A Continuation Method for Weakly Contractive Mappings under the Interior Condition
© D. Ariza-Ruiz and A. Jiménez-Melado. 2009
Received: 29 July 2009
Accepted: 8 October 2009
Published: 18 October 2009
Recently, Frigon proved that, for weakly contractive maps, the property of having a fixed point is invariant by a certain class of homotopies, obtaining as a consequence a Leray-Schauder alternative for this class of maps in a Banach space. We prove here that the Leray-Schauder condition in the aforementioned result can be replaced by a modification of it, the interior condition. We also show that our arguments work for a certain class of generalized contractions, thus complementing a result of Agarwal and O'Regan.
and is a strict set-contraction or, more generally, condensing, then has a fixed point in (see, e.g.,  or ). The first continuation method in the setting of a complete metric space for contractive maps comes from the hands of Granas , in 1994, who gave a homotopy result for contractive maps (for more information on this topic see, e.g., [4, 5] or ).
We remark that the condition (I-C) by itself cannot be a substitute for the condition (L-S), and an additional assumption on the domain of needs to be made in order to guarantee the existence of a fixed point for . The class of sets that we need is defined as follows: suppose that is an open neighborhood of the origin. We say that is strictly star shaped if for any we have that . It was shown in  that if is bounded and strictly star shaped and is a condensing mapping satisfying the condition (I-C), then has a fixed point. Of course, this result includes the case of a contractive map (i.e., a map for which there exists such that for all ), but our aim in this note is, following the pattern of Granas  and Frigon et al. , to give a continuation method for weakly contractive mappings, in the setting of a complete metric space, under some conditions on the homotopy which are the counterpart of the condition (I-C) and the notion of a strictly star shaped set in a space without a vector structure. Finally, in the last section we show that our arguments also work for a class of generalized contractions, thus complementing a result of Agarwal and O'Regan .
2. Weakly Contractive Maps
In this chapter we deal with the concept of weakly contractive maps, as it was introduced by Dugundji and Granas in .
It was shown in  that any weakly contractive map defined on a complete metric space has a unique fixed point. Some years later, Frigon  proved that, for weakly contractive maps, the property of having a fixed point is invariant by a certain class of homotopies, obtaining as a consequence a Leray-Schauder alternative for weakly contractive maps in the setting of a Banach space. We prove here that the Leray-Schauder condition in the aforementioned result can be replaced by the condition (I-C), and it will also be obtained as a consequence of a continuation method. The definition of homotopy that we need for our purposes is the following.
In the proof of the main result of this chapter we shall make use of the following lemma (see Frigon ).
and notice that is nonempty since has a fixed point in , that is, . We will show that is both open and closed in , and hence, by connectedness, we will have that . As a result, will have a fixed point in , which establishes a contradiction.
Observe that , because if then , which contradicts the fact that does not have any fixed point in . Notice that , because, otherwise, we would have , that is, and since , by (P5), we have that . However, since , and for all , there exists such that for all . Hence, since for all , applying (P2), we have that for all , that is, for all . Taking limits, we arrive to the contradiction .
As an immediate consequence of the previous theorem, we obtain the following fixed point result of the Leray-Schauder type for weakly contractive maps under the condition (I-C).
Suppose that is an open and strictly star shaped subset of a Banach space , with , and that is a weakly contractive map with being bounded. If satisfies the condition (I-C), then has a fixed point in .
Since satisfies the condition (I-C), there exists such that for and with . We may assume that for every , because otherwise we are finished. Define as , and let be the zero map. Notice that has a fixed point in , that is, and also that and are two weakly contractive mappings. So, the result will follow from Theorem 2.4 once we prove that is (I-C)-homotopic to . Let us check it.
3. A Class of Generalized Contractions
In this section we give a homotopy result for this class of mappings under the condition (I-C). In the proof of our theorem we shall use the following result .
The proof of the following theorem is very similar to the proof of Theorem 2.4, and we give a sketch of it.
To show that is closed in , consider a sequence in , with as , and show that ; that is, that there exists with . To prove that exists, take any sequence in with , prove that is Cauchy, and define as the limit of , as .
That is a Cauchy sequence, as well as , follows from standard arguments which can be seen in [11, Theorem 3.1]. It remains to show that . To prove this, suppose that it is not true and arrive to a contradiction as follows: we have that , and also that , because does not have any fixed point in . Then, by (P5) . On the other hand, because for large enough. To be convinced of it, just apply (P2): since , and for all , there exists such that for all . Then, for all since .
As an immediate consequence, we obtain the following result, whose proof is omitted because it is analogous to the proof of Theorem 2.5.
This research is partially supported by the Spanish (Grant MTM2007-60854) and regional Andalusian (Grants FQM210, FQM1504) Governments.
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