On existence of solutions to geometrically nonlinear problems for shallow shells of the Timoshenko type with free edges

We study the geometrically nonlinear physically linear boundary-value problems solvability for shallow isotropic elastic shells within the framework of S. P. Timoshenko shift model. The method of study consists in reducing of the equilibrium equations reference system to one nonlinear differential equation relative to deflection. In this case the significant role is played by integral representations for the tangential shifts and the angle of rotations, which are reduced with the attraction of the general solutions of the inhomogeneous Cauchy–Riemann equation. The solvability of equation relative to deflection is established with the use of of principle of contraction mappings.