Principal Solution in Weyl–titchmarsh Theory for Second Order Sturm–liouville Equation on Time Scales

A connection between the oscillation theory and the Weyl–Titchmarsh theory for the second order Sturm–Liouville equation on time scales is established by using the principal solution. In particular, it is shown that the Weyl solution coincides with the principal solution in the limit point case, and consequently the square integrability of the Weyl solution is obtained. Moreover, both limit point and oscillatory criteria are derived in the case of real-valued coefficients, while a generalization of the invariance of the limit circle case is proven for complex-valued coefficients. Several of these results are new even in the discrete time case. Finally, some illustrative examples are provided.


Introduction
In this paper we continue in the development of the Weyl-Titchmarsh theory for the second order Sturm-Liouville dynamic equation Here λ ∈ C and [a, ∞) T := [a, ∞) ∩ T, where T denotes a time scale (i.e., any nonempty closed subset of R), which is bounded from below with a := min T and unbounded from above.The coefficients p(•), q(•), and w(•) are (if not specified otherwise) real-valued piecewise rdcontinuous functions on [a, ∞) T (i.e., they belong to C prd ) and satisfy (i) inf

P. Zemánek
Observe that there is no restriction on the sign of p(•).Let us emphasize that the first condition in (1.1) cannot be replaced by the weaker assumption p(t) = 0 on [a, ∞) T , see [11,Remark 2.2].
We also note that (E λ ) includes several equations of particular interest, especially the second order Sturm-Liouville differential and difference equations.
The history of the Weyl-Titchmarsh theory goes back to the celebrated paper [23] devoted to the second order Sturm-Liouville differential equation.Its extension to equation (E λ ) was given by several authors e.g. in [14,17,22,26,28], see also the references therein.One of the crucial questions of this theory concerns the number of linearly independent solutions of (E λ ), which are square integrable with respect to the weight w(•), i.e., such that ∞ a w(t) |y σ (t, λ)| 2 ∆t < ∞.It can be shown that there exists at least one square integrable solution for every λ ∈ CKR.Moreover, the situation when all solutions of (E λ ) are square integrable (i.e., the limit circle case) is invariant with respect to λ ∈ C.These facts give rise to the dichotomous classification of equation (E λ ) as being in the limit point case (i.e., at least one solution is not square integrable) or in the limit circle case for all λ ∈ C, see Section 2 for more details.In the first result of this paper we derive a generalization of the latter invariance in the case of complex-valued coefficients (see Theorem 2.5).
The existence of a square integrable solution remains open only when equation (E λ ) is in the limit point case and λ ∈ R.But for λ ∈ R equation (E λ ) can be also classified as oscillatory or nonoscillatory and this behavior is partially invariant with respect to λ as a consequence of the Sturmian theory, see e.g.[10].Moreover, the nonoscillatory case is equivalent with the existence of a solution, which is eventually smaller than any other linearly independent solution.This solution is said to be principal and we show that it plays a significant role in the present problem.In particular, we utilize the principal solution of (E λ ) for a development of a limit point criterion (see Theorem 3.1) and we discuss its connection with the Weyl solution and its square integrability in the limit point case (see Theorem 3.5).These results are new in the case T = Z, while in the case T = R they can be found in [5, Section 2].
The paper is organized as follows.In the next section we derive a generalization of the invariance of the limit circle case, recall several results from the Weyl-Titchmarsh theory equation (E λ ), and present basic properties of the principal solution.The main results are established in Section 3.

Preliminaries
For the foundations of the time scale calculus we refer to [3].For brevity, we write only 2 .By a solution of equation (E λ ) we mean a function y(•, λ) defined on [a, ∞) T such that the functions y(•, λ) and p(•) y ∆ (•, λ) are piecewise rdcontinuously delta-differentiable on [a, ∞) T and equation (E λ ) is satisfied for all t ∈ [a, ∞) T , see also [18, pg. 4].
In the first part of this section we consider equation (E λ ) with complex-valued coefficients.For simplicity we summarize the assumptions put on the coefficients of equation (E λ ).
