatures are concerned with how to define a sufficient condition for a nondecreasing sequence on a quasiordered set to have an upper bound. Recently, Du [7] and Lin and Du [3] defined the concepts of sizing-up function and -bounded quasiordered set (see Definitions 1.1 and 1.3 below) to describe a rational condition for a nondecreasing sequence on a quasiordered set to have an upper bound.

Definition 1.1 (see [3, 7]).

Let be a nonempty set. A function defined on the power set of is called - if it satisfies the following properties:

,

if .

Definition 1.2 (see [3, 7]).

Let be a nonempty set and a sizing-up function. A multivalued map with nonempty values is said to be of if, for each and , there exists a such that .

Definition 1.3 . (see [3, 7]).

A quasiordered set with a sizing-up function , in short , is said to be - if every nondecreasing sequence in satisfying

(1.1)

has an upper bound.

In [7] (see also [3]), Lin and Du established the following abstract maximal element principle in a -bounded quasiordered set with a sizing-up function .

Theorem LD [see [3, 7]]

Let be a -bounded quasiordered set with a sizing-up function . For each , let be defined by . If is of type , then, for each , there exists a nondecreasing sequence in and such that

(i) is an upper bound of ,

(ii),

(iii).

It is well known that Ekeland's variational principle is equivalent to Caristi's fixed point theorem, to Takahashi's nonconvex minimization theorem, to the drop theorem, and to the petal theoerm. Many generalizations in various different directions of these results in metric (or quasimetric) spaces and more general in topological vector spaces have been investigated by several authors in the past; for detail, one can refer to [2, 3, 7–9, 11–23]. By applying Theorem LD, Du [7] gave a generalized Brézis-Browder principle, system (vectorial) versions of Ekeland's variational principle and maximal element principle and a vectorial version of Takahashi's nonconvex minimization theorem. Moreover, the author investigated the equivalence between scalar versions and vectorial versions of these results. For more detail, one can see [7].

The paper is divided into four sections. In Section 3, we establish some new critical point theorems for nonlinear dynamical systems which are generalizations of Dancš, Hegedüs and Medvegyev's principles in uniform spaces and metric spaces by applying an abstract maximal element principle established by Lin and Du. We also give some generalizations of Ekeland's variational principle, Caristi's common fixed point theorem for multivalued maps, Takahashi's nonconvex minimization theorem, and common fuzzy fixed point theorem for -functions. Some existence theorems of nonconvex versions of variational inclusion and disclusion problems in metric spaces are also given in Section 4. Our techniques and some results are quite original in the literatures.

2. Preliminaries

Let us begin with some basic definitions and notation that will be needed in this paper. 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., [11, 12, 16, 24–26]). Let be a multivalued map. A point is called a critical point (or stationary point or strict fixed point) [1, 3, 8, 27–29] of if .

Let "" be a quasiorder (preorder or pseudoorder, that is, a reflexive and transitive relation) on . Then is called a quasiordered set. In a quasiordered set , recall that an element in is called a of if there is no element of , different from , such that . Denote by and the set of real numbers and the set of positive integers, respectively. A sequence in is called (resp., ) if (resp., ) for each .

Let be a nonempty set and , any subsets of . Denote by

(2.1)

Recall that a uniform is a nonempty set endowed of a uniformity , with the latter being a family of subsets of and satisfying the following conditions:

() for any ,

()If , , then there exists such that ,

()If , then there exists such that ,

()If and , then .

Two points and of are said to be - whenever and . A sequence in is called a for (-, for short) if, for any , there exists such that and are -close for , . A nonempty subset of is said to be - if every -Cauchy sequence in converges. A uniformity defines a unique topology on . A uniform space is said to be Hausdorff if and only if the intersection of all the reduces to the diagonal of , that is, if for all implies that . This guarantees the uniqueness of limits of sequences.

Let be a metric space. A real-valued function is said to be proper if . Recall that a function is called a -function [9, 18], if the following conditions hold:

() for all ,

()if and in with such that for some , then ,

()for any sequence in with , if there exists a sequence in such that , then ,

()for , and imply that .

It is known that any -distance [15, 18, 19, 21, 22, 30, 31] is a -function; see [18, Remark ].

