ABSTRACT
Lemma 9.5. If C E Loo([-I, I]) and the operator (9.12) is regular in X = L,([-I, I]), then L is a bounded operator from L,,(Q) into C,(X).
Combining the preceding three lemmas we obtain still another reformulation of the problem (9.1)/(9.2):
Theorem 9.2. Let c E Loo([-1,1)), and suppose that the operator (9.12) is regular in X = L,([-I, 1]). Then every solution of the problem (9.1)/(9.2) solves the operator equation
(9.16)
(9.17)
where L is defined by (9.14) and 9 is defined by (9.13). Conversely, every solution x E C,(X} of(9.15}, with L given by (9.14) and 9 given by (9.13), is a solution of the problem (9.1)/(9.2). By Theorem 9.2, we may reduce the solvability problem for the Barbashin equation (9.1) in the space C1(X) = C1(Lp) to that of the operator equation (9.15) in the space C,(X). Let us introduc;e some notation. For 0 ~ t,r ~ T and 0 < s ~ 1 we put
u(t, s) = x(t, s), vet, s) = x(t, -8), e(t,s) = get,s}, q(t,s) = g(t,-s).
