Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators
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In the class of quasi-contractive operators satisfying Zamfirescu’s conditions, the most used fixed point iterative methods, that is, the Picard, Mann, and Ishikawa iterations, are all known to be convergent to the unique fixed point. In this paper, the comparison of the first two methods with respect to their convergence rate is obtained. 1. Introduction In the last three decades many papers have been published on the iterative approximation of fixed points for certain classes of operators, using the Mann and Ishikawa iteration methods, see [4], for a recent survey. These papers were motivated by the fact that, under weaker contractive type conditions, the Picard iteration (or the method of successive approximations), need not converge to the fixed point of the operator in question. However, there exist large classes of operators, as for example that of quasi-contractive type operators introduced in [4, 7, 10, 11], for which not only the Picard iteration, but also the Mann and Ishikawa iterations can be used to approximate the fixed points. In such situations, it is of theoretical and practical importance to compare these methods in order to establish, if possible, which one converges faster. As far as we know, there are only a few papers devoted to this very important numerical problem: the one due to Rhoades [11], in which the Mann and Ishikawa iterations are compared for the class of continuous and nondecreasing functions f : [0,1] → [0,1], and also the author’s papers [1, 3, 5], concerning the Picard and Krasnoselskij iterative procedures in the class of Lipschitzian and generalized pseudocontractive operators. An empirical comparison of Newton, Mann, and Ishikawa iterations over two families of decreasing functions was also reported in [13]. In [4] some conclusions of an empirical numerical study of Krasnoselskij, Mann, and Ishikawa iterations for some Lipschitz strongly pseudocontractive mappings, for which the Picard iteration does not converge, were also presented.
Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:2 (2004) 97–105 2000 Mathematics Subject Classification: 47H10, 54H25 URL: http://dx.doi.org/10.1155/S1687182004311058
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Picard iteration converges faster than Mann iteration
It is the main purpose of this paper to compare the Picard and Mann iterations over a class of quasi-contractive mappings, that is, the ones satisfying the Zamfirescu’s conditions [15]. Theorem 3.1 in the present paper shows that for the aforementioned class of operators, considered in uniformly convex Banach spaces, the Picard iteration always converges faster than the Mann iterative procedure. Moreover, Theorem 3.3 extends this result to arbitrary Banach spaces and also to Mann iterations defined by weaker assumptions on the sequence {αn }. 2. Some fixed point iteration procedures Let E be a normed linear space and T : E → E a given operator. Let x0 ∈ E be arbitrary and {αn } ⊂ [0,1] a sequence of real numbers. The sequence {xn }∞ n=0 ⊂ E defined by
xn+1 = 1 − αn xn + αn T
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