A general stability result in a memory-type Timoshenko system

In this paper we consider the following Timoshenko system \begin{eqnarray*} \varphi _{t t}-(\varphi _{x}+\psi )_{x}=0,\quad (0,1)\times R^+\\ \psi _{t t}-\psi _{x x}+\varphi _{x}+\psi +\int_0^t g(t-\tau )\psi_{x x}(\tau )d\tau =0,\quad (0,1)\times R^{+} \end{eqnarray*} with Dirichlet boundary conditions where $g$ is a positive nonincreasing function satisfying \begin{eqnarray*} g'(t)\leq -H(g(t)) \end{eqnarray*} and $H$ is a function satisfying some regularity and convexity conditions. We establish a general stability result for this system.