Some Oscillation Criteria for General Self‐Adjoint Differential Equations

It is not clear how the appraisal in (3.11) is obtained, so the validity of Theorem 1 is in doubt. The theoren can, however, be modified as follows. Replace hypothesis 2 by the following: “2. F(x) ⩾ − f(x)I , ( f ⩾ 0, f ∈ C [0,∞)), and there exists a continuous function e(x) , x > 0, and a non empty set 5 of unit vectors such that E(x) ⩾ e(x)I(S) , with∫ 0 x( e ( t ) − f ( t ) )   d t →   ∞ with x ”. Then the desired conclusion holds. Indeed, the original proof can be used up to (3.11) whereupon one reasons as follows. Recall T + N ⩾ I for x ⩾ x 2 ; thus (T + N) −1 exists for these x and from (3.8) we see (∗) (T + N) −1 (T' + N') (T + N) −1 = E + (T+N) −1 F(T + N) −1 since N ' = F . Then (T + N) −1 F (T + N) −1 ⩾ − f(x) (T +N) −2 since (T + N) −1 is symmetric. But (T + N) −1 ⩾ I for x ⩾ x 2 implies (T + N) −2 ⩽ I , x ⩾ x 2 so (T +N) −1 F(T + N) −1 ⩾ − f(x)I . Integrating (∗) from x 2 to x > x 2 gives − (T(x) + N(x)) −1 ⩾ − (T(x 2 ) + N(x 2 )) −1 + ( ∫ x 2 x( e ( t ) − f ( t ) )   d t ) I(S) Thus there exists x 3 > x 2 , by use of hypothesis 2 above, such that (T(x 3 )+N(x 3 )) −1 < 0 (S) , an impossibility since T + N ⩾ I, x ⩾ x 2 . This proves the theorem. A similar modification of hypotheses must be made in Theorem III.