On the existence of solutions to boundary value problems for nonlinear equilibrium equations of shallow anisotropic shells of Timoshenko type in Sobolev space

The existence of solutions to the boundary value problem for a system of five nonlinear partial differential equations of the second order under given nonlinear boundary conditions is proved, which describes the equilibrium state of elastic shallow inhomogeneous anisotropic shells with unfixed edges in the framework of the Timoshenko shear model. The boundary value problem is reduced to a nonlinear operator equation in Sobolev space, the decidability of which is established using the principle of contracted mappings.