  In this paper, we consider a doubly nonlinear parabolic equation $ \partial
_t \beta (u) - \nabla \cdot \alpha (x , \nabla u) \ni f$ with the homogeneous
Dirichlet boundary condition in a bounded domain, where $\beta : \mathbb{R} \to
2 ^{ \mathbb{R} }$ is a maximal monotone graph satisfying $0 \in \beta (0)$ and
$ \nabla \cdot \alpha (x , \nabla u )$ stands for a generalized $p$-Laplacian.
Existence of solution to the initial boundary value problem of this equation
has been investigated in an enormous number of papers for the case where
single-valuedness, coerciveness, or some growth condition is imposed on $\beta
$. However, there are a few results for the case where such assumptions are
removed and it is difficult to construct an abstract theory which covers the
case for $1 < p < 2$. Main purpose of this paper is to show the solvability of
the initial boundary value problem for any $ p \in (1, \infty ) $ without any
conditions for $\beta $ except $0 \in \beta (0)$. We also discuss the
uniqueness of solution by using properties of entropy solution.
