- Research Article
- Open Access
Critical Point Theorems for Nonlinear Dynamical Systems and Their Applications
- Wei-Shih Du^{1}Email author
https://doi.org/10.1155/2010/246382
© Wei-Shih Du. 2010
- Received: 9 February 2010
- Accepted: 18 April 2010
- Published: 23 May 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.
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 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]).
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 .
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.
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.
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.
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.
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.
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.
Then (H1) in Theorem 3.1 holds with .
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) .
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.
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.
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 .
it follows that for all . So the conclusion (b) holds.
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.
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.
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.
Declarations
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.
Authors’ Affiliations
References
- Dancš S, Hegedűs M, Medvegyev P: A general ordering and fixed-point principle in complete metric space. Acta Scientiarum Mathematicarum 1983,46(1–4):381–388.MathSciNetMATHGoogle Scholar
- Hyers DH, Isac G, Rassias TM: Topics in Nonlinear Analysis and Applications. World Scientific, River Edge, NJ, USA; 1997:xiv+699.View ArticleMATHGoogle Scholar
- Lin L-J, Du W-S: From an abstract maximal element principle to optimization problems, stationary point theorems and common fixed point theorems. Journal of Global Optimization 2010,46(2):261–271. 10.1007/s10898-009-9423-1MathSciNetView ArticleMATHGoogle Scholar
- Bishop E, Phelps RR: The support functionals of a convex set. In Proc. Sympos. Pure Math., Vol. VII. American Mathematical Society, Providence, RI, USA; 1963:27–35.Google Scholar
- Brézis H, Browder FE: A general principle on ordered sets in nonlinear functional analysis. Advances in Mathematics 1976,21(3):355–364. 10.1016/S0001-8708(76)80004-7MathSciNetView ArticleMATHGoogle Scholar
- Brøndsted A: On a lemma of Bishop and Phelps. Pacific Journal of Mathematics 1974, 55: 335–341.MathSciNetView ArticleMATHGoogle Scholar
- Du W-S: On some nonlinear problems induced by an abstract maximal element principle. Journal of Mathematical Analysis and Applications 2008,347(2):391–399. 10.1016/j.jmaa.2008.06.020MathSciNetView ArticleMATHGoogle Scholar
- Kang BG, Park S: On generalized ordering principles in nonlinear analysis. Nonlinear Analysis: Theory, Methods & Applications 1990,14(2):159–165. 10.1016/0362-546X(90)90021-8MathSciNetView ArticleMATHGoogle Scholar
- Lin L-J, Du W-S: On maximal element theorems, variants of Ekeland's variational principle and their applications. Nonlinear Analysis: Theory, Methods & Applications 2008,68(5):1246–1262. 10.1016/j.na.2006.12.018MathSciNetView ArticleMATHGoogle Scholar
- Yuan GX-Z: KKM Theory and Applications in Nonlinear Analysis, Monographs and Textbooks in Pure and Applied Mathematics. Volume 218. Marcel Dekker, New York, NY, USA; 1999:xiv+621.Google Scholar
- Amemiya M, Takahashi W: Fixed point theorems for fuzzy mappings in complete metric spaces. Fuzzy Sets and Systems 2002,125(2):253–260. 10.1016/S0165-0114(01)00046-XMathSciNetView ArticleMATHGoogle Scholar
- Chang SS, Luo Q: Caristi's fixed point theorem for fuzzy mappings and Ekeland's variational principle. Fuzzy Sets and Systems 1994,64(1):119–125. 10.1016/0165-0114(94)90014-0MathSciNetView ArticleMATHGoogle Scholar
