- Research Article
- Open Access
Equivalent Extensions to Caristi-Kirk's Fixed Point Theorem, Ekeland's Variational Principle, and Takahashi's Minimization Theorem
© The Author(s). 2010
- Received: 26 September 2009
- Accepted: 24 November 2009
- Published: 9 December 2009
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.
- Fixed Point Theorem
- Error Bound
- General Distance
- Multivalued Mapping
- Nondecreasing Function
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
EVP has been shown to have many equivalent formulations such as Caristi-Kirk fixed point theorem, the drop theorem , 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 ). 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  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 . 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.
For convenience, we recall the concepts of -distance and -distance and some properties which will be used in the paper.
Definition 2.1 (see ).
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.;
are -distances on . Note that for each with . For more examples, we see .
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
is a bounded -distance on .
The following proposition shows that we can construct another -distance from a given -distance under certain conditions.
is a -distance. In particular, if is bounded on , then is a -distance.
In addition, is obviously lower semicontinuous in its second variable.
Now, for each there exists such that
from which it follows that . Similarly, we have . Thus . Therefore, is a -distance on
Next, if is bounded on , then there exists such that
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
Definition 2.3 (see ).
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;
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  plays an important role in extending existing results from a single-valued mapping to a multivalued mapping.
Theorem 3.1 (see [11, Proposition ]).
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
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.
so, applying Theorem 3.2 to the above and with for , we obtain as in Theorem 3.2.
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.
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
so we can apply the conclusion in the previous paragraph to to get the same conclusion.
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.
As applications of Theorems 3.4 and 3.5, several generalizations of EVP will be presented in this section.
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.
From the proof of Theorem 3.5, we see that the function defined by
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 ).
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
that is, , which is a contradiction.
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 .
If we apply Theorem 4.1 directly, then the factor on the right-hand side of the inequality
When , is a -distance on , and is a nondecreasing function such that
and , we arrive at the following conclusion, from which (by taking ) we can obtain [17, Theorem ], a generalization of EVP.
Note that there exist nondecreasing functions satisfying
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.
is a -distance. Applying Theorem 4.1 to the -distance, the desired conclusion follows.
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.
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.
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.
then there exists such that .
Let (with ) be fixed. Since is l.s.c., the set is nonempty and complete. Thus, by Theorem 4.1, there exists such that
Similarly, we can use Theorem 4.2 to establish the following result.
then there exists such that .
where for and .
As a conclusion in this paper, the following result states that these seven theorems are equivalent.
Theorems 3.2–3.5, 4.1-4.2, and 5.1-5.2 are all equivalent.
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,
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
This gives and hence a contradiction to the assumption.
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.
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 .
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.
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.
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
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
If , then this inequality holds for each . Therefore satisfies the generalized condition of Hamel.
As stated in , 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
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.
The proof is immediate from Theorem 6.2.
For an l.s.c. function denote
We say that (or ) has a generalized local error bound if there exist and a nondecreasing function such that
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  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.
that is, This must be true. Otherwise, if then for the set would be empty. This contradicts the assumption.
for each with However, once is nonempty, there exists such that as stated below.
Then for each with there exists such that In particular, there exists such that .
By Theorem 7.2, the set is nonempty.
Now for , since (no matter whether or ), there exists such that and
from which we obtain and . Similarly, we have such that and . This, with implies . Thus , which is a contradiction.
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 .
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
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.
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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