The problem of determining the stress-strain state rigidly fixed sandwich plate with transversely soft core in the presence of constraints (i.e., material nonlinearity) corresponding to the ideal elastic-plastic model for the core material is considered. Solvability of the generalized statement of the problem as a problem of finding the saddle point of some functionals is investigated.

The convergence of iterative methods for solving variational inequalities with monotonetype operators in Banach spaces is studied. Such inequalities arise in the description of deformation processes of soft rotational network shells. Certain properties of these operators, such as coercivity, potentiality, bounded Lipschitz continuity, pseudomonotonicity, and inverse strong monotonicity, are determined. An iterative method for solving these variational inequalities is proposed, its convergence is investigated, and the boundedness of the iterative sequence is proved. Moreover, it is proved that any weakly convergent subsequence of the iterative sequence converges to a solution of the original variational inequality.

By using the two-layer iterative method we obtain the basic characteristics of the equilibrium position of sandwich plate with a transversely soft filler in geometrical nonlinear one-dimensional statement. To solving problem we previously construct its finite-difference approximation Analysis of the results of numerical experiments is performed. Obtained data testify to the effectiveness of the proposed method.

We consider a spatial equilibrium problem of a soft network shell in the presence of several external point loads concentrated at some pairwise distinct points. A generalized statement of the problem is formulated in the form of integral identity. Then we introduce an auxiliary problem with the right-hand side given by the delta function. For the auxiliary problem we are able to find the solution in an explicit form. Due to this, the generalized statement of the problem under consideration is reduced to finding the solution of the operator equation. We establish the properties of the operator of this equation (boundedness, continuity, monotonicity, and coercitivity), which makes it possible to apply known general results from the theory of monotone operatorsfor the proof of the existence theorem. It is proved that the set of solutions of the generalized problem is non-empty, convex, and closed.

We construct the finite-dimensional approximations for mixed variational inequalities with pseudomonotone operators and convex non-differentiable functionals in Banach spaces. Such variational inequalities arise in the mathematical description of the processes of an established filtration and in the problems of determining the equilibrium of soft shells. The convergence of these approximations are investigated.