Uniform asymptotic stability in functional differential equations with infinite delay

On the qualitative analyses solutions of new mathematical models of integro‐differential equations with infinite delay

“…In Tunç and Tunç [20], new explicit qualitative conditions are established for solutions of this scalar nonlinear IDE of second order with infinite delay. Xu [21] studies the uniform asymptotic of the zero solution of the scalar IDE…”

“…Tunç and Tunç [20] deal with the scalar nonlinear IDE of second order with infinite delay: $$$$ {\displaystyle \begin{array}{cc}\hfill {x}^{\prime \prime }& +a(t)F\left(t,x,{x}^{\prime}\right)+b(t)G\left(x,{x}^{\prime}\right)+c(t)H\left({x}^{\prime}\right)+d(t)Q(x)\hfill \\ {}\hfill & +\int_{-\infty}^t\exp \left(-\left(t-s\right)\right)U\left(s,{x}^{\prime }(s)\right)\mathrm{d}s=E\left(t,x,{x}^{\prime}\right).\hfill \end{array}} $$$$ In Tunç and Tunç [20], new explicit qualitative conditions are established for solutions of this scalar nonlinear IDE of second order with infinite delay. Xu [21] studies the uniform asymptotic of the zero solution of the scalar IDE $$$$ {x}^{\prime }=a(t)x+\int_{-\infty}^tD\left(t,s\right)x(s) ds. $$$$ By using the direct method of Lyapunov, the author obtains sufficient conditions for the stability of the zero solut...…”