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The paper is devoted to the theory of traces in the spaces and to the inhomogeneous Dirichlet problem for elliptic equations of infinite order. A trace criterion is established, simple sufficient trace conditions are adduced, and the correctness of the indicated Dirichlet problem is proved on the basis of the results obtained.Bibliography: 6 titles.

In this paper we obtain necessary and sufficient conditions for nontriviality of the spaces in the cases where the functions are defined on or are periodic with period with respect to variables.Bibliography: 5 titles.

For a sequence of Banach spaces , a concept of limit is introduced that is a natural generalization of the concept of the limit of a monotonically decreasing numerical sequence. Necessary and sufficient conditions are obtained for an imbedding and for a compact imbedding. Applications are given to the Sobolev spaces of infinite order . Necessary and sufficient conditions bearing an algebraic character are established for the imbedding . Sufficient algebraic imbedding conditions are obtained for the spaces for any .Bibliography: 8 titles.

In the study of the Cauchy-Dirichlet problem (1)(2)infinite order Sobolev spaces naturally arise, where and are numerical sequences. In this paper criteria for the nontriviality of are established and the problem (1), (2) is investigated. Further, a theorem is obtained on the existence of the limit (as ) of solutions of nonlinear 2th order boundary value problems of elliptic and hyperbolic type, from which, in particular, follows the solvability of the mixed problem for the nonlinear hyperbolic equation , , where 0$ SRC=http://ej.iop.org/images/0025-5734/27/2/A01/tex_sm_2506_img11.gif/> is arbitrary.Bibliography: 9 titles.

On the half-line we investigate the following equation in a Banach space: (1)where are closed operators which commute with . We consider the following classes of equations: parabolic, inverse parabolic, hyperbolic, quasi-elliptic, and quasi-hyperbolic. We present boundary value problems for these classes and prove that they are well-posed. The proofs are based on a solvability theorem for the operator equation , where is a closed operator.Bibliography: 20 items.

In this paper nonlinear equations A(u) = h are studied, where the operator does not necessarily satisfy the well-known condition of coerciveness. With the equation A(u) = h, which is in general not solvable for an arbitrary right side h, we associate a certain equation of the form B*A(u) = h, which is always solvable. Then the original equation A(u) = h is solvable up to Ker B*. This gives a description of the domain of values of the original operator A(u). Bibliography: 9 titles.