In the article, the problems of unique solvability and determination of the redefinition coefficient function in the initial inverse problem for a nonlinear Whitham type partial differential equation with impulse effects are studied. The modified method of characteristics allows partial differential equations of the first order to be represented as ordinary differential equations that describe the change of unknown function along the line of characteristics. The unique solvability of the initial inverse problem is proved by the method of successive approximations and contraction mappings. The determination of the unknown coefficient is reduced to solving the nonlinear integral equation. KEYWORDS inverse problem, Whitham type equations, determination of the coefficient function, method of successive approximations, unique solvability.

An inverse problem for a second order system of ordinary integro-differential equations with impulsive effects, mixed maxima and two redefinition vectors is investigated. A system of nonlinear functional integral equations is obtained by applying some transformations. The existence and uniqueness of the solution of the nonlinear inverse problem is reduced to the unique solvability of the system of nonlinear functional integral equations in Banach space P C ([0, T ], R n ). The method of successive approximations in combination with the method of compressing mapping is used in the proof of unique solvability of the nonlinear functional integral equations. Then values of redefinition vectors are founded. KEYWORDS inverse problem, second order system, impulsive integro-differential equations, two-point nonlinear boundary value conditions, two redefinition vectors, mixed maxima, existence and uniqueness of solution. FOR CITATION Yuldashev T.K., Fayziyev A.K. Inverse problem for a second order impulsive system of integrodifferential equations with two redefinition vectors and mixed maxima.