Abstract
A refinement of the Hille–Wintner comparison theorem is obtained for two half-linear differential equations of the second order. As a consequence, some new nonoscillation tests for such equations are derived by means of this improved comparison technique. In most of our results coefficients and their integrals do not need to be nonnegative and are allowed to oscillate in any neighborhood of infinity.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Two basic comparison principles in the theory of linear differential equations which relate oscillation (resp. nonoscillation) of all solutions of a pair of equations
and
where \(q_i: [a,\infty ) \rightarrow \mathbb {R}\), \( i = 1, 2\), are continuos functions, are the Sturm comparison theorem which asserts that nonoscillation of (\(\hbox {L}_2\)) implies that of (\(\hbox {L}_1\)) (or, equivalently, oscillation of (\(\hbox {L}_1\)) implies that of (\(\hbox {L}_2\))) provided that
for all sufficiently large t, and the Hille–Wintner theorem in which the same conclusion is obtained under the condition that \(q_1\) and \(q_2\) are integrable on \([a,\infty )\) and the pointwise inequality (1.1) between coefficients is replaced by the integral inequality
holding for all t large enough (see Hille [8] and Wintner [25, 26]).
Both results have been extended to the pair of nonlinear differential equations of the form
and
where \(\alpha > 0\) is a given constant and \(q_1\) and \(q_2\) are as before. See Mirzov [20] and Elbert [4, 5] for generalization of the Sturm theorem and Kusano et al. [9, 14,15,16] for extension of the Hille–Wintner integral comparison theorem.
Here, by a solution of (\(\hbox {E}_i\)) for a fixed i we understand a real-valued function y which is continuously differentiable on \([t_y,\infty )\) for some \(t_y \ge a\) together with \(|y'|^{\alpha -1}y'\) and satisfies (\(\hbox {E}_i\)) on \([t_y,\infty )\). In this paper we consider only solutions of (\(\hbox {E}_i\)) which are not identically zero in any neighborhood of infinity. We call such a solution oscillatory if it has arbitrarily large zeros in \([t_y, \infty )\); otherwise we say that it is nonoscillatory. Since it is well-known that if one solution of (\(\hbox {E}_i\)) is oscillatory (resp. nonoscillatory), then all of them are so, it is natural to call equation (\(\hbox {E}_i\)) itself oscillatory (resp. nonoscillatory) if one (and so all) of its solutions enjoy the respective property.
A variety of sufficient conditions for nonoscillation of (\(\hbox {E}_1\)) can be obtained by application of any of the above comparison theorems in particular situations where (\(\hbox {E}_2\)) is a suitable nonoscillatory equation. For example, if (\(\hbox {E}_2\)) is the nonoscillatory Euler type equation
then (1.2) gives the Hille’s nonoscillation criterion
which is assumed to hold on some \([T,\infty )\), \(T \ge a\).
However, one shortfall of the classical Hille–Wintner theorem and its half-linear extension is that it does not imply another well-known nonoscillation test which improves (1.3), namely, the Hille-Nehari criterion which assumes the satisfaction of the condition
for all large t (see Došlý [1]).
Thus, one of our main purposes here is to extend and refine integral comparison criterion (1.2) so that it would not only imply (1.4), but also unify other earlier results and produce a series of new sufficient conditions for nonoscillation of equation (\(\hbox {E}_1\)). We emphasize that in our approach we do not suppose apriori that the coefficient \(q_1(t)\) and/or its integral \(\int _t^\infty q_1(s)ds\) (if it exists) is nonnegative in some neighborhood of infinity. Some of our results were motivated by their linear prototypes which were obtained by Kamenev [11,12,13] and Wong [27].
A survey of generalizations and extensions of the Hille-Wintner theorem in the half-linear settings published before the year 2005 as well as an overwiev of methods and techniques used in their proofs can be found in the monograph Došlý et al [2]. A number of useful nonoscillation criteria for half-linear differential equations of the second order generalizing the classical linear results of Hille, Potter, Moore, Willett and others have been obtained in Li and Yeh [18, 19]. For more recent results concerning this topic we refer to Došlý and Pátiková [3], Fišnarová and Mařik [6], Hasil and Veselý [7], Kandelaki, Lomtatidze and Ugulava [10], Li and Yeah [19], Naito [21], Pátiková [22], Sugie and Wu [24] and Yang and Lo [28].
2 Main results
The following necessary and sufficient condition for nonoscillation of (\(\hbox {E}_1\)) which will be used in the proof of our main theorem is a direct consequence of the (generalized) Sturm comparison theorem. For the proof see Skhalykho [23] or Li and Yeh [17].
Lemma 2.1
Eq. \({(\hbox {E}_1)}\) is nonoscillatory if and only if there exists a function \(u \in C^1([t_1,\infty ), \mathbb {R})\) for some \(t_1 \ge a\) such that
for \(t \ge t_1\).
