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New almost periodic type functions and solutions of differential equations. (English) Zbl 0880.43009

The authors consider the classes AΠ c :=Π c +Π 0 and AΠ r :=Π r +Π 0 , where Π 0 is either C 0 (J,X)={φ:JX continuous with φ(t)0 as |t|}, PAP 0 (J,X):={φC b (J,X):M(|φ|):=lim t 0 t |φ(s)|ds=0} or WAP 0 (J,X):={φ weakly Eberlein almost periodic C b (J,X):M(|φ|)=0}, with C b = bounded continuous functions, J= or [0,), X Banach space; Π c (,X) denotes any translation invariant closed linear subspace of C b (,X) containing all constants and with φ also γ λ ·φ, γ λ (t):=e iλt , λ, and such that φφ[0,) is isometric; Π r denotes a Π 0 space containing only recurrent functions; Π c (J,X):=Π c (,X)J. These classes subsume various generalized almost periodic and almost automorphic (aa) functions, e.g. Π r = almost periodic functions or =AA:= uniformly continuous bounded Levitan - a.p. functions. It is shown that the + in AΠ is direct, AΠ r and Π c +C 0 are closed subspaces of C b and closed with respect to convolution with L 1 . If φAΠ r and Pφ(t):= 0 t φds is ergodic, then PφAΠ r . When the Beurling spectrum of a uniformly continuous bounded φ has positive distance from 0, then if φAΠ r (J,X) [resp. all differences φ(·+s)-φ(·)AΠ r ], then PφAΠ r [resp. φAΠ r ].

As an application, if y is a bounded solution on J of y '' +a 1 y ' +a 0 y=φAΠ(J,X) with (iμ) 2 +iμa 1 +a 0 0 for all real μ, then yAΠ r (J,X); for systems Y ' =AY+φ+μG(t,Y(t)) where the matrix A has no purely imaginary eigenvalues and φ, G are suitably asymptotic aa, the existence of a unique aaa solution for sufficiently small μ is shown. Similarly, a bounded solution u of the quasilinear heat equation u xx =u t +ψ(x,t,u) with initial data u| t=0 aaa is again aaa. (In theorem 4.6, p. 1146, instead of “(4.3)” it should read “(4.4) with λ=iν and ψ instead of φ”).

MSC:
43A60Almost periodic functions on groups, etc.; almost automorphic functions
28B05Vector-valued set functions, measures and integrals (measure theory)
43A45Spectral synthesis on groups, semigroups, etc.
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
35K55Nonlinear parabolic equations