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A Continuation Method for Weakly Kannan Maps
Fixed Point Theory and Applications volume 2010, Article number: 321594 (2010)
Abstract
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.
1. Introduction
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 [1] 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 [2]). A similar generalization of the contractive condition was considered by Dugundji and Granas [3], 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 [4] 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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ1_HTML.gif)
Recently, Kirk [5] 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 [6], 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 [6] (for more information on this topic see, for instance, [7–9]). After Granas, Frigon [8] gave a similar result for weakly contractive maps.
A variant of the Banach contraction principle was given by Kannan [10], 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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ2_HTML.gif)
In this note, following the pattern of Dugundji and Granas [3], 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 [3] to introduce the concept of weakly Kannan maps.
Definition 2.1.
Let be a metric space,
, and
. Therefore
is a weakly Kannan map if there exists
, with
for every
such that, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ3_HTML.gif)
Remark 2.2.
Clearly, any weakly Kannan map has at most one fixed point: if
and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ4_HTML.gif)
Remark 2.3.
Notice that if is a weakly Kannan map and we define
on
as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ5_HTML.gif)
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.
Remark 2.4.
Although Kannan showed that the concept of Kannan map is independent of the concept of contractive map, Janos [11] observed that any contractive map whose Lipschitz constant defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ6_HTML.gif)
is less than is a Kannan map. Next, we exhibit an example of a weakly Kannan map
, with
, which is not a Kannan map, thus showing that the constant
in the aforementioned result by Janos is sharp.
Example 2.5.
Consider the metric space with the usual metric
, and let
be the function defined as
. Then,
and
is a weakly Kannan map, but not a Kannan map.
The equality follows from the fact that
for all
together with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ7_HTML.gif)
We also have that is not a Kannan map because
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ8_HTML.gif)
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 [3], 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.
Let be a complete metric space and suppose that
is a map such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ9_HTML.gif)
for all , where
. If, in addition, there exists a sequence
in
with
, then
has a fixed point in
.
Observe that relation (2.7) can be written in the following more general form:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ10_HTML.gif)
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.
Theorem 2.6.
Let be a complete metric space and assume that
is a bounded function satisfying the following condition: for any sequence
in
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ11_HTML.gif)
Assume also that is a map such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ12_HTML.gif)
for all . If there exists a sequence
in
with
, then
has a unique fixed point
in
, and
.
Proof.
Since is bounded, there exists
such that
for all
. Suppose that
is a sequence in
with
and use (2.9) to obtain that, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ13_HTML.gif)
This implies that is a Cauchy sequence. Since
is complete, the sequence
is convergent, say to
. Then
because
. Thus, by (*),
.
That is a consequence of the following relation and the fact that
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ14_HTML.gif)
Finally, is the unique fixed point of
because if
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ15_HTML.gif)
Corollary 2.7.
Let be a complete metric space and suppose that
is a weakly Kannan map. Then,
has a unique fixed point
and, for any
, the sequence of iterates
converges to
.
Proof.
Since is a weakly Kannan map, there exists a function
with
for all
, satisfying (2.1) for all
. Hence, the function
given as
is bounded and satisfies the conditions (*) and (2.9).
Consider any and define
,
We may assume that
because otherwise we have finished. We will prove that
and hence, by Theorem 2.6,
will converge to a point
which is the unique fixed point of
.
First of all, observe that the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ16_HTML.gif)
holds for all . In fact, it is a consequence of the following one, which is true by (2.1):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ17_HTML.gif)
From (2.13) we obtain that the sequence is nonincreasing, for
, and then it is convergent to the real number
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ18_HTML.gif)
To prove that , suppose that
and arrive to a contradiction as follows: use
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ19_HTML.gif)
and the definition of to obtain
for all
This, together with (2.13), gives that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ20_HTML.gif)
for all , which is impossible since
and
.
Remark 2.8.
We do not know whether Theorem A is, or not, a particular case of Theorem 2.6, although that is the case if the functions satisfy the additional assumption
. To see this, suppose that the map
is in the conditions of Theorem A, that is,
satisfies relation (2.7) for some given functions
,
, and suppose also that the functions
satisfy in addition
. Define
as
, where
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ21_HTML.gif)
Let us see that, with this function ,
satisfies the hypotheses of Theorem 2.6. Indeed,
is clearly bounded and also satisfies (*); if
is a sequence in
and
, with
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ22_HTML.gif)
Since we also have that , we obtain that
.
