Print version ISSN 0041-6932
Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca June 2009
Abstract. Many functional versions of the Caristi-Kirk fixed point theorem are nothing but logical equivalents of the result in question.
2000 Mathematics Subject Classification. Primary 47H10. Secondary 54H25.
Key words and phrases. Metric space; lsc function; Fixed/periodic point; Normality; Local boundedness; Feng-Liu property; Semimetric; Maximal element; Cauchy and asymptotic sequence; Regularity; Non-expansive map.
Let be a complete metric space; and , some function from to with
(1a) is -lsc over (, whenever ).
Let also be a selfmap of ; and put , ; each point of the former (latter) will be called fixed (periodic) under . The following 1975 fixed point result in Caristi and Kirk  is basic for us.
(1b) , for all .
hence, in particular,
The original proof of this result is by transfinite induction; see also Wong . (It works as well for highly specialized versions of Theorem 1; cf. Kirk and Saliga ). Note that, in terms of the associated (to ) order
the contractivity condition (1b) reads
(1c) , for all (i.e.: is progressive).
So, by the Bourbaki meta-theorem , the underlying result is logically equivalent with the Zorn maximality principle subsumed to the precise order; i.e., with Ekeland's variational principle . This tells us that the sequential type argument used in its proof (cf. Section 6) is also working in our framework; see also the paper of Pasicki . A proof of Theorem 1 involving the chains of the structure may be found in Turinici ; and its sequential translation has been developed in Dancs, Hegedus and Medvegyev . Further aspects involving the general case may be found in Brunner  and Manka ; see also Taskovic , Valyi , Nemeth  and Isac .
Now, the Caristi-Kirk fixed point theorem found (especially via Ekeland's approach) some basic applications to control and optimization, generalized differential calculus, critical point theory and normal solvability; see the above references for details. So, it must be not surprising that, soon after its formulation, many extensions of Theorem 1 were proposed. (These involve its standard version related to (1.2); and referred to as Theorem 1(st). But, only a few are concerned with the extended version of the same, related to (1.1); and referred to as Theorem 1(ex)). For example, in the 1982 paper by Ray and Walker , the following result of this type was obtained. Call the function , semi-normal when
(b1) is decreasing and ;
and normal provided (in addition)
(c1) ; where .
(Further aspects involving these notions will be developed in Section 2).
(1d) , for all .
Then, has at least one fixed point in .
Clearly, Theorem 2 includes Theorem 1(st), to which it reduces when . The reciprocal inclusion also holds (cf. Park and Bae ). Summing up, Theorem 2 is but a logical equivalent of Theorem 1 (st). (For a different proof of this, we refer to Section 4 below).
On the other hand, in the 2005 paper by Turinici  the following fixed point result was established. Call the function , right locally bounded from above at when
(d1) , for some .
If this holds for each , we say that is right locally bounded from above (on ). Further, let us say that is locally bounded (above) in case: the image of each bounded part in is bounded (in ).
(1e) , .
Then, conclusions of Theorem 2 are retainable.
This result extends the one in Bae, Cho and Yeom ; see also Bae . But, as precise there, all these extend Theorem 1(st); hence, so does Theorem 3. (In fact, a direct verification is available, via ). The converse inclusion is also true, by the provided proof; wherefrom Theorem 3 is a logical equivalent of Theorem 1 (st). For a technical extension of such facts we refer to Section 4.
Further, in Section 5, we show that our developments include as well the fixed point statement (comparable with Theorem 1(ex)) due to Feng and Liu . And in Section 6, some extensions of Theorem 1 are given, in terms of maximality principles over metrical structures comparable with the 1976 Brezis-Browder's . In fact, the obtained statements are extendable beyond the metrizable context; we shall discuss them elsewhere.
(A) Let be a semi-normal function (cf. (b1)). In particular, it is Riemann integrable on each compact interval of and
Some basic facts involving the couple (where the primitive is the one of (c1)) are being collected in
(ii) is a topological order isomorphism from to ; hence so is (between and )
(iii) is almost concave: is decreasing on ,
(iv) is concave: , for all with and all
(v) is sub-additive and is super-additive.
The proof is evident, by (2.1) above; so, we do not give details. Note that iii) and iv) are equivalent to each other, under ii). This follows from the (non-differential) mean value theorem in Bantas and Turinici .
(B) Now, let be a normal function (cf. (b1)+(c1)); note that, in such a case
Further, let be a metric space; and , some function with
Given the function let us attach it the function from to as
(Here, by convention, ]). This may be viewed as an "explicit" formula; its "implicit" version is given as
The definition is consistent, via (2.2); moreover, is -lsc if and only if is -lsc. The following variational completion of these is available.