- Göpfert A, Tammer C, Zălinescu C: On the vectorial Ekeland's variational principle and minimal points in product spaces. Nonlinear Analysis: Theory, Methods & Applications 2000,39(7):909–922. 10.1016/S0362-546X(98)00255-7MathSciNetView ArticleMATHGoogle Scholar
- Granas A, Horvath CD: On the order-theoretic Cantor theorem. Taiwanese Journal of Mathematics 2000,4(2):203–213.MathSciNetMATHGoogle Scholar
- Kada O, Suzuki T, Takahashi W: Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Mathematica Japonica 1996,44(2):381–391.MathSciNetMATHGoogle Scholar
- Lee GM, Lee BS, Jung JS, Chang SS: Minimization theorems and fixed point theorems in generating spaces of quasi-metric family. Fuzzy Sets and Systems 1999,101(1):143–152. 10.1016/S0165-0114(97)00034-1MathSciNetView ArticleMATHGoogle Scholar
- Lin L-J, Chuang C-S: Existence theorems for variational inclusion problems and the set-valued vector Ekeland variational principle in a complete metric space. Nonlinear Analysis: Theory, Methods & Applications 2009,70(7):2665–2672. 10.1016/j.na.2008.03.053MathSciNetView ArticleMATHGoogle Scholar
- Lin L-J, Du W-S: Ekeland's variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces. Journal of Mathematical Analysis and Applications 2006,323(1):360–370. 10.1016/j.jmaa.2005.10.005MathSciNetView ArticleMATHGoogle Scholar
- Lin L-J, Du W-S: Some equivalent formulations of the generalized Ekeland's variational principle and their applications. Nonlinear Analysis: Theory, Methods & Applications 2007,67(1):187–199. 10.1016/j.na.2006.05.006MathSciNetView ArticleMATHGoogle Scholar
- Lin L-J, Du W-S: Systems of equilibrium problems with applications to new variants of Ekeland's variational principle, fixed point theorems and parametric optimization problems. Journal of Global Optimization 2008,40(4):663–677. 10.1007/s10898-007-9146-0MathSciNetView ArticleMATHGoogle Scholar
- Suzuki T: Generalized distance and existence theorems in complete metric spaces. Journal of Mathematical Analysis and Applications 2001,253(2):440–458. 10.1006/jmaa.2000.7151MathSciNetView ArticleMATHGoogle Scholar
- Takahashi W: Nonlinear Functional Analysis, Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar
- Tammer Chr: A generalization of Ekeland's variational principle. Optimization 1992,25(2–3):129–141. 10.1080/02331939208843815MathSciNetView ArticleMATHGoogle Scholar
- Bose RK, Sahani D: Fuzzy mappings and fixed point theorems. Fuzzy Sets and Systems 1987,21(1):53–58. 10.1016/0165-0114(87)90152-7MathSciNetView ArticleMATHGoogle Scholar
- Chang SS: Fixed point theorems for fuzzy mappings. Fuzzy Sets and Systems 1985,17(2):181–187. 10.1016/0165-0114(85)90055-7MathSciNetView ArticleMATHGoogle Scholar
- Chang SS, Zhu YG: On variational inequalities for fuzzy mappings. Fuzzy Sets and Systems 1989,32(3):359–367. 10.1016/0165-0114(89)90268-6MathSciNetView ArticleMATHGoogle Scholar
- Moţ G, Petruşel A: Fixed point theory for a new type of contractive multivalued operators. Nonlinear Analysis: Theory, Methods & Applications 2009,70(9):3371–3377. 10.1016/j.na.2008.05.005MathSciNetView ArticleMATHGoogle Scholar
- Petruşel G: Existence and data dependence of fixed points and strict fixed points for multivalued -contractions. Carpathian Journal of Mathematics 2007,23(1–2):172–176.MathSciNetMATHGoogle Scholar
- Rus IA: Strict fixed point theory. Fixed Point Theory 2003,4(2):177–183.MathSciNetMATHGoogle Scholar
- Guran L: Fixed points for multivalued operators with respect to a -distance on metric spaces. Carpathian Journal of Mathematics 2007,23(1–2):89–92.MathSciNetMATHGoogle Scholar
- Guran L, Petruşel A: Existence and data dependence for multivalued weakly Ćirić-contractive operators. Acta Universitatis Sapientiae. Mathematica 2009,1(2):151–159.MathSciNetMATHGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. 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.