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 $F_{h}(\lambda_{h},u_{h})-\varrho_{h} = 0$.

翻译：考虑无穷维Banach空间上对应于稳态分支情形的非线性问题$F(\lambda,u)=0$。文献中仅在特定情形下能找到近似问题$F_{h}(\lambda_{h},u_{h})=0$的分支点。我们证明：对于任意逼近$F$的$F_{h}$，总存在一个近似问题$F_{h}(\lambda_{h},u_{h})-\varrho_{h} = 0$，其分支点与$F(\lambda,u)=0$的分支点具有相同性质。首先，针对定义在一般Banach空间上的函数$\widehat{F}$，我们给出其方程存在特定类型分支点的充分条件。该结论的证明需运用变分分析方法、Graves定理及其推论，以及集值映射的压缩映射原理。这些技术使我们能够证明超定扩展系统中存在某些期望分量为零的解，进而得到常数（或函数）$\widehat{\varrho}$的存在性，使得方程$\widehat{F}(\lambda,u)-\widehat{\varrho} = 0$具有特定类型分支点，且该方程是原方程$\widehat{F}(\lambda,u) = 0$的摄动形式。同时揭示了一类在分支点处$C^{p}$-等价（右等价）于$\widehat{F}(\lambda,u)-\widehat{\varrho}$的映射族。随后，针对$F(\lambda,u)=0$的逼近问题，我们给出了联系精确函数与近似函数的条件。作为一般Banach空间上定理的应用，我们建立了近似方程$F_{h}(\lambda_{h},u_{h})-\varrho_{h} = 0$存在性的条件表述。