The homotopy principle in the existence level for maps with only singularities of types A, D and E

Let N and P be smooth manifolds of dimensions n and p (n \geq p \geq 2) respectively. Let \Omega(N,P) denote an open subspace of J^{infty}(N,P) which consists of all regular jets and jets with prescribed singularities of types A_{i}, D_{j} and E_{k}. An \Omega-regular map f:N \to P refers to a smooth map having only singularities in \Omega(N,P) and satisfy the transversality condition. We will prove the homotopy principle in the existence level for \Omega-regular maps.