时间：2022-09-07 06:16:10
[a] Mathematics Department, University of Lagos, Lagos, Nigeria.
*Corresponding author.
Received 15 March 2012; accepted 11 June 2012
Abstract
We introduce the Jungck-multistep-SP iteration and prove some convergence as well as stabiilty results for a pair of weakly compatible generalized contractive-like inequality operators defined on a Banach space. As corollaries, the results show that the Jungck-SP and Jungck-Mann iterations can also be used to approximate the common fixed points of such operators. The results are improvements, generalizations and extensions of the work of Chugh and Kumar (2011). Consequently, several results in literature are generalized.
Key words: Jungck-multistep-SP iteration; Banach space; Stability; Convergence
It is known that the operators satisfying (1.6) are generalizations of Kannan maps (Kannan, 1969) and Chatterjea maps (Chatterjea, 1972). Zamfirescu (1972) proved that the Zamfirescu operator has a unique fixed point which can be approximated by Picard iteration (1.1). Berinde (2004) showed that Ishikawa iteration can be used to approximate the fixed point of a Zamfirescu operator when X is a Banach space while it was shown by Olareru (2006) that if X is generalised to a complete metrizable locally convex space (which includes Banach spaces), the Mann iteration can be used to approximate the fixed point of a Zamfirescu operator. Several researchers have studied the convergence rate of these iterations with respect to the Zamfirescu operators. For example, it has been shown that the Picard iteration (1.1) converges faster than the Mann iteration (1.2) when dealing with the Zamfirescu operators. For example, see Popescu (2007). It is still a subject of research as to conditions under which the Mann iteration will converge faster than the Ishikawa or vice-versa when dealing with the Zamfirescu operators.
Jungck was the first to introduce an iteration scheme, which is now called Jungck iteration scheme (Jungck, 1976) to approximate the common fixed points of what is now called Jungck contraction maps. Singh et al. (2005) introduced the Jungck-Mann iteration procedure and discussed it’s stability for a pair of contractive maps. Olatinwo and Imoru (2008), Olatinwo (2008) built on that work to introduce the Jungck-Ishikawa and Jungck-Noor iteration schemes and used their convergences to approximate the coincidence points (not common fixed points) of some pairs of generalized contractive-like operators with the assumption that one of each of the pairs of maps is injective. However, a coincidence point for a pair of quasicontractive maps need not be a common fixed point. In 2010, Olaleru & Akewe (2010) introduced the Jungck-multistep iteration and show that its convergence can be used to approximate the common fixed points of thosepairs of contractive-like operators without assuming the injectivity of any of the operators. Hence the iterative sequence considered in Olaleru and Akewe (2010) is a generalization of the those used in Olatinwo and Imoru (2008) and Olatinwo (2008). The fact that the injectivity of any of the maps is not assumed in Olaleru and Akewe (2010) and the common fixed points of those maps are approximated and not just the coincidence points make the corollary of the results in Olaleru & Akewe (2010) an improvement of the results of Olatinwo (2008), Olatinwo and Imoru (2008). Consequently, a lot of results dealing with convergence of Picard, Mann, Ishikawa and multistep iterations for single quasicontractive operators on Banach spaces were generalized. Several stability results are proved in literature, some of the authors whose stability results are of paramount importance in fixed point iterative processes are: Bhagwati & Ritu (2011); Chugh & Kumar (2011); Olatinwo (2008); Osilike (1995); Singh et al. (2005).
Theorem 4.1 yields the following corollaries:
Corollary 4.2. Let X be a Banach space and for an arbitrary set Y such that (2.14) holds and . For any and let be the Jungck-SP iterative process defined by (2.7) converging to p (that is ) with and all n. Then the Jungck-SP iterative process defined by (2.7) is (S,T)-stable.
Corollary 4.3. Let X be a Banach space and for an arbitrary set Y such that (2.14) holds and . For any and let be the Jungck-Mann iterative process defined by (2.3) converging to p (that is ) with and all n. Then the Jungck-Mann iterative process defined by (2.3) is (S,T)-stable.
Remark 4.4. Weaker versions of Theorem 4.1 are the the stability results in Chugh and Kumar (2011) where S is assumed injective and the stability result is not to the common fixed point but to the coincidence point of S,T. Furthermore, the Jungck-multistep-SP iteration used in Theorem 4.1 is more general than the Jungck-SP used in Chugh and Kumar (2011).
Acknowledgment
The author is thankful to Prof. J. O. Olaleru for his useful comments/suggestions leading to the improvement of this paper and for supervising his Ph.D Thesis.
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