Estimations of Solutions of the Sturm– Liouville Equation with Respect to a Spectral Parameter

AbstractThis paper is concerned with estimations of solutions of the Sturm–Liouville equation
$$\big(p(x)y'(x)\big)'+\Big(\mu^2 -2i\mu d(x)-q(x)\Big)\rho(x)y(x)=0, \ \ x\in[0,1],$$(p(x)y'(x))'+(μ2-2iμd(x)-q(x))ρ(x)y(x)=0,x∈[0,1],where $${\mu\in\mathbb{C}}$$μ∈C is a spectral parameter. We assume that the strictly positive function $${\rho\in L_{\infty}[0,1]}$$ρ∈L∞[0,1] is of bounded variation, $${p\in W^1_1[0,1]}$$p∈W11[0,1] is also strictly positive, while $${d\in L_1[0,1]}$$d∈L1[0,1] and $${q\in L_1[0,1]}$$q∈L1[0,1] are real functions. The main result states that for any r > 0 there exists a constant cr such that for any solution y of the Sturm–Liouville equation with μ satisfying $${|{\rm Im}\, \mu|\leq r}$$|Imμ|≤r, the inequality $${\|y(\cdot,\mu)\|_C\leq c_r\|y(\cdot,\mu)\|_{L_1}}$$∥y(·,μ)∥C≤cr∥y(·,μ)∥L1 is true. We apply our results to a problem of vibrations of an inhomogeneous string of length one with damping, modulus of elasticity and potential, rewritten in an operator form. As a consequence, we obtain that the operator acting on a certain energy Hilbert space is the generator of an exponentially stable C0-semigroup.


Introduction
The investigation of numerous physical problems requires the analysis of various spectral boundary-value problems of Sturm-Liouville type, which explains the necessity of studying estimations or asymptotic with respect to a spectral parameter of their solutions. For an extensive bibliography on these topics, see [6]. In particular, in recent years much attention has been given to the investigation of non-self-adjoint spectral problems related to oscillations of an inhomogeneous string under various conditions of damping, potential, 566 L. Rzepnicki IEOT etc. (see, e.g., [10] and the references therein, or [1,11,15,18,22]). This problem can be used as a model for much more complicated phenomena in physics such as vibrating strings and membranes in vehicle dynamics or aircraft. The focus of the present work is on the equation (1. 1) In what follows we write C = C[0, 1] as the space of continuous functions with the supremum norm · C . The symbol W k p [0, 1], where p ≥ 1 and k ∈ N, stands for the Sobolev space with the k-th derivative in L p [0, 1] which is a classical Lebesgue space. For convenience, we introduce the notation

Estimations with Respect to a Spectral Parameter
The proof of Theorem 1.2 depends on some auxiliary estimations, the most important of which are stated in the lemma below. This lemma, apart from being the main tool in the proof of Theorem 1.2, is also useful in many applications as we will show in the next section. where τ = Im μ and m g = m p m ρ , and The proof of Lemma 2.1 is taken care of two steps. The first is the transformation as in [13] of the Eq. (1.1) into a system of first order differential 568 L. Rzepnicki IEOT equations (see also [12]). Then we will apply the following proposition, which is a particular case of [3, Section 3, §4].
a.e.. Then for any solution Y of the vector differential equation the following estimations hold for any x ∈ [0, 1] where · denotes the Euclidean norm in where is the integral modulus of continuity, with ω 1 (g, ) → 0 when → 0.
We now move on to the proof of Lemma 2.1.
Proof. First consider the case when ρ ∈ L 1 [0, 1] has the representation Therefore, only η is bounded, not ρ itself. Let y = y(x, μ) be a solution of (1.1) for μ = 0. Denote so that where the positive constants are

Vol. 76 (2013)
Estimations for Solutions of the Sturm-Liouville 569 Let y be a solution of (1.1). Define the functions w j = w j (x, μ) for j = 1, 2 by setting Then the following relations are true: Differentiating (2.10), using both the fact that y is a solution of (1.1) and the identities (2.11), we obtain a system of first order differential equations where the matrix M consists of the elements from L 1 [0, 1] given by: . where

