Mapping Properties of Mixed Fractional Differentiation Operators in Hölder Spaces Defined by Usual Hölder Condition

Abstract

We study mixed fractional derivative in Marchaud form of function of two variables in Hölder spaces of different orders in each variables. We consider Hölder spaces defined both by first order differences in each variable and also by the mixed second order difference, the main interest being in the evaluation of the latter for the mixed fractional derivative in the cases Hölder class.

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“…≤ − , Which was required to prove. We now state the difference problem corresponding to the problem (3)- (7).…”

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“…Direct extension of the Riemann-Liouville fractional integro-differentiation operations to the case of many variables, when these operators are applied for each variable or some of them, gives the so-called partial and mixed fractional integrals and derivatives. They are known [1], as well as [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. Thus, in [2], using the two-dimensional Laplace transform, a solution of the two-dimensional Abel integral equation was obtained.…”

“…Верхний створ для каждой реки располагался в предгорно-низкогорной зоне, а замыкающий в равнинной части республики. Содержание NO2 -, NO3и NH4 + в водах рек определяли методами химического анализа [11][12][13].…”