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We consider a mean value iteration for a family of functions, which corresponds to the Mann iteration with lim⁡n→∞ αn≠0. We prove convergence results for this iteration when applied to strongly pseudocontractive or strongly accretive maps.

We show that the Ishikawa iteration and the corresponding Mann iteration are equivalent when applied to ψ-uniformly pseudocontractive or ψ-uniformly accretive maps.

We show that the convergence of Mann iteration is equivalent to the convergence of Ishikawa iteration for various classes of non-Lipschitzian operators.

We prove a "collage'' theorem for a generalized contractive type operators.

We generalize the classical "collage'' theorem, due to Barnsley, to contractive like operators.

We prove a "collage'' theorem for weak contractive maps and we use it for inverse problems.

We prove the equivalence between the \(T\)-stabilities of Krasnoselskij and Ishikawa iterations; a consequence is the equivalence with the \(T\)-stability of Picard-Banach iteration.

We establish the convergence of Krasnoselskij iteration for various classes of non-Lipschitzian operators.

We prove the equivalence between the T-stabilities of the Krasnoselskij and the Mann iterations; a consequence is the equivalence with the T-stability of the Picard-Banach iteration.

The stochastic processes of finite length defined by recurrence relations request additional relations specifying the first terms of the process analogously to the initial conditions for the differential equations. As a general rule, in time series theory one analyzes only stochastic processes of infinite length which need no such initial conditions and their properties are less difficult to be determined. In this paper we compare the properties of the order 1 autoregressive processes of finite and infinite length and we prove that the time series length has an important influence mainly if the serial correlation is significant. These different properties can manifest themselves as transient effects produced when a time series is numerically generated. We show that for an order 1 autoregressive process the transient behavior can be avoided if the first term is a Gaussian random variable with standard deviation equal to that of the theoretical infinite process and not to that of the white noise innovation.

New proofs of existence and uniqueness results for the solution of the Cauchy problem with delay are obtained by use of Mann-Ishikawa iteration.

We prove an inequality which is crucial in the proof of the main result from B.E. Rhoades and ?tefan M. ?oltuz, The Equivalence between the Convergences of Ishikawa and Mann Iterations for an Asymptotically Pseudocontractive Map, J. Math. Anal. Appl., 283 (2003), 681-688.

We shall prove that Krasnoselskij iteration converges if and only if Mann-Ishikawa iteration converges, for certain classes of strongly pseudocontractive mappings.

We show that all \(T\)-stabilities of Mann-Ishikawa iterations and modified Mann-Ishikawa iterations are equivalent.

We study the errors estimation for the implicit Mann iteration..

We show the convergence of an implicit mean value iteration when applied to uniformly pseudocontractive maps. Remarks about other implicit mean value iterations are given.

We show that the Mann and Ishikawa iterations are equivalently used to approximate fixed points of generalized contractions.

We prove that the associated sequence of Mann iteration is decreasing and hence bounded provided that the operator satisfies minimal assumptions. In particular we obtain for a nonexpansive operator that the associated sequence of Ishikawa iteration is decreasing for a nonexpansive operator. Applications to the convergence of Mann iteration are given.

The Mann-Ishikawa iterations and the Mann-Ishikawa iterations with errors are equivalent models for several classes of non-Lipschitzian operators.

We show the equivalence bewteen the convergences of Mann and Ishikawa iterations dealing with various classes of non-Lipschitzian operators.

In this note we show that a result previously obtained by us [An equivalence between the convergences of Ishikawa, Mann and Picard iterations, Math. Commun., 8, pp.~15--22, 2003], holds under weaker assumptions.

In order to prove the convergence of Ishikawa and Mann iterations, the convergence of one type of sequences is needed. Our purpose, in this note, is to give a new proof of the convergence for one of them. We also give generalizations for the sequences.

We show that certain Mann and Ishikawa iteration schemes are equivalent for various classes of functions.

We give an affirmative answer to the following question: are Mann and Ishikawa iterations equivalent under the assumptions that \(\lim_{n\rightarrow\infty }\alpha_{n}\neq0\;\) and \(\lim_{n\rightarrow\infty}\beta_{n}\neq0\)?