Measures of noncompactness and superposition operator in the space of regulated functions on an unbounded interval
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Measures of noncompactness and superposition operator in the space of regulated functions on an unbounded interval Szymon Dudek1
· Leszek Olszowy1
Received: 13 November 2019 / Accepted: 4 July 2020 © The Author(s) 2020
Abstract In this paper, we formulate necessary and sufficient conditions for relative compactness in the space BG(R+ , E) of regulated and bounded functions defined on R+ with values in the Banach space E. Moreover, we construct four new measures of noncompactness in the space BG(R+ , E). We investigate their properties and we describe relations between these measures. We provide necessary and sufficient conditions so that the superposition operator (Niemytskii) maps BG(R+ , E) into BG(R+ , E) and, additionally, be compact. Keywords Space of regulated functions · Criterion of relative compactness · Measure of noncompactness · Nemytskii operator Mathematics Subject Classification 47H30 · 46E40
1 Introduction The measures of noncompactness and fixed point theorems are often chosen to investigate solvability of the nonlinear equations. Using a suitable function space together with convenient measures of noncompactness we can obtain elegant existence theorems. In recent years, there have appeared a lot of papers concerning the space G(J , E) of regular functions defined on a bounded interval J and with values in the Banach space E [2,4–11,14]. On the other hand, there have not been any papers focused on the space of regular functions on an unbounded interval R+ . Throughout this paper we are going to fill this gap. We will investigate the space BG(R+ , E) of regular and bounded functions defined on R+ and with values in the Banach space E. In Sect. 3 we formulate necessary and sufficient relative compactness conditions in the space BG(R+ , E). In the sequel we construct four new and convenient measures of noncompactness in BG(R+ , E) and investigate their properties.
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Szymon Dudek [email protected] Leszek Olszowy [email protected]
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Faculty of Mathematics and Applied Physics, Rzeszów University of Technology, al. Powsta´nców Warszawy 8, 35-959 Rzeszów, Poland 0123456789().: V,-vol
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S. Dudek, L. Olszowy
In Sect. 4 we present necessary and sufficient conditions for superposition operator so that it maps BG(R+ , E) into BG(R+ , E). Moreover, we formulate necessary and sufficient conditions so that the superposition operator F f : BG(R+ , E) → BG(R+ , E) is compact. Finally, Sect. 5 shows the applicability one of the mentioned measures of noncompactness to the existence result for some nonlinear integral equation.
2 Notation, definitions and auxiliary facts This section is devoted to recalling some facts which will be used in our further investigations. Assume that E is a real Banach space with the norm · and the zero element θ . Denote by B E (x, r ) the closed ball centered at x and with radius r . The symbol B E (r ) stands for the ball B E (θ, r ). We write X , convX , ConvX to denote the closure, convex hull and the convex closure of a set X , respectively. Furt
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