Open Access

Some Combined Relations between Contractive Mappings, Kannan Mappings, Reasonable Expansive Mappings, and -Stability

Fixed Point Theory and Applications20092009:815637

https://doi.org/10.1155/2009/815637

Received: 13 May 2009

Accepted: 31 August 2009

Published: 27 September 2009

Abstract

In recent literature concerning fixed point theory for self-mappings in metric spaces , there are some new concepts which can be mutually related so that the inherent properties of each one might be combined for such self-mappings. Self-mappings can be referred to, for instance, as Kannan-mappings, reasonable expansive mappings, and Picard -stable mappings. Some relations between such concepts subject either to sufficient, necessary, or necessary and sufficient conditions are obtained so that in certain self-mappings can exhibit combined properties being inherent to each of its various characterizations.

1. Introduction

As it is wellknown fixed point theory and related techniques are of increasing interest for solving a wide class of mathematical problems where convergence of a trajectory or sequence to some equilibrium set is essential, (see, e.g., [17]). Some of the specific topics recently covered in the field of fixed point theory are, for instance as follows.

(1)The properties of the so-called -times reasonably expansive mapping are investigated in [1] in complete metric spaces as those fulfilling the property that for some real constant . The conditions for the existence of fixed points in such mappings are investigated.

(2)Strong convergence of the wellknown Halpern's iteration and variants is investigated in [2, 8] and several the references therein.

(3)Fixed point techniques have been recently used in [4] for the investigation of global stability of a wide class of time-delay dynamic systems which are modeled by functional equations.

(4)Generalized contractive mappings have been investigated in [5] and references therein, weakly contractive and nonexpansive mappings are investigated in [6] and references therein.

(5)The existence of fixed points of Liptchitzian semigroups has been investigated, for instance, in [3].

(6)Picard's -stability is discussed in [9] related to the convergence of perturbed iterations to the same fixed points as the nominal iteration under certain conditions in a complete metric space.

(7)The so-called Kannan mappings in [10] are recently investigated in [11, 12] and references therein.

Let be a metric space. Consider a self-mapping . The basic concepts used through the manuscript are the subsequent ones:

(1) is -contractive, following the contraction Banach's principle, if there exists a real constant such that
(1.1)
(2) is -Kannan, [1012], if there exists a real constant such that
(1.2)

(3) is ( )-times reasonable expansive self-mapping if there exists a real constant such that ; , , [1],

(4)Picard's -stability means that if is a complete metric space and Picard's iteration satisfies as for then , that is, q is a fixed point of , [9]. It is proven in [9] that, if the self-mapping satisfies a property, referred to through this manuscript as the property for some real constants and (see Definition 1.2 in what follows), then Picard's iteration is -stable if .

The following result is direct.

Proposition 1.1.

If a self-mapping is -contractive, then it is also -contractive; .

If a self-mapping is -Kannan, then it is also -Kannan; .

The so- called the -property is defined as follows.

Definition 1.2.

A self-mapping with possesses the -property for some real constants and if ; , .

The above property has been introduced in [9] to discuss the -stability of Picard's iteration. If the -property is fulfilled in a complete metric space and, furthermore, , then Picard's iteration is -stable defined as as as . The main results obtained in this paper rely on the following features.

(1)In fact -contractive mappings are -Kannan self-mappings and vice-versa under certain mutual constraints between the constants and , [1012]. A necessary and sufficient condition for both properties to hold is given. Some of such constraints are obtained in the manuscript. The existence of fixed points and their potential uniqueness is discussed accordingly under completeness of the metric space, [14, 810, 13].

(2)If is ( )-times reasonable expansive self-mapping then it cannot be contractive as expected but it is -Kannan under certain constraints. The converse is also true under certain constraints. Some of such constraints referred to are obtained explicitly in the manuscript. The existence of fixed points is also discussed for two types of ( )-times reasonable expansive self-mappings proposed in [1].

(3)The -property guaranteeing Picard's -stability of iterative schemes, under the added condition , is compatible with both contractive self-mappings and -Kannan ones under certain constraints. A sufficient condition that as self-mapping possessing the -property is -Kannan is also given. It may be also fulfilled by ( )-times reasonable expansive self-mappings.

