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In this paper we discuss the bifurcation problem for the abstract operator equation of the form $F(u,\lambda )=\theta$ with a parameter $\lambda$, where $F:X\times R\to Y$ is a $C^1$ mapping, $X,Y$ are Banach spaces. By the bounded linear generalized inverse $A^{+}$ of $A=F_u(u_0,{\lambda }_0)$, an abstract bifurcation theorem for the case $dimN\left(F_u(u_0,{\lambda }_0)\right)\ge codimR\left(F_u(u_0,{\lambda }_0)\right)=1$ has been obtained.