Well-posed linear initial value problems of the form \(d^ 2u(t)/dt^ 2=Au(t)\), \(u(0)=x\), \((du/dt)(0)=0\) have unique solutions given by \(u(t)=C(t)x\) where the cosine function C satisfies \[ \begin{cases} C(t+s)+C(t- s)= 2C(t)C(s) \quad (t,s\in {\mathbb{R}})\\ C(0)=I.\end{cases} \tag{*} \] Linear cosine function theory is analogous to \((C_ 0)\) semigroup theory, and while a theory of one parameter nonlinear semigroups has been developed, a nonlinear cosine function theory does not exist. Our paper is devoted to the construction of some examples of nonlinear cosine functions, i.e. nonlinear operators \(C(t)\), on a Banach space, satisfying (*).