On the Solvability of Nonlinear Equilibrium Problems for Shallow Anisotropic Timoshenko-type Shells in Curvilinear Coordinates

We prove the existence of solutions to the boundary value problem for a system of five nonlinear second order partial differential equations, with given nonlinear boundary conditions, describing the equilibrium state of elastic shallow inhomogeneous anisotropic shells with free edges within the framework of Timoshenko’s shear model, referred to arbitrary curvilinear coordinates. The boundary value problem is reduced to a nonlinear operator equation for generalized displacements in the Sobolev space, the solvability of which is established using the contraction mapping principle.