Abstract
We develop the Weyl‐Titchmarsh theory for time scale symplectic systems. We introduce the M(λ)‐function, study its properties, construct the corresponding Weyl disk and Weyl circle, and establish their geometric structure including the formulas for their center and matrix radii. Similar properties are then derived for the limiting Weyl disk. We discuss the notions of the system being in the limit point or limit circle case and prove several characterizations of the system in the limit point case and one condition for the limit circle case. We also define the Green function for the associated nonhomogeneous system and use its properties for deriving further results for the original system in the limit point or limit circle case. Our work directly generalizes the corresponding discrete time theory obtained recently by S. Clark and P. Zemánek (2010). It also unifies the results in many other papers on the Weyl‐Titchmarsh theory for linear Hamiltonian differential, difference, and dynamic systems when the spectral parameter appears in the second equation. Some of our results are new even in the case of the second‐order Sturm‐Liouville equations on time scales.
Highlights
In this paper we develop systematically the Weyl-Titchmarsh theory for time scale symplectic systems
In the present paper we develop the Weyl-Titchmarsh theory for more general linear dynamic systems, namely, the time scale symplectic systems xΔ tAtxtBtut, Sλ uΔ tCtxtDtut − λW t xσ t, t ∈ a, ∞ Ì, where A, B, C, D, W are complex n × n matrix functions on a, ∞ Ì, W t is Hermitian and nonnegative definite, λ ∈, and the 2n × 2n coefficient matrix in system Sλ satisfies
As it is known in 52, Theorem 5.8 and discussed in 54, Remark 3.8, for a fixed t0 ∈ a, b Ì and a piecewise rd-continuous n × n matrix function A · on a, b Ì which is regressive on a, t0 Ì, the initial value problem yΔ tAtyt for t ∈ a, ρ b Ì with y t0 y0 has a unique solution y · ∈ C1prd on a, b Ì for any y0 ∈ n
Summary
In this paper we develop systematically the Weyl-Titchmarsh theory for time scale symplectic systems. The results for linear Hamiltonian difference systems were generalized in 1, 2 to discrete symplectic systems xk 1 Akxk Bkuk, uk 1 Ckxk Dkuk λWkxk 1, k ∈ 0, ∞ , 1.4 where Ak, Bk, Ck, Dk, Wk are complex n × n matrices such that Wk is Hermitian and nonnegative definite and the 2n × 2n transition matrix in 1.4 is symplectic, that is, Sk : Ak Bk , Ck Dk. In the unifying theory for differential and difference equations—the theory of time scales—the classification of second-order Sturm-Liouville dynamic equations yΔΔ t q t yσ t λyσ t , t ∈ a, ∞ Ì, 1.6 to be of the limit point or limit circle type is given in 4, 43. A certain uniqueness result is proven for the limit point case
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