In this paper, we establish compactness and continuous dependence on parameters for solution-set of the second order differential inclusion including self-adjoint operator in the form \begin{align*} \left\{ \begin{gathered} \frac{\partial^2}{\partial t^2} u(t,x) +2\mathcal{A} \frac{\partial}{\partial t}u(t,x)+\mathcal{A}^{2} u(t,x) \in F(t,u(t),\mu),\,\, \hfill (t,x)\in [0,T)\times\Omega \\ u(0,x)=\frac{\partial }{\partial t}u(0,x)=0, \,\, \hfill x \in \Omega, \\ %u(T,x) = h(x), \,\, \hfill x\in\Omega, \end{gathered}\right. %\label{MainProblem} \end{align*} where $\mathcal A$ is a self-adjoint operator. We use the spectral theory on Hilbert spaces to obtain formulation for mild solutions. Using the mild solution formula together with a measure of non-compactness with values in an ordered space, we construct useful bounds for solution operators. Then, we establish necessarily upper semi-continuous and condensing settings, which mainly help to obtain the global existence of mild solutions and the compactness of the mild solution set. Finally, we provide a brief discussion on the continuous dependence of the solution-set on parameter $\mu$.