1.1. Notation

Assume that and are the sets of integer and real numbers, , , , .

If is a self mapping in a metric space , then denotes the set of fixed points of .

2. Combined Compatible Relations of -Contractive Mappings, -Kannan Mappings, and the -Property

It is of interest to establish when a -contractive mapping is also -Kannan and viceversa.

Theorem 2.1.

The following properties hold:

(i)if is -contractive with then it is -Kannan with ,

(ii) is -contractive and -Kannan if and only if
(2.1)
(iii)if is -contractive and -Kannan with and then the inequality
(2.2)

cannot hold for all , y in ,

(iv)if is -contractive and -Kannan with , and then the inequalities:
(2.3)

are feasible for all , y in .

Proof.
  1. (i)
    Since is -contractive, then
    (2.4)
     
from the triangle inequality property of the distance in metric spaces. Since , then
(2.5)
so that is -Kannan with provided that . As a result, if is -contractive with , then it is also -Kannan.
  1. (ii)

    It is direct if is -contractive and -Kannan with and . For , the result holds trivially.

     
  2. (iii)

    Proceed by contradiction. Assume that the inequality holds for with where is the (empty or nonempty) set of fixed points of . Since , the inequality leads to . This implies that since . However, ; , what is a contradiction. Therefore, the inequality cannot cold in X.

     
  3. (iv)
    The first inequality can potentially hold even for the set of fixed points. Furthermore, one gets from the triangle inequality for the distance :
    (2.6)
     

for all . Also, by using , one gets . As a result, the second inequality follows by combining both partial results. The third inequality follows from the second one and Property (i). Property (iv) has been proven.

Theorem 2.1(ii) leads to the subsequent result.

Corollary 2.2.

If is -contractive and -Kannan, then
(2.7)

Proof.

One gets from Theorem 2.1(ii) for that ; and ; . Both inequalities together yield the result.

The following two results follows directly from Theorem 2.1(iii) for .

Corollary 2.3.

If is -contractive and -Kannan with , then the inequality cannot hold .

Corollary 2.4.

If is -contractive and -Kannan with , then the inequality cannot hold for .

The following three results follows directly from Theorem 2.1(iv) for .

Corollary 2.5.

If is -contractive and -Kannan with , then the inequality is feasible .

Proof.

The proof follows since
(2.8)

is feasible from the first feasible inequality in Theorem 2.1(ii) and .

Corollary 2.6.

If is -contractive and -Kannan with , then the inequality

Proof.

The proof follows since
(2.9)

is feasible from the second feasible inequality in Theorem 2.1(ii) and .

Corollary 2.7.

If is -contractive and -Kannan with , then the inequality

Proof.

The proof follows directly since
(2.10)

are feasible from the third feasible inequality in Theorem 2.1(ii) and .

Remark 2.8.

It turns out from Definition 1.2 that if has the property for some real constants and , then it has also the ; , . The subsequent result is concerned with some joint , -Kannan and -contractiveness of a self-mapping .

Theorem 2.9.

The following properties hold:

(i) is -Kannan if it has the -property for any real constants L and m which satisfy the constraints , ,

(ii)assume that is -contractive. Then, it is also -Kannan and it possesses the -property for any real constant m which satisfies ,

(iii)assume that is -Kannan and . Then has the -property with and ,

(iv)assume that is -contractive with and . Then is -Kannan and it has the -property with and .

Proof.
  1. (i)
    If has the -property, one has from the triangle inequality for distances
    (2.11)
     
since . The above inequality together with the triangle inequality leads to
(2.12)
Thus, is -Kannan with which holds if and . Property (i) is proven. Furthermore, if is -contractive then it is also -Kannan if with from Theorem 2.1(ii). Then, is -contractive, -Kannan, and it has the -property if which holds for if and which is already fulfilled since is -Kannan with the -property. Property (ii) has been proven.
  1. (iii)
    By using the triangle inequality for distances and taking and , one gets
    (2.13)
     
for any real constant after using the subsequent relation:
(2.14)
which follows directly from the -Kannan property. Furthermore, since , the relation (2.14) leads to
(2.15)
(2.16)
Then, the substitution of (2.16) into (2.13) yields
(2.17)

which proves Property (iii). Property (iv) is a direct consequence of Properties (ii)-(iii) since is -Kannan with .