Our main result now follows. As can be seen from its subsequent applications in the nonoscillation theory of secod order half-linear differential equations, it significantly improves the classical Hille–Wintner comparison theorem.
Theorem 2.1
Let \(Q_1(t)\) and \(Q_2(t)\) be continuously differentiable real-valued functions such that
on \([a,\infty )\) and equation \({(\hbox {E}_2)}\) be nonoscillatory. If there exists a \( T \ge a\) such that
for \(t \ge T\), where v is a solution of
on \([T,\infty )\), then equation \({(\hbox {E}_1)}\) is also nonoscillatory.
Proof
. Define
Then
for all large t because of (2.1) and the assertion follows from Lemma 2.1. \(\square \)
There exists a large class of primitives of \(-q_1(t)\) and \(-q_2(t)\) which can be used in Theorem 2.1 for \(Q_1(t)\) and \(Q_2(t)\), respectively. For example, if \(q_1\) and \(q_2\) are integrable on \([a,\infty )\) (possibly only conditionally), we can take
Or, if there exist finite limits
we can take
We remark that if \(\lim _{t \rightarrow \infty } \int _a^t q_i(s)ds\) exist and are finite, then
On the other hand, there are functions for which the limit in (2.6) does not exist as a finite number, but at the same time (2.4) is satisfied. An example of this type of functions is given, for example, in [10].
It is known that if (2.4) holds for a fixed i and equation (\(\hbox {E}_i\)) is nonoscillatory, then the solution \(v_i\) of the associated Riccati equation \(v_i' + \alpha |v_i|^{1+\frac{1}{\alpha }} + q_i(t) = 0\) can be expressed as
on \([T,\infty )\) for some \(T \ge a\), where \(Q_i\) is given by (2.5) (see Kandelaki et al. [10]).
Remark 2.1
Clearly, if both limits in (2.4) exist as finite numbers and the functions \(Q_1(t)\) and \(Q_2(t)\) defined by (2.5) satisfy
for all large t, then from (2.7) it follows that \(v(t) - Q_2(t) \ge 0\) and
for any (nonnegative) solution v of (2.2), so that the Hille-Wintner criterion is contained in the new result.
Corollary 2.1
Let \(k > 0\) be a given constant. If \(Q_1\) is a continuously differentiable function such that \(Q_1'(t) = -q_1(t)\) on \([a,\infty )\) and
for all sufficiently large t, then equation \({(\hbox {E}_1)}\) is nonoscillatory.
Proof
Compare (\(\hbox {E}_1\)) through (2.1) with the nonoscillatory equation
for which \(v(t) = k^{\alpha /(\alpha +1)}t^{-\alpha }\) is the exact solution of the corresponding Riccati equation. \(\square \)
Remark 2.2
The right-hand side of the second inequality in (2.8) as the function of k assumes its maximum at \(k = \big (\alpha /(\alpha +1)\big )^{\alpha +1}\). With this value of k the condition (2.8) becomes the Hille–Nehari criterion
which extends (1.4) (where \(Q_1(t) = \int _t^\infty q_1(s)ds\)) to the larger class of coefficients satisfying (2.4) (cf. with Theorem 1.6 from Kandelaki [10]).
Theorem 2.2
Suppose that (2.4) holds and Eq. \({(\hbox {E}_2)}\) is nonoscillatory. If
for \(t \ge T \ge a\), where v is the solution of (2.2) on \([T,\infty )\), then Eq. \({(\hbox {E}_1)}\) is also nonoscillatory.
Proof
Since (2.4) holds and (\(\hbox {E}_2\)) is supposed to be nonoscillatory, we can express the solution v of (2.2) as \(v(t) = Q_2(t) + \alpha \int _t^\infty |v(s)|^{1+\frac{1}{\alpha }} ds\). Inserting this integral expression for v into the left-hand side of (2.1) and using (2.9), we find that all conditions of Theorem 2.1 are satisfied, and so equation (\(\hbox {E}_1\)) is nonoscillatory. \(\square \)
A class of explicitly solvable Riccati equations of type (2.2) which can be used in Theorems 2.1 and 2.2 includes equations
and
with the exact solutions \(v = f\) and \(v = -f\), respectively. In particular, if in (2.9) we use (2.10) and define \(Q_2\) by (2.3) where \(q_2(t) = - \alpha |f(t)|^{1+\frac{1}{\alpha }} - f'(t)\), we obtain
Theorem 2.3
Suppose that for \(q_1\) the limit in (2.4) exists as a finite number and define \(Q_1\) by (2.5). If there exists a function \(f \in C^1([a,\infty ), \mathbb {R})\) such that \(\lim _{t \rightarrow \infty }f(t) = 0\), \(\int _a^\infty |f(t)|^{1+{1 \over \alpha }} dt < \infty \) and
for all large t, then Eq. \({(\hbox {E}_1)}\) is nonoscillatory.
