  In this paper we prove the well-posedness and we study the asymptotic
behavior of nonoscillatory $L^p$-solutions for a third order nonlinear scalar
differential equation. The equation consists of two parts: a linear third order
with constant coefficients part and a nonlinear part represented by a
polynomial of fourth order in three variables with variable coefficients. The
results are obtained assuming three hypotheses: (i) the characteristic
polynomial associated with the linear part has simple and real roots, (ii) the
coefficients of the polynomial satisfy asymptotic integral smallness
conditions, and (iii) the polynomial coefficients are in $L^p([t_0,\infty[)$.
These results are applied to study a fourth order linear differential equation
of Poincar\'e type and a fourth order linear differential equation with
unbounded coefficients. Moreover, we give some examples where the classical
theorems can not be applied.
