SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR FRACTIONAL-ORDER INTEGRO-DIFFERENTIAL EQUATIONS WITH INVOLUTION

This paper considers a boundary value problem for a fractional-order integro-differential equation 0<α<1 with an involutivity transformation. To investigate the solvability of the boundary value problem, we use the parameterization method proposed by Professor D. Dzhumabaev. This is done by introducing a parameter μ = x(0) and changing variables. x (t ) = u (t ) +μ This change of variables formally divides the problem into two parts: a Cauchy problem for a fractional-order integro-differential equation with an involutive transformation 0 < 𝛼𝛼 < 1 and a linear equation with respect to the introduced parameter. By determining the solution to the Cauchy problem for a fractional-order integro-differential equation with an involutive transformation 0 < 𝛼𝛼 < 1 and substituting it into the boundary condition, we obtain a linear equation with respect to the introduced parameter. Assuming that the coefficient of this equation is nonzero, we find a unique solution to the boundary value problem. A relationship is established between the solvability of the problem and the coefficient of the resulting equation.

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