A pseudoparabolic type quasilinear integro-differential equation with degenerate kernel and integral condition

Nonlocal mixed-value problem for a quasinlinear pseudoparabolic type integro-differential equation with degenerate kernel and reflective first argument is considered. The questions of one-value solvability of such equations are examined. For equations of the third order the method of degenerate kernel is developed. Fourier method of variables’ separation is used. After some denoting the examined equation is reduced to a system of countable algebraic equations’ systems with complex right-hand side. After simple transformation countable system of nonlinear integral equations is obtained. One-value solvability of this system is proved by method of successive approximations combined with method of contractive mappings.