A nonlinear generalized Degasperis-Procesi equation is investigated. Assuming that the strong solution of the equation is bounded in the sense of $L^{\infty }\left(R\right)$-norm and the initial data belong to the space $L^{\mathrm{1}}(R)\cap L^{\mathrm{2}}(R)$, we prove that the solutions are stable in the space $L^{\mathrm{1}}(R)$.