The authors deal with the initial-boundary value problem for a Timoshenko system in a form \[ \begin{cases} \rho_1\varphi_{tt}(x,t)-k_1(\varphi_x+\psi)_x=0,\\ \rho_2\psi_{tt}(x,t)-k_2\psi_{xx}+\int_0^t g(t-\tau)(a(x)\psi_{x}(\tau))_x\,d\tau+ k_1(\varphi_x+\psi)+b(x)h(\psi_t)=0,\\ \varphi(0,t)=\varphi(L,t)=\psi(0,t)=\psi(L,t)=0,\;t>0,\\ \varphi(x,0)=\varphi_0(x),\;\varphi_t(x,0)=\varphi_1(x),\;\psi(x,0)=\psi_0(x),\;\psi_t(x,0)=\psi_1(x),\;x\in (0,L) \end{cases} \]
for the case of equal speeds of propagation \((k_1/\rho_1=k_2/\rho_2)\). They establish a general stability estimate using the multiplier method and some properties of convex functions. Without imposing any growth condition on \(h\) at the origin, they show that the energy of the system is bounded above by a quantity, depending on \(g\) and \(h\) tending to zero as time goes to infinity.