The following lemma guarantees the existence and uniqueness of the solution of any initial value problem associated with equation (E λ ), see [3,Theorem 5.8].Moreover, it shows an intimate connection between equation (E λ ) with real-valued coefficients and the scalar symplectic dynamic system, i.e., the system of the form where S(•, λ) : [a, ∞) T → C 2×2 is a piecewise rd-continuous function satisfying for all λ ∈ C and all t ∈ [a, ∞) T the symplectic-type identity see also [18,Theorem 3.4].Here S * (t, λ) = [S(t, λ)] * = [S(t, λ)] , i.e., * stands for the conjugate transpose.The later fact was used e.g. in [17], where some results of the Weyl-Titchmarsh theory for equation (E λ ) were obtained as a special case of general results for system (S λ ) established in [18], see also [19,20].Finally, we note that system (S λ ) is closely related to the linear Hamiltonian dynamic system, which leads to system (S λ ) with polynomial dependence on λ, see [21].In addition, system (S λ ) reduces to the linear Hamiltonian differential system if T = R.
Lemma 2.2.Let Hypothesis 2.1 be satisfied.Equation (E λ ) is equivalent with the the first order system system of the form as in (S λ ), where .
The matrix-valued function S( We denote by Φ(t, λ) the fundamental matrix of systems of the form as in (S λ ) determined by the initial value condition Φ(a, λ) = I, i.e., , where φ 1 (t, λ) and φ 2 (t, λ) are linearly independent solutions of equation (E λ ) such that .
The following lemma extends [25,Theorem 7.2.1] to any time scale.This result is new even in the case T = Z.Observe that its proof does not rely on the symplectic-type identity (2.1), which may be violated under Hypothesis 2.1, compare with the proof of [19,Theorem 6.1].
The following theorem generalizes [25,Theorem 7.2.2] to any time scale, see also [19,Theorem 6.1 and Remark 6.2(ii)], [26,Theorem 3.2], and more generally [20].In its proof we utilize the Euclidean (or 2 ) vector norm on C 2 , i.e., and also the spectral matrix norm on C 2×2 , i.e., for A ∈ C 2×2 we put For brevity, we also employ the condensed notation M σ * (t Theorem 2.5.Let Hypothesis 2.1 be satisfied and assume that there exists λ 0 ∈ C such that all solutions of (E λ 0 ) satisfy (2.4).Then equation (E λ ) possesses the same property for any λ ∈ C.
Moreover, Theorem 2.5 and Remark 2.6 immediately yield the following sufficient condition for the invariance concerning solutions of (E λ ) and their quasi-derivatives.
Corollary 2.7.Let Hypothesis 2.1 be satisfied and assume that (2.12) Then all solutions of (E λ ) and their quasi-derivatives satisfy respectively, for any λ ∈ C.
Proof.According to Theorem 2.5 and Remark 2.6 it suffices to show that all solutions of equation (E 0 ) and their quasi-derivatives satisfy (2.13) with λ = 0. From the first condition in (2.12) we get for some α > 0. Upon using similar arguments as in the proof of Theorem 2.5 with Ψ(t) and Ψ(t), respectively, we obtain the conclusion.
Henceforward we restrict our attention only to equation (E λ ) with the coefficients satisfying the following hypothesis, although we will not repeat it explicitly.
We denote by L 2 w and N (λ) the linear spaces consisting of all square integrable functions with respect to the weight w(•) and of all square integrable solutions of (E λ ), respectively, i.e., Moreover, for brevity, by n(λ) we mean the number of (nontrivial) linearly independent square integrable solutions of equation (E λ ), i.e., n(λ) := dim N (λ).
More precisely, the number n(λ) satisfies 1 ≤ n(λ) ≤ 2 for any λ ∈ CKR by [17, Theorem 3.10], which upon combining with Theorem 2.9 yields the famous dichotomy for equation (E λ ) as stated in Theorem 2.10 below.The latter estimate is obtained by using the so-called Weyl circles, which are nested and converge to a circle (n(λ) = 2) or a point (n(λ) = 1), see e.g.[19,Sections 3 and 4].This geometrical background naturally motivates the limit circle and limit point terminology.Finally, we note that Theorem 2.9 is known as the invariance of the limit circle case and Theorem 2.10 below as the Weyl alternative.Theorem 2.10.Only one of the following statements is true.
If equation (E λ ) is in the limit point case and λ ∈ CKR, then the unique square integrable solution (up to a constant multiple) corresponds to the so-called Weyl solution X (•, λ), which is of the form where ϕ(•, λ) and ψ(•, λ) are linearly independent solutions of (E λ ) determined by the initial conditions for α ∈ [0, π) and m + (λ) can be defined as the limit ∆τ . (2.18) Similarly for α = π/2 we get Similarly, in the case α = 0 we have Moreover, according to (2.14), we get the Weyl solution X (t, λ) = e − √ −λ t if α = π/2, and Nevertheless one easily observes that these two expressions differ only by a constant multiple and they both satisfy X (•, λ) ∈ L 2 w .Finally, we point out that the Weyl solution and m + (λ) are well defined even for any λ ∈ CK(0, ∞) and the property X (•, λ) ∈ L 2 w remains valid on λ ∈ CK[0, ∞), see Theorem 3.5 for more details.
In the last part of this section we focus on the principal solution of equation (E R λ ), i.e., equation (E λ ) with λ ∈ R. In addition, without loss of generality, we consider only realvalued solutions of ( T if some nontrivial solution has infinitely many generalized zeros on [a, ∞) T .As a consequence of the Sturmian theory, see e.g.[10], it follows that in the latter case every solution does as well.In the opposite case equation (E R λ ) is said to be nonoscillatory, i.e., if there exists a solution such that p(t) y σ (t, λ) y(t, λ) > 0 for all t ∈ [a, ∞) T large enough.In other words, (E R λ ) is nonoscillatory if it is eventually disconjugate.Remark 2.12.As a consequence of the Sturmian theory it also follows that if (E λ 0 ) is nonoscillatory for some λ 0 ∈ R, then (E λ ) is nonoscillatory for all λ ≤ λ 0 .The simple equation −y ∆∆ (t, λ) = λ y(t, λ) illustrates that equation (E λ ) can be oscillatory for some values of λ ∈ R and nonoscillatory for another values λ ∈ R. On the other hand, it is well known in the special case T = R that the oscillatory/nonoscillatory behavior is invariant in the limit circle case, i.e., equation (E λ ) being in the limit circle case is either oscillatory or nonoscillatory for all λ ∈ R.An elegant proof based on the existence of the finite limit of Υ(t, λ, ν) discussed in Remark 2.4 can be found in [25,Theorem 7.3.1].A similar statement on a general time scale remains open and its solution is closely connected with the problem discussed in Remark 3.6(ii), see also Corollary 3.3.
Following [6], a nontrivial solution y(•, λ) of (E R λ ) is called principal if there exists t 0 ∈ [a, ∞) T such that p(t) y σ (t, λ) y(t, λ) > 0 for all t ∈ [t 0 , ∞) T , and it satisfies lim t→∞ y(t, λ) ỹ(t, λ) = 0 for any solution ỹ(•, λ) of (E R λ ) which is linearly independent of y(•, λ).Any solution linearly independent of the principal solution is said to be nonprincipal, see also [1].The existence of the principal solution of (E R λ ) is equivalent with its nonoscillatory behavior, see [6, Theorem 3.1].Moreover, the principal solution is determined uniquely up to a nonzero constant multiple and satisfies while for any nonprincipal solution ỹ(•, λ) we have where t 0 , t 1 ∈ [a, ∞) T are such that the denominators are positive on the intervals of integration.The following statement will be useful in the proof of Theorem 3.1 and it can be verified by direct calculations.
Theorem 2.13.Let λ ∈ R and assume that equation is the principal solution of (E λ ), where t 1 ∈ [a, ∞) T is such that the denominator is positive on [t 1 , ∞) T .

Main results
As a simple consequence of the existence of the principal solution we obtain the following limit point criterion.If T = R it reduces to [16, Theorem 4.1], see also [9] and [8,Theorem 11.6], while in the case T = Z and w(t) ≡ 1 it can be found in [15,Theorem 5].
Theorem 3.1.Let us assume that there exists ν ∈ R such that equation (E ν ) is nonoscillatory and the corresponding principal solution ŷ(•, ν) satisfies then equation (E λ ) is in the limit point case for all λ ∈ C.
Proof.Let (3.1) hold and ν ∈ R be such that the assumptions are satisfied.With respect to Theorem 2.10 it suffices to show that there exists a solution y(•, ν) ∈ L 2 w .Since (E ν ) is nonoscillatory, it possesses the principal solution ŷ(•, ν) and we define ỹ(t, ν) := ŷ(t, ν) < ∞, where we used the Cauchy-Schwarz inequality in the last two steps, see [3,Theorem 6.15].But this yields a contradiction with the assumption (3.1).Hence there exists a nontrivial solution of (E ν ), which is not in L 2 w , i.e., equation (E ν ) is in the limit point case.Therefore (E λ ) is in the limit point case for all λ ∈ C by Theorem 2.10.
