On the problem of solvability of nonlinear boundary value problems for arbitrary isotropic shallow shells of the Timoshenko type with free edges

The solvability of a geometrically nonlinear boundary value problem for elastic shallow arbitrary isotropic inhomogeneous shells with free edges in the framework of the S.P. Timoshenko shear model is being investigated. The research method is based on integral representations for generalized displacements containing arbitrary holomorphic functions. Holomorphic functions are found so that the generalized displacements satisfy the given boundary conditions. As a result, the original problem reduces to one nonlinear operator equation for generalized displacements in Sobolev space, the solvability of which is established using the principle of contraction mappings.