A study of the convergence of a recursive process for solving a stationary problem of the theory of soft shells (Q1915668)

Let \(\overline u= (u_1, u_2)\), \(({d\overline u\over dx})^2= ({du_1\over dx})^2+ ({du_2\over dx})^2\), \(\overline f= (f_1, f_2)\) and \(Q=(\begin{smallmatrix} 0\\ q_2(x)\end{smallmatrix} \begin{smallmatrix} q_1(x)\\ 0\end{smallmatrix})\). The boundary value problem \[ - {d\over dx} \Biggl(g \Biggl( \Biggl|{d\overline u\over dx} \Biggr|^2 {d\overline u\over dx}\Biggr)\Biggr)+ Q(x) {d\overline u\over dx}= \overline f(x),\tag{1} \] \(0< x< 1\), \(\overline u(0)= \overline u(1)= 0\), arises in the stationary problem of the theory of soft shells and the function \(g(\xi^2)\xi\) which characterizes the material of the shell is assumed to be absolutely continuous, non-decreasing for \(\xi\geq 0\) and linear at infinity. The notion of a `weak' solution leads to the consideration of the equation (2) \(A_0\overline u+ L \overline u= \overline f\), where \(A_0\) and \(L\) are operators generated by the forms \[ \langle A_0 \overline u, \eta\rangle= \int^1_0 g\Biggl( \Biggl|{d\overline u\over dx} \Biggr|\Biggr) \Biggl({d\overline u\over dx}, {d\overline \eta\over dx}\Biggr) dx,\;\langle L\overline u, \overline \eta\rangle= \int^1_0 \Biggl( Q(x) {d\overline u\over dx}, \overline \eta\Biggr) dx \] and transform \(V= \mathring W^{(1)}_2(0, 1) \times \mathring W^{(1)}_2(0, 1)\) into \(V^*= W^{(- 1)}_2(0, 1)\times W^{(- 1)}_2(0, 1)\). The problem (2) is seen to be equivalent to a certain variational inequality and a two-tiered recursive process is then introduced. The sequence \(u^{(n)}\) of approximations constructed by the recursive process is bounded in \(V\) and from this sequence, it is possible to select a subsequence that converges weakly in \(V\) to a solution of (2).