Abstract
In this paper, we discuss the existence and asymptotic stability of the time periodic solution for the evolution equation with multiple delays in a Hilbert space H u ′ ( t ) + A u ( t ) = F ( t , u ( t ) , u ( t − τ 1 ) , … , u ( t − τ n ) ) , t ∈ R , where A : D ( A ) ⊂ H → H is a positive definite selfadjoint operator, F : R × H n + 1 → H is a nonlinear mapping which is ω-periodic in t, and τ 1 , τ 2 , … , τ n are positive constants. We present essential conditions on the nonlinearity F to guarantee that the equation has ω-periodic solutions or an asymptotically stable ω-periodic solution. The discussion is based on analytic semigroups theory and an integral inequality with delays.
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