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Generic WellPosedness for a Class of Equilibrium Problems
Journal of Inequalities and Applications volumeÂ 2008, ArticleÂ number:Â 581917 (2008)
Abstract
We study a class of equilibrium problems which is identified with a complete metric space of functions. For most elements of this space of functions (in the sense of Baire category), we establish that the corresponding equilibrium problem possesses a unique solution and is wellposed.
1. Introduction
The study of equilibrium problems has recently been a rapidly growing area of research. See, for example, [1â€“3] and the references mentioned therein.
Let be a complete metric space. In this paper, we consider the following equilibrium problem:
where belongs to a complete metric space of functions defined below. In this paper, we show that for most elements of this space of functions (in the sense of Baire category) the equilibrium problem (P) possesses a unique solution. In other words, the problem (P) possesses a unique solution for a generic (typical) element of [4â€“6].
Set
Clearly, is a complete metric space.
Denote by the set of all continuous functions such that
We equip the set with the uniformity determined by the base
where . It is clear that the space with this uniformity is metrizable (by a metric ) and complete.
Denote by the set of all for which the following properties hold.
(P1) For each , there exists such that for all .
(P2) For each , there exists such that for all satisfying .
Clearly, is a closed subset of . We equip the space with the metric and consider the topological subspace with the relative topology.
For each and each subset , put
For each and each , set
Assume that the following property holds.
(P3) There exists a positive number such that for each and each pair of real numbers satisfying , there is such that
In this paper, we will establish the following result.
Theorem 1.1.
There exists a set which is a countable intersection of open everywhere dense subsets of such that for each , the following properties hold:

(i)
there exists a unique such that
(1.6)

(ii)
for each , there are and a neighborhood of in such that for each and each satisfying , the inequality holds.
In other words, for a generic (typical) , the problem (P) is wellposed [7â€“9].
2. An Auxiliary Density Result
Lemma 2.1.
Let and . Then there exist and such that and for all .
Proof.
By (P1) there is such that
Set
For each , there is such that
For each , there is such that
For each , there is such that
For each , set
For any , put
and for any , put
Clearly, is an open covering of . Since any metric space is paracompact, there is a continuous locally finite partition of unity subordinated to the covering . Namely, for any , is a continuous function and there exist such that and that
Define
Clearly, is well defined, continuous, and satisfies
Let . Then
Assume that and that . Then
If , then in view of (2.5), (2.6), and (2.13), , a contradiction (see (2.12)). Then , and by (2.7),
Since this equality holds for any satisfying , it follows from (2.10) that
for all .
Relations (2.1), (2.2), and (2.15) imply that
By (1.2), (2.7), (2.8), and (2.10)
Assume that
Then in view of (2.2) and (2.18), Together with (2.7) and (2.10), this implies that
Combined with (2.11), this implies that
for all
Let
and assume that
Then in view of (2.22),
By (2.23) and the choice of (see (2.3)â€“(2.6)), and by (2.4), (2.6), (2.7), and (2.8),
Since the inequality above holds for any satisfying (2.22), the relation (2.10) implies that
Together with (2.11), (2.12), and (2.15), this implies that for all
By (2.17), . In view of (2.16), possesses (P1). Since possesses (P2), it follows from (2.7), (2.8), and (2.10) that possesses (P2). Therefore and Lemma 2.1 now follows from (2.16) and (2.26).
3. A Perturbation Lemma
Lemma 3.1.
Let , , and let satisfy
Then there exist and such that
and if satisfies then .
Proof.
By (P2) there is a positive number
such that
Set
Define
Clearly, is continuous and
By (3.6) and (3.7),
Let . We estimate . If , then by (3.6) and (3.7),
Assume that
By (3.3) and (3.11),
By (3.5), (3.6), (3.7), (3.11), and (3.12),
Together with (3.10) this implies that
Assume that . In view of (P3) and (3.3), there is such that
It follows from (3.15) and (3.9) that
for all . Set
Clearly, the function is continuous and
In view of (3.1), (3.18), and (3.6),
Since the function possesses (P2), it follows from (3.9), (3.20), and (3.18) that possesses the property (P2). Thus .
By (3.6), (3.14), and (3.18) for all
Assume that
If , then by (3.6) and (3.18),
and together with (3.17), this implies that
This inequality contradicts (3.22). The contradiction we have reached proves that
This completes the proof of the lemma.
4. Proof of Theorem 1.1
Denote by the set of all for which there exists such that for all . By Lemma 2.1, is an everywhere dense subset of .
Let and be a natural number. There exists such that
By Lemma 3.1, there exist and such that
and the following property holds.
(P4) For each satisfying the inequality holds.
Denote by the open neighborhood of in such that
Assume that
By (1.3), (4.3), and (4.4),
In view of (4.5) and (P4),
Thus we have shown that the following property holds.
(P5) For each and each satisfying (4.4), the inequality holds.
Set
Clearly, is a countable intersection of open everywhere dense subset of . Let
Choose a natural number . There exist and an integer such that
The property (P4), (4.3), and (4.9) imply that for each satisfying
we have
Thus we have shown that the following property holds.
(P6) For each satisfying (4.10), the inequality holds.
By (P1) there is a sequence such that
In view of (4.12) and (P6) for all large enough natural numbers , we have
Since is any positive number, we conclude that is a Cauchy sequence and there exists
Relations (4.12) and (4.14) imply that for all
We have also shown that any sequence satisfying (4.12) converges. This implies that if satisfies for all , then . By (P6) and (4.15),
Let and satisfy (4.4). By (P5), . Together with (4.16), this implies that
Theorem 1.1 is proved.
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Zaslavski, A.J. Generic WellPosedness for a Class of Equilibrium Problems. J Inequal Appl 2008, 581917 (2008). https://doi.org/10.1155/2008/581917
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DOI: https://doi.org/10.1155/2008/581917