Periodic solution of generalized Abel integral equation of the first kind

Background. The problem of existence of periodic solutions of Volterra integral equations has not been sufficiently studied even in the linear case. In the academic literature little attention is paid to this problem. Therefore, there is a need to create a methodology for investigating the existence of periodic solutions of exactly integral equations that considers the specificity of these mathematical objects. In the article this problem is solved for the class of generalized Abel integral equations of the first kind. At the present time these equations have remained relevant as objects of the research. Firstly, they have many important applications. Secondly, the research of Abel integral equations greatly contributed to the emergence and development of a whole mathematical direction such as fractional calculus, which is very popular in Russia and abroad. Materials and methods. Methods of mathematical analysis, functional analysis, and the theory of differential and integral equations are used to solve the problems posed in the paper. Results. The criterion of the existence and uniqueness of a continuous periodic solution of the generalized Abel integral equation of the first kind is proved. The cases of natural and positive real exponents of the kernel are considered. The formulas of periodic solutions are obtained, and their main periods are found. Conclusions. The theorems formulated in the article characterize the image of the class of continuous periodic functions under a linear map, given by the Riemann - Liouville operator. These theorems can be useful for research in the field of fractional integro-differentiation. Abel integral equations with the natural and real exponents of the kernels are in obvious relation, namely, the first equation is a particular case of the second equation, but the proved criterions of the periodicity of their solutions are not in such relation. This is due to the fact that the nonlocal Riemann - Liouville operators of the natural and fractional orders of integration differ from each other by the properties. The first operator has a differential operator as an inverse, which is a local operator. The locality of differential operators and the nonlocality of Volterra integral operators also explain the differences in the research of the problem of the existence of periodic solutions of appropriate equations.