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

The aim of the paper is to introduce the Banach space consisting of real functions defined on a locally compact and countable at infinity metric space and having increments tempered by a modulus of continuity. We are going to provide a condition that is sufficient for the relative compactness in the Banach space in question. A few particular cases of that Banach space will be discussed.

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

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.

On the Existence of the Solutions of a Fredholm Integral Equation with a Modified Argument in Hölder Spaces

This article concerns the entity of solutions of a quadratic integral equation of the Fredholm type with an altered argument, x ( t ) = p ( t ) + x ( t ) ∫ 0 1 k ( t , τ ) ( T x ) ( τ ) d τ , where p , k are given functions, T is the given operator satisfying conditions specified later and x is an unknown function. Through the classical Schauder fixed point theorem and a new conclusion about the relative compactness in Hölder spaces, we obtain the existence of solutions under certain assumptions. Our work is more general than the previous works in the Conclusion section. At the end, we introduce several tangible examples where our entity result can be adopted.

On a measure of noncompactness in the space of regulated functions and its applications

In this paper we formulate a criterion for relative compactness in the space of functions regulated on a bounded and closed interval. We prove that the mentioned criterion is equivalent to a known criterion obtained earlier by D. Fraňkova, but it turns out to be very convenient in applications. Among others, it creates the basis to construct a regular measure of noncompactness in the space of regulated functions. We show the applicability of the constructed measure of noncompactness in proving the existence of solutions of a quadratic Hammerstein integral equation in the space of regulated functions.

Experimental Study of Effects of Operation Parameters on Efficiency of Vibratory Pile Driving

Vibratory pile driving is a common application method of piles foundation construction. A test bed was designed and built according to the principle of vibratory pile driving. The test bed is composed of excitation system, machinery parts and signal acquisition system. At first soil samples with different relative compactness and different saturation were made. Then tests of vibratory pile driving have been done by using the soil samples and the test bed. For each soil sample, time of pile sinking to the bottom of the container was recorded respectively under the condition of the exciting force with same amplitude. The time represents efficiency of vibratory pile driving. The former is shorter and the latter is higher. Relationships between efficiency of vibratory pile driving and exciting frequency, relative compactness of soil, saturation of soil were researched respectively by single factor method. The experimental results have guide meaning to vibratory pile driving construction.

On the Space of Functions with Growths Tempered by a Modulus of Continuity and Its Applications

We are going to study the space of real functions defined on a bounded metric space and having growths tempered by a modulus of continuity. We prove also a sufficient condition for the relative compactness in the mentioned function space. Using that condition and the classical Schauder fixed point theorem, we show the existence theorem for some quadratic integral equations of Fredholm type in the space of functions satisfying the Hölder condition. An example illustrating the mentioned existence result is also included.