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Equivalent Extensions to Caristi-Kirk's Fixed Point Theorem, Ekeland's Variational Principle, and Takahashi's Minimization Theorem

Fixed Point Theory and Applications20092010:970579

DOI: 10.1155/2010/970579

Received: 26 September 2009

Accepted: 24 November 2009

Published: 9 December 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

(1.1)

(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

(1.2)

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
(2.1)
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
(2.2)

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

(2.3)

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 ,
(2.4)
then the function defined by
(2.5)

is a -distance. In particular, if is bounded on , then is a -distance.

Proof.

Since is nondecreasing, for ,
(2.6)

In addition, is obviously lower semicontinuous in its second variable.

Now, for each there exists such that

(2.7)
Taking such that
(2.8)
we obtain that, for in with and ,
(2.9)

from which it follows that . Similarly, we have . Thus . Therefore, is a -distance on

Next, if is bounded on , then there exists such that

(2.10)

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

(2.11)

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
(2.12)
and imply
(2.13)

and imply

(2.14)

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, 1416], 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
(3.1)

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
(3.2)

where

Proof.

For each with , the set
(3.3)
is a nonempty closed subset of since is lower semicontinuous and
(3.4)
for some . Thus is a complete metric space. By Theorem 3.1, there exists such that . Since
(3.5)
there exists such that . Thus , , and
(3.6)

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
(3.7)
and the function for . Obviously . For any , , and we have
(3.8)

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
(3.9)
for some . If for each with there exists such that
(3.10)
then there exists such that
(3.11)

where

Proof.

For each with , by assumption, there exists such that
(3.12)

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
(3.13)
then there exists such that
(3.14)
where with
(3.15)

Proof.

For , define . Then for the case where is nondecreasing we have
(3.16)

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

(3.17)

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
(4.1)
for some . Then there exists such that and
(4.2)

Proof.

Take . Then is a nonempty complete metric space. We claim that there must exist such that
(4.3)
Otherwise for each the set
(4.4)

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

(4.5)

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
(4.6)
Then there exists such that and
(4.7)
If also and , is l.s.c. in its second variable, then there exists satisfying the above property and the following inequality:
(4.8)

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

(4.9)
is nonempty and complete. Note that is nondecreasing and . Applying the conclusion of the first part to the function on , we obtain such that
(4.10)
for all with . For , we still have the inequality. Otherwise, there would exist such that and
(4.11)
This with and the triangle inequality yield
(4.12)

that is, , which is a contradiction.

Remark 4.3.
  1. (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 .

     
  2. (ii)

    If we apply Theorem 4.1 directly, then the factor on the right-hand side of the inequality

     
(4.13)
in Theorem 4.2 can be replaced with .
  1. (iii)

    When , is a -distance on , and is a nondecreasing function such that

     
(4.14)
applying Theorem 4.2 to the -distance
(4.15)

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
(4.16)
Then for each there exists such that
(4.17)

Note that there exist nondecreasing functions satisfying

(4.18)

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
(4.19)
for some . If is a nondecreasing function and for some and each there holds
(4.20)
then there exists such that and
(4.21)

Proof.

Proposition 2.2 shows that the function defined by
(4.22)

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
(4.23)
Then for with and
(4.24)
there exists such that
(4.25)

Proof.

Let satisfy
(4.26)
It is easy to see that is a bounded -distance on and hence
(4.27)
is a -distance. By Theorem 4.2, there exists such that
(4.28)
for all with and
(4.29)

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
(5.1)
for some . If for each with there exists such that and
(5.2)

then there exists such that .

Proof.

Denote
(5.3)

Let (with ) be fixed. Since is l.s.c., the set is nonempty and complete. Thus, by Theorem 4.1, there exists such that

(5.4)
The point must satisfy . Otherwise, we suppose that
(5.5)
By the assumption, there exists a point with such that
(5.6)
which implies and hence contradicts the inequality
(5.7)

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
(5.8)

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
(5.9)

where for and .

Remark 5.4.

Up to now, beginning with Theorem 3.1, we have established the following results with the proof routes:
(5.10)

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,

(5.11)

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

(5.12)
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
(5.13)

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
(6.1)

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
(6.2)
Then for the case it suffices to prove that the set is nonempty for each with , where
(6.3)

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

(6.4)
The point must be in . Otherwise, if were not in , then
(6.5)
By the assumption, there exists a point with such that
(6.6)
from which and the inequality we obtain
(6.7)

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

(6.8)

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

(7.1)

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
(7.2)

Proof.

The proof is immediate from Theorem 6.2.

For an l.s.c. function denote

(7.3)

We say that (or ) has a generalized local error bound if there exist and a nondecreasing function such that

(7.4)

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
(7.5)
Then is nonempty and
(7.6)

Proof.

Let be given. Since is l.s.c. and bounded below with and by Theorem 7.1, it suffices to prove
(7.7)

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,
(7.8)

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
(7.9)

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
(7.10)
and denote . Then
(7.11)

By Theorem 7.2, the set is nonempty.

Now for , since (no matter whether or ), there exists such that and

(7.12)

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
(7.13)
Suppose that is nonempty for some . If for all , then there exists such that
(7.14)

Proposition 7.7.

Let be a complete nonempty subset of a metric space , , , and let be a -distance on . Denote
(7.15)
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
(7.16)

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

(7.17)

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.

Authors’ Affiliations

(1)
Department of Mathematical Sciences, Xi'an Jiaotong-Liverpool University

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