Proof. Let the points be as in the premise of this implication. By Lemma 1 (i) and the implicit formula (b2), this gives ; or equivalently (by a simple re-arrangement) . On the other hand, (2a) yields ; so, by Lemma 1(iii): . A simple combination with the previous relation gives . It suffices now taking (2.2 ) into account to get the desired conclusion.
In particular, the nonexpansivity condition (2a) holds under
(c2) , for some .
Let be a complete metric space; and be a -lsc function (cf. (1a)). Further, let be semi-normal (in the sense of (b1)); and be nonexpansive (cf. (2a)). Given the self-map , we are interested in establishing sufficient conditions under which (1.1) or (1.2) be retainable. There are two possible answers, according to the normality condition being or not fulfilled.
(A) The former of these requires the normality setting (of (c1)).
(3a) , for each .
Then, is strongly fp-admissible (); hence fp-admissible ().
Proof. Let stand for the associated (to ) function. By the remarks in Section 2, is -lsc too. On the other hand, (3a) and Lemma 2 give (1b) (modulo ). This shows that Theorem 1 applies to and ; wherefrom, the desired conclusion is clear.
As in Theorem 1, we have a couple of fixed point statements under this formulation; referred to as Theorem 4(ex) and Theorem 4(st). Clearly, Theorem 4 includes Theorem 1 (for ). The reciprocal is also valid, by the argument above. Hence, this result is nothing but a logical equivalent of Theorem 1. On the other hand, when is taken as in (c2), Theorem 4(st) is just Theorem 2 above. Further aspects may be found in Zhong, Zhu and Zhao ; see also Lin and Du .
(B) Another answer to the same is to be stated in the original semi-normality setting (with (c1) not accepted). Roughly speaking, this is to be obtained via "surrogates" of (c1); like, e.g.,
(3b) there exists with .
(Here, as already precise, ]).
Proof. Without loss, one may assume ; because gives (by (3a)) . Denote . Clearly, is nonempty closed (by the assumptions about and the continuity of ). Further, take some arbitrary fixed . By Lemma 1(i) and the choice of one gets (via (3a))
This shows that ; hence is -invariant. In addition (for each ) ; wherefrom (cf. (3b)) . This (by (b1)) gives ; hence (again by (3a)) we finally derive (1b) over (modulo ). Summing up, Theorem 1(st) applies to and ; wherefrom, the desired conclusion is clear.
Now, Theorem 5 includes Theorem 1(st) (for and as in (c2)). The reciprocal is also true, by the argument above; hence, Theorem 5 is a logical equivalent of Theorem 1 (st). Combining with a preceding fact, one therefore derives that Theorem 4 (st) includes Theorem 5 . Concerning this aspect, note that the function given as (for some )
Let again be a complete metric space; and be some -lsc function. Given the self-map , the "dual" way of getting (1.2 ) is by using contractivity condition like in Theorem 3. A natural extension of these is as below. Let be a mapping. We say that the function is right locally -proper at if there exists and a strictly increasing continuous function with
(a4) , when .
If is generic in such a convention, we say that is right locally -proper (on ). Assume that is endowed with this last property. The following fixed point result is available.
Proof. If for some then (1e) gives ; wherefrom . So, without loss, one may assume that
(4a) , for each ;
where, as usually, . By the right local -properness of at , there must be some and a strictly increasing continuous function in such a way that (a4) is retainable (with ). Take some with (possibly, by the choice of ); and put . This is a nonempty part of (since it contains ); which, in addition, is closed (by the choice of ) and -invariant (from (4a)). Further, define the function (from to ) ; it is -lsc over , by the -lsc property of and the choice of . Finally, (1e) yields (via (a4)+(4a)) the evaluation (1b) on (modulo ). Summing up, Theorem 1 (st) is applicable to and ; wherefrom, the conclusion is clear.
Now, Theorem 6 includes Theorem 1(st), to which it reduces when , . The reciprocal inclusion also holds, by the argument above. Summing up, Theorem 6 is logically equivalent with Theorem 1(st); hence to Theorem 3 as well. Concerning this last aspect, note that Theorem 6 includes directly Theorem 3 (by taking as a linear function). However, the reciprocal inclusion is not "easily" obtainable. In fact, let us consider the choice , ; as well as (for )
Clearly, is right locally -proper at origin [just take (as attached function) ]; hence, right locally -proper (on ); wherefrom, Theorem 6 works here. On the other hand, Theorem 3 is not (directly) applicable when ; hence the claim.
