On generalized impulsive piecewise constant delay differential equations
This paper deals with the following impulsive differential equations with piecewise constant delay of generalized type: \[ \begin{aligned} & x^{\prime }(t)=a(t)x(t)+b(t)x(\beta (t)), x(\tau )=c_{0},~t\neq t_{k},\\ & \left.\Delta x\right | _{t=t_{k}}=d_{k}x(t_{k}^{-}), k\in \mathbb{Z}, \end{aligned}\tag{1} \] \[ \begin{aligned} & y^{\prime }(t)=a(t)y(t)+b(t)y(\beta (t))+g(t),~y(\tau )=c_{0}, t\neq t_{k},\\ & \left. \Delta y\right | _{t=t_{k}}=d_{k}y(t_{k}^{-}), k\in\mathbb{Z},\end{aligned}\tag{2} \] \[ \begin{aligned} & z^{\prime }(t)=a(t)z(t)+b(t)z(\beta (t))+f(t,z(t),z(\beta (t))),~t\neq t_{k},\\ & \left. \Delta z\right | _{t=t_{k}}=J_{k}z(t_{k}^{-}), k\in\mathbb{Z},z(\tau )=c_{0}.\end{aligned}\tag{3} \] First, a representation formula for solutions of equation (1) is obtained. Later, Green's function is established and a representation formula which corresponds to variation of parameters formula for solutions of equation (2) is stated. Similarly, variation of parameters formula for equation (3) is also established. Moreover, Gronwall type integral inequalities are obtained. Finally, oscillatory properties as well as stability of solutions are investigated and some examples are given to illustrate the results.
piecewise constant delay of generalized type
scientific article; zbMATH DE number 6498015
scientific article