Solving the Hammerstein integral equation in the irregular case by successive approximations

The branches of a solution of the nonlinear integral equation
$$ u(x)=\int_a^bK(x,s)q(s,u(s),\lambda)\,ds, $$
where $q(s,u,\lambda)=u(s)+\sum_{i=0}^\infty\sum_{k=1}^\infty q_{ik}(s)u^i\lambda^k$ and $\lambda$ is a parameter, are constructed by successive approximations. Under consideration is the case when unity is a characteristic number of the kernel $K(x,s)$ of rank $n\ge1$, and $\lambda=0$ is a bifurcation point. The principal term of the asymptotic expansion constructed is used as an initial approximation. The uniform convergence is established in some neighborhood about the bifurcation point on using the implicit function theorem and the Schmidt lemma.