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); 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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© 1980 Springer-Verlag
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Becker, R.I. (1980). Periodic solutions of semilinear functional differential equations in a Hilbert space. In: Izé, A.F. (eds) Functional Differential Equations and Bifurcation. Lecture Notes in Mathematics, vol 799. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0089307
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DOI: https://doi.org/10.1007/BFb0089307
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-09986-4
Online ISBN: 978-3-540-39251-4
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