  In this paper, we establish the averaging principle for stochastic functional
partial differential equations (SFPDEs) characterized by H\"older coefficients
and infinite delay. Firstly, we rigorously establish the existence and
uniqueness of strong solutions for a specific class of finite-dimensional
systems characterized by H\"older continuous coefficients and infinite delay.
We extend these results to their infinite-dimensional counterparts using the
variational approach and Galerkin projection technique. Subsequently, we
establish the averaging principle (the first Bogolyubov theorem) for SFPDEs
with infinite delay, subject to conditions of linear growth and H\"older
continuity. This is achieved through classical Khasminskii time discretization
and reductio ad absurdum, illustrating the convergence of solutions from the
original Cauchy problem to those of the averaged equation across the finite
interval [0, T]. To illustrate our findings, we present two applications:
stochastic generalized porous media equations and stochastic reaction-diffusion
equations with H\"older coefficients.
