- Research Article
- Open access
- Published:
Equivalent Extensions to Caristi-Kirk's Fixed Point Theorem, Ekeland's Variational Principle, and Takahashi's Minimization Theorem
Fixed Point Theory and Applications volume 2010, Article number: 970579 (2009)
Abstract
With a recent result of Suzuki (2001) we extend Caristi-Kirk's fixed point theorem, Ekeland's variational principle, and Takahashi's minimization theorem in a complete metric space by replacing the distance with a -distance. In addition, these extensions are shown to be equivalent. When the
-distance is l.s.c. in its second variable, they are applicable to establish more equivalent results about the generalized weak sharp minima and error bounds, which are in turn useful for extending some existing results such as the petal theorem.
1. Introduction
Let be a complete metric space and
a proper lower semicontinuous (l.s.c.) bounded below function. Caristi-Kirk fixed point theorem [1, Theorem
] states that there exists
for a relation or multivalued mapping
if for each
with
there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ1_HTML.gif)
(see also [2, Theorem ] or [3, Theorem
]) while Ekeland's variational principle (EVP) [4, 5] asserts that for each
and
with
there exists
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ2_HTML.gif)
EVP has been shown to have many equivalent formulations such as Caristi-Kirk fixed point theorem, the drop theorem [6], the petal theorem [3, Theorem ], Takahashi minimization theorem [7, Theorem
], and two results about weak sharp minima and error bounds [8, Theorems
and
]. Moreover, in a Banach space, it is equivalent to the Bishop-Phelps theorem (see [9]). EVP has played an important role in the study of nonlinear analysis, convex analysis, and optimization theory. For more applications, EVP and several equivalent results stated above have been extended by introducing more general distances. For example, Kada et al. have presented the concept of a
-distance in [10] to extend EVP, Caristi's fixed point theorem, and Takahashi minimization theorem. Suzuki has extended these three results by replacing a
-distance with a
-distance in [11]. For more extensions of these theorems, with a
-distance being replaced by a
-function and a
-function, respectively, the reader is referred to [12, 13].
Theoretically, it is interesting to reveal the relationships among the above existing results (or their extensions). In this paper, while further extending the above theorems in a complete metric space with a -distance, we show that these extensions are equivalent. For the case where the
-distance is l.s.c. in its second variable, we apply our generalizations to extend several existing results about the weak sharp minima and error bounds and then demonstrate their equivalent relationship. In particular, when the
-distance reduces to the complete metric, our results turn out to be equivalent to EVP and hence to its existing equivalent formulations.
2.
-Distance and
-Distance
For convenience, we recall the concepts of -distance and
-distance and some properties which will be used in the paper.
Definition 2.1 (see [10]).
Let be a metric space. A function
is called a
-distance on
if the following are satisfied:
for all
;
for each ,
is l.s.c.;
for each there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ3_HTML.gif)
From the definition, we see that the metric is a
-distance on
. If
is a normed linear space with norm
, then both
and
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ4_HTML.gif)
are -distances on
. Note that
for each
with
. For more examples, we see [10].
It is easy to see that for any and
-distance
, the function
is also a
-distance. For any positive
and
-distance
on
, the function
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ5_HTML.gif)
is a bounded -distance on
.
The following proposition shows that we can construct another -distance from a given
-distance under certain conditions.
Proposition 2.2.
Let ,
a
-distance on
, and
a nondecreasing function. If, for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ6_HTML.gif)
then the function defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ7_HTML.gif)
is a -distance. In particular, if
is bounded on
, then
is a
-distance.
Proof.
Since is nondecreasing, for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ8_HTML.gif)
In addition, is obviously lower semicontinuous in its second variable.
Now, for each there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ9_HTML.gif)
Taking such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ10_HTML.gif)
we obtain that, for in
with
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ11_HTML.gif)
from which it follows that . Similarly, we have
. Thus
. Therefore,
is a
-distance on
Next, if is bounded on
, then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ12_HTML.gif)
Thus is also a
-distance on
.
