A fixed point theorem for $L$-contractions

Our goal is to introduce a new fixed point theorem for operators acting on the space $C([0,T];X)$. This result can be considered as a generalization of the celebrated Banach Contraction Principle.
Let $X$ be a Banach space, $T > 0$ and consider the space $C([0,T];X)$ of continuous $X$-valued functions from the segment $I=[0,T]$ to $X$ equipped with the uniform norm:

\begin{equation*} ||{u}||=\max_{t\in [0,T]} ||u(t)||_{X}. \end{equation*}

Let $F$ be a closed subset of $C([0,T];X)$. Consider a continuous non-linear operator $N\colon F\to F$ that maps $F$ to itself.
We say that the operator $N$ is $L$-contraction on $F$ if for any $u,v\in F$ it satisfies the so called $L$-condition:

\begin{equation*} ||N(u)(t)-N(v)(t)||_{X} \leq L (||u(t)-v(t)||_{X}), \end{equation*}
where $L\colon C[0,T]\to C[0,T]$ is a linear positive monotone operator acting on the space $C([0,T]; \mathbb R)$ of the real-valued continuous functions and having the spectral radius $\rho(L) < 1$.
Our main result is the following theorem.
Theorem. Suppose that an operator $N$ is $L$-contraction on $F$. Then $N$ has a fixed point in $F$.