Measures of noncompactness and superposition operator in the space of regulated functions on an unbounded interval

Abstract

In this paper, we formulate necessary and sufficient conditions for relative compactness in the space $$BG({\mathbb {R}}_+,E)$$ B G ( R + , E ) of regulated and bounded functions defined on $${\mathbb R}_+$$ R + with values in the Banach space E. Moreover, we construct four new measures of noncompactness in the space $$BG({\mathbb {R}}_+,E)$$ B G ( 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 $$BG({\mathbb {R}}_+,E)$$ B G ( R + , E ) into $$BG({\mathbb {R}}_+,E)$$ B G ( R + , E ) and, additionally, be compact.

Existence and Attractivity Results for Fractional Differential Inclusions via Nocompactness Measures

“…For measures of nocompactness in the space of regulated functions see [15]. Consider the following inclusion…”

Solvability for two dimensional functional integral equations via Petryshyn’s fixed point theorem

“…MNC allow us to obtain the existence of solutions of FIEs. A nice MNC to prove that certain map has a fixed point in a set is provided in [17]. They are often applied to the theories of Hadamard-type fractional integral operators and they find relations of the obtained results with earlier results about integral operators on different function spaces as well as the operator theory and geometry of Banach spaces (see [9,23,24,[30][31][32]).…”

Existence and Uniqueness of Solutions for a Third‐Order Three‐Point Boundary Value Problem via Measure of Noncompactness

“…A new direction of research with the presentation of the notion of a measure of noncompactness was opened up in 1930 by Kuratowski, which can be applied to prove the existence results related to various integral, differential equations, integro-differential equations as well as their systems. As an important application of this measure, Darbo's fixed point theorem generalizes the Schauder-fixed point theorem and Banach contraction principle, especially in proving the existence of solutions for classes of nonlinear equations (see [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]).…”