The following result is crucial in this paper.

Lemma 2.1.

Let be a metric space and let be a function. Assume that satisfies condition . If a sequence in with , then is a Cauchy sequence in .

Proof.

Let in with . We claim that is a Cauchy sequence. For each , let . Then is nonincreasing and so exists. If , then there exist sequences and with such that for . On the other hand, since , by , we have , a contradiction. Therefore which shows that is a Cauchy sequence in .

Remark 2.2.

Notice that the function was assumed a -function in [18, Lemma 2.1] and the proof of [18, Lemma 2.1] was incomplete since only was demonstrated if any sequence in satisfied

3. New Critical Point Theorems in Uniform Spaces and Metric Spaces

In this section, we will establish some new critical point theorems for nonlinear dynamical systems which are generalizations of Dancš-Hegedüs-Medvegyev's principle with common fuzzy fixed point in uniform spaces and metric spaces.

Theorem 3.1.

Let be a nonempty set, and let and be functions. Let be a nonempty subset of and a multivalued map with nonempty values. Suppose the following:

(H1) for all and all ,

(H2)for any and , there exists such that for all .

Then there exists a sizing-up function such that is of type .

Proof.

Define by

(3.1)

Then is a sizing-up function. We will claim that is of type . Let and be given. By (H1) and (H2), there exists such that

(3.2)

Hence is of type .

Theorem 3.2.

Let be a uniform space, and let and be functions. Let be a sequentially -complete nonempty subset of and a multivalued map with nonempty values. Suppose that conditions (H1) and (H2) in Theorem 3.1 hold and further assume that

(H3)for each , and is closed in ,

(H4), with implies that ,

(H5)for each , there exists such that , with and implies that .

Then there exist a quasiorder on and a sizing-up function such that is a -bounded quasiordered set.

Proof.

Put a binary relation on by

(3.3)

and let be defined by . Clearly, for each and is a quasiorder from (H3) and (H4). Let be the same as in Theorem 3.1. From the proof of Theorem 3.1, we know that is a sizing-up function and is of type . We want to show that is a -bounded quasiordered set. Let be a nondecreasing sequence in satisfying

(3.4)

Since for , with , . Let and choose such that . By (H5), there exists such that , with and imply that . Since , there exists such that for all , with . It implies that and hence for all , with . Since , we have and for . Therefore, is a nondecreasing -Cauchy sequence in . By the sequential -completeness of , there exists such that as . For each , since is closed from (H3) and

(3.5)

we obtain or . Hence is an upper bound of . Therefore is a -bounded quasiordered set.

Theorem 3.3.

Let be a Hausdorff uniform space, and let and be functions. Let be a sequentially -complete nonempty subset of , a map, and a multivalued map with nonempty values. Let be any index set. For each , let be a fuzzy map on . Suppose the conditions (H1), (H2), (H3), and (H5) in Theorem 3.2 hold and further assume

, with implies that and ;

(H6)for any , there exists such that .

Then there exists such that

(a) for all ,

(b).

Proof.

Applying Theorem 3.1 and Theorem 3.2, is of type and is a -bounded quasiordered set, where , , and are the same as in Theorems 3.1 and 3.2. By Theorem LD, for each , there exists such that . Then it follows from the definition of , , and that for all . We want to prove that . Since for all and all , by (H5), we have for all and all . Since is a Hausdorff uniformity,

(3.6)

and hence we have . For each , by (H6), . On the other hand, by , we have . Therefore . The proof is completed.

Theorem 3.4.

Let , , , , and be the same as in Theorem 3.3. Assume that the conditions (H1), (H2), (H3), , and (H5) in Theorem 3.3 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

(3.7)

where is the characteristic function for an arbitrary set . Note that for . Then for any , there exists such that . So (H6) in Theorem 3.3 holds and hence all conditions in Theorem 3.3 are satisfied. Therefore the result follows from Theorem 3.3.

Remark 3.5.

Let be a complete metric space. For each , let

(3.8)

It is easy to see that the family is a Hausdorff uniformity on and is -complete.

Lemma 3.6.