L. Rzepnicki IEOT
Simple calculations reveal that the eigenvalues of M R are (2.13) Note that due to (2.10), for Y = [w 1 , w 2 ] T and μ = 0, we get It now follows from the above equation and Proposition 2.2 that for any solution of (1.1) with μ = 0 and ρ satisfying (2.7) and (2.8), the following estimations hold (2.14) On the other hand, for α ± we have . (2.15) This and the inequality |s| + |τ | ≤ 2|μ| yields (2.17) The above relations and (2.14) imply the following inequalities: Vol. 76 (2013) Estimations for Solutions of the Sturm-Liouville 571 For the second step of the proof assume that ρ ∈ L ∞ [0, 1] and m ρ ≤ ρ(x) ≤ M ρ for a.e. x ∈ [0, 1]. It follows from Proposition 2.3 applied with δ ∈ (0, 1) that we can find a representation (2.20) such that We want to estimate (2.17) in terms of ρ δ and ρ 1,δ with the use of (2.21).
The expression (2.17) can be estimated for a fixed δ ∈ (0, 1) by setting m g = m p m ρ and using the above inequality and (2.21): then under assumptions of Lemma 2.1 the following inequality is true
Proof. It suffices to consider the inequality (2.2) with the conditions (2.26). IEOT Remark 2.5. It is possible to obtain estimations analogous to Lemma 2.1 and Corollary 2.4 when ρ satisfies only (1.2) and is not of bounded variation. It suffices to take δ = |μ| −1 in (2.23) and then the aforementioned inequalities follow only with a factor containing the modulus of continuity and |μ|.
The proof of the main result -Theorem 1.2 -relies also on the following lemma, which is concerned with estimations of solutions, for μ from the whole complex plane. A part of this will be useful in further applications too.
, then for any solution y of the Eq. (1.1) the following inequality holds for any μ ∈ C \ {0} and x 1 , Proof. Write y instead of y(·, μ) for convenience. The proof of the estimate (2.28) uses the integral representation for functions from W 1 For a solution y of (1.1) set g = py and g = (2iμd + q − μ 2 )ρy. Using the integral representation (2.31), we obtain Integration of the last term in the equality (2.32) yields Using (2.33), we finally obtain which gives (2.29). It remains to show (2.30). If y is a solution of (1.1), then We will now estimate the first integral from (2.34). Due to where in the last inequality we used the fact f C ≤ f L2 for f ∈ W 1 2 [0, 1]. Analogously, one can obtain: Combining the above estimations together concludes the proof.
Note that when y is the solution of (1.1) and |μ| ≤ R we have where the constants c 1 and c 2 depend on R. On the other hand, using Cauchy inequality and then (2.28), we obtain for |μ| ≤ R and > 0 Note that we can choose such small that 2c 2 0 c 2 (R + 1) 2 ≤ 1 2 and then we have where c 3 is a constant dependent on R and . Using (2.37) and (2.38) in (2.36), we obtain where constantsc 1 ,c 2 and c 3 depend only on R. This gives 1 2 We can now prove Theorem 1.2.

