Abstract
In this paper, we deal with a vectorial form of Ekeland-type variational principle for multivalued bioperator whose domain is a complete metric space and its range is a subset of a locally convex Hausdorff topological space. From this theorem, Caristi-Kirk fixed point theorem for multivalued maps is established in a more general setting and our techniques allow us to improve and to extend their results in (Ansari in J. Math. Anal. Appl. 335: 561-575, 2007; Bednarczuk and Zagrodny in Arch. Math. 93: 577-586, 2009; Bianchi, Kassay and Pini in J. Math. Anal. Appl. 305: 502-512, 2005).
MSC:49K30, 90C29.
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1 Introduction
It is well known that Ekeland’s variational principle (for short, EVP), a very important result first presented by Ekeland [10] in 1974, is as follows:
Theorem 1.1 (Ekeland [10, 11])
Let X be a complete metric space with a metric d. Let f be a function from X into which is proper lower semicontinuous and bounded from below. Then for and , there exists such that
-
(i)
,
-
(ii)
for every .
It is well known that the primitive EVP is a powerful tool on many applications in optimization, nonlinear analysis, mathematical economy and mathematical programming. Moreover, EVP is equivalent to the Caristis fixed point theorem [5, 8] and nonconvex minimization theorem according to Takahashi [21]. The studies of several forms of Ekeland’s variational principle for vector valued functions were obtained by many authors, for instance, Nemeth [20], Tammer [22] and Isac [16, 17].
To begin with, let X be a (real) linear space, E be a (real) topological vector space, K be a cone in E and , be two map**s. Under mild conditions of f and e, Nemeth [20] got the conclusion that for and there exists a such that
-
(i)
,
-
(ii)
; whenever ,
where is a complement of K.
Next, let Y be a topological vector space, , X be a real Banach space and . Tammer [22] showed that under mild assumptions on f and K, for , there exists an such that
-
(i)
, ,
-
(ii)
,
-
(iii)
, ,
where is a weakly e-minimal solution of f on K and .
On the other hand, by assuming that is a complete metric space, Y is a locally convex Hausdorff space and is a normal cone, Isac [16] proved that for and satisfying mild conditions, there exists and such that
-
(i)
,
-
(ii)
,
-
(iii)
, ,
where .
Recently, there have been many new formulation cases of EVP in [1–3, 6, 7]. In 2007, a generalization of the Ekeland-type variational principle for vector valued functions in the setting of complete quasi-metric spaces with ω-distance was introduced by Ansari [2]. Let be a complete quasi-metric space, C be a proper, closed and convex cone in a locally convex Hausdorff topological vector space Y. satisfies mild conditions. For every and , there exists such that
-
(i)
,
-
(ii)
, for all , ,
where .
By an approach similar to [2], Araya et al. [3] obtained a vectorial version of Ekeland’s variational principle related to equilibrium problem. In 2008, Al-Homidan et al. [1] established Ekeland-type variational principles in the setting of quasi-metric spaces with a Q-function. Recently, Bednarczuk and Zagrodny [6] introduced an Ekeland-type vector variational principle for monotonically semicontinuous map**s with perturbations given by a convex bounded subset of directions multiplied by the distance function, and they proved EVP for vector-valued map**s by combining topological and set-theoretic methods. Very recently, Khanh and Quy [19] have proposed a very weak type of generalized distances and used it to weaken the assumptions about lower semicontinuity in the existing versions of Ekeland’s variational principle on the complete metric space: to find such that, for all
where K is a convex cone in the Hausdorff locally convex space Y, , p is a weak τ-function and .
Motivated by the above mentioned works, we establish a vectorial form of Ekeland-type variational principle for multivalued bioperator whose domain is a complete metric space and its range is a subset of a locally convex Hausdorff topological space by using the set theoretic methods. We also consider Caristi-Kirk fixed point theorem in a more general setting and our techniques allow us to improve and to extend their results in [2, 6, 7].
2 Preliminaries
This section provides the preliminary terminology and notation used throughout this paper. Let be a complete metric space, Y be a locally convex space (i.e., a linear topological space with a local base consisting of convex neighborhoods of the origin, see [15]) and K be a closed convex cone in Y. For any we define
Now, we define the concept of a ω-distance for a metric space which has been introduced by Kada et al. [18].
Definition 2.1 Let be a metric space. A function is called a w-distance on X if the following conditions are satisfied:
-
(i)
for any ;
-
(ii)
for any fixed , is lower semi continuous;
-
(iii)
for any , there exists such that and implying .
