Weighted differential inequality and oscillatory properties of fourth order differential equations

In this paper, we investigate the oscillatory properties of two fourth order differential equations in dependence on boundary behavior of its coefficients at infinity. These properties are established based on two-sided estimates of the least constant of a certain weighted differential inequality.


Introduction
Let I = (0, ∞) and 1 < p < ∞. Let u be a positive function continuous on I. Suppose that v is a positive function sufficiently times continuously differentiable on the interval I.
In the paper, we investigate the oscillatory properties of the following half-linear and linear fourth order differential equations: and v(t)y (t)u(t)y(t) = 0, t ∈ I.
One of the directions of the qualitative theory of differential equations is the investigation of their oscillatory properties, which have important applications in physics, technology, medicine, biology, and in other scientific applications. Therefore, the oscillatory properties of various models described by linear, quasilinear, and nonlinear differential equations, including delay differential equations, are intensely studied. Recently, [14,15], and [16] have investigated the oscillatory properties of second order impulsive differential equations, the research of which has been significantly developed in recent decades. Most of the results regarding the oscillatory properties of differential equations relate to second order equations. In particular, there are fairly simple methods for establishing the oscillatory properties of second order linear and half-linear equations given in a symmetric form (see, e.g., [3]). At present, the oscillatory properties of such equations are studied mainly through three methods. The first method considers the equation as a perturbation of an Euler-type equation, the solutions of which are known. The second method is based on reducing of the equation to a Hamiltonian system. The third method, called the variational principle, is based on establishing inequality (3). The first and second methods are difficult to extend to equations of the fourth or higher order. Therefore, in the recent works [4,5,17], and [18], to overcome these difficulties that arise when equations are fourth or higher order, at least one of the coefficients has to be taken as a power function. The third (variational) method is more flexible in its extension to fourth or higher order equations.
The key idea of this method is to characterize a suitable inequality and estimate its least constant. Thus, in the papers [1,7,8], and [13] restrictions on the coefficients have been removed using the variational method.
In this paper, we also use the variational method: based on the variational Lemma A (see Sect. 4), we equivalently relate nonoscillation of equations (1) and (2) with the value of the least constant C in inequality (3), then we obtain estimates for the least constant C in inequality (3), and from the obtained estimates, in terms of the coefficients, we derive the oscillatory properties of equations (1) and (2). One of the main and most technically difficult problems in the theory of inequalities is finding of the exact values of their least constants. Unfortunately, in this paper we have not found the exact value of the least constant in inequality (3), we have been able to obtain its two-sided estimates, which is currently the best possible.
Let T ≥ 0. Denote by W 2 p,v (T, ∞) the space of functions f : (T, ∞) → R having generalized derivatives up to the second order on the interval (T, ∞), for which f p,v < ∞, By the conditions on the function v, we have thatM p (T, ∞) ⊂ W 2 p,v (T, ∞). Denote bẙ W 2 p,v (T, ∞) the closure of the setM p (T, ∞) with respect to the norm f p,v . Let us consider the following second order differential inequality: On the basis of Lemma 3.1 of the work [7], we have the following statement connecting the oscillatory properties of equation (2) to the least constant C T in inequality (3).

Lemma 1 Let C T be the least constant in (3).
(i) Equation (2) is nonoscillatory if and only if there exists a number T > 0 such that is oscillatory if and only if for any number T > 0 we have that C T > 1.
In this paper, we establish an analogue of Lemma 1 also for equation (1). Since from Lemma 1 it follows that the oscillatory properties of equations (1) and (2) depend on the least constant C T in (3), we first find two-sided estimates of C T of independent interest. Then, on the basis of the obtained estimates, we study the oscillatory properties of equations (1) and (2). This paper is organized as follows. Section 2 contains all the auxiliary statements necessary to prove the main results. In Sect. 3, we find two-sided estimates of the least constant C T . In Sect. 4, on the basis of the obtained results, we get nonoscillation conditions of equations (1) and (2), and then oscillation conditions of equation (2). Section 5 contains criteria of strong nonoscillation and strong oscillation of equation (2).
In the sequel, χ (a,b) (·) stands for the characteristic function of the interval (a, b) ⊂ I. Moreover, p = p p-1 .

Auxiliary statements
Let 0 ≤ a < b ≤ ∞. In the book [9], there is the following statement.

holds if and only if
We also need a statement from the works [6] and [12]. Let 0 < τ ≤ ∞ and where C is the least constant in (6).
Assume that From the results of the paper [10] we have one more statement.
hold, then hold, then Let us note that the second conditions in (7) and (9) cover all the possible singularities of the function v at infinity.

