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Boundary value problem for differential equation with fractional order derivatives with different origins. (Russian. English summary) Zbl 1413.34017
Summary: We study a spectral problem for an ordinary differential equation with composition of fractional order differentiation operators in Riemann-Liouville and Caputo senses with different origins. We prove that for the problem under study there exist infinite sequences of eigenvalues and eigenfunctions. All of the eigenvalues are real and positive, and the eigenfunctions form an orthogonal basis in \(L_{2}\left(0,1\right)\).

34A08 Fractional ordinary differential equations
34L05 General spectral theory of ordinary differential operators
34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34B08 Parameter dependent boundary value problems for ordinary differential equations
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