When is unbounded on
, the condition in Proposition 2.2 may not be satisfied. However, if
is a nondecreasing function satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ13_HTML.gif)
then the function in Proposition 2.2 is a
-distance (see [11, Proposition
]), a more general distance introduced by Suzuki in [11] as below.
Definition 2.3 (see [11]).
is said to be a
-distance on
provided that
for all
and there exists a function
such that
and
for all
, and
is concave and continuous in its second variable;
and
imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ14_HTML.gif)
and
imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ15_HTML.gif)
and
imply
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ16_HTML.gif)
Suzuki has proved that a -distance is a
-distance [11, Proposition
]. If a
-distance
satisfies
and
for
, then
(see [11, Lemma
]). For more properties of a
-distance, the reader is referred to [11].
3. Fixed Point Theorems
From now on, we assume that is a complete metric space and
is a proper l.s.c. and bounded below function unless specified otherwise. In this section, mainly motivated by fixed point theorems (for a single-valued mapping) in [10, 11, 14–16], we present two similar results which are applicable to multivalued mapping cases. The following theorem established by Suzuki's in [11] plays an important role in extending existing results from a single-valued mapping to a multivalued mapping.
Theorem 3.1 (see [11, Proposition ]).
Let be a
-distance on
. Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ17_HTML.gif)
Then for each with
there exists
such that
In particular, there exists
such that
.
Based on Theorem 3.1, [11, Theorem ] asserts that a single-valued mapping
has a fixed point
in
when
holds for all
(which generalizes [10, Theorem
] by replacing a
-distance with a
-distance). We show that the conclusion can be strengthened under a slightly weaker condition (in which
holds on a subset of
instead) for a multivalued mapping
Theorem 3.2.
Let be a
-distance on
and
a multivalued mapping. Suppose that for some
there holds
for each
with
. Then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ18_HTML.gif)
where
Proof.
For each with
, the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ19_HTML.gif)
is a nonempty closed subset of since
is lower semicontinuous and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ20_HTML.gif)
for some . Thus
is a complete metric space. By Theorem 3.1, there exists
such that
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ21_HTML.gif)
there exists such that
. Thus
,
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ22_HTML.gif)
Clearly, [8, Thoerem ] follows as a special case of Theorem 3.2 with
. In addition, when
and
is a single-valued mapping, Theorem 3.2 contains [11, Theorem
]. The following simple example further shows that Theorem 3.2 is applicable to more cases.
Example 3.3.
Consider the mapping defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ23_HTML.gif)
and the function for
. Obviously
. For any
,
, and
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ24_HTML.gif)
so, applying Theorem 3.2 to the above and
with
for
, we obtain
as in Theorem 3.2.
Motivated by [16, Theorem ] and [14, Theorem
], we further extend Theorem 3.2 as follows.
Theorem 3.4.
Let be a
-distance on
and
a multivalued mapping. Let
and
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ25_HTML.gif)
for some . If for each
with
there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ26_HTML.gif)
then there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ27_HTML.gif)
where
Proof.
For each with
, by assumption, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ28_HTML.gif)
based on the inequalities and
. Upon applying Theorem 3.2 to the lower semicontinuous function
on
which is complete, we arrive at the conclusion.
Next result is immediate from Theorem 3.4.
Theorem 3.5.
Let be a
-distance on
,
either nondecreasing or upper semicontinuous
, and
a multivalued mapping. If for some
and each
with
there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ29_HTML.gif)
then there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ30_HTML.gif)
where with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ31_HTML.gif)
Proof.
For , define
. Then for the case where
is nondecreasing we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ32_HTML.gif)
Thus the conclusion follows from Theorem 3.4.
For the case where is u.s.c., we define
by
. Since
is u.s.c.,
is well defined and nondecreasing. Now, for some
and each
with
there exists
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ33_HTML.gif)
so we can apply the conclusion in the previous paragraph to to get the same conclusion.