Let f be a function from X to Y. It is said to be
-
(i)
C-bounded below if there exists such that , for all ;
-
(ii)
-lower semicontinuous if for all , closed;
-
(iii)
-upper semicontinuous if for all , closed;
-
(iv)
-continuous if it is both -lower semicontinuous and -upper semicontinuous;
-
(v)
-lower semicontinuous if for all , closed;
-
(vi)
-upper semicontinuous if for all , closed;
-
(vii)
-continuous if it is both -lower semicontinuous and -upper semicontinuous;
-
(viii)
C-lower semicontinuous if for all , closed;
-
(ix)
C-upper semicontinuous if for all , closed;
-
(x)
C-continuous if it is both C-lower semicontinuous and C-upper semicontinuous.
Remark 2.3 It is easy to see that the C-lower (respectively upper) semicontinuity of f implies the -lower (respectively upper) semicontinuity.
Definition 2.4 (Holmes [15])
Let X be a linear topological space over the field .
-
(1)
A sequence is bounded if whenever in .
-
(2)
A set is bounded if every sequence in A is bounded.
Let C be a convex cone in a linear topological space Y with and D a convex subset of C such that . In order to show the main results, let us give the following definition.
Definition 2.5 A generalized nonlinear scalarization function is defined by
Remark 2.6 If with , then Definition 2.5 reduces to the definition of the Gerstewitz function [14]
Lemma 2.7 For , we set . Then the following hold:
-
(i)
If for some , for each .
-
(ii)
For each , there exists a real number λ such that .
-
(iii)
Let . If for some , for each .
Proof (i) Let for some and . We note that
This implies .
-
(ii)
Assume there exists such that for all , . From (i), we have for all . Then we see that
Note that
Since D is convex, we have
which is a contradiction to .
-
(iii)
Let and if for some . Assume that for some , . From (i), we have , a contradiction. □
Proposition 2.8 The function is well defined.
Proof For any , define
It is sufficient to show that K is bounded from below.
Assume that for each , there exists such that and .
By Lemma 2.7 (ii), there exists such that .
By Lemma 2.7 (iii), we have for each , a contradiction. Then K is bounded from below. □
Let us recall the basic set-theoretical concepts and tools which are used in the sequel. Let X be a nonempty set and a relation. By we mean that and we write if and only if there are finite elements such that
The relation is said to be the transitive closure of s, and if s is transitive. We say that the element is maximal with respect to a relation (i.e., x is s-maximal) if for every ,
Definition 2.9 (See [13])
A set X with a relation is countably orderable with respect to s if for every nonempty subset there exists a well-ordered relation μ on W such that
implies that W is at most countable.
Theorem 2.10 (See [13])
Let X be a countably orderable set by a relation . Assume that for any sequence satisfying
there are a subsequence and an element x such that
Then an -maximal element of X exists.
Moreover, if s is transitive, then there exists an s-maximal element of X.
3 Main theorem
In this section, we will present the following vectorial form of an equilibrium version of vector Ekeland’s principle in setting complete metric spaces and ω-distance.
Theorem 3.1 Let X be a complete metric space, be a ω-distance on X, Y be a locally convex space, C be a closed and convex cone in Y and D be a closed convex and bounded subset of C such that . Let be a function satisfying the following conditions:
-
(i)
for all ;
-
(ii)
for every ;
-
(iii)
for each , the function is (D,C)-lower semicontinuous;
-
(iv)
for each fixed , is C-bounded below.
Then for every , there exists such that
-
(i)
;
-
(ii)
for all .
Proof Let be a relation defined as follows: For any ,
We will first show that r is transitive. Suppose that and . Thus, we have
This implies that
By assumption (ii), we obtain
Therefore, by the convexity of D, we have
Indeed, if , we are done. If , for , we have
So, we have
Hence , which implies that .
By the definition of ω-distance, . Therefore, there is a real number such that
From (3.1), (3.2) and (3.3), we have
This implies that .
We define by
It is easy to see that , and so is nonempty for all . By assumption (iii), we note that is a closed set for all . We now show that is a countably orderable set by a relation .
Let
where for all .
Let W be any nonempty subset of A which is well ordered by a relation s satisfying
Then, for any with , we note that
because r is transitive. Since , , thus , which implies that . Moreover,
Thus is well ordered by the relation “<” and hence is at most countable. Since V is one-to-one map** on W, W is at most countable.
For any , we let with for all . We next show that there is an element such that for all .
