ODE to $L^p$ norms

In this paper we relate the geometry of Banach spaces to the theory of differential equations, apparently in a new way. We will construct Banach function space norms arising as weak solutions to ordinary differential equations of first order. This provides as a special case a new way of defining varying exponent $L^p$ spaces, different from the Orlicz type approach. We explain heuristically how the definition of the norm by means of the particular ODE is justified. The resulting class of spaces includes the classical $L^p$ spaces as a special case. We present an ODE-free means of defining the norms investigated.