Remark 3.6.
When and
is a single-valued mapping, Theorem 3.4 reduces to [16, Theorem
] while Theorem 3.5 to [16, Theorems
and
]. If also
for all
, then Theorem 3.5 reduces to [14, Theorem
] (when
is nondecreasing) and [15, Theorem
] (when
is upper semicontinuous). In the later case, it also extends [14, Theorem
].
Furthermore, we will see that the relaxation of from a single-valued mapping (as in several existing results stated before) to a multivalued one (as in Theorems 3.2–3.5) is more helpful for us to obtain more results in the next section.
4. Extensions of Ekeland's Variational Principle
As applications of Theorems 3.4 and 3.5, several generalizations of EVP will be presented in this section.
Theorem 4.1.
Let be a
-distance on
,
,
satisfy
, and
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ34_HTML.gif)
for some . Then there exists
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ35_HTML.gif)
Proof.
Take . Then
is a nonempty complete metric space. We claim that there must exist
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ36_HTML.gif)
Otherwise for each the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ37_HTML.gif)
would be nonempty and . As a mapping from
to
,
satisfies the conditions in Theorem 3.4, so there exists
such that
. This is a contradiction.
Now, for each , since
and
, inequality (4.3) still holds.
It is worth noting that in the above proof is a multivalued mapping to which Theorem 3.4 is directly applicable, in contrast to [11, Theorem
] and [16, Theorem
].
From the proof of Theorem 3.5, we see that the function defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ38_HTML.gif)
satisfies the condition in Theorem 4.1 when is a nondecreasing or u.s.c. function. So, based on Theorem 4.1 or Theorem 3.5, we obtain next result (from which [11, Theorem
] follows by taking
).
Theorem 4.2.
Let be a
-distance on
,
,
satisfy
, and
either nondecreasing or u.s.c.. Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ39_HTML.gif)
Then there exists such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ40_HTML.gif)
If also and
, is l.s.c. in its second variable, then there exists
satisfying the above property and the following inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ41_HTML.gif)
Proof.
Similar to the proof of Theorem 4.1, the first part of the conclusion can be derived from Theorem 3.5.
Now, let and
l.s.c. in its second variable. Then the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ42_HTML.gif)
is nonempty and complete. Note that is nondecreasing and
. Applying the conclusion of the first part to the function
on
, we obtain
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ43_HTML.gif)
for all with
. For
, we still have the inequality. Otherwise, there would exist
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ44_HTML.gif)
This with and the triangle inequality yield
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ45_HTML.gif)
that is, , which is a contradiction.
Remark 4.3.
-
(i)
For the case where
is nondecreasing, the function
in the proof of Theorem 4.2 reduces to
. From the proof we can further see that the nonemptiness and the closedness of
imply the existence of
in
such that
.
-
(ii)
If we apply Theorem 4.1 directly, then the factor
on the right-hand side of the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ46_HTML.gif)
in Theorem 4.2 can be replaced with .
-
(iii)
When
,
is a
-distance on
, and
is a nondecreasing function such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ47_HTML.gif)
applying Theorem 4.2 to the -distance
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ48_HTML.gif)
and , we arrive at the following conclusion, from which (by taking
) we can obtain [17, Theorem
], a generalization of EVP.
Corollary 4.4.
Let ,
a
-distance on
,
and
satisfy
and
. Let
be a nondecreasing function such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ49_HTML.gif)
Then for each there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ50_HTML.gif)
Note that there exist nondecreasing functions satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ51_HTML.gif)
For example, and
. Clearly, Corollary 4.4 is not applicable to these examples. For these cases, we present another extension of EVP by using Theorem 4.1 and Proposition 2.2.
Theorem 4.5.
Let be a
-distance on
,
,
satisfy
, and
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ52_HTML.gif)
for some . If
is a nondecreasing function and for some
and each
there holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ53_HTML.gif)
then there exists such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ54_HTML.gif)
Proof.
