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Periodic solutions of delay impulsive differential equations. (English) Zbl 1242.34134
The problem studied is the following class of delay differential equations in a Banach space \((X,\|\cdot\|)\), \[ u'(t)+Au(t)= f(t,u(t),u_t),\quad t >0, \quad t \neq t_k, \] subject to the initial value \(u_0=\phi\) and the impulse conditions \(\Delta u(t_i)=I_i(u(t_i))\), \(i=1,2,\dots,\) where \(0<t_1<t_2<\dots <\infty\), \(A\) is an unbounded operator, \(r>0\), \(u_t(s)=u(t+s)\), for \(s \in [-r,0]\) and \(\Delta u(t_i)\) denotes the jump of \(u\) at the instant \(t_i\).
For a \(T\)-periodic function in the first variable \(f\) it is proved that, if the solutions to the above-mentioned problems are ultimately bounded, then there exists a \(T\)-periodic solution for a certain initial function \(\phi\). This result is deduced from the Arzelà-Ascoli theorem, which guarantees compactness for a certain operator of interest, and Horn’s fixed point theorem, by imposing suitable conditions on the function \(f\), the impulse functions \(I_i\) and the impulse instants \(t_i\), and assuming some compactness hypotheses and the existence and uniqueness of mild solutions for each initial value problem on the interval \([0,\infty)\).
The study extends some previous results about non-impulsive equations and similar impulsive ordinary differential equations.

MSC:
34K30 Functional-differential equations in abstract spaces
34K13 Periodic solutions to functional-differential equations
34K45 Functional-differential equations with impulses
47N20 Applications of operator theory to differential and integral equations
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