Finally, to see that satisfies relation (2.9), use relation (2.7) with
, together with the same relation interchanging the roles of
and
, and the fact that
, to obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ23_HTML.gif)
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.
Corollary 2.9.
Assume that is a complete metric space,
, and
is a weakly Kannan map with associated function
satisfying (2.1). If
is defined as usual, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ24_HTML.gif)
then has a fixed point.
Proof.
In view of Corollary 2.7, it suffices to show that the closed ball is invariant under
. To prove it, consider any
and obtain the relation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ25_HTML.gif)
from which, having in mind that ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ26_HTML.gif)
To end the proof, obtain that through the above inequality by considering two cases: if
, then
because
. Otherwise, we would have
, and consequently
, from which
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ27_HTML.gif)
3. A Homotopy Result
In 1974 Ćirić [13] 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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ28_HTML.gif)
Then, has a fixed point in
.
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 [14] considered a certain class of quasicontractions: those maps , where
is a metric space, for which there exists
such that, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ29_HTML.gif)
and gave the following homotopy result.
Theorem 3 B.
Let be a complete metric space,
an open subset of
, and
satisfying the following properties:
(i) for all
and all
,
(ii)there exists such that for all
and
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ30_HTML.gif)
(iii) is continuous in
, uniformly for
.
If has a fixed point in
, then
also has a fixed point in
for all
.
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.
Theorem 3.1.
Let be a complete metric space,
an open subset of
, and
satisfying the following properties:
(P1) for all
and all
,
(P2)there exists such that for all
and
one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ31_HTML.gif)
and for all
,
(P3)there exists a continuous function such that, for every
and
,
.
If has a fixed point in
, then
also has a fixed point in
for all
.
Proof.
Consider the nonempty set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ32_HTML.gif)
We will prove that , and for this it suffices to show that
is both closed and open in
.
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
.
That is a Cauchy sequence is a consequence of the following relation, where we have used (P2), (P3), and the fact that
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ33_HTML.gif)
Write and let us see that
and also that
. That
is a consequence of the following relation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ34_HTML.gif)
and that is straightforward from (P1).
Next we prove that is open in
: suppose that
and let us show that
, for some
. Since
, there exists
with
. Consider
with
and use the continuity of
to obtain
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ35_HTML.gif)
for all .
To show now that any is also in
, it suffices to prove that the map
has a fixed point. And this is true by Corollary 2.9, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ36_HTML.gif)
Remark 3.2.
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.
Remark 3.3.
The counterpart to Theorem 3.1 for weakly contractive maps was proved by Frigon [8]. In that result, it was assumed, in place of our (3.3), an equivalent formulation of the following condition (H'):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ37_HTML.gif)
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.
Example 3.4.
Consider the metric space , where
and
, and let
be the map given as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ38_HTML.gif)
First of all, we will see that the map does not satisfy condition (Q). Define, for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ39_HTML.gif)
Then, for , we have that
, since
. Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ40_HTML.gif)
showing that no can be found to satisfy (Q).
Secondly, observe that is not weakly contractive, since any weakly contractive map is continuous.
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ41_HTML.gif)
Thus, take and assume that
, with
. If any of the points
equals
, for example
, then use
and
to obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ42_HTML.gif)
Otherwise, we would have that . In this case, since
, then we may assume additionally that
, and we claim that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ43_HTML.gif)
To be convinced of this, check the following chain of inequalities having in mind that for all
, that
, and also that
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ44_HTML.gif)
Next, define by
and let us see that
satisfies (P1), (P2), and (P3).
It is obvious that satisfies (P1). To check (P2), observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ45_HTML.gif)
for all and all
, and hence, if
is the function previously defined, we have that, for all
and all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F321594/MediaObjects/13663_2009_Article_1257_Equ46_HTML.gif)
Finally, (P3) is trivially satisfied with .
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Acknowledgments
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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Ariza-Ruiz, D., Jiménez-Melado, A. A Continuation Method for Weakly Kannan Maps. Fixed Point Theory Appl 2010, 321594 (2010). https://doi.org/10.1155/2010/321594
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DOI: https://doi.org/10.1155/2010/321594