ON THE SPECTRUM OF WEAKLY ALMOST PERIODIC SOLUTIONS OF CERTAIN ABSTRACT DIFFERENTIAL EQUATIONS

In a sequentially weakly complete Banach space, if the dual operator of a linear operator A satisfies certain conditions, then the spectrum of any weakly almost periodic solution of the differential equation u’ Au + f is identical with the spectrum of f except at the origin, where f is a weakly almost periodic function.


INTRODUCTION.
Suppose Z is a Banach space and X* is the dual space of X.Let J be the interval t A continuous function f J X is said to be strongly almost periodic if, given > O, there is a positive real number ;- () such that any interval of the real line of length contains at least one point T for which supllf(t+T)-f(t)II . (

1.1) tJ
We say that a function f J X is weakly almost periodic if the scalar- valued function <x*, f(t)> x*f(t) is almost periodic for each x* X*.
It is known that, if X is sequentially weakly complete, f J X is weakly almost periodic, and is a real number, then the weak limit m(e-i}'tf(t)) w-lim e-itf(t)dt exists in X and is different from the null element c) of X for at most a count- able set {n}=l called the spectrum of f(t) (see Theorem 6, p. 43, Amerio-Prouse [I]).We denote by o(f(t)) the spectrum of f(t). 2.

RESULTS
Our first result is as follows (see Theorem 9, p. 79, Amerio-Prouse [I] for the spectrum of an sl-almost periodic function).
THEOREM 1. Suppese X is a sequentially weakly complete Banach space, A is a densely defined linear operator with domain D(A) and range R(A) in X, and the dual operator A* is densely defined in X*, with R(i A*) being dense in X* for all real O.Further, suppose f J X is a weakly almost periodic (or an sl-almost periodic continuous) function.If a differentiable function u" J D(A) is a weakly almost periodic solution of the oifferential equation u'(t) Au(t) + f(t) ( on J, with u' being weakly continuous on J, then o(u(t) PROOF OF THEOREM I. First we note that u is bounded on J, since u is weakly almost periodic.Hence, for x* X*, we have }'tu(t)dt} x*m(e-itf(t)) (ix A*x*)m(e-itu(t)).REMARK I.The conclusion of Theorem remains valid if D(A*) is total and R(i A*) is total for all real O, instead of dense in X*.We indicate the proof of the following result.
THEOREM 2. In a sequentially weakly complete Banach space X, suppose A is a densely defined linear operator, the dual operator A* is densely defined in X*, with R( 2 + A*) being dense in X* for all real x O, and f J X is a weakly almost periodic (or an S I -almost periodic continuous) function.If a twice differen- tiable function u J D(A) is a weakly almost periodic solution of the differen- tial equation u"(t) Au(t) + f(t) (3.1) on J, with u" being weakly continuous and u' bounded on J, then o(u(t)) \ {0} o(f(t)) \ {0}.x*m(e-tf(t)). (3.3)Thus we have x*m(e -itf(t) (2x* + A*x*)m(e-iXtu(t)).
Now the rest of the proof parallels that of Theorem I.
REMARK 2. The conclusion of Theorem 2 also remains valid if D(A*) is total and R( 2 + A*) is total for all real O, instead of dense in X*.REEARK 3. If X is a Hilbert space and A is a nonnegative self-adjoint operator, then the hypotheses on A in Theorem 2 are verified (see Corollary 2, p. 208, Yosida [2]) and so Theorem 2 is a generalization of a result of Zaidman [3].
NOTE.As a consequence of our Theorem 1, we have the following result: THEOREM 3. In a Hilbert space H, suppose A is a self-adjoint operator and f J H is a weakly almost periodic (or an sl-almost periodic continuous) function.
If a differentiable function u J D(A) is a weakly almost periodic solution of the differential equation u'(t) Au(t) + f(t) on J, with u' being weakly continuous on J, then (u(t)) \ {0} o(f(t)) \ {0}.