HARTMAN-TYPE COMPARISON THEOREMS FOR HALF-LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER

Comparison theorem of the Hartman type for a continuous family of non-linear differential equations of the form p(t, λ)ϕ(u ′) for s = 0 and ϕ(0) = 0, is proved with the help of the generalized Mingarelli's identity.

It is easy to see that (E 0 ) and (E 1 ) can be embedded in a continuous family of equations The basic Sturm's comparison results when re-formulated for a family of nonlinear equations (E λ ) read as follows.
The purpose of this article is to prove analogues of the Sturm's comparison theorems for a family of half-linear differential equations with a parameter λ by comparing three underlying equations (instead of two), and replacing the assumption of monotonicity (in λ) of the coefficient functions p(t, λ) and q(t, λ), and the initial function c(λ) by the assumption that for any fixed t the function The main tool utilized in this work is a "half-linear" generalization of the identity obtained by A. Mingarelli [8] (see also Kuks [6]) in his extension of multiple comparison principle of the Sturm type developed by P. Hartman [3].
Also, denote by Φ α the form defined for X, Y ∈ R and α > 0 by EJQTDE, 2012 No. 84, p. 3 From the Young inequality it follows that Φ α (X, Y ) ≥ 0 for all X, Y ∈ R and the equality holds if and only if and Also, for given 0 The following lemma which is the main tool in this paper can be verified easily by a direct computation.
Lemma 2.1 Let x i and p i ϕ(x i ), i = 1, , , , n, be continuously differentiable functions on an interval I.Then, for 1 ≤ m < n and 1 provided that x i+j (t) with j = m − 1 do not vanish in I.
If, for fixed values 0 ≤ h 1 < h 2 ... < h n , x i = u i = u(t, h i ), i = 1, ..., n, are respective solutions of half-linear differential equations (E h i ), then (2.1) reduces to If m = 1, then for i = 1, ..., n − 1 we get the (n − 1)-tuple of generalized Picone's identities EJQTDE, 2012 No. 84, p. 4 derived by the present author et al. in [4][5] and, finally, if α = 1, m = 1 and i = 1, then (2.4) reduces to the classical Picone's formula (2.5) (see [10]).Development of the qualitative theory of linear and half-linear differential equations in the last decades has proven that the identities (2.4) and (2.5) are very useful tools in obtaining comparison theorem, uniqueness, factorization of operators and bounds for eigenvalues for equations under study, and they have been generalized and extended to various classes of ordinary and partial differential equations of the second and the higher (even) orders (see [1]).

Comparison theorems
An analogue of the Sturm's fundamental comparison theorem is the following result.
Remark 3.1.If the function p(t, λ) is concave in λ for a fixed t ∈ [a, b] and (the so-called midpoint concavity property).Similarly, if q(t, λ) is strictly convex in λ on [0, ∞) for a fixed t ∈ [a, b] and Proof of Theorem 3.1.Suppose that for some value of parameter λ 2 > 0 the solution u 2 = u(t, λ 2 ) has consecutive zeros at t = c and t = d, but there is an ε 0 > 0 with The left-hand side of (3.3) is zero, while the integral on the right-hand side is positive.This contradiction proves that at least one of u 1 = u(t, λ 2 − ε 0 ) and u 3 = u(t, λ 2 + ε 0 ) must have a zero in (c, d).

Proof. Let h
), i = 1, 2, 3, δ > 0, be fixed and let t m be the zero next before t 0 .Then it must be a zero of u 2 = u(t, h 2 ) and not of u 3 = u(t, h 3 ), because between a and t m there are not less than m (and by assumption exactly m) zeros of u(t, h 3 ).The formula (2.2) (with n = 3, m = 1 and i = 1) integrated between t m and t 0 shows that from which the desired inequality readily follows.If u(t, λ), λ 1 < λ < λ 2 , have no zeros in the interval (a, t 0 ), then the proof of the theorem can be done in a similar way by integrating the identity (2.2) between a and t 0 .depending on parameter λ, where c(λ) is convex and d(λ) is concave on [0, ∞), has an increasing sequence of eigenvalues 0 < µ 1 < λ 1 < µ 2 < λ 2 < ..., and that the k-th eigenfunction has exactly k zeros in the interval (a, b).The results of this sort will be the subject of the forthcoming paper.

Remark 3 . 3 .
As in the classical linear Sturmian theory, under some additional conditions, Theorems 3.1-3.3can be used to prove that the eigenvalue problem consisting of the differential equation (E λ ) and the two-point boundary conditions u ′ (a) − c(λ)u(a) = 0, u ′ (b) − d(λ)u(b) = 0,