Separation and the existence theorem for second order nonlinear differential equation 1

Sufficient conditions for the invertibility and separability in L2(−∞,+∞) of the degenerate second order differential operator with complex-valued coefficients are obtained, and its applications to ...


Introduction and main results
A concept of the separability was introduced in the fundamental paper [1].The Sturm-Liouville's operator Jy = −y ′′ + q(x)y, x ∈ (a, +∞), is called separable [1] in L 2 (a, +∞), if y, −y ′′ + qy ∈ L 2 (a, +∞) imply −y ′′ , qy ∈ L 2 (a, +∞).From this it follows that the separability of J is equivalent to the existence of the estimate y ′′ L 2 (a,+∞) + qy L 2 (a,+∞) ≤ c Jy L 2 (a,+∞) + y L 2 (a,+∞) , y ∈ D(J), where D(J) is the domain of J.In [1] (see also [2,3]) some criteria of the separability depended on a behavior q and its derivatives has been obtained for J.Moreover, an example of non-separable operator J with non-smooth potential q was shown in this papers.Without differentiability condition on function q the sufficient conditions for the separability of J has been obtained in [4,5].In [6,7] so-called Localization Principle of the proof for the separability of higher order binomial elliptic operators was developed in Hilbert space.In [8,9] it was shown that local integrability and semiboundedness from below of q are enough for separability of J in L 1 (−∞, +∞).Valuation method of Green's functions [1][2][3]8,9] (see also [10]), parametrix method [4,5], as well as method of local estimates for the resolvents of some regular operators [6,7] have been used in these works.
Sufficient conditions of the separability for the Sturm-Liouville's operator have been obtained in [11][12][13][14][15], where Q is an operator.A number of works were devoted to the separation problem for the general elliptic, hyperbolic and mixed-type operators.
An application of the separability estimate (1.1) in the spectral theory of J has been shown in [15][16][17][18], and it allows us to prove an existence and a smoothness of solutions of nonlinear differential equations in unbounded domains [11,[17][18][19][20]. Brown [21] has shown that certain properties of positive solutions of disconjugate second order differential expressions imply the separation.The connection of separation with concrete physical problems has been noted in [22].
We denote L 2 := L 2 (R), R = (−∞, +∞), the space of square integrable functions.Let l is a closure in L 2 of the expression l 0 y = −y ′′ + r(x)y ′ + s(x)ȳ ′ defined in the set C ∞ 0 (R) of all infinitely differentiable and compactly sapported functions.Here r and s are complex -valued functions, and ȳ is the complex conjugate to y.
In this report we investigate some problems for the operator l.Although the operator l, similarly to the Sturm-Liouville operator J, is a singular differential operator of second order, their properties are different.The theory of the Sturm-Liouville operator J, in contrast to the operator l, developing a long time, while the idea of research is often based on the positivity of the potential q(x) (see, eg, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]).Because of the coefficients r and s, are the methods developed for the Sturm-Liouville problems are often not applicable to the study of the operator l.The spectral properties for selfadjoint singular differential operators of second order, without the free term, have been to a certain extent investigated; a review of literature can be found in [23,24].Note that the differential operator l is used, in particular, in the oscillatory processes in the medium with resistance depended on velocity [25, pp. 111-116].
The operator l is said to be separable in L 2 if the following estimate holds: , where • 2 is the L 2 -norm.In the present communication the sufficient conditions for the invertibility and separability of the differential operator l are obtained.Moreover, spectral and approximate results for the inverse operator l −1 are achieved.Using a separation theorem, which is obtained for the linear case, the solvability of the degenerate nonlinear second order differential equation −y ′′ + r(x, y)y ′ = F (x ∈ R) is proved.
Let's consider the degenerate differential equation ly = −y ′′ + r(x)y ′ + s(x)ȳ ′ = f. (1. 2) The function y ∈ L 2 is called a solution of (1.2) if there exists a sequence {y n } +∞ n=1 such that y n − y 2 → 0, ly n − f 2 → 0 as n → +∞.If the operator l is separable, then the solution y of (1.2) belongs to the weighted Sobolev space W 2 2 (R, |r| + |s|) with the norm y ′′ 2 + (|r| + |s|)y ′ 2 .So, the study of the qualitative behavior of solutions of (1.2) and spectral and approximative properties of l can be reduced to the investigation of embedding W 2 2 (R, |r| + |s|) ֒→ L 2 .We denote where g and h are given functions.By C (1) loc (R) we denote the set of functions f such that ψf ∈ C (1) (R) for all ψ ∈ C ∞ 0 (R).Theorem 1.Let functions r and s satisfy the conditions (1.3) EJQTDE, 2012 No. 66, p. 2 Then l is invertible and l −1 is defined in all L 2 .Theorem 2. Assume that functions r and s satisfy the conditions (1.4) Then for y ∈ D(l) the estimate holds, i.e. the operator l is separable in L 2 .
We use the statement of Theorem 2 for proof of the following Theorems 3-5.
Theorem 3. Assume that functions r and s satisfy (1.4) and let lim We assume that the conditions of Theorem 3 hold, and consider a set be the Kolmogorov's widths of the set M in L 2 .Here {Σ k } is a set of all subspaces Σ k of L 2 whose dimensions are not greater than k.Through N 2 (λ) denote the number of widths d k which are not smaller than a given positive number λ. Estimates of the width's distribution function N 2 (λ) are important in the approximation problems of solutions of the equation ly = f .The following statement holds.
Theorem 4. Assume that the conditions of Theorem 3 be fulfilled, and let a function q satisfy γ q,Re r < ∞.Then the following estimates hold: Example.Assume that r = (1 + x 2 ) β (β > 0) and let s = 0. Then the conditions of Theorem 2 are satisfied if β ≥ 1/2.If β > 1/2, then the conditions of Theorem 4 are satisfied and the following estimates hold:

Consider the following nonlinear equation
where x ∈ R, r is a real-valued function and f ∈ L 2 .
A function y ∈ L 2 is called a solution of equation (1.6), if there exists a sequence of twice continuously differentiable functions Theorem 5. Let the function r be continuously differentiable with respect to both arguments and satisfy the following conditions (1.7) Then there exists a solution y of (1.6), and

Auxiliary statements
The next statement is a corollary of the well known Muckenhoupt's inequality [26].
Lemma 2.1.Let functions g and h such that γ g,h < ∞.Then for all y ∈ C ∞ 0 (R) the following inequality holds: Moreover, if C is a smallest constant for which estimate (2.1) holds, then The following lemma is a particular case of Theorem 2.2 [23].
Lemma 2.2.Let the given function h satisfy conditions Then the set Denote by L a closure in L 2 -norm of the differential expression defined on the set C ∞ 0 (R).Lemma 2.3.Assume that functions r and s satisfy condition (1.3).Then the operator L is boundedly invertible in L 2 .
EJQTDE, 2012 No. 66, p. 4 Proof.Let L λ = L + λE, where λ ≥ 0, and E be the identity map of L 2 to itself.Note that L is separable if and only if L λ = L + λE is separable for some λ.If z is a continuously differentiate function with a compact support, then Therefore Re T = 0 and from (2.3) it follows that We estimate the left-hand side of inequality (2.4) by using the Holder's inequality.
Then by (1.3) we have L λ z 2 ≥ δ z 2 .This estimate implies that L λ is invertible.
Let us proof that L −1 λ is defined in all L 2 .Assume the contrary.Let R(L λ ) = L 2 .Then there exists a non-zero element z 0 ∈ L 2 such that z 0 ⊥ R(L λ ).According to operator's theory z 0 satisfies the equality where On the other hand using the expression L * λ ψ we have Hence by (2.6) the following estimate holds.Choose the function θ such that Here ξ > 0. Then it follows from (2.7) Since z 0 ∈ L 2 , passing to the limit as ξ → +∞ in the last inequality, we have z 0 2 = 0. Then z 0 = 0. We obtain the contradiction, which gives that R(L λ ) = L 2 .The lemma is proved.2 Lemma 2.4.Assume that functions r and s satisfy condition (1.4).Then L is separable in L 2 and for z ∈ D(L) the following estimate holds: (2.8) (2.9) It is easy to show that (2.9) holds for all z from D(L λ ).
Let ∆ j = (j − 1, j + 1) (j ∈ Z) and let {ϕ j } +∞ j=−∞ be a sequence of functions from We continue r(x) and s(x) from ∆ j to R so that its continuations r j (x) and s j (x) are bounded and periodic functions with period 2. Denote by L λ,j the closure in L 2 (R) of the differential operator −z ′ + [r j (x) + λ]z + s j (x)z defined on C ∞ 0 (R).Using the method which was applied for L λ one can proof that L λ,j are invertible and L −1 λ,j are defined in all L 2 .In addition, the following inequality (Re r j + λ) holds.From estimate (2.10) by (1.4) it follows

.11)
Let us introduce the operators B λ and M λ : At any point x ∈ R the sums of the right-hand side in these terms contain no more than two summands, therefore B λ and M λ is defined on all L 2 .It is easy to show that (2.12) Using (2.11) and properties of ϕ j (j ∈ Z) we find that lim λ→+∞ B λ = 0, hence there exists a number λ 0 such that B λ ≤ 0.5 for all λ ≥ λ 0 .Then it follows from (2.12) (2.13) EJQTDE, 2012 No. 66, p. 6 Using (2.13) and properties of ϕ j (j ∈ Z) we have From the last inequalities and (2.14) we obtain (Re r + λ)z 2 ≤ c 7 L λ z 2 , z ∈ D(L λ ), therefore it follows from condition (1.4) When λ = 0 from this inequality we have estimate (2.8).The lemma is proved.2 Lemma 2.5.Assume that functions r and s satisfy condition (1.3).Then for y ∈ D(l) the estimate y ′ 2 + y 2 ≤ c ly 2 .
Integrating by parts, we have

.16)
Since we see Re A = 0. Therefore, it follows from (2.16) Hence, using the Holder's inequality, the condition γ 1,Re r < ∞ in (1.3) and Lemma 2.1 we obtain (2.15) for any y ∈ C ∞ 0 (R).If y is an arbitrary element of D(l), then there exists a sequence The estimate (2.15) holds for y n .From (2.15) passing to the limit as n → ∞ we obtain the same estimate for y.The lemma is proved.
if there exists a sequence Lemma 2.6.If junctions r and s satisfy condition (1.3), then the equation (2.17) has a unique solution.
EJQTDE, 2012 No. 66, p. 7 Proof.From (2.15) it follows that the solution y of (2.17) is unique and belongs to W 1 2 (R).Lemma 2.3 shows that L −1 is defined in all L 2 .Then by the construction (2.17) is solvable.The proof is complete.2