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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.
34K45 Functional-differential equations with impulses
34G20 Nonlinear differential equations in abstract spaces
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