About Power Absolute Integrated of the Solution of Volterra System of Linear Second Kind Integral Equations on Half-axis

Sufficient conditions are established such as the sign of the functions of absolute and quadratic integrability on the half-axis of the components of the solution of a linear system of Volterra-type integral equations of the second kind in the new formulation, namely, without the assumption that the components of the free term possess the properties under study. To achieve the intended goals, the matrix method of weight functions, the matrix method of cutting functions, the matrix method of partial cutting of the author, and the method of integral inequalities of Ya. V. Bykov are developed. The results obtained in this paper are generalizing for linear scalar and vector integral equations previously considered by the author. Note that some matrix cutting functions introduced introduce a quadratic trinomial of some expressions from the terms of the kernel and the free term in such a way that the core actually absorbs the bad properties of the free term (external force). Note that growth, nonsmoothness, alternating terms of the terms of the kernel and the free term are cut off. It should also be emphasized that the kernel and the free term of the considered system of equations consist additively of two kernels, the terms of the first kernel are cut off by both arguments, the terms of the second kernel are cut off by the second argument, i.e. in part, it is shown that the conditions imposed on these two cores are different, i.e. do not intersect. The introduction of a certain matrix weight function helps to establish the absolute integrability on the half-axis of the solution component of the system under consideration, and it plays a decisive role. At the end, the simplest illustrative example is given, which shows that the results of the present article are also new for the scalar linear integral equation of the Volterra type of the second kind on the semiaxis.

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