Let be a metric space, a map and a multivalued map with nonempty values. Suppose that

(h1)for each , ,

(h2), with implies that and ,

(h3)if a sequence in satisfies for each , then .

Then there exist functions and such that the conditions (H1) and (H2) in Theorem 3.1 hold.

Proof.

Define and by

(3.9)

Then (H1) in Theorem 3.1 holds with .

Let us verify (H2). Let and be given. Then there exists such that

(3.10)

Note first that for some . Indeed, on the contrary, suppose that for all . Take . Thus . Hence there exists such that . Since , there exists such that . Continuing in the process, we can obtain a sequence such that, for each ,

(i),

(ii).

So, we have which contradicts condition (h3). Therefore there exists such that . Let . Choose such that

(3.11)

Let and assume that is already known. Then, by induction, we obtain a sequence in such that and

(3.12)

It follows that

(3.13)

By (h2) and (h3), we have . So there exists such that

(3.14)

Since for each , we have

(3.15)

From (3.12) and (3.15), we obtain

(3.16)

Let . Hence, combining (3.10), (3.14), and (3.16), we have

(3.17)

Let . Thus, by (3.13) and (3.17), and . On the other hand, from the definition of , we have

(3.18)

Finally, in order to complete the proof, we need to show that for all . Let . Then and . For any , since , we get

(3.19)

and hence it implies that . Therefore (H2) can be satisfied.

Theorem 3.7.

Let be a complete metric space, a map, and a multivalued map with nonempty values. Let be any index set. For each , let be a fuzzy map on . Suppose that conditions (h2) and (h3) in Theorem 3.4 hold and further assume

for each , and is closed,

(h4)for any , there exists such that .

Then there exists such that

(a) for all ,

(b).

Proof.

For each , define

(3.20)

Then is a Hausdorff uniformity on and is -complete. Clearly, conditions (H3), , and (H6) in Theorem 3.3 hold. By Lemma 3.6, (H1) and (H2) in Theorem 3.1 holds. Let for . Take . If , with and , then which means that . So (H5) in Theorem 3.2 holds. Therefore the conclusion follows from Theorem 3.3.

Theorem 3.8.

Let , , , and be the same as in Theorem 3.7. Assume that the conditions , (h2) and (h3) in Theorem 3.7 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 ,

(b).

Remark 3.9.

Theorems 3.3–3.8 all generalize and improve the primitive Dancš-Hegedüs-Medvegyev's principle.

4. Some Applications to Nonlinear Problems

The following result is a generalization of Ekeland's variational principle and Takahashi's nonconvex minimization theorem for -functions with common fuzzy fixed point theorem.

Theorem 4.1.

Let be a complete metric space, a proper l.s.c. and bounded from below function, a nondecreasing function, and a -function on with being l.s.c. for each . Let be any index set. For each , let be a fuzzy map on . Suppose that, for each and any , there exists such that and . Then for each and any with and , there exists such that

(a),

(b) for all with ,

(c) for all .

Moreover, if one further assumes that

(H)for each and any with , there exists with such that ,

then .

Proof.

Take as an identity map. Let be given and let with and . Put

(4.1)

By the lower semicontinuity of and , is a nonempty closed set in . So is a complete metric space. Define by

(4.2)

Then for each , we have and is closed. It is easy to see that if , with , then . By our hypothesis, for each , there exists such that .

We will prove that if a sequence in satisfies for each , then . Let satisfy for each . Then is a nonincreasing sequence. Since is bounded below, exists. We claim that . Let , . For with , since is nondecreasing, we have

(4.3)

Then for each . Since , we obtain and . By Lemma 2.1, is a Cauchy sequence in , and hence we have . So all the conditions of Theorem 3.7 are satisfied. Applying Theorem 3.7, there exists such that

(4.4)

(4.5)

Since , we have the conclusion (a). From (4.5), for all with . For any , since

(4.6)

it follows that for all . So the conclusion (b) holds.

Moreover, assume that condition (H) holds. On the contrary, if , then there exists with such that . But, by (b), we have

(4.7)

a contradiction. Therefore . The proof is completed.

By using Theorem 4.1, we can immediately obtain the following -function version of generalized Ekeland's variational principle, generalized Takahashi's nonconvex minimization theorem, and generalized Caristi's common fixed point theorem for multivalued maps.

