# A Continuation Method for Weakly Contractive Mappings under the Interior Condition

- David Ariza-Ruiz
^{1}and - Antonio Jiménez-Melado
^{1}Email author

**2009**:809315

https://doi.org/10.1155/2009/809315

© D. Ariza-Ruiz and A. Jiménez-Melado. 2009

**Received: **29 July 2009

**Accepted: **8 October 2009

**Published: **18 October 2009

## Abstract

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.

## 1. Introduction

and is a strict set-contraction or, more generally, condensing, then has a fixed point in (see, e.g., [1] or [2]). The first continuation method in the setting of a complete metric space for contractive maps comes from the hands of Granas [3], in 1994, who gave a homotopy result for contractive maps (for more information on this topic see, e.g., [4, 5] or [6]).

*Interior Condition ( I-C )*, if there exists such that

where (some generalizations of this result can be found in [8, 9]).

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 [7] 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 [3] and Frigon et al. [10], 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 [11].

## 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 [12].

Definition 2.1.

It was shown in [12] that any weakly contractive map defined on a complete metric space has a unique fixed point. Some years later, Frigon [5] 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.

Definition 2.2.

Let be a complete metric space, and an open subset of . Let be two weakly contractive maps. We say that is (I-C)-homotopic to if there exists with the following properties:

(P2) there exists such that for every , with , and , where ;

(P3) there exists a compactly positive function such that for every , and ;

(P4) there exists a continuous function such that, for every and , ;

In the proof of the main result of this chapter we shall make use of the following lemma (see Frigon [5]).

Lemma 2.3.

Let , , and weakly contractive. If , then has a fixed point.

Theorem 2.4.

Let be two weakly contractive maps. Suppose that is homotopic to and is bounded. If has a fixed point in , then has a fixed point in .

Proof.

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 .

Therefore, and, consequently, .

Using Lemma 2.3, we obtain that has a fixed point in for every such that . Thus for any , and therefore is open in .

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).

Theorem 2.5.

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 .

Proof.

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.

(P2) Since satisfies the condition (I-C), we have that for with and . Hence, for every , with , and .

where is the continuous function defined as .

Then, it is easy to see that , which contradicts that is strictly star shaped, since we also have that .

## 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 [11].

Lemma 3.1.

The proof of the following theorem is very similar to the proof of Theorem 2.4, and we give a sketch of it.

Theorem 3.2.

Let be a complete metric space, and an open subset of . Let be two maps such that there exists with the following properties:

(P2) there exists such that for every , with , and , where ;

(P4) there exists a continuos function such that, for every and , ;

If has a fixed point in , then has a fixed point in .

Proof.

We will arrive to a contradiction by showing that , and for this we only need prove that is closed and open in .

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 .

To prove that is open argue as in Theorem 2.4, use Lemma 3.1 instead of Lemma 2.3.

As an immediate consequence, we obtain the following result, whose proof is omitted because it is analogous to the proof of Theorem 2.5.

Theorem 3.3.

If satisfies the condition (I-C), then has a fixed point in .

## Declarations

### Acknowledgment

This research is partially supported by the Spanish (Grant MTM2007-60854) and regional Andalusian (Grants FQM210, FQM1504) Governments.

## Authors’ Affiliations

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