Further results concerning -Kannan mappings follow below.

Theorem 2.10.

Assume that is -Kannan. Then, the following properties hold:

(i) ; , ,

(ii)if is -Kannan and -contractive, then

(ii.1) ,

(ii.2) , , ,

(ii.3) ; , ,

(iii)if is -contractive for some , then
(2.18)

also, ; , ,

(iv)if is a complete metric space and is -contractive for some or if it is -Kannan and -contractive, then is independent of ; , so that consists of a unique fixed point.

Proof.

Proceed by complete induction by assuming that ; , . Since is -Kannan, take so that one gets from the triangle inequality for distances and the above assumption for that
(2.19)

Since ; , then so that ; and the proof of Property (i) is complete.

Property (ii.1) follows from Property (i), since is -Kannan, by taking into account that it is -contractive Property (ii.2) follows directly from Property (i) and Theorem 2.1(i). Property (ii.3) follows from
(2.20)

Property (iii) follows again directly from Property (i) and Theorem 2.1(i) and the first part of Property (ii) for .

Property (iv) follows directly from Properties (ii) and (iii) from the uniqueness of the fixed point Banach's contraction mapping principle since is a strict contraction.

Proposition 2.11.

If is -Kannan, then ; . If, in addition, is -contractive, then .

Proof.

It holds that
(2.21)
for all by using the triangle property of distances and Theorem 2.10(i). The first part of the result has been proven. The second part of the result follows since
(2.22)

if is -contractive.

Remark 2.12.

If is -contractive and -Kannan, it follows from Corollary 2.2 and Proposition 2.11 that ; .

Proposition 2.13.

If is -Kannan then ; .

Proof.

It follows from Proposition 2.11 and Theorem 2.10(i) since
(2.23)

Proposition 2.14.

If is -Kannan for some , then
(2.24)

Proof.

The upper-bound for has been obtained in Proposition 2.11. Its lower-bound follows from Theorem 2.10(i) subject to which holds if and only if . The proof is complete.

3. Combined Compatible Results about the -Property, -Kannan-Mappings, and a Class of Expansive Mappings

Definition 3.1 ([see [1]]).

Let be a complete metric space. Also, is said to be an ( )-times reasonable expansive self-mapping if there exists a real constant such that
(3.1)

Theorem 3.2.

Let be a complete metric space. Assume that is a continuous surjective self-mapping which is continuous everywhere in and -Kannan while it also satisfies for some real constant , some , (i.e., is ( ) times reasonable expansive self-mapping). Then, the following properties hold if :

(i) ; ,

(ii) has a unique fixed point in ,

(iii) has a fixed point in even if it is not -Kannan.

Proof.

Since is -Kannan and it satisfies ; some real constant , some , , then
(3.2)
Since and , then
(3.3)
and Property (i) has been proven. Also,
(3.4)
The last expression can be rewritten as
(3.5)

where is the identity mapping on X; that is, ; , is defined by ; (and then it is a surjective mapping since is surjective) and the functional is defined as . It turns out that is continuous everywhere on its definition domain (and then lower semicontinuous bounded from below as a result) since the distance mapping is continuous on . Then, has a fixed point in in [1, Lemma  2.4], even if is not -Kannan, since f is surjective on is the identity mapping on , and is lower semicontinuous bounded from below. The fixed point is unique since is a complete metric space. Properties (ii)-(iii) have been proven.

The subsequent result gives necessary conditions for Theorem 3.2 to hold as well as a sufficient condition for such a necessary condition to hold.

Theorem 3.3.

Let ( ) be a complete metric space. Assume that is a surjective self-mapping which is continuous everywhere in X which satisfies for some real constant , some , . The following holds. (i) The following zero limit exists
(3.6)
  1. (ii)
    If is -Kannan then a sufficient condition for Property (i) to hold is:
    (3.7)
     
and a necessary condition for the above sufficient condition to hold is:
(3.8)
  1. (iii)
    If is -Kannan then two joint necessary conditions for Property (i) to hold are:
    (3.9)
     

and such limits superior and inferior coincide as existing limits and are zero.

Proof.
  1. (i)

    Assume that Property (i) does not hold. Then, has not a fixed point in X what contradicts Theorem 3.2(iii). Thus, Property (i) holds.

     
  2. (ii)
    The condition ; together with the -Kannan property yield:
    (3.10)
     
for all . If
(3.11)
then
(3.12)
with , since
(3.13)
for all so that as is a a sufficient condition for Property (i) to hold. The necessary condition for the above sufficient to hold follows directly from the constraint ; .
  1. (iii)
    It follows since the subsequent constraints follow directly from the hypotheses and has a fixed point
    (3.14)
     

Theorem 3.2 may be generalized by generalizing the inequality to eventually involve other powers of , not necessarily being respectively identical to ( ) and , as follows.

Theorem 3.4.

Let be a complete metric space. Then, the following properties hold.
  1. (i)
    assume that is a surjective self-mapping which is continuous everywhere in X and satisfies:
    (3.15)
     
for some real constants ; some , , then, has at least a fixed point in X and it may eventually possess = card fixed points in X.
  1. (ii)
    if Property (i) holds for then has at least a fixed point in X and, furthermore,
    (3.16)
     
Proof.
  1. (i)
    From the statement constraints, it follows that
    (3.17)
     
so that
(3.18)
, where each functional ; is defined by
(3.19)
and the functions and are defined, respectively, as , ; . Note that , is continuous, and then lower semicontinuous, on X; since and are both continuous in X. Since is surjective then is also surjective so that and are also surjective . From [1, Lemma  2.4], they have a coincidence point since (3.18) holds and , is continuous. Then, there exists for some for each so that so that with provided that .
  1. (ii)

    It follows directly from Property (i), (3.18) and ; .

     

Remark 3.5.

Note that although if , it is not proven that since some of the existing fixed points for can mutually coincide or even more than one fixed point can eventually exist for each .

It is wellknown that nonexpansive and asymptotically non-expansive mappings can have fixed points as contractions have. See, for instance, [1, 2, 6, 1418]. However, and generally speaking, ( )-times reasonable expansive self-mappings do not necessarily have a fixed point, although they might have them, [1]. It has been proven in [1] that continuous and surjective ( )-times reasonable expansive self-mappings in complete metric spaces have a fixed pointing X if they fulfil the property:
(3.20)

Proposition 3.6.

Assume that ; , , some real constant . Then,
(3.21)

for all , some real constant .

Proof.

It follows directly from (3.20) by interchanging and in (3.20).

Proposition 3.7.

If (3.20) holds , , then
(3.22)

for all , .

Proof.

Take in (3.20).

Proposition 3.8.

Assume that is an ( )-times reasonable expansive self-mapping which satisfies (3.20) and is a complete metric space. Then
(3.23)

Proof.

If follows from (3.20) and Proposition 3.6 that
(3.24)
for provided that , and
(3.25)

for provided that .

Assume that (3.23) holds. Since is an ( )-times reasonable expansive self-mapping, there exists a real constant such that ; which is impossible since . Instead of (3.24) one can have:
(3.26)
for provided that . Since is an ( ) times reasonable expansive self-mapping, there exists a real constant such that
(3.27)
Then, either so that , or
(3.28)

and the proof is complete.

Proposition 3.8 may be rewritten in a more clear equivalent form as follows:

Proposition 3.9.

A necessary condition for a self-mapping in complete metric space to be an ( ) times reasonable expansive self-mapping which satisfies Property  (3.20) is that (3.23) holds.

Theorem  2.10 of [1] may be reformulated subject to the above necessary condition as follows.

Theorem 3.10.

Assume that is a complete metric space and that is a continuous surjective ( )-times reasonable expansive self-mapping which satisfies the constraint (3.20) and the necessary condition of Proposition 3.9. Then has a fixed point in X.

If the self-mapping satisfies Theorem 3.10 and it is also -Kannan, then the subsequent result holds:

Theorem 3.11.

Assume that Theorem 3.10 holds. Then, is in addition -Kannan if and only if
(3.29)
and the existing fixed point is unique.
  1. (ii)
    The following inequalities also hold:
    (3.30)
     

Proof.

The proof follows from (3.28) and the -Kannan-property.
(3.31)

which, together with (3.28), yields (3.29) since . The fixed point of (Theorem 3.10) is unique since is a complete metric space. Property (i) has been proven. Property (ii) is a direct result from Property (i) and (3.28).

Remark 3.12.

It is interesting to compare Theorem 3.2 with Theorem 3.11, subject to Proposition 3.9, and their respective guaranteed inequalities for distances in X for the case when is simultaneously -Kannan and -times reasonable expansive self-mapping. Note that Theorem 3.2 is based on the fulfilment of the inequality ; , for some for some real constant while Theorem 3.11 is based on ; for some real constants .

It is also of interest to investigate when being a continuous surjective ( )-times reasonable expansive self-mapping (Definition 3.1) satisfying either Theorem 3.10 or Theorem 3.2 has also the -property for some real constants and (Definition 1.2). Note that if either Theorem 3.10 or Theorem 3.2 are fulfilled then so that Definition 1.2 is well-posed.

Theorem 3.13.

The following properties hold:
  1. (i)
    assume that is a nonempty complete metric space and that is a continuous surjective ( )-times reasonable expansive self-mapping according to Theorem 3.10 so that it has a fixed point in X . Then, also possesses the -property for some real constants and if
    (3.32)
     
Two necessary conditions for the above condition to hold are:
(3.33)
provided that ; and
(3.34)
  1. (ii)
    assume that is a nonempty complete metric space and that is a continuous surjective ( ) times reasonable expansive self-mapping which satisfies Theorem 3.2. Then, also possesses the -property for some real constants and if and only if
    (3.35)
     
Two necessary conditions for the above necessary and sufficient condition to hold are,
(3.36)

provided that ; and ; , some real constant .

Proof.

It follows from (3.28) and the -property under direct calculations.

Remark 3.14.

Note from direct inspection of Definition 1.2 and the triangle property of distances that if has the -property then for any .
(3.37)
The subsequent results are focused on the combinations of one of the two conditions below, compatible under extra conditions with the existence of fixed points, with the -property in a metric space for some real constant :
  1. (a)
    (3.38)
     
where (see Theorem 3.2)
  1. (b)
    (3.39)
     
where being a surjective -times reasonable expansive self-mapping (see Propositions 3.7 and 3.8). Note by direct inspection that (3.40) is equivalent to
(3.40)

Theorem 3.15.

The following properties hold:(i) fulfils simultaneously (3.38) and the -property for some if
(3.41)

A necessary condition for (3.41) to hold is ; , .

Another necessary condition for (3.41) to hold is

(3.42)

provided that and

(ii) fulfils simultaneously (3.38) and the -property for some , if

(3.43)

Proof.

The sufficiency parts of Properties (i) and (ii) follow directly from Remark 3.14 and (3.38) and (3.39), respectively. The first necessary condition of Property (i) is a direct need for the lower-bound of in (3.41) do not exceed its upper-bound, . The second necessary condition is proven as follows. From the first necessary condition and the triangle inequality for distances, one gets:
(3.44)

if .

Theorem 3.16.

The following properties hold:
  1. (i)

    A necessary condition for (3.39) to hold with being an -times reasonable expansive self-mapping is

     
(3.45)
  1. (ii)

    A necessary condition for to possess, in addition, the -property for some and is

     
(3.46)
Proof.
  1. (i)
    Take , so that
    (3.47)
     
There are two potential possibilities for each , since (3.39) holds, namely, either:
  1. (a)
    (3.48)
     
for some real constants since, in addition, is ( )-times reasonable expansive, so that Property (i) holds directly, or
  1. (b)
    (3.49)
     

what leads to the contradiction . Thus, the above result of logic implications cannot hold if , as a result, if (3.39) holds then (3.48) is a necessary condition for to be an -times reasonable expansive self-mapping. Property (i) has been proven.

  1. (ii)
    Property (i) is equivalent to
    (3.50)
     
so that if satisfies, in addition, the -property for some and then:
  1. (a)
    (3.51)
     
  1. (b)
    (3.52)
     
(3.53)
provided that
(3.54)

, provided that . The combination of (3.52) to (3.54) proves the result.