Finally, combining this with the construction of Theorem 4, we may state a "hybrid" fixed point result as follows. In addition to (subject to the precise conditions) take a couple ; where is normal (cf. (b1)+(c1)) and is nonexpansive (according to (2a)).
Then, has at least one fixed point in .
This result extends Theorem 6; hence, Theorem 1(st) as well. On the other hand, Theorem 7 follows from Theorem 1(st); because, so does Theorem 4(st). Summing up, Theorem 7 is but a logical equivalent of Theorem 1(st); note that this conclusion also includes the fixed point statement in Suzuki . Further, we may ask whether the construction in Theorem 5 may be used as well in deriving a fixed point result extending Theorem 6. The answer is positive; further aspects will be delineated elsewhere. Some extensions of the obtained facts to multivalued maps are immediate; note that, in such a way, one extends the fixed point results in Mizoguchi and Takahashi ; see also Petrusel and Sîntamarian .
Let be a complete metric space; and , some -lsc function (cf. (1a)). Further, let be a selfmap of . In the 2006 paper by Feng and Liu , an interesting fixed point result (comparable with Theorem 1 ) was established. Let be a function with
(a5) is continuous, increasing, subadditive and ;
it will be referred to as a Feng-Liu function. Clearly, , in view of (a5) (the last part). We also note the useful property
For, if this fails, there must be an such that: for each , there exists with . But then (as is increasing) we necessarily have , contradiction; hence the claim.
(5a) , for all .
Then, conclusions of Theorem 1 are retainable.
Summing up, Theorem 8 is but a logical equivalent of Theorem 1. Further technical aspects may be found in Jachymski , Petrusel , Bîrsan  and Rozoveanu ; see also Kada, Suzuki and Takahashi .
(A) Let be a nonempty set; and , some quasi-order (i.e.: reflexive and transitive relation) over it. By a pseudometric on we shall mean any map . If, in addition, is reflexive , triangular  and sufficient [ implies ], we say that it is an almost metric (over ). Suppose that we fixed such an object. Call , -maximal, in case: and imply . Existence results involving such points may be viewed as (almost) "metrical" versions of the Zorn-Bourbaki maximality principle . To state one of these, one may proceed as below. Call the ascending sequence in , -Cauchy when: , such that ; and -asymptotic, provided: , as . Clearly, each (ascending) -Cauchy sequence is -asymptotic too. The reciprocal is also true when all such sequences are involved; i.e., the global conditions below are equivalent each other:
(6a) each ascending sequence is -Cauchy
(6b) each ascending sequence is -asymptotic.
Either of these will be referred to as: is regular (modulo ). The following answer to the posed question obtained in Turinici  is available.
(6c) is sequentially inductive:
each ascending sequence is bounded above (modulo ).
Then, for each there exists a -maximal with .
Note that the non-sufficient version of this result extends the Brezis-Browder ordering principle . Further statements in the area were obtained in Altman  and Anisiu ; see also Kang and Park . However, all these are (mutually) equivalent; see Turinici  for details.
(B) A basic application of these facts may be given along the following lines. Let be an almost metric space; and be a function. The regularity condition below is considered
(6d) is descending complete: for each -Cauchy sequence with descending there exists in such a way that and .
Theorem 9. Let the precise condition be admitted; and let be a selfmap with the property (1b). Then, (1.1) is retainable in the stronger sense: for each there exists in with
Proof. Let stand for the order (i.e.: antisymmetric quasi-order) given by (a1) (modulo ). We claim that Proposition 1 applies for ; wherefrom, all is clear. Let be an ascending sequence in
(6e) , whenever .
The sequence is descending in ; hence a Cauchy one. As a consequence, is -Cauchy; wherefrom is regular (modulo ). Putting these together there must be (via (6d)) some with and , for all . Fix some rank . By (6e) and the triangular property of ,
This, by the relation above, yields (passing to limit as )
As was arbitrarily chosen, is an upper bound of in ; which tells us that is sequentially inductive; hence the claim.
Now, the regularity condition (6d) holds whenever is complete and
(6f) is descending -lsc: for all , whenever is descending and .
In particular, (6f) holds when is -lsc. Note that, in such a case, Theorem 9 is just the fixed point statement in Caristi and Kirk  (Theorem 1). Some related aspects may be found in Isac  and Nemeth .
Acknowledgement. The author is very indebted to the referee for the careful reading of the manuscript and a number of useful suggestions.
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"A. Myller" Mathematical Seminar
"A. I. Cuza" University
11, Copou Boulevard
700506 Iasi, Romania
Recibido: 5 de marzo de 2008
Aceptado: 31 de mayo de 2009