On the stability of hyperbolic difference equations with unbounded delay term

Abstract The paper studies the unconditionally stable difference scheme for the approximate solution of the hyperbolic differential equation with unbounded delay term $$\begin{aligned} \ \left\{ \begin{array}{l} v_{tt}(t)+A^{2}v(t)=a\left( v_{t}(t-w )+Av(t-w )\right) +f(t),t\in (0,\infty ), \\ v(t)=\varphi (t),t\in [-w,0] \end{array} \right. \end{aligned}$$ v tt ( t ) + A 2 v ( t ) = a v t ( t - w ) + A v ( t - w ) + f ( t ) , t ∈ ( 0 , ∞ ) , v ( t ) = φ ( t ) , t ∈ [ - w , 0 ] in a Hilbert space H with a self-adjoint positive definite operator A. The main theorem on unconditionally stability estimates for the solutions of this problem are established. Numerical results and explanatory illustrations are presented show the validation of the theoretical results.