Proposition 2.2 shows that the function defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ55_HTML.gif)
is a -distance. Applying Theorem 4.1 to the
-distance, the desired conclusion follows.
Remark 4.6.
We have obtained Theorem 4.5 from Theorem 4.1. Conversely, when is a
-distance, Theorem 4.1 follows from Theorem 4.5 by taking
for all
. In this case they are equivalent results. If also
holds for some
and all
, Theorem 4.5 is obviously applicable. In particular, when we take
for certain point
, the condition in Theorem 4.5 about
can be deleted.
Theorem 4.7.
Let be a
-distance on
,
,
either nondecreasing or u.s.c., and
nondecreasing. Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ56_HTML.gif)
Then for with
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ57_HTML.gif)
there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ58_HTML.gif)
Proof.
Let satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ59_HTML.gif)
It is easy to see that is a bounded
-distance on
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ60_HTML.gif)
is a -distance. By Theorem 4.2, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ61_HTML.gif)
for all with
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ62_HTML.gif)
from which we obtain and hence
. Thus the desired conclusion follows.
Upon taking and
in Theorem 4.7 and replacing
with
, we obtain (ii) of [10, Theorem
], which is also an extension to EVP.
5. Nonconvex Minimization Theorems
In this section we mainly apply the extensions of EVP obtained in Section 4 to establish minimization theorems which generalize [11, Theorem ] (an extension to [10, Theorem
] and [7, Theorem
]). From these results we also derive Theorem 3.2. Consequently, seven theorems established in Sections 3–5 are shown to be equivalent.
Firstly, we use Theorem 4.1 to prove the following result.
Theorem 5.1.
Let be a
-distance on
,
, and
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ63_HTML.gif)
for some . If for each
with
there exists
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ64_HTML.gif)
then there exists such that
.
Proof.
Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ65_HTML.gif)
Let (with
) be fixed. Since
is l.s.c., the set
is nonempty and complete. Thus, by Theorem 4.1, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ66_HTML.gif)
The point must satisfy
. Otherwise, we suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ67_HTML.gif)
By the assumption, there exists a point with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ68_HTML.gif)
which implies and hence contradicts the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ69_HTML.gif)
Similarly, we can use Theorem 4.2 to establish the following result.
Theorem 5.2.
Let be a
-distance on
,
, and
either nondecreasing or u.s.c.. If for each
with
there exists
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ70_HTML.gif)
then there exists such that
.
Example 5.3.
Consider the function for
. Obviously,
attains its minimum at
. For this simple example, we can also apply Theorem 5.2 to conclude that there exists
such that
since for any
and each
we have
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ71_HTML.gif)
where for
and
.
Remark 5.4.
Up to now, beginning with Theorem 3.1, we have established the following results with the proof routes:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ72_HTML.gif)
As a conclusion in this paper, the following result states that these seven theorems are equivalent.
Theorem 5.5.
Theorems 3.2–3.5, 4.1-4.2, and 5.1-5.2 are all equivalent.
Proof.
By Remark 5.4, it suffices to show that Theorems 5.1-5.2 both imply Theorem 3.2.
Suppose that for some and for each
with
there exists
such that
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ73_HTML.gif)
If there exists with
such that
, then, since there exists
such that
,
,
In this case, Theorem 3.2 follows.
Next we claim that there must exist such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ74_HTML.gif)
Otherwise, suppose that for each
with
. By Theorem 5.1 or Theorem 5.2 there exists
such that
. Since
for
, according to the property that
and
imply
,
is a singleton. This implies that there exists
such that
and
, from which and the triangle inequality we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ75_HTML.gif)
This gives and hence a contradiction to the assumption.
6. Generalized
-Conditions of Takahashi and Hamel
The condition in Theorem 5.2 is sufficient for to attain minimum on
. In this section we show that such a condition implies more when the
-distance
(on
) is l.s.c. in its second variable. For convenience we introduce the following notions.
Definition 6.1.
