ON THE STEPANOV ALMOST PERIODIC SOLUTION OF A SECOND-ORDER INFINITESIMAL GENERATOR DIFFERENTIAL EQUATION

The Stepanov almost periodic solution of a certain second-order differential equation in a reflexive Banach space is shown to be almost periodic.


INTRODUCTION.
Let X be a Banach space and J the interval < t < A continuous function f:J-X is said to be (Bochner or strongly) almost periodic if, given e > 0 there exists a positive real number r such that any interval of the real line of length r contains at least one point % for which sup f(t + %) f(t) e t6J (i.I) A function f 6 LPoc (J;X) with 1 p < is said to be Stepanov- bounded or S p-bounded on J if f sP-t6J A function f e LPoc (J;X) with 1 p < is said to be Stepanov almost periodic or S p-almost periodic if, given e > 0 there is a positive real number r r(6) such that any interval of the real line of length r contains at least one point % for which sup [f+ t6J f(s + %)-f(s) ds] / < (1.3) We denote by L(X,X) the set of all bounded linear operators on X into itself, with the uniform operator topology.
An operator-valued function T:J L(X,X) is called a strongly continuous group if T(t +. tm) T(t) T(t_) for all t, T(0) I the identity operator on X for each x 6 X, T( t) x, t6J X is continuous. (1.4) (1.5) (1.6) The infinitesimal generator A of a strongly continuous group T:J A(X,X) is a closed linear operator, with domain D(A) dense in X defined by Ax lim T(t) x-x t for xeD(A) i. 7 t-0 (see Dunford and Schwartz [3]).
The group T is said to be almost periodic if T(t)x, t6J-X is almost periodic for each x6X NOTE i.
Suppose A and B are two densely-defined closed linear operators, having their domains and ranges in a Banach space X and a function f:J-X is continuous.
Then a solution of the differential equation ul/(t) Au/(t) + Bu(t) + f(t) a.e. on J (1.8) is a twice differentiable function u(t) with u/(t) 6D(A), u(t) 6D(B) for all t6J and satisfying the equation (1.8) a.e.(almost everywhere) on J Our result i as follows. THEOREM.
Suppose X is a reflexive Banach space, f'.J X is an S Ialmost periodic continuous function, and A is the infinitesimal generator of an almost periodic group T: J-A (X, X) Further, suppose that u:J-X, with its derivative uI(t) 6D(A) for ali t6J is a strong solution of the differential equation u//(t) Au/(t) + B(t) u(t) + f(t) a.e. on J, (1.9) where B:J A(X,X) is almost periodic with respect to the norm of L(X,X).
If u is S I-almost periodic and u is Sbounded on J then u and u are both almost periodic from J to X 2. LEMMAS.
If g:J X is an almost periodic function, and if G:J-L(X,X) is an almost periodic group, then G(t)g(t), t6J-X is an almost periodic function X a Banach space). PROOF.
LEMMA 3. Let X be a reflexive Banach space, h:J X an S 1almost periodic continuous function, and Then H is almost periodic if it is S I-bounded on J PROOF.
See Notes (ii) of Rao [4].Now, given e > 0 we may choose z to be an e-almost period of B and also an e-S I-almost period of u (see pp. I0, 77 and 78, Amerio and Prouse [i]).
Then we have v(t + s) ds for any h > 0 (3.5)Since v is S I-almost periodic, it follows easily that vh(t) is almost periodic for each fixed h > 0 As shown for scalar-valued functions in Besicovitch [2], pp.80-81, we can prove that v h -v as h-0+ in the t) T(t) u/(O) + T(t-s) [B(s) u(s) + f(s)] ds on J.(2.1) PROOF.For an arbitrary but fixed t6J we have[r(t-s) u'(s)] r(t-s) [u #(s) Au'(s)] by (1.2) and(3.3).