  We consider a nonlinear problem $F(\lambda,u)=0$ on infinite-dimensional
Banach spaces that correspond to the steady-state bifurcation case. In the
literature, it is found again a bifurcation point of the approximate problem
$F_{h}(\lambda_{h},u_{h})=0$ only in some cases. We prove that, in every
situation, given $F_{h}$ that approximates $F$, there exists an approximate
problem $F_{h}(\lambda_{h},u_{h})-\varrho_{h} = 0$ that has a bifurcation point
with the same properties as the bifurcation point of $F(\lambda,u)=0$. First,
we formulate, for a function $\widehat{F}$ defined on general Banach spaces,
some sufficient conditions for the existence of an equation that has a
bifurcation point of certain type. For the proof of this result, we use some
methods from variational analysis, Graves' theorem, one of its consequences and
the contraction mapping principle for set-valued mappings. These techniques
allow us to prove the existence of a solution with some desired components that
equal zero of an overdetermined extended system. We then obtain the existence
of a constant (or a function) $\widehat{\varrho}$ so that the equation
$\widehat{F}(\lambda,u)-\widehat{\varrho} = 0$ has a bifurcation point of
certain type. This equation has $\widehat{F}(\lambda,u) = 0$ as a perturbation.
It is also made evident a class of maps $C^{p}$ - equivalent (right equivalent)
at the bifurcation point to $\widehat{F}(\lambda,u)-\widehat{\varrho}$ at the
bifurcation point. Then, for the study of the approximation of
$F(\lambda,u)=0$, we give conditions that relate the exact and the approximate
functions. As an application of the theorem on general Banach spaces, we
formulate conditions in order to obtain the existence of the approximate
equation $F_{h}(\lambda_{h},u_{h})-\varrho_{h} = 0$.
