Abstract
Abstract Some inequalities connected to measures of noncompactness in the space of regulated function R(J, E) were proved in the paper. The inequalities are analogous of well known estimations for Hausdorff measure and the space of continuous functions. Moreover two sufficient and necessary conditions that superposition operator (Nemytskii operator) can act from R(J, E) into R(J, E) are presented. Additionally, sufficient and necessary conditions that superposition operator Ff : R(J, E) → R(J, E) was compact are given.
Highlights
When studying solvability of various non-linear equations, it is signi cant to properly choose the space in which the equation is considered
Some inequalities connected to measures of noncompactness in the space of regulated function R(J, E) were proved in the paper
Except stating general properties of this space [1, 5, 7,8,9,10,11,12] it is possible to use formulas for measures of noncompactness, conditions su cient for the superposition operator Ff to act from R(J, E) into R(J, E), and conditions for continuity of this operator [2,3,4,5,6, 12,13]
Summary
When studying solvability of various non-linear equations, it is signi cant to properly choose the space in which the equation is considered. Su cient and necessary conditions that superposition operator is compact in the space of regulated functions will be given (Theorem 4.9). Let ME denote the family of all nonempty and bounded subsets of E and NE its subfamily consisting of all relatively compact sets. For a given nonempty bounded subset X of E, we denote by βE(X) the so-called Hausdor measure of noncompactness of X. This quantity is de ned by formula βE(X) := {r > : X has a nite r−net in E}. We are going to recall the construction of a measure of noncompactness in the space R(J, E) To this end, let us take a set X ∈ MR(J,E). Construction of the measure (2.1) addresses inaccuracies existent in the construction of measures given in [3, 4]
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