On the existence of solutions of one nonlinear boundary-value problem for shallow shells of Timoshenko type with simply supported edges

Solvability of one system of nonlinear second order partial differential equations with given initial conditions is considered in an arbitrary field. Reduction of the initial system of equations to one nonlinear operator equation is used to study the problem. The solvability is established with the use of the principle of contracting mappings. The method used in these studies is based on the integral representations for the displacements. These representations are constructed with the use of general solutions to the inhomogeneous Cauchy-Riemann equation.


Introduction
Let us introduce in the plane bounded domain Ω and consider a system of nonlinear differential equations in the form , , 2 1 2 2 2 1 The system (1) together with the boundary conditions (2)-(4) describes the state of equilibrium isotropic elastic homogeneous shell with simply supported edges within the framework of Timoshenko shear model [1, pp. 168-170, 269]. Here are stresses ( , = 1,3 ̅̅̅̅ ); ( = 1,2) and 3 are tangential and normal displacements of the points of 0 ; ( = 1,2) are rotation angles of normal cross-section of 0 ; are components of the external forces acting on the shell; = is the Poisson coefficient, = is Young's modulus, 1 , 2 = are principal curvatures; 2 = is the shear coefficient; ℎ = is the shell width; 1 , 2 are the Cartesian coordinates of the points in the domain Ω.
There are a number of works devoted to the solvability of nonlinear problems in the framework of the Timoshenko displacement model [2][3][4][5][6][7]. For this purpose the theory of problems Rimann-Hilbert for holomorphic functions in the unit circle is used. Therefore, the field assumed from the beginning the unit circle [2][3][4][5][6], or conformal mappings on the unit circle [7]. At the present time, on the unit circle existence theorems of solutions of nonlinear problems for Timoshenko-type shell with rigidly clamped edges [2], with free edges [3] and with simply supported edges [4][5][6] are obtained. In [7] the system (1) is studied for shells of Timoshenko type with free edges in an arbitrary field Ω. The method of works [3], [4], [7] is developing on the case of arbitrary elastic shell with simply supported edges in this paper.
Consider boundary-value problem (1)-(4) in a generalized formulation. Let the following conditions hold true: (a) Ω is a simply connected domain with the boundary in what follows is a generalized solution to the problem (1)-(4) if the vector satisfies almost everywhere the equations of system (1) and it satisfies boundary conditions (2)-(4) in pointwise fashion.
is a Sobolev space. Let us note that due to embedding theorems for Sobolev spaces

Solution to problem (1)-(4) with respect to tangential displacements and angles of rotation
Let us consider the first two equations in (1) and initially assume that 3 w is fixed. In terms of the complex function   these equations can be represented in the form (6) is an inhomogeneous Cauchy-Riemann equation. It has general solution [8]: is an arbitrary holomorphic function that belongs to the space where the integral exists in the principal value sense of Cauchy (almost everywhere when Sf is a linear bounded operator in (7) can be also rewritten in the form of an inhomogeneous Cauchy-Riemann equation is an arbitrary holomorphic function of the class  .
We differentiate relation (13) with respect to z , we find We substitute relations for the tangential displacements we have a Rimann-Hilbert problem with the boundary condition (15) in an arbitrary field Ω. Using conformal mapping, we have a Rimann-Hilbert problem for the holomorphic We substitute expression (23) into (13) to obtain Consider tangential displacements 1 w and 2 w that satisfy the first two equations (1) and conditions (2), (3). Upon substituting (23), (25) into (10) and assuming that condition (21) is true, we obtain (1). These functions should satisfy boundary conditions (2), (4).
Let us note that the structure of left-hand sides in the last two equations (1) coincides with the structure of left-hand sides in boundary conditions (2) and (4). Relations for tangential displacements differ only in the right-hand sides. Therefore at fixed right-hand sides for rotation angles we obtain (5)