Generalized solutions of quasilinear elliptic differential-difference equations

A Dirichlet problem for a functional-differential equation the operator of which is represented by the product of a quasilinear differential operator and a linear shift operator is considered. The nonlinear operator has differentiable coefficients. A sufficient condition for the strong ellipticity of the differential-difference operator is proposed. For a Dirichlet problem with an operator satisfying the strong ellipticity condition, the existence and uniqueness of a generalized solution is proved. The situation is considered in which the differential-difference operator belongs to the class of pseudomonotone ${(S)}_+$ operators; in this case, a generalized solution of the Dirichlet problem exists. As an example, a nonlocal problem with a Bitsadze–Samarskii boundary condition is considered.