In this paper, we first consider a broad class of nonlinear mappings
containing the class of contractive mappings in a metric space. Let $(X,d)$ be a metric space.
A mapping $T:X\rightarrow X$ is called
contractively generalized hybrid if there are $\alpha,\beta\in \mathbb{R}$ and $r\in [0,1)$ such that
\begin{equation*}
  \alpha d(Tx,Ty) + (1-\alpha) d(x,Ty) 
  \leq
  r\{ \beta d(Tx,y) + (1-\beta) d(x,y)\}
\end{equation*} 
for all $x,y \in X$.
Then,
we deal with fixed point theorems for these
nonlinear mappings in a complete metric space. Using the results, 
we prove well-known fixed point theorems in a complete metric space.
Furthermore, we obtain an estimating expression for contractively generalized hybrid mappings in a Banach space.