Inequalities and exponential stability and instability in finite delay Volterra integro-differential equations

We use Liapunov functionals to obtain sufficient conditions that ensure exponential stability of the nonlinear Volterra integro-differential equation $$x^{\prime }(t)=p(t)x(t)-\int \limits _{t-\tau }^{t}q(t,s)x(s)ds,$$ where the constant $\tau $ is positive, the function $p$ does not need to obey any sign condition and the kernel $q$ is continuous. Our results improve the results obtained in literature even in the autonomous case. In addition, we give a new criteria for instability.