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

- M. De la Sen
^{1}Email author

**2009**:815637

**DOI: **10.1155/2009/815637

© M. De la Sen. 2009

**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., [1–7]). 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:

(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 , [10–12]. 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, [1–4, 8–10, 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 ,

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.

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.

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.

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.

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
**.**

- (iii)

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:

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

(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.

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

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.

Remark 2.12.

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

Proposition 2.13.

Proof.

Proposition 2.14.

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]]).

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 :

(ii) has a unique fixed point in ,

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

Proof.

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.

- (ii)

- (iii)

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

- (iii)

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.

**;**some , , then, has at least a fixed point in X and it may eventually possess = card fixed points in X.

- (ii)

*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 .

- (ii)

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 .

*X*if they fulfil the property:

Proposition 3.6.

for all , some real constant .

Proof.

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

Proposition 3.7.

Proof.

Proposition 3.8.

Proof.

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.

Proof.

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.

- (i)

- (ii)

provided that ; and ; , some real constant .

Proof.

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

Remark 3.14.

- (a)

Theorem 3.15.

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

Another necessary condition for (3.41) to hold is

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

Proof.

Theorem 3.16.

- (i)

- (b)

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.

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

## 4. Examples

Example 4.1.

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.

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.

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

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.

what is a contradiction.

Example 4.4.

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

## References

- Chen CF, Zhu CX:
**Fixed point theorems for times reasonable expansive mapping.***Fixed Point Theory and Applications*2008,**2008:**-6.Google Scholar - Hu L-G:
**Strong convergence of a modified Halpern's iteration for nonexpansive mappings.***Fixed Point Theory and Applications*2008,**2008:**-9.Google Scholar - Saeidi S:
**Approximating common fixed points of Lipschitzian semigroup in smooth Banach spaces.***Fixed Point Theory and Applications*2008,**2008:**-17.Google Scholar - De la Sen M:
**About robust stability of dynamic systems with time delays through fixed point theory.***Fixed Point Theory and Applications*2008,**2008:**-20.Google Scholar - Latif A, Abdou AAN:
**Fixed points of generalized contractive maps.***Fixed Point Theory and Applications*2009,**2009:**-9.Google Scholar - Xiao J-Z, Zhu X-H:
**Common fixed point theorems on weakly contractive and nonexpansive mappings.***Fixed Point Theory and Applications*2008,**2008:**-8.Google Scholar - Karpagam S, Agrawal S:
**Best proximity point theorems for -cyclic Meir-Keeler contractions.***Fixed Point Theory and Applications*2009,**2009:**-9.Google Scholar - De la Sen M:
**Stability and convergence results based on fixed point theory for a generalized viscosity iterative scheme.***Fixed Point Theory and Applications*2009,**2009:**-19.Google Scholar - Qing Y, Rhoades BE:
**-stability of Picard iteration in metric spaces.***Fixed Point Theory and Applications*2008,**2008:**-4.Google Scholar - Kannan R:
**Some results on fixed points. II.***The American Mathematical Monthly*1969,**76**(4):405–408. 10.2307/2316437MathSciNetView ArticleMATHGoogle Scholar - Kikkawa M, Suzuki T:
**Some similarity between contractions and Kannan mappings.***Fixed Point Theory and Applications*2008,**2008:**-8.Google Scholar - Enjouji Y, Nakanishi M, Suzuki T:
**A Generalization of Kannan's fixed point theorem.***Fixed Point Theory and Applications*2009,**2009:**-10.Google Scholar - Subrahmanyam PV:
**Completeness and fixed-points.***Monatshefte für Mathematik*1975,**80**(4):325–330. 10.1007/BF01472580MathSciNetView ArticleMATHGoogle Scholar - Halpern B:
**Fixed points of nonexpanding maps.***Bulletin of the American Mathematical Society*1967,**73**(6):957–961. 10.1090/S0002-9904-1967-11864-0MathSciNetView ArticleMATHGoogle Scholar - Shioji N, Takahashi W:
**Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*1998,**34**(1):87–99. 10.1016/S0362-546X(97)00682-2MathSciNetView ArticleMATHGoogle Scholar - Dominguez Benavides T, Acedo GL, Xu H-K:
**Construction of sunny nonexpansive retractions in Banach spaces.***Bulletin of the Australian Mathematical Society*2002,**66**(1):9–16. 10.1017/S0004972700020621MathSciNetView ArticleMATHGoogle Scholar - Medghalchi A, Saeidi S:
**Weak and strong convergence for some of nonexpansive mappings.***Taiwanese Journal of Mathematics*2008,**12**(9):2489–2499.MathSciNetMATHGoogle Scholar - Goebel K, Kirk WA:
**A fixed point theorem for asymptotically nonexpansive mappings.***Proceedings of the American Mathematical Society*1972,**35**(1):171–174. 10.1090/S0002-9939-1972-0298500-3MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.