On stability and oscillation of equations with a distributed delay which can be reduced to difference equations

For the equation with a distributed delay $$ x'(t) + ax(t)+ int_0^1 x(s+[t-1])d R(s)=0 $$ we obtain necessary and sufficient conditions of stability, exponential stability and oscillation. These results are applied to some particular cases, such as integro-differential equations and equations with a piecewise constant argument. Well known results for equations with a piecewise constant argument are obtained as special cases.