For a completely continuous nonlinear equation $\lambda x = \mathcal{K}(x)$ with $\mathcal{K}(0) = 0$, approximate equations $\lambda x_n = \mathcal{K}_n (x_n )$, with $\mathcal{K}_n (0) = 0$, are analyzed. The family $\{ {\mathcal{K}_n } \}$ is collectively compact and pointwise convergent to $\mathcal{K}$. The existence and convergence of bifurcating solutions for $\lambda x = \mathcal{K}(x)$ and $\lambda x_n = \mathcal{K}_n (x_n )$ is investigated. A numerical example is given using Nekrasov’s equation.