Nonlocal boundary value problem for an equation with fractional derivatives with different origins

We consider a linear ordinary differential equation of fractional order with a composition of left and right-sided fractional derivative operators in the principal part. Equations containing a composition of fractional order differentiation operators with different origins appear when modeling various physical and geophysical phenomena. Their appearance is caused by the use of the concept of the effective rate of change in the parameters of the simulated processes. In particular, equations of the type considered in this work arise when describing dissipative oscillatory systems. Fractional differentiation is understood in the sense of Riemann-Liouville and Gerasimov-Caputo. For the equation under study, a nonlocal boundary value problem is investigated. The nonlocal boundary condition is specified in the form of an integral operator of the desired solution. Under a certain condition on the kernel of the operator appearing in the nonlocal condition, the problem under consideration is equivalently reduced to the Fredholm integral equation of the second kind. Sufficient conditions for the unique solvability of the problem under study are found, including an integral constraint on the variable potential. As a corollary, the Lyapunov inequality for solutions to the nonlocal problem under consideration is obtained. It is shown that the condition on the kernel of the integral operator from the nonlocal condition that arises in the solution of the problem is necessary in the sense that if this condition is violated, the uniqueness of the solution to the problem is lost.