Remark 2.3
. Similarly as in the linear case (see Kamenev [12]) it can be shown that the existence of a function f with the properties stated in Theorem 2.3 is also a necessary condition for nonoscillation of (\(\hbox {E}_1\)).
Corollary 2.2
Suppose that (2.4) holds. If
and for all sufficiently large t the inequality
holds, then Eq. \({(\hbox {E}_1)}\) is nonoscillatory.
Proof
It follows from Theorem 2.3 where \(f(t) = (\alpha +1)\big |Q_1(t)\big |\). \(\square \)
Remark 2.4
. Under the additional restriction \(Q_1(t) \ge 0\) on \([a,\infty )\), condition (2.11) in Corollary 2.2 reduces to the Opial type nonoscillation criterion
for all sufficiently large t. This results can be generalized further as follows.
Theorem 2.4
Let \(Q_1: [a,\infty ) \rightarrow [0,\infty )\) be a continuously differentiable function such that \(Q_1'(t) = -q_1(t)\) on \([a,\infty )\). If there exists a function \(\beta (t)\) such that
for all large t, then equation \({(\hbox {E}_1)}\) is nonoscillatory.
Proof
Define
Then
and since \(|u(t)|^{1+\frac{1}{\alpha }} \le |Q_1(t) + \beta (t) |^{1+\frac{1}{\alpha }}\) by (2.13), we finally obtain
for all t large enough. Conclusion now follows from Lemma 2.1. \(\square \)
Remark 2.5
. If, in Theorem 2.4, the function \(\beta (t)\) is set equal to \(\alpha Q_1(t)\), then (2.13) reduces to Opial’s criterion (2.12). The advantage of more general condition (2.13) is illustrated by the following example.
Example 2.1
Consider equation (\(\hbox {E}_1\)) with the coefficient
which changes its sign infinitely often on any interval of the form \([T,\infty ), T \ge 1\). It is easy to verify that in this case we can take
To apply Theorem 2.4, we choose
and show without difficulty that (2.13) holds for all large t if \(c \ge 3^{\alpha +1}\), so that equation (\(\hbox {E}_1\)) with \(q_1\) given by (2.14) is nonoscillatory in this case. It is to be remarked that Opial’s condition (2.12) is not satisfied for any \(c > 0\).
The proof of the following theorem is similar to that of Theorem 2.4 and is omitted.
Theorem 2.5
Let \(Q_1\) be a continuously differentiable real-valued function such that \(Q'_1(t) =-q_1(t)\) and \(Q_1(t) \le 0\) on \([a,\infty )\). If there exists a function \(\beta (t)\) such that
for all large t, then equation \({(\hbox {E}_1)}\) is nonoscillatory.
Example 2.2
Consider equation (\(\hbox {E}_1\)) with the oscillatory coefficient as in (2.14), but with the opposite sign, i.e.
In this case \(Q_1(t) \le 0\) on \([1,\infty )\) and neither (2.12) nor Theorem 2.4 are applicable. However, if we take \(\beta (t) = 1/(ct^\alpha )\) with \( c \ge 3^{\alpha +1}\), then it is not difficult to verify that all conditions of Theorem 2.5 are satisfied, and so equation (\(\hbox {E}_1\)) with coefficient \(q_1\) given by (2.15) must be nonoscillatory.
Motivated by Wong’s Theorem 4 in [27] we unify the above two theorems in a way which enables to handle also the case where \(Q_1(t)\) is not eventually nonnegative or eventually nonpositive.
Theorem 2.6
Let \(Q_1 \in C\big ([a,\infty ),\mathbb {R}\big )\) be such that \(Q'_1(t) = - q_1(t)\) on \([a,\infty )\). If there exists a function \(\beta (t)\) such that
for all sufficiently large t, then equation \({(\hbox {E}_1)}\) is nonoscillatory.
The proof of this theorem is left to the reader.
