New decay results for a viscoelastic-type Timoshenko system with infinite memory

This paper is concerned with the following memory-type Timoshenko system $$\begin{aligned} {\left\{ \begin{array}{ll} \rho _1 \varphi _{tt}-K(\varphi _x+\psi )_x =0,\\ \rho _2\psi _{tt}-b\psi _{xx}+K(\varphi _x+\psi )+\displaystyle \int \limits _0^{+\infty } g(s)\psi _{xx}(t-s){\mathrm{d}}s=0,\\ \end{array}\right. } \end{aligned}$$ with Dirichlet boundary conditions, where g is a positive nonincreasing function satisfying, for some nonnegative functions $$\xi $$ and G, $$\begin{aligned}g'(t)\le -\xi (t)G(g(t)),\qquad \forall t\ge 0.\end{aligned}$$ Under appropriate conditions on $$\xi $$ and G, we establish some new decay results that generalize and improve many earlier results in the literature such as Mustafa (Math Methods Appl Sci 41(1): 192–204, 2018), Messaoudi et al. (J Integral Equ Appl 30(1): 117–145, 2018) and Guesmia (Math Model Anal 25(3): 351–373, 2020). We consider the equal speeds of propagation case, as well as the nonequal-speed case. Moreover, we delete some assumptions on the boundedness of initial data used in many earlier papers in the literature.