Boundary value problems for second order differential equations of Sobolev type (Q1110694)

The author considers the boundary value problem of Sobolev type \[ u''(t,x)=f[t,x,u(t,x),u'(t,x)],\quad t,x\in J=[a,b], \] \[ u(a,x)=\alpha (x),\quad u(b,x)=\beta (x),\quad x\in J, \] where \(f\in C(J\times J\times R^ n\times R^ n,R^ n)\), \(\alpha,\beta \in C(J,R^ n)\). In particular, he provides necessary and sufficient conditions for the existence and uniqueness of a solution to this problem and sufficient conditions for the convergence of the Picard and approximate Picard methods to this solution.