A function is said to satisfy the generalized
-condition of Takahashi(Hamel) if for some
, some nondecreasing function
, and each
with
there exists
(
) such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ76_HTML.gif)
where In particular, for the case
the generalized
-condition of Takahashi
Hamel
is called the generalized condition of Takahashi
Hamel
.
When , the above concepts, respectively, reduce to
-condition of Takahashi
Hamel
and the condition of Takahashi
Hamel
in [8].
It is clear that for any the generalized
-condition of Takahashi implies the generalized
-condition of Takahashi and the generalized
-condition of Hamel implies the generalized
-condition of Hamel. For any
the generalized
-condition of Takahashi and the generalized
-condition of Hamel are, respectively, weaker than that of Takahashi and of Hamel. For example, when
, the function
satisfies the generalized
-conditions of Takahashi and Hamel for any
but it does not satisfy that of Takahashi nor of Hamel. Furthermore, the generalized
-condition of Hamel always implies that of Takahashi. Next result asserts that the converse is also true in a complete metric space.
Theorem 6.2.
Let be a
-distance on
such that
is l.s.c. on
for each
. For
,
satisfies the generalized
-condition of Takahashi if and only if
satisfies the generalized
-condition of Hamel.
Proof.
The sufficiency is obvious, so it suffices to prove the necessity. Let satisfy the generalized
-condition of Takahashi and let
be the corresponding nondecreasing function in the definition. Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ77_HTML.gif)
Then for the case it suffices to prove that the set
is nonempty for each
with
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ78_HTML.gif)
Let with
be fixed. Since
and
are both l.s.c., the set
is nonempty and complete. Thus, by Theorem 4.1 or Theorem 4.2, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ79_HTML.gif)
The point must be in
. Otherwise, if
were not in
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ80_HTML.gif)
By the assumption, there exists a point with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ81_HTML.gif)
from which and the inequality we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ82_HTML.gif)
that is, . And hence
This is a contradiction. Therefore,
.
Next, we suppose that satisfies the generalized condition of Takahashi. For each
, the function
satisfies the generalized
-condition of Takahashi, so
satisfies the generalized
-condition of Hamel. This implies that
is nonempty. For each
with
, if
, then
for some
In this case we can find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ83_HTML.gif)
If , then this inequality holds for each
. Therefore
satisfies the generalized condition of Hamel.
7. Generalized Weak Sharp Minima and Error Bounds
As stated in [8], the -condition of Takahashi is one of sufficient conditions for an inequality system to have weak sharp minima and error bounds. With Theorem 6.2 being established, the generalized
-condition of Takahashi plays a similar role for the generalized weak sharp minima and error bounds introduced below.
For a proper l.s.c. and bounded below function we say that
has generalized local(global) weak sharp minima if the set
of minimizers of
on
is nonempty and if for some
and some nondecreasing function
and each
with
there holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ84_HTML.gif)
where
Due to the equivalence stated in Theorem 6.2, the generalized -condition of Takahashi is sufficient for
to have generalized local (global) weak sharp minima.
Theorem 7.1.
Let be a
-distance on
such that
is l.s.c. on
for each
. If, for some
,
satisfies the generalized
-condition of Takahashi, then the set
of minimizers of
on
is nonempty and for every
with
and each
there holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ85_HTML.gif)
Proof.
The proof is immediate from Theorem 6.2.
For an l.s.c. function denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ86_HTML.gif)
We say that (or
) has a generalized local error bound if there exist
and a nondecreasing function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ87_HTML.gif)
where The function
is said to have a generalized global error bound if the above statement is true for
.
When and
, the study of generalized error bounds has received growing attention in the mathematical programming (see [18] and the references therein). Now, using Theorem 7.1, we present the following sufficient condition for an l.s.c. inequality system to have generalized error bounds.
Theorem 7.2.
