Mathematics & Physics 2016, 9(2), 131–143 УДК 517.958 Solvability of One Nonlinear Boundary-value Problem for a System of Differential Equations of the Theory of Shallow Timoshenko-type Shells Marat G.Ahmadiev Samat N. Timergaliev∗ Lilya S. Kharasova† Naberezhnye Chelny Institute, Kazan Federal University Suumbike, 10A, Naberezhnye Chelny, 423812 Russia Received 05.09.2015, received in revised form 11.01.2016, accepted 19.02.2016 Solvability of a system of nonlinear second order partial differential equations with given initial conditions is considered in the paper. <...> Reduction of the initial system of equations to one nonlinear operator equation is used to study the problem. <...> Keywords: system of nonlinear differential equations, equilibrium equations, integral representations, existence theorem. <...> The system (1) together with the boundary conditions (2)–(5) describes the state of equilibrium of shallow isotropic elastic homogeneous shell with simply supported edges within the framework of Timoshenko shear model [1]. <...> Here Tij are stresses, Mij are moments; γk ij(i, j = 1, 3, k = 0, 1) are components of deformation of the shell middle surface S0 that is homeomorphic to Ω; wi(i = 1, 2) and w3 are tangential and normal displacements of the points of S0; ψi(i = 1, 2) are rotation angles of normal cross-section of S0; a is the vector of generalized displacements; Rj(j = 1, 3), Lk(k = 1, 2),N2,P2, P3 are components of the external forces acting on the shell; µ = const is the Poisson coefficient, E = const is Young‘s modulus, k1, k2 = const are principal curvatures; k2 = const is the shear coefficient; h = const is the shell width; α1,α2 are the Cartesian coordinates of the points in the domain Ω. <...> It is linear with respect to tangential displacements w1,w2, rotation angles ψ1,ψ2 and it is a nonlinear system with respect to deflection w3. <...> Solvability of the system of nonlinear differential equations that describes shell equilibrium in the framework of the Kirchhoff-Love model has been well studied [2–5]. <...> The questions of the existence of solutions of nonlinear problems <...>