On unilaterally supported viscoelastic von Kármán plates with a long memory

We deal with the system consisting of the nonlinear integro-differential variational inequality for the deflection and the nonlinear quasistationary equation for the Airy stress function. The system describes moderately large deflections with an inner obstacle of a thin viscoelastic plate made of a long memory material. The corresponding Volterra type canonical integro-differential variational inequality is solved using a semidiscrete approximation transforming the problem into the sequence of stationary variational inequalities of von Kármán type. The existence of a solution as well as the convergence of a semidiscrete approximation to a solution of the Volterra variational inequality with a nonlinear main part are verified.