In case for some , we can put and so we have done. Then, it is enough to consider the case . Since for each , we obtain
From (3.4) and assumption (ii), we observe that
for all . Since F is C-bounded below, there exists such that
By the convexity of D, we have
for any . Therefore, it follows from (3.5) and (3.6) that
Since 0 , by the Separation theorem, there exists such that
This implies that for some , and for any , . Hence and ≥ 0 for any . Hence, for each , we have
for some and . Since for any , it follows that
Since , we have that is bounded above by . Moreover, is a monotone sequence then the series converges. This implies that . It is easy to see that is a Cauchy sequence in . By the completeness of X and closedness of , converges to a certain . Since r is transitive and , then for all , and so . This entails that satisfies the condition in Theorem 2.10. Now, the proof includes applying Theorem 2.10 to show that has an r-maximal element . Let us observe that for , any r-maximal element of is an r-maximal element of X. Hence, (i) holds for . Finally, we show that satisfies (ii). Assume that for some . Since r is transitive and is r-maximal, . Consequently, and , a contradiction. Hence satisfies (ii). □
Remark 3.2 We see in the proof that we do not use the symmetry condition of the metric. So, the conclusion in Theorem 3.1 still holds if we replace the word ‘‘metric space’’ with ‘‘quasi-metric space’’.
By setting for all in Theorem 3.1, we obtain the following Corollary which is proven by Ansari [2].
Corollary 3.3 (Theorem 3.1 in [2])
Let be a complete quasi-metric space, a ω-distance on X, Y be a locally convex Hausdorff topological vector space, C be a proper, closed and convex cone in Y with apex at origin and , and be a fixed vector such that . Let be a function satisfying the following:
-
(i)
, for all ;
-
(ii)
for all ;
-
(iii)
for each fixed , the function is -lower semicontinuous and C-bounded below.
Then for every and for every , there exists such that
-
(a)
,
-
(b)
, for all , .
If , where is lower semicontinuous and bounded below, then we have the following result.
Corollary 3.4 Let X be a complete metric space, be a ω-distance on X, Y be a locally convex space, C be a closed and convex cone in Y and D be a closed convex and bounded subset of C such that . Let be (D,C)-lower semicontinuous and C-bounded below. Then for every there exists such that
-
(i)
;
-
(ii)
for all .
We obtain that Corollary 3.4 is the extension of the following.
Corollary 3.5 Let X be a complete metric space, be a ω-distance on X, Y be a locally convex space, C be a closed and convex cone in Y and D be a closed convex and bounded subset of C such that . Let be (D,C)-lower semicontinuous and C-bounded below. Then for every there exists such that
-
(i)
;
-
(ii)
for all .
Proof By all conditions of Corollary 3.4, we have for every there exists such that
From 3.7, we have (i) holds.
If (ii) were not satisfied, we would have for some . Then there are and such that
Since 0 , by the Separation theorem, there exists such that for some , and . Hence and 0 for any .
From (3.9), we obtain that
Using the same method of (3.9), we conclude that , a contradiction. Consequently (ii) holds □
If we set , and for in Theorem 3.1, we have the following result which is a well-known Ekeland’s variational principle in a more general setting.
Corollary 3.6 Let X be a complete metric space, be a ω-distance on X, be a function satisfying the following conditions:
-
(i)
for all ;
-
(ii)
for every ;
-
(iii)
for each the function is lower semicontinuous and bounded below.
Then for every and , there exists such that
-
(i)
;
-
(ii)
for all .
Remark 3.7 By setting and , where is lower semicontinuous and bounded below in Corollary 3.6, we obtain Theorem 1.1 proven by Ekeland [10, 11].
The following theorem provides the equivalence between the equilibrium version of Ekeland-type variational principle, the equilibrium problem, Caristi-Kirk type fixed point theorem and Oettli and Théra type theorem
Theorem 3.8 Let X be a complete metric space, be a ω-distance on X, Y be a locally convex space, C be a closed and convex cone in Y and D be a closed convex and bounded subset of C. Let and be a function satisfying the following conditions:
-
(i)
for all ;
-
(ii)
for every ;
-
(iii)
for each , the function is -lower semicontinuous;
-
(iv)
for each fixed , is C-bounded below;
-
(v)
for each , there is such that and .
Then T has at least one fixed point, i.e., there exists such that .
Proof By assumption (i)-(iv) applied to Theorem 3.1, there exists such that
On the other hand by assumption (v), there exists such that
Then we see that , and so , that is, T has at least one fixed point. □
Remark 3.9 We set , , and replace ω-distance by d-distance in Theorem 3.8, we obtain Theorem 3.1 in [3] and Theorem 4.1 in [4] (vectorial Caristi-Kirk fixed point theorem).
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Acknowledgement
The first author would like to thank the Office of the Higher Education Commission of Thailand for supporting by grant fund under the program Strategic Scholarships for the Join Ph.D. Program Thai Doctoral degree for this research.
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The work presented here was carried out in collaboration between all authors. SP designed theorems and methods of the proof and interpreted the results. KS proved the theorems, interpreted the results and wrote the paper. All authors read and approved the final manuscript.
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Sitthithakerngkiet, K., Plubtieng, S. Vectorial form of Ekeland-type variational principle. Fixed Point Theory Appl 2012, 127 (2012). https://doi.org/10.1186/1687-1812-2012-127
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DOI: https://doi.org/10.1186/1687-1812-2012-127