Definition

Another system of interest is modelled by the discrete evolution equation of the form $$x(t_{i+1})=x(t_{i})+f(t_{i},x(t_{i}))(g(t_{i+1})-g(t_{i})) \ \ i=0,\dots N-1$$“In the limit” as $(t_{0},\dots,t_{N})\in[0,1]$ gets finer we obtain $$x(t)=x(0)+\int\limits _{0}^{t} f(s,x(s))\, dg(s), \ \ t\in[0,1]$$where the integral is the Riemann-Stieltjes Integral where $$g:[0,1]\to \mathbb{R}$$ ## Note If $g\in C^1$ then then integral becomes $$\int\limits _{0}^{t} f(s,x(s))g'(s) \, ds$$