Open Access

A Continuation Method for Weakly Contractive Mappings under the Interior Condition

Fixed Point Theory and Applications20092009:809315

DOI: 10.1155/2009/809315

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

Suppose that is a Banach space, that is an open bounded subset of , containing the origin, and that is a mapping. It is well known that if satisfies the Leray-Schauder condition defined as
(L-S)

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]).

On the other hand, it has been recently shown in [7] that, for condensing mappings, the condition (L-S) can be replaced by a modification of it which we call the interior condition, and is defined as follows: a mapping satisfies the Interior Condition ( I-C ), if there exists such that
(I-C)

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.

Let be a complete metric space and an open subset of . A function is said to be weakly contractive if there exists compactly positive (i.e., for every ) such that
(2.1)
If is a compactly positive function, we define for
(2.2)

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:

(P1) and for every ;

(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 , ;

(P5) if and , with , then .

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.

We argue by contradiction. Suppose that does not have any fixed point in , and let be a homotopy between and , in the sense of Definition 2.1. Consider the set
(2.3)

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.

To show that is closed, suppose that is a sequence in converging to and let us show that . Since , there exists with . Fix . Using that is bounded and that is continuous on the compact interval , it is easy to show that there exists such that , and hence for all . Define and let be such that for all , . Then for all because, otherwise, we would have for some , and then
(2.4)
which is a contradiction. Then is a Cauchy sequence and, since is complete, there exists such that as . In addition, since for all we have that
(2.5)

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, .

Next we show that is open in . Let . Then there exists with . Let be such that , and let such that for every with . Then, if ,
(2.6)

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.

(P1) For all , and .

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

(P3) Since is weakly contractive, there exists a compactly positive function such that for every . Then, if and ,
(2.7)
(P4) Since is bounded, there exists such that for all . Hence,
(2.8)

where is the continuous function defined as .

(P5) Suppose that for some and we have that . Then, since , and is open. Let us see that : suppose, on the contrary, that , that is, and define
(2.9)

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

A multitude of generalizations and variants of Banach's contractive condition have been given after Banach's theorem (see, e.g., Rhoades [13]) and, recently, Agarwal and O'Regan [11] have given a homotopy result (thus generalizing a fixed point theorem of Hardy and Rogers [14]) under the following generalized contractive condition: there exists such that for all
(3.1)

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.

Let be a complete metric space, , , and . Suppose that there exists such that for one has
(3.2)

Then there exists with .

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:

(P1) and for every ;

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

(P3) there exists such that for all and one has
(3.3)

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

(P5) if and , with , then .

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

Proof.

Suppose that does not have any fixed point in and consider the nonempty set
(3.4)

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.

Suppose that is an open and strictly star shaped subset of a Banach space , with , and that is map with being bounded. Assume also that there exists such that for all and one has
(3.5)

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

(1)
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga

References

  1. Petryshyn WV: Fixed point theorems for various classes of 1-set-contractive and 1-ball-contractive mappings in Banach spaces. Transactions of the American Mathematical Society 1973, 182: 323–352.MathSciNetMATHGoogle Scholar
  2. Reich S: Fixed points of condensing functions. Journal of Mathematical Analysis and Applications 1973, 41: 460–467. 10.1016/0022-247X(73)90220-5MathSciNetView ArticleMATHGoogle Scholar
  3. Granas A: Continuation method for contractive maps. Topological Methods in Nonlinear Analysis 1994,3(2):375–379.MathSciNetMATHGoogle Scholar
  4. Agarwal RP, Meehan M, O'Regan D: Fixed Point Theory and Applications, Cambridge Tracts in Mathematics. Volume 141. Cambridge University Press, Cambridge, UK; 2001:x+170.View ArticleMATHGoogle Scholar
  5. Frigon M: On continuation methods for contractive and nonexpansive mappings. In Recent Advances on Metric Fixed Point Theory (Seville, 1995). Volume 48. Edited by: Dominguez Benavides T. Universidad de Sevilla, Seville, Spain; 1996:19–30.Google Scholar
  6. O'Regan D, Precup R: Theorems of Leray-Schauder Type and Applications, Series in Mathematical Analysis and Applications. Volume 3. Gordon and Breach Science, Amsterdam, The Netherlands; 2001:x+206.MATHGoogle Scholar
  7. Jiménez-Melado A, Morales CH: Fixed point theorems under the interior condition. Proceedings of the American Mathematical Society 2006,134(2):501–507.MathSciNetView ArticleMATHGoogle Scholar
  8. González C, Jiménez-Melado A, Llorens-Fuster E: A Mönch type fixed point theorem under the interior condition. Journal of Mathematical Analysis and Applications 2009,352(2):816–821. 10.1016/j.jmaa.2008.11.032MathSciNetView ArticleMATHGoogle Scholar
  9. Shaini P, Singh N: Fixed point theorems for mappings satisfying interior condition. International Journal of Mathematical Analysis 2008,3(1–4):45–54.MathSciNetMATHGoogle Scholar
  10. Frigon M, Granas A, Guennoun ZEA: Alternative non linéaire pour les applications contractantes. Annales des Sciences Mathématiques du Québec 1995,19(1):65–68.MathSciNetMATHGoogle Scholar
  11. Agarwal RP, O'Regan D: Fixed point theory for generalized contractions on spaces with two metrics. Journal of Mathematical Analysis and Applications 2000,248(2):402–414. 10.1006/jmaa.2000.6914MathSciNetView ArticleMATHGoogle Scholar
  12. Dugundji J, Granas A: Weakly contractive maps and elementary domain invariance theorem. Bulletin de la Société Mathématique de Grèce. Nouvelle Série 1978,19(1):141–151.MathSciNetMATHGoogle Scholar
  13. Rhoades BE: A comparison of various definitions of contractive mappings. Transactions of the American Mathematical Society 1977, 226: 257–290.MathSciNetView ArticleMATHGoogle Scholar
  14. Hardy GE, Rogers TD: A generalization of a fixed point theorem of Reich. Canadian Mathematical Bulletin 1973, 16: 201–206. 10.4153/CMB-1973-036-0MathSciNetView ArticleMATHGoogle Scholar

Copyright

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

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.