Inequality (3)
The problem of studying inequality (3) is of independent interest, since it can be applied to study the spectral properties of fourth order differential operators, as well as to obtain a priori estimates for differential equations and considered as an embedding to solve various problems of analysis. Therefore, we investigate it in the following more general form In the paper [11], inequality (11) was studied for different zero boundary conditions at the endpoints of I. However, the obtained estimates of the least constant in (11) are somewhat cumbersome, and they are not suitable for applying them to establishing the oscillatory properties of equations (1) and (2). In this paper, we modify the method of studying inequality (3) given in [11] and find two-sided estimates of the least constant C in (11) suitable for establishing simple conditions of the oscillatory properties of equations (1) and (2). Inequality (11) under conditions (7) is well known. Since due to (8) inequality (11) is equivalent to inequality (6), for τ = ∞, estimates of the least constant in (11) have the form Now, we consider inequality (11) under conditions (9). Let 0 < τ ≤ ∞ and Let v 1-p ∈ L 1 (I), then for any τ ∈ I there exists k τ > 0 such that where k τ is increasing in τ , lim τ →0 + k τ = 0 and lim τ →∞ k τ = ∞. Moreover, for k τ 1 = 1, we have that Equality (13) is used below to prove the main theorem of this section. The arbitrariness of the parameter τ : 0 < τ < ∞ allows to get the required estimates of the least constant C in (11) in contrast to what was in [11], where the parameter τ is fixed and equal to τ 1 .
By the condition of Theorem 1, we have that v 1-p ∈ L 1 (I). Therefore, for any τ ∈ I, there exists k τ such that (13) holds. We define a strictly decreasing function ρ : (0, τ ) → (τ , ∞) from the equalities where ρ -1 is the inverse function to the function ρ.
From the condition v 1-p ∈ L 1 (I) it follows that g ∈ L 1 (I). For any τ ∈ I, integrating both sides of (23) from τ to ∞ and both sides of (25) from 0 to τ , we establish that ∞ 0 g(t) dt = 0. Hence, we constructed the function g ∈ L p,v from the functions g 1 ∈ L 1 and g 2 ∈ L 2 . Substituting the constructed function in (11) and using (18), we obtain where all the terms in the left-hand side are nonnegative. Let the function g ∈ L p,v (I) be constructed from the function g 1 ∈ L 1 . Then from (26) and (27) Due to the arbitrariness of the function g 1 ∈ L 1 , by Theorem B, from the inverse Hölder inequality the latter gives The last two estimates imply that Similarly, for the function g ∈ L p,v (I) constructed from the function g 2 ∈ L 2 , due to (26) and (27), we obtain From (28) and (29) we deduce that This gives the left-hand estimate of (15). Moreover, from (29) we also deduce the left-hand estimate of (16). The proof of Theorem 1 is complete.
Let us begin with equation (1). From Theorem 9.4.4 of the book [3], where the variational method of nonoscillation is established for half-linear higher order equation, we have the following statement.
Due to the compactness of the set supp f for f ∈M p (T, ∞), inequality (30) coincides with the inequality From (31) and the density ofM p (T, ∞) inW 2 p,v (T, ∞) we have the following lemma.
holds with the least constant C T : 0 < C T < 1, then equation (1) is nonoscillatory.
Proof From (33) and (34), in view of the upper limit definition, there exist T 1 > 0 and T 2 > 0 such that These inequalities hold for T = max{T 1 , T 2 }, hence where B(T, ∞) is equal to B(∞) from (12) for p = q and the interval (T, ∞) instead of the interval (0, ∞). Then from (35) and (12) it follows that C T < 1, and by Lemma 2 equation (1) is nonoscillatory. The proof of Theorem 2 is complete.
Let positive functions a and b belong to C n (I). In the oscillation theory of differential equations there is known the reciprocity principle [2]: the equation (-1) n (a(t)y (n) (t)) (n) = b(t)y(t) is nonoscillatory if and only if the equation (-1) n ( 1 b(t) y (n) (t)) (n) = 1 a(t) y(t) is nonoscillatory.
Now, let us assume that together with the function v the function u is also sufficiently times continuously differentiable on the interval I. Then, by the reciprocity principle, equation (51) is nonoscillatory if and only if the equation u -1 (t)y (t)λv -1 (t)y(t) = 0, t ∈ I, is nonoscillatory. The statement equivalent to the above statement is as follows: equation (51) is oscillatory if and only if equation (63) is oscillatory. Thus, on the basis of the reciprocity principle, from Theorem 6 we have the following theorem.