Theorem 4.2.

Let , , , and be the same as in Theorem 4.1. Let be any index set. For each , let be a multivalued map with nonempty values such that, for each and any , there exists such that . Then for each and with and , there exists such that

(a),

(b) for all with ,

(c) is a common fixed point for the family .

Moreover, if one further assumes that

(H)for each and any with , there exists with such that ,

then .

Remark 4.3.

Theorem 4.2 extends some results in [2, 8, 14, 15, 19, 22] and references therein.

The following result is an existence theorem of nonconvex version of variational disclusion problem with common fuzzy fixed point theorem in metric spaces.

Theorem 4.4.

Let be a complete metric space, a nonempty set with , and a multivalued map. Let be any index set. For each , let be a fuzzy map on . Assume that

()for each , the set or is a closed subset of ,

() with and implies that ,

()if a sequence in satisfies for each , then as ,

()for any , there exists such that and .

Then there exists such that

(a) for all ,

(b) for all .

Proof.

Take as an identity map. Define by

(4.8)

Clearly, , (h3), and (h4) in Theorem 3.7 hold. To see (h2), let , with . We need to consider the following two possible cases:

Case 1.

If , then is obvious.

Case 2.

If , then . For any , if , one has . Otherwise, if , then it follows from and () that . So . Therefore .

By Cases 1 and 2, we prove that (h2) holds. Applying Theorem 3.7, there exists such that

() for all ,

().

From (2), we obtain for all .

Remark 4.5.

Theorem 4.4 generalizes [17, Theorems ] which is one of the main results of Lin and Chuang [17].

Here, we give an example illustrating Theorem 4.4.

Example 4.6.

Let with the metric for , . Then is a complete metric space. Let and let be defined by

(4.9)

Let and , for every , and define a fuzzy map by

(4.10)

Clearly, for each . Note that, for each , or is nonempty and closed in . So () and () hold. To see (), let with and . It is easy to see that holds. Finally, let be a sequence in satisfing for each . So is a nondecreasing sequence and for each . Thus converges in and hence as . So () also holds. By Theorem 4.4, there exists (in fact, we take ) such that and for all .

The following conclusion is immediate from Theorem 4.4.

Theorem 4.7.

Let , , , , , , and be the same as in Theorem 4.4. Let be any index set. For each , let be a multivalued map with nonempty values. Suppose that for each , there exists such that .

Then there exists such that

(a) is a common fixed point for the family ,

(b) for all .

Following a similar argument as in Theorem 4.4, we can easily obtain the following existence theorem of nonconvex version of variational inclusion problem in metric spaces.

Theorem 4.8.

In Theorem 4.4, if conditions and are replaced by the condition and , where

for each , the set or is a closed subset of ,

with and implies that ,

then there exists such that

(a) for all ,

(b) for all .

Theorem 4.9.

Let , , , , , , and be the same as in Theorem 4.8. Let be any index set. For each , let be a multivalued map with nonempty values. Suppose that for each , there exists such that .

Then there exists such that

(a) is a common fixed point for the family ,

(b) for all .

The following existence theorem of nonconvex version of variational inclusion and disclusion problem in the Ekeland's sense is immediate from Theorem 4.4.

Theorem 4.10.

Let be a complete metric space, a -function on with being l.s.c. for each , a topological vector space with origin , a multivalued maps, and . Let be any index set. For each , let be a fuzzy map on . Assume that

(S1)for each , the set or is closed in ,

(S2) with and implies that ,

(S3) if a sequence in satisfies for each , then as ,

(S4)for any , there exists such that and .

Then for each with and , there exists such that

(i),

(ii) for all ,

(iii) for all .

Proof.

Let be given and defined by for . Put . Since , . By (S1), be a complete metric space. It is not hard to see that all conditions in Theorem 4.4 are satisfied from (S1)–(S4). Applying Theorem 4.4, there exists such that for all and for all or, equivalently,

(a),

(b) for all .

For any , if , then, by (S2) and (a), we have , which is a contradiction. Therefore for all .

Remark 4.11.

Theorem in [17] is a special case of Theorem 4.10.