Applications
The focus of this section is on the application of results of Sect. 2 to the problem of an inhomogeneous, damped string of length one. We assume that the string has left end fixed and right one moving with damping. The density of the string will be denoted as ρ. The string is characterized by the modulus of elasticity, which is called p, while the function q is the potential. We also claim presence of viscous damping with the friction coefficient 2d.
Denote the vertical position of the string in time t ∈ [0, ∞) on the interval [0, 1] by v = v(x, t). Then small vibrations are described by the wave equation with the boundary conditions: and the initial conditions: The functions v 0 and v 1 are initial position and velocity, respectively.
Parameter h ∈ C in (3.2) is allowed to be complex, for then we can deal with a broader class of physical phenomena related with the string equation (see, e.g., instance [20] and the references therein). This is why we regard v : [0, 1] × [0, ∞) → C as a complex-valued function.
In this section we assume: and is of bounded variation. The remaining functions are real valued and satisfy Note that the assumptions in this section allows to use the results of the previous section.
In what follows we understand as L 2 [0, 1] the weighted L 2 [0, 1] space with the scalar product  (3.9) which, in view of assumptions on ρ, p and q, induces the norm equivalent to the classical one: The problem (3.1)-(3.3) can be transformed into an abstract Cauchy problem (3.12) in the Hilbert space 1]. It suffices to take V (t) = [v(·, t), v t (·, t)] T and the linear operator B h : (3.14) The first component u of the domain is W 2 1 [0, 1] since it should be coherent with Definition 1.1.
Our main aim in this section is to prove that B h is a generator of the exponentially stable C 0 -semigroup e B h t , t ≥ 0, of contractions. We will follow the methods used in [19], where the problem (3.1)-(3.3) was considered with p = 1, d = 0 and q = 0. Particular cases of the problem of a damped string and its energy with different types of boundary conditions were investigated for instance in [1,2,12], but in each case some of coefficients were zero. The main advantage of our approach is the presence of nontrivial functions p, d, q and weak assumptions on them. The fact that ρ is of bounded variation is also important, since usually it is claimed that ρ is differentiable or an element of a Sobolev space. Extensive literature on the problem of a damped string can be found in [10]. For some applications of this problem see [20,21]. An interesting approach to the problem for homogeneous string can be found in [5,16,17], where a damping which is indefinite occurs.
The norm in the space H was chosen in such a way that the square of the norm of a solution of the problem (3.11)-(3.12) is the physical energy of the string. Therefore H is called the energy space. In this norm the inequality Re B h x, x H ≤ 0 holds for x ∈ D(B h ), hence B h is dissipative. One can show that B h has no dissipative extensions, thus B h is maximal dissipative. This property is important in theorems of C 0 -semigroup generation.
When B h is the generator of an exponentially stable C 0 -semigroup of contractions, the solution of the problem (3.11)-(3.12) exists and converges exponentially to zero with respect to the energy norm. Consequently, physical energy of the string decays exponentially in time.
We want to investigate spectral properties of the operator B h given by the expressions (3.13)-(3.14). In particular, we will establish that B h is closed, densely defined and invertible. For convenience we introduce an operator   (3.19) and v is expressed by the formula The problem l μ (y 1 ) = 0 with appropriate initial conditions can be transformed into a first order system where Z = [y 1 , z] T , z = py 1 and R(μ) is given by Proceeding in the same way as in [24, Theorem 1.2.1], one can prove the existence of unique solution of the above system which is analytic with respect to μ. Therefore there exists the solution y 1 = y 1 (x, μ) of the equation l μ (y 1 ) = 0 such that y 1 is analytic with respect to μ. Analogous result can be obtained for y 2 . Note that in view of (3.21) we have the following identity for Wronskian: W (y 1 , y 2 , μ) = p(0) y 1 (0, μ)y 2 (0, μ) − y 1 (0, μ)y 2 (0, μ) = p(0).
Then the particular solution y 0 = y 0 (x, μ) ∈ W 2 1 [0, 1] of the inhomogeneous Eq. (3.17) is given by the formula Let u(x, μ) = Cy 1 (x, μ) + y 0 (x, μ). By using boundary conditions (3.18), we obtain μ). (3.24) Vol. 76 (2013) Estimations for Solutions of the Sturm-Liouville 579 Note that u ∈ W 2 1 [0, 1]. Consider the operator Q : H → D(A h ) which acts as follows Q(f, g) = (u, v) where u is given by (3.24) and v by (3.20). We will prove that this operator is bounded on H whenever the expression (3.24) is well defined i.e. U [y 1 ](μ) = 0. The expression (3.23) for y 0 can be considered as an operator which assign 1]. In view of the formula for the derivative (3.26) Consequently, the first part of (3.24) is a one-dimensional compact operator acting on H → W thus it consists of at most a countable number of nonzero points with the accumulation point at infinity. It follows that A h is closed. Moreover, simple calculations reveal that the eigenfunction corresponding to an eigenvalue μ is given by [y 1 (·, μ), iμy 1 (·, μ)] T .
We now want to provide that A h is densely defined. Note that for all  Obviously, all of those results are also true for B h . Other properties of the operator B h such as base properties were studied in [22,23], but with stronger assumptions on p, d, q and ρ.
According to Proposition 3.2, first we need to show that B h is the generator of a C 0 -semigroup of contractions. Then we should establish the existence of the resolvent of the operator B h on the imaginary axis and find its form, which is easy to estimate. This is where the inequalities for the solutions of the Sturm-Liouville equation are needed.
It is well-known that a densely defined, maximal dissipative operator generates a contraction C 0 -semigroup (see [4, Chapter II, Theorem 3.15]). We will use this fact to prove our first result. . Moreover, let ρ be as in (3.4)  According to Theorem 3.3 what is left to show is the existence of the resolvent of B h on the imaginary axis and its estimation. Instead of this we will establish that the spectrum of A h is separated from the real axis. Additionally, we provide a lower bound for the imaginary part of eigenvalues of A h . Lemma 3.5. Let the functions p, d, q be as in (3.5), (3.6), (3.7). Furthermore, let ρ be of bounded variation V and satisfying (3.4). Whenever Re h > 0, then there exists a constant c > 0 such that for any eigenvalue μ of the operator A h , the following inequality holds Im μ ≥ c > 0.
Proof. It was shown that if Re h > 0 then the operator B h is dissipative and Im A h w, w H ≥ 0, hence for μ ∈ σ(A h ) we have Im μ ≥ 0. We will show that eigenvalues are separated from the real axis. If μ is an eigenvalue of A h with eigenfunction w = (u, v), then from Assumptions on ρ, p, d and q give us the inequality