# Critical Point Theorems for Nonlinear Dynamical Systems and Their Applications

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## 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 Open image in new window -functions. Some applications to the existence theorems of nonconvex versions of variational inclusion and disclusion problems in metric spaces are also given.

## Keywords

Fixed Point Theorem Existence Theorem Nonlinear Dynamical System Uniform Space Nondecreasing Sequence## 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 Open image in new window be a complete metric space. Let Open image in new window be a multivalued map with nonempty values. Suppose that the following conditions are satisfied:

(i)for each Open image in new window , we have Open image in new window and Open image in new window is closed,

(ii) Open image in new window , Open image in new window with Open image in new window implies that Open image in new window ,

(iii)for each Open image in new window and each Open image in new window , we have Open image in new window .

Then there exists Open image in new window such that Open image in new window .

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, 7, 8, 9, 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 Open image in new window -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.

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

Open image in new window if Open image in new window .

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

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

has an upper bound.

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

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

(i) Open image in new window is an upper bound of Open image in new window ,

(ii) Open image in new window ,

(iii) Open image in new window .

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, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 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 Open image in new window -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 Open image in new window be a nonempty set. A fuzzy set in Open image in new window is a function of Open image in new window into Open image in new window . Let Open image in new window be the family of all fuzzy sets in Open image in new window . A fuzzy map on Open image in new window is a map from Open image in new window into Open image in new window . This enables us to regard each fuzzy map as a two-variable function of Open image in new window into Open image in new window . Let Open image in new window be a fuzzy map on Open image in new window . An element Open image in new window of Open image in new window is said to be a fuzzy fixed point of Open image in new window if Open image in new window (see, e.g., [11, 12, 16, 24, 25, 26]). Let Open image in new window be a multivalued map. A point Open image in new window is called a *critical point* (or *stationary point* or *strict fixed point*) [1, 3, 8, 27, 28, 29] of Open image in new window if Open image in new window .

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

Recall that a *uniform* Open image in new window is a nonempty set Open image in new window endowed of a uniformity Open image in new window , with the latter being a family of subsets of Open image in new window and satisfying the following conditions:

() Open image in new window for any Open image in new window ,

()If Open image in new window , Open image in new window , then there exists Open image in new window such that Open image in new window ,

()If Open image in new window , then there exists Open image in new window such that Open image in new window ,

()If Open image in new window and Open image in new window , then Open image in new window .

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

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

() Open image in new window for all Open image in new window ,

()if Open image in new window and Open image in new window in Open image in new window with Open image in new window such that Open image in new window for some Open image in new window , then Open image in new window ,

()for any sequence Open image in new window in Open image in new window with Open image in new window , if there exists a sequence Open image in new window in Open image in new window such that Open image in new window , then Open image in new window ,

()for Open image in new window , Open image in new window and Open image in new window imply that Open image in new window .

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

The following result is crucial in this paper.

Lemma 2.1.

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

Proof.

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

Remark 2.2.

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

## 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 Open image in new window be a nonempty set, and let Open image in new window and Open image in new window be functions. Let Open image in new window be a nonempty subset of Open image in new window and Open image in new window a multivalued map with nonempty values. Suppose the following:

(H1) Open image in new window for all Open image in new window and all Open image in new window ,

(H2)for any Open image in new window and Open image in new window , there exists Open image in new window such that Open image in new window for all Open image in new window .

Then there exists a sizing-up function Open image in new window such that Open image in new window is of type Open image in new window .

Proof.

Hence Open image in new window is of type Open image in new window .

Theorem 3.2.

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

(H3)for each Open image in new window , Open image in new window and Open image in new window is closed in Open image in new window ,

(H4) Open image in new window , Open image in new window with Open image in new window implies that Open image in new window ,

(H5)for each Open image in new window , there exists Open image in new window such that Open image in new window , Open image in new window with Open image in new window and Open image in new window implies that Open image in new window .

