© 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.
(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.
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.
The following properties hold:
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.
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:
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 :
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.
and such limits superior and inferior coincide as existing limits and are zero.
and the proof is complete.
Proposition 3.8 may be rewritten in a more clear equivalent form as follows:
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.
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.
Another necessary condition for (3.41) to hold is
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.
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.
Then the following hold.
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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