Fixed point iterations for real functions

Borwein, David; Borwein, Jonathan M.

doi:10.1016/0022-247x(91)90139-q

Cited by 45 publications

(50 citation statements)

References 6 publications

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“…Other extensions of Ishikawa's result have been obtained by imposing stronger conditions on T , see for example [3]. In similar spirit, Yuan Qing and Liu Qihou [9] have also extended a result of Borwein and Borwein [1] to noncompact intervals.…”

Section: Preliminaries and Notationmentioning

confidence: 80%

Convergence of Ishikawa iterations on noncompact sets

Mutangadura¹

2014

Quaestiones Mathematicae

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Recall that Ishikawa's theorem [4] provides an iterative procedure that yields a sequence which converges to a fixed point of a Lipschitz pseudocontrative map T : C → C, where C is a compact convex subset of a Hilbert space X. The conditions on T and C, as well as the fact that X has to be a Hilbert space, are clearly very restrictive. Modifications of the Ishikawa's iterative scheme have been suggested to take care of, for example, the case where C is no longer compact or where T is only continuous. The purpose of this paper is to explore those cases where the unmodified Ishikawa iterative procedure still yields a sequence that converges to a fixed point of T , with C no longer compact. We show that, if T has a fixed point, then every Ishikawa iteration sequence converges in norm to a fixed point of T if C is boundedly compact or if the set of fixed points of T is "suitably large". In the process, we also prove a convexity result for the fixed points of continuous pseudocontractions.

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Section: Preliminaries and Notationmentioning

confidence: 80%

Convergence of Ishikawa iterations on noncompact sets

Mutangadura¹

2014

Quaestiones Mathematicae

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“…The purpose of this article is to give a necessary and sufficient condition for the strong convergence of the W extend and improve the corresponding results of Rhoades [5], Borwein and Borwein [6], Qing and Qihou [7], Phuengrattana and Suantai [4], and many others.…”

Section: Introductionmentioning

confidence: 89%

“…There are many articles have been published on the iterative methods using for approximation of fixed points of nonlinear mappings, see for instance [1][2][3][4][5][6][7]. However, there are only a few articles concerning comparison of those iterative methods in order to establish which one converges faster.…”

Section: Rate Of Convergence and Numerical Examplesmentioning

confidence: 99%

“…He also proved the Ishikawa iteration converges faster than the Mann iteration for the class of continuous and nondecreasing functions. Later in 1991, Borwein and Borwein [6] proved the convergence of the Mann iteration of continuous functions on a bounded closed interval. In 2006, Qing and Qihou [7] extended their results to an arbitrary interval and to the Ishikawa iteration and gave some control conditions for the convergence of Ishikawa iteration on an arbitrary interval.…”

Section: Introductionmentioning

confidence: 99%

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