A novel efficient method for nonlinear boundary value problems (Q1678588)

The authors consider the nonlinear fourth-order boundary value problem \(u^{(4)}(x) = f(x, u(x) , u''(x))\), \(u(0) = u(1) = u''(0) = u''(1)\), on the interval \([0,1]\). By rewriting this as a system of two second-order boundary value problems it is possible to apply a standard Banach fixed point argument. This allows to find sufficient conditions for arbitrary solutions or positive solutions to exist uniquely and to derive the usual Picard iteration scheme for finding a sequence of functions that converges against the solution (i.e., a theoretical approximation method for the solution).