On the $(u,p)$ problem in the theory of elasticity

A problem of the theory of elasticity is considered for a body with vectors of displacements $u$ and loads $p$ simultaneously defined on one part of the body and with undefined conditions on the remaining part of the body. For a doubly connected domain, where the vectors $u$ and $p$ are set on one of its boundaries (inner or outer), an iterative method based on reduction of the initial problem to a sequence of mixed problems is justified.