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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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##### References:
 [1] Liu, J., Bounded and periodic solutions of finite delay evolution equations, Nonlinear anal., 34, 101-111, (1998) · Zbl 0934.34066 [2] Amann, H., Periodic solutions of semi-linear parabolic equations, (), 1-29 [3] Burton, T., Stability and periodic solutions of ordinary differential equations and functional differential equations, (1985), Academic Press, Inc. · Zbl 0635.34001 [4] Massera, J., The existence of periodic solutions of systems of differential equations, Duke math. J., 17, 457-475, (1950) · Zbl 0038.25002 [5] Xiang, X.; Ahmed, N., Existence of periodic solutions of semilinear evolution equations with time lags, Nonlinear anal., 11, 1063-1070, (1992) · Zbl 0765.34057 [6] Yoshizawa, T., Stability theory by liapunov’s second method, Math. soc. Japan, (1966), Tokyo · Zbl 0144.10802 [7] Ezzinbi, K.; Liu, J.; Minh, N., Periodic solutions of impulsive evolution equations, Int. J. evol. equ., 4, 103-111, (2009) · Zbl 1198.34106 [8] Liang, J.; Liu, J.H.; Xiao, T.J., Nonlocal impulsive problems for nonlinear differential equations in Banach spaces, Math. comput. modelling, 49, 798-804, (2009) · Zbl 1173.34048 [9] Guo, D.; Liu, X., Extremal solutions of nonlinear impulsive integrodifferential equations in Banach spaces, J. math. anal. appl., 177, 538-552, (1993) · Zbl 0787.45008 [10] Liu, J., Nonlinear impulsive evolution equations, Dyn. contin. discrete impuls. syst., 6, 77-85, (1999) · Zbl 0932.34067 [11] Rogovchenko, Y., Impulsive evolution systems: main results and new trends, Dyn. contin. discrete impuls. syst., 3, 57-88, (1997) · Zbl 0879.34014 [12] Yang, T., Impulsive control theory, (2001), Springer-Verlag Berlin, Heidelberg [13] Huang, F.L.; Liang, J.; Xiao, T.J., On a generalization of horn’s fixed point theorem, J. math. anal. appl., 164, 34-39, (1992) · Zbl 0763.47027 [14] Liang, J.; Huang, F.L., Horn-type theorem in Fréchet spaces and applications, Chin. ann. math. ser. B, 12, 131-136, (1991) · Zbl 0747.47039 [15] Liang, J.; Xiao, T.J., Functional-differential equations with infinite delay in Fréchet space, Sichuan daxue xuebao, 26, 382-390, (1989) · Zbl 0725.34084
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