A measure of noncompactness in the space of functions with tempered increments on the half-axis and its applications

The Space of Functions with Tempered Increments on a Locally Compact and Countable at Infinity Metric Space

“…The simple example of such a metric space is the set R + with the natural metric d(t, s) = |t − s|. Such a case was considered in paper [3].…”

“…, m. This implies (cf. Lemma 2.6 in [3]) that there exists a finite ε−net of the set A that consists of functions y 1 , y 2 , . .…”

“…Particularly, we can look both for criteria for relative compactness in the space C ω (R k ) and measures of noncompactness constructed on the basic of such criteria which will be more convenient in applications to the theory of integral equations as well as the theory of partial differential equations considered in the space C ω (R k ). In paper [3], we investigated an integral equation in the space C ω (R k ), where ω(ε) = ε α was the Hölder modulus of continuity. It is worthwhile mentioning that the integral equation considered in [3] has a rather very simple form.…”

“…In paper [3], we investigated an integral equation in the space C ω (R k ), where ω(ε) = ε α was the Hölder modulus of continuity. It is worthwhile mentioning that the integral equation considered in [3] has a rather very simple form. Moreover, the integral equation investigated in paper [5] has also rather simple form.…”

“…Finally, let us pay attention to the fact that the Banach function space described above with the metric space taken as R + finds a lot of applications in the theory of functional integral equations (cf. [1][2][3][4][5], among others). We expect that also the Banach space studied in this paper finds some applications in the theory of functional, differential, and integral equations.…”