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

• Ronald I. Becker
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 799)

## Abstract

The existence of solutions in a weak sense of the following boundary value problem is proved:
$$\begin{gathered}\frac{{dx}}{{dt}} = Ax\mathop \sum \limits_{i = 1}^k B_i (t,\bar x(t))x(t + \omega _i ) + \int_{ - \underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{r} }^o {B_o (t,\bar x(t),\theta )x} (t + \theta )d\theta + f(t,x_t ) \hfill \\x_o = x_p \hfill \\\end{gathered}$$
where x(t) takes values in a Hilbert space H;
$$\bar x(t) = (x(t + v_1 ),...,x(t + v_m ), \int_{ - \underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{r} }^o {C(t,\theta )x} (t + \theta )d\theta ){\text{and}}$$
C(t,ϑ) is a bounded linear operator on H, uniformly bounded in (t,ϑ); xt is the function on [−r,0] defined by xt(ϑ) = x(t+ϑ); the ωi and νi lie in [−r,0]; p > r > 0; Bi (i=0, …, k) are bounded linear operators on H, uniformly bounded in their arguments, continuous in the second argument in Hm+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,xt) 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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