A stability result of a Timoshenko beam system with a delay term in the internal fractional feedback

In this article, we consider the Timoshenko beam system with a delay term in the internal fractional feedback in a bounded domain: \(\begin{aligned} \left\{ \begin{array}{ll} \rho _1 \varphi _{tt}-K\left( \varphi _{x}+\psi \right) _x=0, \\ \rho _2 \psi _{tt}-b\psi _{xx}+K \left( \varphi _x+\psi \right) +\mu _1 \psi _t+\mu _2\partial _{t}^{\alpha ,\beta }\psi (x,t-\tau )=0. \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\rho }_1{\phi }_{\mathit{tt}}-K{\left({\phi }_x,+,\psi \right)}_x=0,\hfill \\ {\rho }_2{\psi }_{\mathit{tt}}-b{\psi }_{\mathit{xx}}+K\left({\phi }_x,+,\psi \right)+{\mu }_1{\psi }_t+{\mu }_2{\partial }_t^{\alpha ,\beta }\psi (x,t-\tau )=0.\hfill \end{array}\right)\end{array}$$ Under a hypothesis between the weight of the delay term in the fractional feedback and the weight of the term without delay, using the energy method combined with the Faedo–Galerkin procedure, we prove the global existence of the solutions. Under the imposed constraints on the weights of the feedbacks and the wave speeds, general decay results of the energy are proved via suitable Lyapunov functionals.