MSC: 26A18 47H10 54C05Keywords: Continuous functions Convergence theorem Fixed point Nondecreasing functions Rate of convergence a b s t r a c tIn this paper, we propose a new iteration, called the SP-iteration, for approximating a fixed point of continuous functions on an arbitrary interval. Then, a necessary and sufficient condition for the convergence of the SP-iteration of continuous functions on an arbitrary interval is given. We also compare the convergence speed of Mann, Ishikawa, Noor and SP-iterations. It is proved that the SP-iteration is equivalent to and converges faster than the others. Our results extend and improve the corresponding results of Borwein and Borwein [D. Borwein, J. Borwein, Fixed point iterations for real functions, J. Math. Anal. Appl. 157 (1991) 112-126], Qing and Qihou [Y. Qing, L. Qihou, The necessary and sufficient condition for the convergence of Ishikawa iteration on an arbitrary interval, J. Math. Anal. Appl. 323 (2006) 1383-1386], Rhoades [B.E. Rhoades, Comments on two fixed point iteration methods, J. Math. Anal. Appl. 56 (1976) 741-750], and many others. Moreover, we also present numerical examples for the SP-iteration to compare with the Mann, Ishikawa and Noor iterations.

In this paper, weak and strong convergence theorems are established for a three-step iterative scheme for asymptotically nonexpansive mappings in Banach spaces. Mann-type and Ishikawa -type iterations are included by the new iterative scheme. The results obtained in this paper extend and improve the recent ones announced by Xu and Noor, Glowinski and Le Tallec, Noor, Ishikawa, and many others.  2005 Elsevier Inc. All rights reserved.

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Necessary and sufficient conditions under which the Cesàro-Orlicz sequence space ces φ is nontrivial are presented. It is proved that for the Luxemburg norm, Cesàro-Orlicz spaces ces φ have the Fatou property. Consequently, the spaces are complete. It is also proved that the subspace of order continuous elements in ces φ can be defined in two ways. Finally, criteria for strict monotonicity, uniform monotonicity and rotundity (= strict convexity) of the spaces ces φ are given.

The classical James constant and the nth James constants, which are measure of B-convexity for the Cesàro sequence spaces ces p and the Cesàro-Orlicz sequence spaces ces M , are calculated. These investigations show that ces p , ces M are not uniformly non-square and even they are not B-convex. Therefore the classical Cesàro sequence spaces ces p are natural examples of reflexive spaces which are not B-convex.Moreover, the James constant for the two-dimensional Cesàro space ces(2) 2 is calculated.

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In this work, we propose a new version of inertial relaxed CQ algorithms for solving the split feasibility problems in the frameworks of Hilbert spaces. We then prove its strong convergence by using a viscosity approximation method under some weakened assumptions. To be more precisely, the computation on the norm of operators and the metric projections is relaxed. Finally, we provide numerical experiments to illustrate the convergence behavior and to show the effectiveness of the sequences constructed by the inertial technique.

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