# zbMATH — the first resource for mathematics

Periodic solutions to operational differential equations with finite delay and impulsive conditions. (English) Zbl 1330.34119
Summary: We study the following semilinear operational differential equation with finite delay and impulsive condition \begin{aligned} &u'(t)+ Au(t) = f (t,u(t),u_t),\;t > 0, t \neq t_i, \\ &u(s) =\phi(s),\;s\in [-r, 0],\\ &\Delta u(t_i) = I_i(u(t_i)),\;i = 1,2,\dots, 0 < t_1 < t_2 <\dots < \infty,\end{aligned} in a Banach space $$(X,\|\cdot\|)$$ with an unbounded operator $$A$$, where $$r > 0$$ is a constant and $$u_t(s) = u(t+s)$$, $$s\in[-r, 0]$$, which constitutes a finite delay, and $$\Delta u(t_i) = u(t^+_i)-u(t^-_i)$$ constitutes an impulsive condition which can be used to model more physical phenomena than the traditional initial value problems. We assume that $$f(t,u,w)$$ is $$T$$-periodic in $$t$$ and then prove under some conditions that if solutions of the equation are ultimate bounded, then the operational differential equation has a $$T$$-periodic solution. The new result obtained here improves the corresponding result of [J. Liang et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 17, 6835–6842 (2011; Zbl 1242.34134)] by eliminating an assumption. Moreover, our arguments of proving the result are suitable for many other problems associated with impulsive conditions.
##### MSC:
 34K45 Functional-differential equations with impulses 34G20 Nonlinear differential equations in abstract spaces
##### Keywords:
periodic solutions; impulsive conditions; delay
Full Text: