Abstract
The main object of this paper is to establish the classification and some criteria of the limit cases for singular second-order linear equations with complex coefficients on time scales. According to the number of linearly independent solutions in suitable weighted square integrable spaces, this class of equations is classified into cases I, II, and III. Moreover, the exact dependence of cases II and III on the corresponding half-planes is given and some criteria of the limit cases are established.
Highlights
In this paper, we consider the classification and criteria of limit cases for the following singular second-order linear equations with complex coefficients:– p(t)y (t) + q(t)yσ (t) = λw(t)yσ (t), t ∈ ρ( ), +∞ ∩ T, ( . )where p and q are complex-valued rd-continuous functions, w is a real rd-continuous function; p(t) = and w(t) > for all t ∈ [ρ( ), +∞) ∩ T; p– is -integrable on [ρ( ), +∞) ∩ T; λ ∈ C is the spectral parameter; T is a time scale with ρ( ) ∈ T and sup T = +∞; σ (t) and ρ(t) are the forward and backward jump operators in T; y is the -derivative; and yσ (t) := y(σ (t))
Singular spectral problems of selfadjoint scalar second-order difference equations over infinite intervals were first studied by Atkinson [ ]
Where –∞ < a < b ≤ +∞, p and q are complex-valued functions, w is a weight function, p(t) = and w(t) > a.e. t ∈ [a, b), p– (t), q(t), and w(t) are locally integrable on [a, b), λ is a spectral parameter. They divided the equations into three cases by using the m-function, which was defined on a collection of rotated half-planes
Summary
We consider the classification and criteria of limit cases for the following singular second-order linear equations with complex coefficients:. In , Weyl gave a dichotomy of the limit-point and limit-circle cases for singular formally self-adjoint second-order linear differential equation [ ]. – p(t)y (t) + q(t)y(t) = λw(t)y(t), t ∈ [a, b), where –∞ < a < b ≤ +∞, p and q are complex-valued functions, w is a weight function, p(t) = and w(t) > a.e. t ∈ [a, b), p– (t), q(t), and w(t) are locally integrable on [a, b), λ is a spectral parameter They divided the equations into three cases by using the m-function, which was defined on a collection of rotated half-planes. In , we employed Weyl’s method to divide the following formally self-adjoint second-order linear equations on time scales into limit-point and limit-circle cases [ ]:.
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