A general second-order nonsymmetric differential equation M[f] = λwf and its formal adjoint M+[g] = λ̄wg (λ∈C) are considered. L2w-(a,b) solutions for this equation is analyzed under sufficient conditions on the complex-value coefficients of M[.].
Amos proved in [1] that all solutions of the second-order ordinary differential
equation M[y] = λwy (λ ∈ C) are in L
2
w(a, ∞), when M is a second-order symmetric ordinary
differential expression in the form M[f] = −(pf0
)
0 + qf on [a,∞) (0 ≡
d
dx), under sufficient conditions
on the coefficients p and q. The case in which not all solutions are in L
2
w(a,∞) was considered by
Atkinson and Evans in [2, Theorem 1]. Here we are concerned with the L
2
w(a, ∞)-solutions of the
general second-order nonsymmetric differential equations M[f] = λwf and M+[g] = λwg ¯ , where M[.]
is defined by
M[f] = −(p(f
0 − rf))0 + up(f
0 − rf) + qf on [a, b) (1.1)
for a suitable complex-valued function f and its formal adjoint is
M+[g] = −(p(g
0 + uf))0 + rp(g
0 + uf) + qg on [a, b). (1.2)
The coefficients p, r, u, and q are complex-valued functions Lebesgue measurable on the interval [a, b)
of the real axis, −∞ < a < b ≤ ∞, and satisfy the following conditions:
p(x) 6= 0 for almost all x ∈ [a, b),
1
p
, r, u, q ∈ Lloc(a, b), (1.3)
where Lloc(a, b) denotes the space of all complex-valued functions integrable over every compact
subinterval of [a, b).
Our objective in this paper is to extend the results in [1] and [2] to a general second-order nonsymmetric differential expression M under sufficient conditions on the complex-valued coefficients
of M.
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