Correctness conditions for high-order differential equations with unbounded coefficients

We give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

Since the coefficients ρ and r j (j = 0, k) are smooth functions, the operator L 0 is a closable operator (see [1,Sect. 6 of Chap. 2]). We denote by L the closure of L 0 .
A function y(x) is called a solution of differential equation (1) if there exists a sequence {y m (x)} ∞ m=1 ⊆ C (k+1) 0 (R) such that y m → y and Ly m → f in the norm of L 2 as m → ∞. It is clear that y ∈ L 2 .
A number of problems of stochastic analysis and stochastic differential equations lead to singular elliptic equations and ordinary differential equations and their systems with unbounded intermediate coefficients. Specific representatives of such equations are the stationary equations of Ornstein-Uhlenbeck (see [5]) and Fokker-Planck-Kolmogorov (see [6]). In the case k = 1, equation (1) is the simplest model of Brownian motion of particles with a covariance matrix determined by the function ρ(x), and r 0 (x) is called the drift coefficient.
For applications of equation (1) to various practical processes, it is important to investigate the correctness of equation (1) with coefficients ρ(x) and r j (x) (j = 0, k -1) from wider classes. In the case that the intermediate coefficients do not depend on the potential and the diffusion coefficient and can grow as a linear function, the correctness of the secondorder singular elliptic equations was studied in [7][8][9][10]. The correctness conditions for the second-order and third-order one-dimensional differential equations with rapidly growing intermediate coefficients were obtained in [11][12][13][14][15][16]. However, in [11][12][13][14][15][16] the condition of weak oscillation is imposed on the intermediate and senior coefficients. In this paper, sufficient conditions for the existence and uniqueness of a solution y(x) of (1) are obtained. Moreover, for the solution, we proved the following inequality: Using this estimate, we obtained compactness conditions for operators θ (x)L -1 and θ (x) d α dx α L -1 (α = 1, k -1). The difference between this result and the results in [7][8][9][10][11][12][13][14][15][16] is that equation (1) is high order, and the coefficients r j (x) (j = 0, k -1) can grow rapidly, and all coefficients can be fluctuating (see Example 4.1). In addition, the leading coefficient can tend to zero at infinity. In other words, the cases of some degenerate equations are covered. Note that the criteria for the existence of positive periodic solutions for differential equations with indefinite singularity and pseudo almost periodic solutions of an iterative functional differential equations, respectively, were found in [17] and [18]. We introduce the following notation: The following statements are the main results of this paper.
be a continuous function, and the following conditions be satisfied: Then, for any f ∈ L 2 , equation (1) has a unique solution y and Changing variable in Theorem 1.1, we obtain the following result.
times continuously differentiable, and r k (x) be a continuous function, and the following conditions be satisfied: Then, for any f ∈ L 2 , equation (1) has a unique solution y and holds.

Some auxiliary statements
Let C (s) . The next lemma is a particular case of Theorem 2.1 in [19].

Lemma 2.1 Suppose that the functions g(x)
, v(x) = 0 (x > 0) are continuous, and for a natural number s, holds. Then, for any y ∈ C (s) where Remark 2.1 If s = 1 and C is the smallest constant for which inequality (5) is valid, then, instead of (6), the inequalitiesT g,v,s ≤ C ≤ 2T g,v,s hold (see [20]).
holds. Moreover, if C 1 is the smallest constant for which (7) is valid, then Proof Changing variable in Lemma 2.1, we obtain the desired result.
Remark 2.2 If s = 1 and C 1 is the smallest constant for which inequality (7) is valid, then the inequalitiesM u,h,s ≤ C 1 ≤ 2M u,h,s hold.
The following statement is proved by application of Lemmas 2.1 and 2.2.

Lemma 2.3 Let continuous functions u(x)
, v(x) = 0 (x ∈ R) satisfy the conditionsT u,v,s < ∞, M u,v,s < ∞ for some natural number s. Then, for any y ∈ C (s) 0 (R), Moreover, if C 2 is the smallest constant for which inequality (8) is valid, then Remark 2.3 If s = 1 and C 2 is the smallest constant for which inequality (8) is valid, then

On a two-term differential operator
Let l 0 be a differential operator from the set C (k+1) 0 (R) to L 2 , which is defined by We denote its closure by l. holds.
Since the operator l and the generalized differentiation operator are closed, we have y ∈ D(l), z = y (k) ∈ D(L) and Thus,L is a closed operator. The proof follows from the following equalities:

Lemma 3.4 Suppose that the functions ρ(x) and r 0 (x) satisfy conditions (a) and (b) of Theorem 1.1. Then l is invertible and its inverse l -1 is bounded.
Proof By Lemma 3.1, l has an inverse l -1 . Since l is a closed operator, using (9), we deduce that R(l) is a closed set. By Lemma 3.3, it suffices to prove R(L) = L 2 . If R(L) = L 2 , then, according to [1, p. 284], there is a nonzero element v(x) ∈ L 2 such that (Lz, v) = z,L * v = 0 (whereL * is the adjoint ofL) for any z ∈ D(L). Since C (1) 0 (R) ⊆ D(L), the set D(L) is dense in L 2 . Therefore, Since v = 0, we have C = 0. Taking into account condition b) of Theorem 1.1, we have that |v(x)| ≥ |C| Hence v / ∈ L 2 . This is a contradiction.

Proofs of the main results
Proof of Theorem 1.1 Set x = mt,ŷ(t) = y(mt),ρ(t) = ρ(mt),r j (t) = r j (mt) (j = 0, k),f (t) = f (mt)m -(k+1) (m > 0). Then equation (1) changes to Letl be a closure ofl 0 , wherel 0 : D(l 0 ) → L 2 is defined bŷ By the conditions of the theorem, we can choose a number m so that Then, according to condition c) of the theorem, Lemma 2.3, and estimate (14), we obtain that, for anyŷ ∈ D(l), By (19) and Lemma 3.1, we get that According to Lemma 3.4 and Remark 3.1, the operatorl is invertible, and its inversel -1 is defined on the whole L 2 . Then, by inequality (20) and the well-known statement on small perturbations [21, Chap. 4, Theorem 1.16], the following operator is also closed and invertible, and the inverse operator P -1 m is defined on the whole space L 2 . So, it follows that, for eachf ∈ L 2 ,ŷ = P -1 mf ∈ D(P m ) andŷ is a solution of equation (17). By (19), we deduce that Using the substitution t = m -1 x, we obtain that the function y(x) =ŷ( 1 m x) is a solution to equation (1). Inequality (21)

implies (3).
Proof of Theorem 1.3 Let the conditions of Theorem 1.1 be satisfied. Without loss of generality, we assume that θ (x) is a real function. Let By Theorem 1.1 and (3), for any y ∈ C (k+1) 0 (R): Ly 2 ≤ 1, we obtain These inequalities are valid for any y ∈ D(L) such that Ly 2 ≤ 1, since L is a closed operator. Therefore, Q j is bounded in L 2 . Let us show that Q j is compact in L 2 . By the Frechet-Kolmogorov theorem, it suffices to show that, for each ε > 0, there is a number N ε such that, for any y ∈ C (k+1) 0 (R), Ly 2 ≤ 1, and N ≥ N ε , the following inequality holds. We have that According to Lemma 2.1, Similarly, using Lemma 2.2, we obtain Set A s,r 0 ,j (N) = max sup t≥N T θ, √ r 0 ,j (x), sup τ ≤-N M θ, √ r 0 ,j (τ ) .