Then there exist a quasiorder Open image in new window on Open image in new window and a sizing-up function Open image in new window such that Open image in new window is a Open image in new window -bounded quasiordered set.

Proof.

we obtain Open image in new window or Open image in new window . Hence Open image in new window is an upper bound of Open image in new window . Therefore Open image in new window is a Open image in new window -bounded quasiordered set.

Theorem 3.3.

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

Open image in new window , Open image in new window with Open image in new window implies that Open image in new window and Open image in new window ;

(H6)for any Open image in new window , there exists Open image in new window such that Open image in new window .

Then there exists Open image in new window such that

(a) Open image in new window for all Open image in new window ,

(b) Open image in new window .

Proof.

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

Theorem 3.4.

Let Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window be the same as in Theorem 3.3. Assume that the conditions (H1), (H2), (H3), Open image in new window , and (H5) in Theorem 3.3 hold. Let Open image in new window be any index set. For each Open image in new window , let Open image in new window be a multivalued map with nonempty values. Suppose that, for each Open image in new window , there exists Open image in new window . Then there exists Open image in new window such that

(a) Open image in new window is a common fixed point for the family Open image in new window (i.e., Open image in new window for all Open image in new window );

(b) Open image in new window .

Proof.

where Open image in new window is the characteristic function for an arbitrary set Open image in new window . Note that Open image in new window for Open image in new window . Then for any Open image in new window , there exists Open image in new window such that Open image in new window . 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.

It is easy to see that the family Open image in new window is a Hausdorff uniformity on Open image in new window and Open image in new window is Open image in new window -complete.

Lemma 3.6.

Let Open image in new window be a metric space, Open image in new window a map and Open image in new window a multivalued map with nonempty values. Suppose that

(h1)for each Open image in new window , Open image in new window ,

(h2) Open image in new window , Open image in new window with Open image in new window implies that Open image in new window and Open image in new window ,

(h3)if a sequence Open image in new window in Open image in new window satisfies Open image in new window for each Open image in new window , then Open image in new window .

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

Proof.

Then (H1) in Theorem 3.1 holds with Open image in new window .

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

(i) Open image in new window ,

(ii) Open image in new window .

and hence it implies that Open image in new window . Therefore (H2) can be satisfied.

Theorem 3.7.

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

for each Open image in new window , Open image in new window and Open image in new window is closed,

(h4)for any Open image in new window , there exists Open image in new window such that Open image in new window .

Then there exists Open image in new window such that

(a) Open image in new window for all Open image in new window ,

(b) Open image in new window .

Proof.

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

Theorem 3.8.

Let Open image in new window , Open image in new window , Open image in new window , and Open image in new window be the same as in Theorem 3.7. Assume that the conditions Open image in new window , (h2) and (h3) in Theorem 3.7 hold. Let Open image in new window be any index set. For each Open image in new window , let Open image in new window be a multivalued map with nonempty values. Suppose that for each Open image in new window , there exists Open image in new window . Then there exists Open image in new window such that

(a) Open image in new window is a common fixed point for the family Open image in new window ,

(b) Open image in new window .

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 Open image in new window -functions with common fuzzy fixed point theorem.

Theorem 4.1.

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

(a) Open image in new window ,

(b) Open image in new window for all Open image in new window with Open image in new window ,

(c) Open image in new window for all Open image in new window .

Moreover, if one further assumes that

(H)for each Open image in new window and any Open image in new window with Open image in new window , there exists Open image in new window with Open image in new window such that Open image in new window ,

then Open image in new window .

Proof.

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

it follows that Open image in new window for all Open image in new window . So the conclusion (b) holds.

a contradiction. Therefore Open image in new window . The proof is completed.

By using Theorem 4.1, we can immediately obtain the following Open image in new window -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 Open image in new window , Open image in new window , Open image in new window , and Open image in new window be the same as in Theorem 4.1. Let Open image in new window be any index set. For each Open image in new window , let Open image in new window be a multivalued map with nonempty values such that, for each Open image in new window and any Open image in new window , there exists Open image in new window such that Open image in new window . Then for each Open image in new window and Open image in new window with Open image in new window and Open image in new window , there exists Open image in new window such that

(a) Open image in new window ,

(b) Open image in new window for all Open image in new window with Open image in new window ,

(c) Open image in new window is a common fixed point for the family Open image in new window .

