Let us consider equations in the formwhereis an open subset of a normed space; for any fixedand H(λ,.) are mappings from the closure Đ of a neighborhood D of the origin in a Banach spaceXinto a Banach space Y with. Let λ be a characteristic value of the pair (T, L) such that T - L(λ.) is a Fredholm mapping with nullity p and index s,p>s≥0 and. We shall introduce new sufficient conditions on L and H for (A, 0) being a bifurcation point of the above equations in the degenerate case when the results obtained in [1], [4], [6], [7]and [8]etc. are not applicable.