Approximation of solutions to integro-differential time fractional wave equations in $L^p-$space

In this paper, we investigate the abstract integro-differential time-fractional wave equation with a small positive parameter $\epsilon$. The $L^p-L^q$ estimates for the resolvent operator family are obtained using the Laplace transform, the Mittag-Leffler operator family, and the $C_0-$semigroup. These estimates serve as the foundation for some fixed point theorems that demonstrate the local-in-time existence of the solution in weighted function space. We first demonstrate that, for acceptable indices $p\in [1,+\mathrm{\infty })$ and $s\in (1,+\mathrm{\infty })$, the mild solution of the approximation problem converges to the solution of the associated limit problem in $L^p((0,T),L^s({\mathbf{R}}^n))$ as $\epsilon \to 0^{+}$. The resolvent operator family and a set of kernel $k(t)$ assumptions form the foundation of the proof's primary methodology for evaluating norms. Moreover, we consider the asymptotic behavior of solutions as $\alpha \to 2^{-}$.