Moreover, if one further assumes that

(H)for each Open image in new window and any Open image in new window with Open image in new window , there exists Open image in new window with Open image in new window such that Open image in new window ,

then Open image in new window .

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

()for each Open image in new window , the set Open image in new window or Open image in new window is a closed subset of Open image in new window ,

() Open image in new window with Open image in new window and Open image in new window implies that Open image in new window ,

()if a sequence Open image in new window in Open image in new window satisfies Open image in new window for each Open image in new window , then Open image in new window as Open image in new window ,

()for any Open image in new window , there exists Open image in new window such that Open image in new window and Open image in new window .

Then there exists Open image in new window such that

(a) Open image in new window for all Open image in new window ,

(b) Open image in new window for all Open image in new window .

Proof.

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

Case 1.

If Open image in new window , then Open image in new window is obvious.

Case 2.

If Open image in new window , then Open image in new window . For any Open image in new window , if Open image in new window , one has Open image in new window . Otherwise, if Open image in new window , then it follows from Open image in new window and ( Open image in new window ) that Open image in new window . So Open image in new window . Therefore Open image in new window .

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

() Open image in new window for all Open image in new window ,

From (2), we obtain Open image in new window for all Open image in new window .

Remark 4.5.

Theorem 4.4 generalizes [17, Theorems Open image in new window ] which is one of the main results of Lin and Chuang [17].

Here, we give an example illustrating Theorem 4.4.

Example 4.6.

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

The following conclusion is immediate from Theorem 4.4.

Theorem 4.7.

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

Then there exists Open image in new window such that

(a) Open image in new window is a common fixed point for the family Open image in new window ,

(b) Open image in new window for all Open image in new window .

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 Open image in new window and Open image in new window are replaced by the condition Open image in new window and Open image in new window , where

for each Open image in new window , the set Open image in new window or Open image in new window is a closed subset of Open image in new window ,

Open image in new window with Open image in new window and Open image in new window implies that Open image in new window ,

then there exists Open image in new window such that

(a) Open image in new window for all Open image in new window ,

(b) Open image in new window for all Open image in new window .

Theorem 4.9.

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

Then there exists Open image in new window such that

(a) Open image in new window is a common fixed point for the family Open image in new window ,

(b) Open image in new window for all Open image in new window .

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

(S1)for each Open image in new window , the set Open image in new window or Open image in new window is closed in Open image in new window ,

(S2) Open image in new window with Open image in new window and Open image in new window implies that Open image in new window ,

(S3) if a sequence Open image in new window in Open image in new window satisfies Open image in new window for each Open image in new window , then Open image in new window as Open image in new window ,

(S4)for any Open image in new window , there exists Open image in new window such that Open image in new window and Open image in new window .

Then for each Open image in new window with Open image in new window and Open image in new window , there exists Open image in new window such that

(i) Open image in new window ,

(ii) Open image in new window for all Open image in new window ,

(iii) Open image in new window for all Open image in new window .

Proof.

Let Open image in new window be given and Open image in new window defined by Open image in new window for Open image in new window . Put Open image in new window . Since Open image in new window , Open image in new window . By (S1), Open image in new window 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 Open image in new window such that Open image in new window for all Open image in new window and Open image in new window for all Open image in new window or, equivalently,

(a) Open image in new window ,

(b) Open image in new window for all Open image in new window .

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

Remark 4.11.

Theorem Open image in new window in [17] is a special case of Theorem 4.10.

## Notes

### Acknowledgments

The author wishes to express his hearty thanks to the anonymous referees for their helpful suggestions and comments improving the original draft. This research was supported by the National Science Council of Taiwan.

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