References
Došlý, O.: Methods of oscillation theory of half-linear second order differential equations. Czechoslovak Math. J. 50(125), 657–671 (2000)
Došlý, O., Řehák, P.: Half-linear Differential Equations. Elsevier, Amsterdam (2005)
Došlý, O., Pátíková, Z.: Hille-Wintner type comparison criteria for half-linear second order differential equations. Arch. Math. (Brno) 42, 185–194 (2006)
Elbert, Á.: A half-linear second order differential equation, Colloquia Mathematica Societatis Janos Bolyai 30: Qualitative Theory of Differential Equations, Szeged, 153–180 (1979)
Elbert, Á.: Oscillation and nonoscillation theorems for some non-linear ordinary differential equations. Leture Notes in Mathematics, vol. 964, pp. 187–212. Springer Verlag, Berlin (1982)
Fišnarová, S., Mařik, R.: Half-linear ODE and modified Riccati equation: comparison theorems, integral characterization of principal solution. Nonlinear Anal. 74, 6427–6433 (2011)
Hasil, P., Veselý, M.: Positivity of solutions of adapted generalized Riccati equation with consequences in oscillation theory. Appl. Math. Lett. 117, 107–118 (2021)
Hille, E.: Non-oscillation theorems. Trans. Am. Math. Soc. 64, 234–252 (1948)
Hoshino, H., Imabayashi, R., Kusano, T., Tanigawa, T.: On second-order half-linear oscillations. Adv. Math. Sci. Appl. 8, 199–216 (1998)
Kandelaki, N., Lomtatidze, A., Ugulava, D.: On oscillation and nonoscillation of a second order half-linear equation. Georgian Math. J. 7, 329–346 (2000)
Kamenev, I.V.: The integral comparison of two second order linear differential equations (Russian). Usp. Mat. Nauk 27, 199–200 (1972)
Kamenev, I.V.: An integral comparison theorem for certain systems of linear differential equations. Differ. Uravn. 8, 778–784 (1972)
Kamenev, I.V.: An integral comparison and the nonoscillation property of solutions of second order linear systems (Russian). Differ. Uravn. 14, 1136–1139 (1978)
Kusano, T., Naito, Y., Ogata, A.: Strong oscillation and nonoscillation of quasilinear differential equations of second-order. Differ. Equ. Dyn. Syst. 2, 1–10 (1994)
Kusano, T., Yoshida, N.: Nonoscillation theorems for a class of quasilinear differential equations of the second-order. J. Math. Anal. Appl. 189, 115–127 (1995)
Kusano, T., Naito, Y.: Oscillation and nonoscillation criteria for second order quasilinear differential equations. Acta Math. Hungar. 76, 81–99 (1997)
Li, H.J., Yeh, C.C.: Sturmian comparison theorem for half-linear second-order differential equations. Proc. R. Soc. Edinburgh 125A, 1193–1204 (1995)
Li, H.J., Yeh, C.C.: Nonoscillation criteria for second-order half-linear differential equations. Appl. Math. Lett. 8, 63–70 (1995)
Li, H.J., Yeh, C.C.: Nonoscillation theorems for second order quasilinear differential equations. Publ. Math. Debrecen 47, 271–279 (1995)
Mirzov, J.D.: On some analogs of Sturm’s and Kneser’s theorems for nonlinear systems. J. Math. Anal. Appl. 53, 418–425 (1976)
Naito, M.: Existence and asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations. Opuscula Math. 43, 221–246 (2023)
Pátiková, Z.: Asymptotic formulas for non-oscillatory solutions of perturbed half-linear Euler equation. Nonlinear Anal. 69, 3281–3290 (2008)
Skhalyakho, Ch.A.: On the oscillatory and nonoscillatory natures of the solutions for a system of nonlinear differential equations (Russian). Differ. Uravn. 16(4), 1523–1526 (1980)
Sugie, J., Wu, F.T.: A new application method for nonoscillation criteria of Hille-Wintner type. Monatsh. Math. 183, 201–218 (2017)
Wintner, A.: On the non-existence of conjugate points. Am. J. Math. 73, 368–380 (1951)
Wintner, A.: On the comparison theorem of Kneser–Hille. Math. Scand. 5, 255–260 (1957)
Wong, J.S.W.: Oscillation and nonoscillation of solutions of second order linear differential equations with integrable coefficients. Trans. Am. Math. Soc. 144, 197–215 (1969)
Yang, X.J., Lo, K.M.: Nonoscillation criteria for quasilinear second order differential equations. J. Math. Anal. Appl. 331, 1023–1032 (2007)
Acknowledgements
The author would like to express his sincere thanks to the referee for the valuable comments and suggestions which helped to improve the presentation of the results. This work was supported by the Slovak Grant Agency VEGA-MŠ project 1/0084/23.
Funding
Open access funding provided by The Ministry of Education, Science, Research and Sport of the Slovak Republic in cooperation with Centre for Scientific and Technical Information of the Slovak Republic.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Gerald Teschl.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Jaroš, J. A refinement of the Hille–Wintner comparison theorem and new nonoscillation criteria for half-linear differential equations. Monatsh Math 204, 893–902 (2024). https://doi.org/10.1007/s00605-024-01949-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-024-01949-z
Keywords
- Half-linear differential equation
- Non-oscillatory solutions
- Hille–Wintner comparison theorem
- Generalized Riccati equation