4. Examples

Example 4.1.

Consider the one -dimensional linear unforced discrete dynamic system
(4.1)
under initial conditions . The distance function is taken as the usual Euclidean norm, namely, ; . It turns out that if then as irrespective of so that is the only stable attractor, which is the only equilibrium point, and the system is globally asymptotically stable. is also the only fixed point of the self-mapping on in the complete metric space ( , ) defined by ; which is -contractive for any real provided that . It is now tested when is -Kannan. Note that
(4.2)
for any sequences ; ; for initial conditions so that by combining the above three relations:
(4.3)

and is -Kannan if which is guaranteed for if which is the condition of Theorem 2.1(i) guaranteeing that if is -contractive, it is also -Kannan.

Example 4.2.

Now consider the -th dimensional linear unforced discrete dynamic system
(4.4)
under initial conditions where is the (or spectral)-norm which coincides with the Euclidean (or Froebenius) norm for vectors. For the matrix , we define the vector-induced -norm by
(4.5)

where is the maximum (real) eigenvalue of . The distance function is taken as the usual Euclidean norm in , namely, ; . Assume that . Define the self-mapping on as ; . It follows that is the only equilibrium point, which is stable, and . The relations obtained for the scalar case still hold with the replacements , , , and the -contractive self-mapping on is also -Kannan if which is still the sufficient condition of Theorem 2.1.

Example 4.3.

Now consider the -th dimensional, perhaps nonlinear, unforced time-varying discrete dynamic system subject to perturbations:
(4.6)
under initial conditions and is a uniformly bounded sequence of real -vectors for any bounded whose elements satisfy . Now consider two solution sequences , under initial conditions . Let be defined from real finite constants , where provided that , that is, all the matrices ; are stability matrices. Consider the distance being the Euclidean norm. If then,
(4.7)
So that the solution sequence is bounded for any bounded initial conditions. Furthermore,
(4.8)

Thus, the self-mapping is -Kannan if , that is if , irrespective of its contractiveness or not. The above condition is guaranteed with and .

Now, assume that the discrete dynamic system is defined by:
(4.9)
for some . Then,
(4.10)
since provided that . In this case, one also has:
(4.11)

Then the following hold.

(1)First, is -contractive with with being its unique stable equilibrium point and its unique fixed point provided that and . The time-varying system is globally asymptotically stable.

(2)If , that is and then the -contractive self-mapping is furthermore -Kannan. Those results still agree with Theorem 2.1. On the other hand, the -property of contractive Kannan self-mappings can be tested for this example according to the formula
(4.12)
from Theorem 2.9 with with , since is -Kannan and -contractive. Note that
(4.13)
with the above lower-bound being reached for , . Note also that . since otherwise, one would have
(4.14)

what is a contradiction.

Example 4.4.

A forced version of the equation of Example 4.1 is
(4.15)
with . If then
(4.16)
independent of the initial condition for any bounded initial condition. Also, it is direct by complete induction the property
(4.17)
On the other hand, if and then
(4.18)
If, in addition, the system is positive and stable, that is , with positive initial conditions and forcing term then is not contractive since for any finite , and
(4.19)
for any real since it holds that
(4.20)
Thus, for , , one has
(4.21)

so that the self-mapping has a fixed point while it is reasonable expansive (see Definition 3.1 and Theorem 3.2). Extensions to the non positive first-order system and the -th order discrete dynamic system can be addressed in the same way. If the system is time-varying with the sequence of parameters fulfilling then as where is the geometric mean of the elements of . Thus, there is still a unique fixed point . Also, if there is a finite subset such that if and only if then there is a unique fixed point since despite the fact that is not contractive.

Declarations

Acknowledgments

The author is grateful to the Spanish Ministry of Education by its partial support of this work through Grant DPI 2009-07197. He is also grateful to the Basque Government by its support through Grants GIC07143-IT-269-07and SAIOTEK S-PE08UN15. The author is also very grateful to the reviewers by their useful comments.

Authors’ Affiliations

(1)
Institute of Research and Development of Processes, Faculty of Science and Technology, University of the Basque Country, Campus de Leioa (Bizkaia)

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© M. De la Sen. 2009

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