Inverse problems of determining an unknown depending on time coefficient for a parabolic equation with involution and periodicity conditions

The solution of the investigated problem with an unknown coefficient in the equation was constructed by using the method of separation of variables. The properties of the induced spectral problem for the second-order differential equation with involution are studied. The dependence on the equation involutive part of the spectrum and its multiplicity as well as the structure of the system of root functions and partial solutions of the problem were investigated. The conditions for the existence and uniqueness of the solution of the inverse problem have been established. To determine the required coefficient, Volterra's integral equation of the second kind was found and solved.