an:05578159
Zbl 1198.34106
Ezzinbi, Khalil; Liu, James H.; Nguyen Van Minh
Periodic solutions of impulsive evolution equations
EN
Int. J. Evol. Equ. 4, No. 1, 103-111 (2009).
00250317
2009
j
34G20 34C25 34A37 47N20 34C11
impulsive evolution equations; periodic solutions
Let \(X\) be a Banach space. The following semilinear equation with impulses
\[
u'(t)+A(t)u(t)=f(t,u(t)),\;t\in (0,\infty) , \;t\neq t_i,\;u(0)=u_0,
\]
\[
\Delta u(t_{i})=I_{i}(u(t_{i})),\;i=1,2,\dots,\;0<t_1<t_2<\dots <\infty,
\]
is studied, where \(A(t)\) and \(f(t,u)\) are \(T\)-periodic in \(t\), \(I_i\), \(i=1,2,\dots,\) are Lipschitzian, \(f(t,u)\) is continuous in \((t,u)\) and Lipschitzian in \(u\). The existence of periodic mild solutions to the above problem is considered. The main result states that if the above equation has an ultimate bounded mild solution, then it has a \(T\)-periodic mild solution. The authors establish this result by means of some compactness assumptions on the semigroup generated by \(A(t)\) and of Horn's fixed point theorem.
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