We study the boundedness of the Hausdorff operator in various types of Lebesgue spaces, e.g. weighted spaces, variable exponent and grand Lebesgue spaces. The results are illustrated by a number of examples. Mathematics subject classification (2010): 28C99, 46E30, 47G10.

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In this paper, we investigate the semi-Markovian random walk processes with jumps and delaying screen in zero. The Laplace transform on time, Laplace-Stieltjes transform on phase of the conditional distribution of semi-Markovian random walk processes with jumps is found. We get a mathematical modeling of the semi-Markov random walk processes with a delaying screen in zero, given in general form by means of integral equation. In this paper, the residence time of the system is given by the gamma distribution with the parameters and resulting in a fractional order integral equation. The purpose of this paper is to reduce the fractional order integral equation to a fractional order differential equation. Finally, we find the exact solution of fractional order differential equation. KEYWORDS fractional order differential equation, gamma distribution, Laplace transform, semi-Markovian random walk process Math Meth Appl Sci. 2018;41:9301-9311.wileyonlinelibrary.com/journal/mma

In this paper a compactness criterion in weighted variable Lebesgue spaces is proved. In particular, are proved a compactness criterion in variable exponent sequence Lebesgue spaces.

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