  In this article we are concerned with the study of the existence and
uniqueness of pathwise mild solutions to evolutions equations driven by a
H\"older continuous function with H\"older exponent in $(1/3,1/2)$. Our
stochastic integral is a generalization of the well-known Young integral. To be
more precise, the integral is defined by using a fractional integration by
parts formula and it involves a tensor for which we need to formulate a new
equation. From this it turns out that we have to solve a system consisting in a
path and an area equations. In this paper we prove the existence of a unique
local solution of the system of equations. The results can be applied to
stochastic evolution equations with a non-linear diffusion coefficient driven
by a fractional Brownian motion with Hurst parameter in $(1/3,1/2]$, which is
particular includes white noise.
