Boundary Value Problems for the Classical and Mixed Integrodifferential Equations with Riemann-Liouville Operators

By method of integral equations, unique solvability is proved for the solution of boundary value problems of loaded third-order integrodifferential equations with Riemann-Liouville operators. 1. Introduction The theory of mixed type equations is one of the principal parts of the general theory of partial differential equations. The interest for these kinds of equations arises intensively because of both theoretical and practical uses of their applications. Many mathematical models of applied problems require investigations of this type of equations. The first fundamental results in this direction were obtained in 1920–1930 by Tricomi [1] and Gellerstedt [2]. The works of M. A. Lavrent’ev, A. V. Bitsadze, F. I. Frankl, M. Protter, and C. Morawetz have had a great impact on this theory, where outstanding theoretical results were obtained and pointed out important practical values. Bibliography of the main fundamental results on this direction can be found, among others, in the monographs of Bitsadze [3], Bers [4], Salakhitdinov and Urinov [5], and Nakhushev [6]. In the recent years, in connection with intensive research on problems of optimal control of the agroeconomical system, long-term forecasting, and regulating the level of ground waters and soil moisture, it has become necessary to investigate a new class of equations called “loaded equations.” Such equations were investigated for the first time in works of N. N. Nazarov and N. Kochin. However, they did not use the term “loaded equation.” For the first time, the term has been used in works of Nakhushev [7], where the most general definition of a loaded equation is given and various loaded equations are classified in detail, for example, loaded differential, integral, integrodifferential, functional equations and so forth, and numerous applications are described [6, 8]. Basic questions of the theory of boundary value problems for partial differential equations are the same for the boundary value problems for the loaded differential equations. However, existence of the loaded operator does not always make it possible to apply directly the known theory of boundary value problems. Works of Nakhushev, M. Kh. Shkhankov, A. B. Borodin, V. M. Kaziev, A. Kh. Attaev, C. C. Pomraning, E. W. Larsen, V. A. Eleev, M. T. Dzhenaliev, J. Wiener, B. Islomov and D. M. Kuriazov, K. U. Khubiev, and M. I. Ramazanov et al. are devoted to loaded second-order partial differential equations. It should be noted that boundary value problems for loaded equations of a hyperbolic, parabolic-hyperbolic, and elliptic-hyperbolic