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：我们考虑无限维Banach空间上对应于稳态分歧情形的非线性问题$F(\lambda,u)=0$。现有文献表明，仅在某些情况下能重新找到近似问题$F_{h}(\lambda_{h},u_{h})=0$的分歧点。我们证明：对任意逼近$F$的$F_{h}$，总存在具有形式$F_{h}(\lambda_{h},u_{h})-\varrho_{h} = 0$的近似问题，该问题拥有与$F(\lambda,u)=0$分歧点性质相同的分歧点。首先，针对定义在一般Banach空间上的函数$\widehat{F}$，我们给出其保持特定类型分歧点的方程存在性的充分条件。证明过程中，我们运用变分分析法、Graves定理及其推论，以及集值映射的压缩映射原理等工具。这些方法使我们能证明某个超定扩展系统存在具有特定零分量的解。进而获得常数（或函数）$\widehat{\varrho}$的存在性，使得方程$\widehat{F}(\lambda,u)-\widehat{\varrho} = 0$具有特定类型的分歧点，且该方程以$\widehat{F}(\lambda,u) = 0$作为扰动项。同时揭示了一类在分歧点处与$\widehat{F}(\lambda,u)-\widehat{\varrho}$保持$C^{p}$等价（右等价）的映射族。关于$F(\lambda,u)=0$的近似问题研究，我们给出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，我们建立获取近似方程$F_{h}(\lambda_{h},u_{h})-\varrho_{h} = 0$存在性的条件。