Some Combined Relations between Contractive Mappings, Kannan Mappings, Reasonable Expansive Mappings, and -Stability
© M. De la Sen. 2009
Received: 13 May 2009
Accepted: 31 August 2009
Published: 27 September 2009
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.
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  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.
(3)Fixed point techniques have been recently used in  for the investigation of global stability of a wide class of time-delay dynamic systems which are modeled by functional equations.
(5)The existence of fixed points of Liptchitzian semigroups has been investigated, for instance, in .
(6)Picard's -stability is discussed in  related to the convergence of perturbed iterations to the same fixed points as the nominal iteration under certain conditions in a complete metric space.
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 ; , , ,
(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 , . It is proven in  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.
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.
A self-mapping with possesses the -property for some real constants and if ; , .
The above property has been introduced in  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 .
(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.
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.
The following properties hold:
(i)if is -contractive with then it is -Kannan with ,
cannot hold for all , y in ,
are feasible for all , y in .
- (i)Since is -contractive, then(2.4)
It is direct if is -contractive and -Kannan with and . For , the result holds trivially.
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.
- (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.
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 .
If is -contractive and -Kannan with , then the inequality cannot hold .
If is -contractive and -Kannan with , then the inequality cannot hold for .
The following three results follows directly from Theorem 2.1(iv) for .
If is -contractive and -Kannan with , then the inequality is feasible .
is feasible from the first feasible inequality in Theorem 2.1(ii) and .
If is -contractive and -Kannan with , then the inequality
is feasible from the second feasible inequality in Theorem 2.1(ii) and .
If is -contractive and -Kannan with , then the inequality
are feasible from the third feasible inequality in Theorem 2.1(ii) and .
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 .
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 .
- (i)If has the -property, one has from the triangle inequality for distances(2.11)
- (iii)By using the triangle inequality for distances and taking and , one gets(2.13)
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.
Assume that is -Kannan. Then, the following properties hold:
(i) ; , ,
(ii)if is -Kannan and -contractive, then
(ii.2) , , ,
(ii.3) ; , ,
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.
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.
If is -Kannan, then ; . If, in addition, is -contractive, then .
if is -contractive.
If is -contractive and -Kannan, it follows from Corollary 2.2 and Proposition 2.11 that ; .
If is -Kannan then ; .
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 ]).
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.
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.
- (ii)If is -Kannan then a sufficient condition for Property (i) to hold is:(3.7)
- (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.
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.
- (ii)The condition ; together with the -Kannan property yield:(3.10)
- (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.
- (i)assume that is a surjective self-mapping which is continuous everywhere in X and satisfies:(3.15)
- (ii)if Property (i) holds for then has at least a fixed point in X and, furthermore,(3.16)
- (i)From the statement constraints, it follows that(3.17)
It follows directly from Property (i), (3.18) and ; .
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 .
for all , some real constant .
It follows directly from (3.20) by interchanging and in (3.20).
for all , .
Take in (3.20).
for provided that .
and the proof is complete.
Proposition 3.8 may be rewritten in a more clear equivalent form as follows:
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  may be reformulated subject to the above necessary condition as follows.
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:
- (ii)The following inequalities also hold:(3.30)
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).
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.
- (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)
- (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)
provided that ; and ; , some real constant .
It follows from (3.28) and the -property under direct calculations.
A necessary condition for (3.41) to hold is ; , .
Another necessary condition for (3.41) to hold is
provided that and
(ii) fulfils simultaneously (3.38) and the -property for some , if
A necessary condition for (3.39) to hold with being an -times reasonable expansive self-mapping is
A necessary condition for to possess, in addition, the -property for some and is
- (i)Take , so that(3.47)
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.
- (ii)Property (i) is equivalent to(3.50)
, provided that . The combination of (3.52) to (3.54) proves the result.
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.
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.
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.
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.
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.
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