Remark 3.2.The additional assumption concerning the convergence of ∞ t 0 w ρ (t) ŷ2 (t, ν) ∆t is trivially satisfied if T = R or if T consists only of isolated points, especially when T = hZ or T = q N .On the other hand, it does not mean y(•, ν) ∈ L 2 w , because σ(ρ(t)) = t for t ∈ [a, ∞) T , which are left-dense and right scattered simultaneously.In particular, it can be shown that one of the integrals ∞ a f (t) ∆t and ∞ a f σ (t) ∆t can be convergent, while the other is divergent, compare with [22,26,27].For example, let us consider the simple time scale Then the integral over T can be written as and similarly we obtain If we define the function f : T → R as and Theorem 3.1 implies that (∆E λ ) is in the limit point case if it is nonoscillatory for some see also [15,Theorem 5].
(ii) By the criterion in [26,Theorem 4.2], see also [13,Theorem 10] Observe that this criterion does not include any oscillatory/nonoscillatory behavior of (∆E λ ) and does not depend on the value of q k , i.e., if (3.4) is satisfied, then equation (∆E λ ) is in the limit point case for any choice of q k .Since conditions (3. = 0 is more interesting.If condition (3.4) holds, then we obtain the same conclusion as before (again the limit point classification does not depend on q k ).But it is also possible that the sum in (3.4) is convergent, while (3.2) is satisfied.For example, let p k ≡ 1, q k ≡ 0, and w k = 1 k 2 +1 , i.e., − ∆ 2 y k (λ) = λ k 2 + 1 y k+1 (λ). (3.5) Then direct calculations show that the sum in (3.4) is convergent, i.e., the assumptions of [26,Theorem 4.2] are not fulfilled, while the sum in (3.2) is divergent.Equation (3.5) with λ = 0 has two linearly independent solutions y [1] k (0) ≡ 1 and y [2] k (0) = k, which are obviously nonoscillatory.Therefore the assumptions of Theorem 3.1 are satisfied, which implies that the equation is in the limit point case.This fact can be verified directly, because the solution y [2] (0) is not square summable with respect to w k .
(iii) Although the criterion of Theorem 3.1 does not include explicitly q k , these coefficients play a significant role in contrast to [26,Theorem 4.2].Let us slightly modify equation (3.5) to the form i.e., with q k ≡ −2.Observe that the coefficients of (3.5) and (3.6) satisfy (3.2), but equation (3.5) is in the limit point case, while (3.6) is in the limit circle case.Indeed, equation (3.6) has for λ = 0 two linearly independent solutions y [1] k (0) = sin(kπ/2) and y [2] k (0) = cos(kπ/2), which are square summable with respect to w k , i.e., it is in the limit circle case for all λ ∈ C by Theorem 2.9.Note that this conclusion does not contradict the result of Theorem 3. As already mentioned, whenever the principal solution ỹ(•, λ) of equation (E R λ ) exists, it is unique up to a nonzero constant multiple.The same is true also for a square integrable solution (being the Weyl solution) of equation (E λ ), which is in the limit point case.In the final part of this paper we establish an intimate connection between these two solutions.
ν) is a nonprincipal solution of (E ν ) by Theorem 2.13.Suppose that the linearly independent solutions ŷ(•, ν) and ỹ(•, ν) belong to L 2 w .Then by the assumptions also ∞ t 0 w ρ (t) ŷ2 (t, ν) ∆t < ∞ for some t 0 ∈ (a, ∞) T and for t 4 ≥ max i=0,1,2,3 {t i } we have [26,integrals (t) ∆t do not converge/diverge at the same time.The contrapositive of Theorem 3.1 yields the following oscillation criterion for (E λ ).(t) y 2 (t) ∆t < ∞.If equation (E λ ) is in the limit circle case for some λ ∈ C, then equation (E λ ) is oscillatory for all λ ∈ R.Several limit point criteria for equation (E λ ) or its delta-nabla counterpart were established in[22, Section 4],[26, Section 4], and [24, Section 3].In the following example we compare Theorem 3.1,[22, Theorem 4.1], and[26, Theorem 4.2]in the case T = Z.We note that the assumptions of[26, Theorem 4.2]are never satisfied if T ∆E λ ) is in the limit point case whether it is oscillatory or not.