A Continuation Method for Weakly Kannan Maps
© D. Ariza-Ruiz and A. Jiménez-Melado. 2010
Received: 25 September 2009
Accepted: 6 December 2009
Published: 26 January 2010
The first continuation method for contractive maps in the setting of a metric space was given by Granas. Later, Frigon extended Granas theorem to the class of weakly contractive maps, and recently Agarwal and O'Regan have given the corresponding result for a certain type of quasicontractions which includes maps of Kannan type. In this note we introduce the concept of weakly Kannan maps and give a fixed point theorem, and then a continuation method, for this class of maps.
Suppose that is a metric space and that is a map. We say that is contractive if there exists such that for all . The well-known Banach fixed point theorem states that has a fixed point if and is complete. In 1962, Rakotch  obtained an extension of Banach theorem replacing the constant by a function of , , provided that is nonincreasing and for all (for a recent refinement of this result see ). A similar generalization of the contractive condition was considered by Dugundji and Granas , who extended Banach theorem to the class of weakly contractive mappings (i.e., , with for all ).
Another focus of attention in Fixed Point Theory is to establish fixed point theorems for non-self mappings. In the setting of a Banach space, Gatica and Kirk  proved that if is contractive, with an open neighborhood of the origin, then has a fixed point if it satisfies the well-known Leray-Schauder condition:
Recently, Kirk  has extended this result to the abstract setting of a certain class of metric spaces: the CAT(0) spaces. In the proof, the author uses a homotopy result due to Granas , which is known as continuation method for contractive maps. In fact, the jump from a Banach space setting to the metric space setting was given by Granas himself in  (for more information on this topic see, for instance, [7–9]). After Granas, Frigon  gave a similar result for weakly contractive maps.
A variant of the Banach contraction principle was given by Kannan , who proved that a map , where is a complete metric space, has a unique fixed point if is what we call a Kannan map, that is, there exists such that, for all ,
In this note, following the pattern of Dugundji and Granas , we extend Kannan theorem to the class of weakly Kannan maps (i.e., , with for all ). This is done in Section 2. In Section 3 we use a local version of the previous result to obtain a continuation method for weakly Kannan maps.
2. Weakly Kannan Maps
In this section we follow the pattern of Dugundji and Granas  to introduce the concept of weakly Kannan maps.
then is well defined, takes values in , satisfies for all (for is smaller than any associated to ), and also satisfies (2.1), with replaced by , for all . Conversely, if is defined as in (2.3) and satisfies the above set of conditions, then is a weakly Kannan map, establishing in this way an equivalent definition for Kannan maps.
To check that is a weakly Kannan map, consider the function given by (2.3). This function is well defined and also takes values in since . Next, assume that and let us see that . To see this, observe that as , so there is such that for all . Observe also that , the restriction of to , is a Kannan map with constant , due to the fact that , for is continuously differentiable on and for all . We will see . To do it, suppose that with and . Then, if , use and that to obtain . Otherwise, we would have and then .
Although the way we have introduced the concept of weakly Kannan map has been by analogy with the work done by Dugundji and Granas in , we would like to mention that this extension may be done in some different ways. For instance, Pathak et al. [12, Theorem 3.1] have proved the following result.
Theorem 2 A.
Observe that relation (2.7) can be written in the following more general form:
for all , where , , and notice that any map satisfying (2.8) also satisfies the relation (2.1) with . In fact, the arguments used by the authors in the proof of Theorem A are also valid for this class of maps. Next, we state this slightly more general result and include the proof for the sake of completeness. Then, we obtain, as a consequence, a fixed point theorem for weakly Kannan maps.
First of all, observe that the inequality
from which the result follows.
To prove the homotopy result of the next section, we will need the following local version of Corollary 2.7.
3. A Homotopy Result
In 1974 Ćirić  introduced the concept of quasicontractions and proved the following fixed point theorem: suppose that is a complete metric space and that is a quasicontraction, that is, there exists such that, for all ,
Observe that any contractive map, as well as any Kannan map, is a quasicontraction; thus, the theorem by Ćirić generalizes the well known fixed point theorems by Banach and Kannan.
On the other hand, Agarwal and O'Regan  considered a certain class of quasicontractions: those maps , where is a metric space, for which there exists such that, for all ,
and gave the following homotopy result.
Theorem 3 B.
The above homotopy result includes the corresponding one for the class of Kannan maps, and in the following theorem we show that an analogous result is true for the wider class of weakly Kannan maps.
We start showing that is closed in : suppose that is a sequence in converging to and let us show that . By definition of , there exists a sequence in with . We will prove that converges to a point with , thus showing that .
A careful reading of the proof shows that hypothesis (P3) in Theorem 3.1 can be easily replaced by the weaker hypothesis (iii) in Theorem B.
Observe that condition (H') means that all the maps , are weakly contractive, and with the same function . Our condition (3.3) is no surprise then. It also means that all the maps are of weakly Kannan type, and with the same function .
We end the section with an example of a homotopy satisfying (P1), (P2), and (P3) but not the hypotheses of Theorem B. In fact, the function will be of weakly Kannan type, but will not satisfy the quasicontractivity condition (Q) (hence, it will not be of Kannan type since any Kannan map satisfies (Q)). Moreover, will not be of weakly contractive type.
Next, let us check that is a weakly Kannan map. Since has as unique fixed point then, the function given by if , , is well defined. We have to check that only takes values in and that for all . In fact, all this will follow if we just show that, for ,
This research was partially supported by the Spanish (Grant no. MTM2007-60854) and regional Andalusian (Grants no. FQM210 and no. FQM1504) Governments.
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