Let be a
-distance on
such that
is l.s.c. on
for each
and
be a proper l.s.c. function. Let
and
be a nondecreasing function. Suppose for each
the set
is nonempty and for each
there exists a point
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ88_HTML.gif)
Then is nonempty and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ89_HTML.gif)
Proof.
Let be given. Since
is l.s.c. and bounded below with
and
by Theorem 7.1, it suffices to prove
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ90_HTML.gif)
that is, This must be true. Otherwise, if
then for
the set
would be empty. This contradicts the assumption.
Remark 7.3.
Note that the nonemptiness of in Theorem 7.2 is not a part of assumption but a part of conclusion. In addition, the condition in Theorem 7.2 implies that
satisfies the generalized
-condition of Takahashi, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ91_HTML.gif)
for each with
However, once
is nonempty, there exists
such that
as stated below.
Theorem 7.4.
Let be a
-distance such that
is l.s.c. on
for each
and
be a nondecreasing function. Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ92_HTML.gif)
Then for each with
there exists
such that
In particular, there exists
such that
.
Proof.
Since both and
are l.s.c., for
with
,
is nonempty complete metric space. Suppose that for each
there held
. Then for each
there exists
such that
. Define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ93_HTML.gif)
and denote . Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ94_HTML.gif)
By Theorem 7.2, the set is nonempty.
Now for , since
(no matter whether
or
), there exists
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ95_HTML.gif)
from which we obtain and
. Similarly, we have
such that
and
. This, with
implies
. Thus
, which is a contradiction.
Remark 7.5.
When and
is a
-distance such that
is l.s.c. on
for each
, we can obtain Theorem 3.1 by applying Theorem 7.4 to the function
. As more applications, the following two propositions are immediate from Theorem 7.4 by taking
,
and
, respectively, on
.
Proposition 7.6.
Let be a complete nonempty subset of a metric space
,
,
, and let
be a
-distance on
such that
is l.s.c. on
for each
. Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ96_HTML.gif)
Suppose that is nonempty for some
. If
for all
, then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ97_HTML.gif)
Proposition 7.7.
Let be a complete nonempty subset of a metric space
,
,
, and let
be a
-distance on
. Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ98_HTML.gif)
Suppose that is l.s.c. in its both variables and
is nonempty for some
. If
for all
, then there exists
such that
In particular, if
and
for all
, then there exists
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ99_HTML.gif)
Remark 7.8.
Upon taking in Propositions 7.6 and 7.7, we obtain [3, Theorem
] which is equivalent to EVP in a complete metric space. In this case EVP implies Theorem 3.1.
Finally, following the statement in Theorem 5.5, on the condition that the -distance
is l.s.c. on
for each
, Theorems 3.1–3.5, 4.1-4.2, 5.1-5.2, 6.2, and 7.1–7.4 turn out to be equivalent since we have further shown that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F970579/MediaObjects/13663_2009_Article_1373_Equ100_HTML.gif)
in Sections 6 and 7. In particular, each theorem stated above is equivalent to Theorem 4.5 (as stated in Remark 4.6) when is a
-distance on
, to [3, Theorem
] and EVP when
(see Remark 7.8), and to the Bishop-Phelps Theorem in a Banach space when
is the corresponding norm. Therefore, we can conclude our paper as below.
Theorem 7.9.
Let be a complete metric space and
a
-distance on
such that
is l.s.c. for each
. Then
(i)Theorems 3.1–3.5, 4.1-4.2, 5.1-5.2, 6.2, and 7.1-7.4 are all equivalent;
(ii)when is a
-distance on
, each theorem in (i) is equivalent to Theorem 4.5;
(iii)when , each theorem in (i) is equivalent to EVP.
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Wu, Z. Equivalent Extensions to Caristi-Kirk's Fixed Point Theorem, Ekeland's Variational Principle, and Takahashi's Minimization Theorem. Fixed Point Theory Appl 2010, 970579 (2009). https://doi.org/10.1155/2010/970579
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DOI: https://doi.org/10.1155/2010/970579