  Let $n$ be a positive integer and $\mathbf{g}=(g_0,g_1,\cdots,g_{n-1})$, with
$g_k\in H(\mathbb{D})$ for $k=0,1,\cdots,n-1$. Let $I_{\mathbf{g}}^{(n)}$ be
the generalized Volterra-type operators on $H(\mathbb{C})$, which is
represented as $$
I_{\mathbf{g}}^{(n)}f=I^n\left(fg_0+f'g_1+\cdots+f^{(n-1)}g_{n-1}\right), $$
where $I$ denotes the integration operator $$(If)(z)=\int_0^zf(w)dw,$$ and
$I^n$ is the $n$th iteration of $I$. This operator is a generalization of the
operator that was introduced by Chalmoukis in \cite{Cn}. In this paper, we
study the boundedness and compactness of the operator $I_{\mathbf{g}}^{(n)}$
acting on Bergman spaces to another. As a consequence of these
characterizations, we obtain conditions for certain linear differential
equations to have solutions in Bergman spaces. Moreover, we study the
boundedness, compactness and Hilbert-Schmidtness of the following sums of
generalized weighted composition operators: Let
$\mathbf{u}=(u_0,u_1,\cdots,u_n)$ with $u_k\in H(\mathbb{D})$ for $0\leq k\leq
n$ and $\varphi$ be an analytic self-map of $\mathbb{D}.$ The sums of
generalized weighted composition operators is defined by
$$L_{\mathbf{u},\varphi}^{(n)}=\sum_{k=0}^nW_{u_k,\varphi}^{(k)},$$ where
$$W_{u_k,\varphi}^{(k)}f=u_k\cdot f^{(k)}\circ\varphi.$$ Our approach involves
the study of new class of Sobolev-Carleson measures for classical Bergman
spaces on unit disk which appears in the first main Theorems \ref{Theorem1.1}
and \ref{Theorem1.2}.
