an:05305432
Zbl 1157.34346
Ezzinbi, Khalil; Liu, James H.; Nguyen Van Minh
Periodic solutions in fading memory spaces
EN
Discrete Contin. Dyn. Syst. 2005, Suppl., 250-257 (2005).
00214479
2005
j
34K13 34K30
Infinite delay; fading memory phase space; bounded and periodic solutions; condensing operators; Hale and Lunel's fixed point theorem
Summary: For \(A(t)\) and \(f(t, x, y) \, T\)-periodic in \(t\), consider the following evolution equation with infinite delay in a general Banach space \(X\),
\[
u'(t) + A(t)u(t) = f\left(t, u\left(t\right), u_{t}\right),\, t > 0,\, u(s) = \phi(s),\, s \leq 0,
\]
where the resolvent of the unbounded operator \(A(t)\) is compact, and \(u_{t} (s) = u(t+s),\, s \leq 0\). We will work with general fading memory phase spaces satisfying certain axioms, and derive periodic solutions. We will show that the related Poincar?? operator is condensing, and then derive periodic solutions using the boundedness of the solutions and some fixed point theorems. This way, the study of periodic solutions for equations with infinite delay in general Banach spaces can be carried to fading memory phase spaces.