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Some convergence results for nonlinear singular integral operators

Karsli H.

Demonstratio Mathematica

2013

Vol. 46, nr 4

729--740

In this paper, we establish some pointwise convergence results for a family of certain nonlinear singular integral operators Tλf of the form (...), acting on functions with bounded (Jordan) variation on an interval [a, b] as λ→λ0. Here, the kernels (...) satisfy some suitable singularity assumptions. We remark that the present study is a continuation and extension of the study of pointwise approximation of the family of nonlinear singular integral operators (1) begun in [18].

Invariance and periodicity

Kupka I.

Demonstratio Mathematica

2007

Vol. 40, nr 2

457-464

In this article is presented a simple concept of invariance. It is used to widely generalise the concept of periodicity.

Graph convergence of set-valued maps and its relationship to other convergences

Del Prete I., Di Iorio M., Holá L.

Journal of Applied Analysis

2000

Vol. 6, nr 2

213-226

The notion of even-outer-semicontinuity for set-valued maps is introduced and compared with related ones from [4] and [11]. The coincidence of these notions provides a new characterization of compactness and of local compactness. The following result is proved: Let X be a topological space, Y a uniform space, {Fσ : σ ∈ ∑} be a net of set-valued maps from X to Y and F be a set valued map from X to Y. Then any two of the following conditions imply the third: (1) the net {Fσ : σ ∈ ∑} is evenly-outer semicontinuous; (2) the net {{Fσ : σ ∈ ∑} is graph convergent to F; (3) the net {Fσ : σ ∈ ∑} is pointwise convergent to F. This theorem generalizes some results from [4] and [11].