# Periodic solutions of semilinear functional differential equations in a Hilbert space

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## Abstract

The existence of solutions in a weak sense of the following boundary value problem is proved: where x(t) takes values in a Hilbert space H; C(t,ϑ) is a bounded linear operator on H, uniformly bounded in (t,ϑ); x

$$\begin{gathered}\frac{{dx}}{{dt}} = Ax\mathop \sum \limits_{i = 1}^k B_i (t,\bar x(t))x(t + \omega _i ) + \int_{ - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r} }^o {B_o (t,\bar x(t),\theta )x} (t + \theta )d\theta + f(t,x_t ) \hfill \\x_o = x_p \hfill \\\end{gathered}$$

$$\bar x(t) = (x(t + v_1 ),...,x(t + v_m ), \int_{ - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{r} }^o {C(t,\theta )x} (t + \theta )d\theta ){\text{and}}$$

_{t}is the function on [−r,0] defined by x_{t}(ϑ) = x(t+ϑ); the ω_{i}and ν_{i}lie in [−r,0]; p > r > 0; B_{i}(i=0, …, k) are bounded linear operators on H, uniformly bounded in their arguments, continuous in the second argument in H^{m+1}and measurable in t,ϑ; f is a function with values in H, continuous on [0,p] × {continuous functions on [−r,0]} and uniformly bounded.The further hypotheses needed are that for each z(t) continuous on [−r,p], the above equation with \(B_i (t,\bar z(t)), B_o (r,\bar z(t),\theta )\) and 0 in place of\(B_i (t,\bar x(t)), B_o (t,\bar x(t),\theta )\) and f(t,x_{t}) has unique solution x ≡ 0, and that A is an operator generating a strongly continuous semigroup which is compact for t > 0.

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