Differential and Integral Equations

Errata Corrige to: "Approximation by means of nonlinear integral operators in the space of functions with bounded $\varphi-$variation"

Laura Angeloni and Gianluca Vinti

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Here the authors want to point out that the term $I_2,$ of Proposition 2 of the original paper ([2]), has to be estimated in a different way. Moreover, now the proof of Theorem 4 of the original paper holds with the new assumption (2) (involving $K_w.3)'$ mentioned below), instead of (6.2) of [2] (involving $K_w.3)$ of the original paper), while, since Theorem 3 (convergence theorem) can be proved with both assumptions $K_w.3)$ and $K_w.3)'$, we prefer here to use directly $K_w.3)'$, in analogy with condition (2). Let us notice that it is easy to see that the two conditions $K_w.3)$ and $K_w.3)'$ cannot be compared. Here we want also to point out that in the convergence theorem of [3] as well as in Lemma 2 of [3], a similar problem occurs and it is solved in the same way proving that $V_{\varphi}[\lambda (H_w \circ f-f)]{\mbox{$\rightarrow$}} 0$, as $w{\mbox{$\rightarrow$}} +\infty$ for sufficiently small $\lambda>0,$ using assumption $K_w.3)$ (see Remark below).

Article information

Differential Integral Equations, Volume 23, Number 7/8 (2010), 795-799.

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A45: Functions of bounded variation, generalizations 26A46: Absolutely continuous functions 41A25: Rate of convergence, degree of approximation 41A35: Approximation by operators (in particular, by integral operators) 47G10: Integral operators [See also 45P05]


Angeloni, Laura; Vinti, Gianluca. Errata Corrige to: "Approximation by means of nonlinear integral operators in the space of functions with bounded $\varphi-$variation". Differential Integral Equations 23 (2010), no. 7/8, 795--799. https://projecteuclid.org/euclid.die/1356019196

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