Common Fixed Points of Weakly Contractive and Strongly Expansive Mappings in Topological Spaces
© M. H. Shah et al. 2010
Received: 17 May 2010
Accepted: 21 July 2010
Published: 4 August 2010
It is well known that if is a compact metric space and is a weakly contractive mapping (see Section 2 for the definition), then has a fixed point in (see [1, p. 17]). In late sixties, Furi and Vignoli  extended this result to -condensing mappings acting on a bounded complete metric space (see  for the definition). A generalized version of Furi-Vignoli's theorem using the notion of weakly -contractive mappings acting on a topological space was proved in  (see also ).
On the other hand, in  while examining KKM maps, the authors introduced a new concept of lower (upper) semicontinuous function (see Definition 2.1, Section 2) which is more general than the classical one. In , the authors used this definition of lower semicontinuity to redefine weakly -contractive mappings and strongly -expansive mappings (see Definition 2.6, Section 2 ) to formulate and prove several results for fixed points.
In this article, we have used the notions of weakly -contractive mappings ( where is a topological space) to prove a version of the above-mentioned fixed point theorem [7, Theorem ] for common fixed points (see Theorem 3.1). We also prove a common fixed point theorem under the assumption that certain iteration of the mappings in question is weakly -contractive. As a corollary to this fact, we get an extension (to common fixed points) of [7, Theorem ] for Banach spaces with a quasimodulus endowed with a suitable transitive binary relation. The most interesting result of this section is Theorem 3.8 wherein the strongly -expansive condition on (with some other conditions) implies that and have a unique common fixed point.
In Section 4, we define a new class of noncommuting self-maps and prove some common fixed point results for this new class of mappings.
Definition 2.1 (see ).
The following lemmas state some properties of lsca mappings. The first one is an analogue of Weierstrass boundedness theorem and the second one is about the composition of a continuous function and a function lsca.
Lemma 2.3 (see ).
Lemma 2.4 (see ).
Remark 2.5 (see ).
Definition 2.6 (see ).
Let . The set of fixed points of (resp., ) is denoted by (resp., ). A point is a coincidence point (common fixed point) of and if . The set of coincidence points of and is denoted by Maps are called (1) commuting if for all , (2) weakly compatible  if they commute at their coincidence points, that is, if whenever , and (3) occasionally weakly compatible  if for some
3. Common Fixed Point Theorems for Commuting Maps
In this section we extend some results in  to the setting of two mappings having a unique common fixed point.
which is a contradiction. Hence the claim follows.
If , then Theorem 3.1(i) reduces to [7, Theorem ].
Corollary 3.2 (see [7, Theorem ]).
We verify the hypothesis of Theorem 3.1.
It is immediate from Theorem 3.1.
which is a contradiction. Hence the claim follows.
The following example illustrates Theorem 3.8.
4. Occasionally Banach Operator Pair and Weak F-Contractions
In this section, we define a new class of noncommuting self-maps and prove some common fixed point results for this new class of maps.
The pair is called a Banach operator pair  if the set is -invariant, namely, . Obviously, commuting pair is a Banach operator pair but converse is not true, in general; see [10–13]. If is a Banach operator pair, then need not be a Banach operator pair.
It is immediate from Theorem 4.6.
Theorem 4.6 holds for a Banach operator pair without condition (4.5) as follows.
- Dugundji J, Granas A: Fixed Point Theory, Vol. 1. PWN, Warsaw, Poland; 1982:209.MATHGoogle Scholar
- Furi M, Vignoli A: A fixed point theorem in complete metric spaces. Bolletino della Unione Matematica Italiana 1969, 4: 505–509.MathSciNetMATHGoogle Scholar
- Bugajewski D: Some remarks on Kuratowski's measure of noncompactness in vector spaces with a metric. Commentationes Mathematicae 1992, 32: 5–9.MathSciNetMATHGoogle Scholar
- Bugajewski D: Fixed point theorems in locally convex spaces. Acta Mathematica Hungarica 2003, 98(4):345–355. 10.1023/A:1022842429470MathSciNetView ArticleMATHGoogle Scholar
- Ćirić LB: Coincidence and fixed points for maps on topological spaces. Topology and Its Applications 2007, 154(17):3100–3106. 10.1016/j.topol.2007.08.004MathSciNetView ArticleMATHGoogle Scholar
- Chen YQ, Cho YJ, Kim JK, Lee BS: Note on KKM maps and applications. Fixed Point Theory and Applications 2006, 2006:-9.Google Scholar
- Bugajewski D, Kasprzak P: Fixed point theorems for weakly -contractive and strongly -expansive mappings. Journal of Mathematical Analysis and Applications 2009, 359(1):126–134. 10.1016/j.jmaa.2009.05.024MathSciNetView ArticleMATHGoogle Scholar
- Jungck G: Common fixed points for noncontinuous nonself maps on nonmetric spaces. Far East Journal of Mathematical Sciences 1996, 4(2):199–215.MathSciNetMATHGoogle Scholar
- Jungck G, Rhoades BE: Fixed point theorems for occasionally weakly compatible mappings. Fixed Point Theory 2006, 7(2):287–296.MathSciNetMATHGoogle Scholar
- Chen J, Li Z: Common fixed-points for Banach operator pairs in best approximation. Journal of Mathematical Analysis and Applications 2007, 336(2):1466–1475. 10.1016/j.jmaa.2007.01.064MathSciNetView ArticleMATHGoogle Scholar
- Hussain N: Common fixed points in best approximation for Banach operator pairs with Ćirić type -contractions. Journal of Mathematical Analysis and Applications 2008, 338(2):1351–1363. 10.1016/j.jmaa.2007.06.008MathSciNetView ArticleMATHGoogle Scholar
- Hussain N, Cho YJ: Weak contractions, common fixed points, and invariant approximations. Journal of Inequalities and Applications 2009, 2009:-10.Google Scholar
- Pathak HK, Hussain N: Common fixed points for Banach operator pairs with applications. Nonlinear Analysis 2008, 69: 2788–2802. 10.1016/j.na.2007.08.051MathSciNetView ArticleMATHGoogle Scholar
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.