Existence solution pendulum equation using the contraction principle

How can I prove that the differential equation $\ddot{x}(t)=-(g/\ell)\sin (x(t))$ must have a solution by using the contraction principle (by Banach).

The numbers $g$ and $\ell$ are fixed constants and the initial conditions are $x(0)=x_0$ and $\dot x(0)=\omega_0$, where $x_0$ and $\omega_0$ are real constants.

According to my textbook on ODE it is possible, but since I am an engineer, I don't know how to formally prove this.

Can someone help me?

The idea is to proof that any ODE which is of the form $$\dot{x}=f(x) \ \ \ \ \ \ x \in \mathbb{R}^n$$ $$x(t_0)=x_0$$ or even for a non-autonomous system (i.e. the right side depends explicitly on time), under some assumptions, (Lipschitz continuity of f with respect to x and perhaps something else(boundedness)). The idea is that the equivalent integral equation $$x(t)=x_0+\int_{t_0}^{t_0+\delta} f(x(t))dt$$ can be written as a fixed point problem by takin $$\varphi(x(t))=x_0+\int_{t_0}^{t_0+\delta} f(x(t))dt$$ and prove that $\varphi^n$ is a contraction with the uniform norm, since the space of continuos functions is complete with that norm there must exist a solution to the original problem