- Research Article
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Critical Point Theorems for Nonlinear Dynamical Systems and Their Applications
Fixed Point Theory and Applications volume 2010, Article number: 246382 (2010)
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.
Let be a nonempty set. A function
defined on the power set
of
is called
-
if it satisfies the following properties:
,
if
.
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ1_HTML.gif)
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
.
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ4_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ7_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ8_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ9_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ10_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ11_HTML.gif)
Then (H1) in Theorem 3.1 holds with .
Let us verify (H2). Let and
be given. Then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ12_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ13_HTML.gif)
Let and assume that
is already known. Then, by induction, we obtain a sequence
in
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ14_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ15_HTML.gif)
By (h2) and (h3), we have . So there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ16_HTML.gif)
Since for each
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ17_HTML.gif)
From (3.12) and (3.15), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ18_HTML.gif)
Let . Hence, combining (3.10), (3.14), and (3.16), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ19_HTML.gif)
Let . Thus, by (3.13) and (3.17),
and
. On the other hand, from the definition of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ20_HTML.gif)
Finally, in order to complete the proof, we need to show that for all
. Let
. Then
and
. For any
, since
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ21_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ22_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ23_HTML.gif)
By the lower semicontinuity of and
,
is a nonempty closed set in
. So
is a complete metric space. Define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ24_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ25_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ26_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ27_HTML.gif)
Since , we have the conclusion (a). From (4.5),
for all
with
. For any
, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ28_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ29_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ30_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ31_HTML.gif)
Let and
, for every
, and define a fuzzy map
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246382/MediaObjects/13663_2010_Article_1235_Equ32_HTML.gif)
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.
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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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Du, WS. Critical Point Theorems for Nonlinear Dynamical Systems and Their Applications. Fixed Point Theory Appl 2010, 246382 (2010). https://doi.org/10.1155/2010/246382
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DOI: https://doi.org/10.1155/2010/246382