A note on oscillation of perturbed half-linear differential equations

In this chapter we shall present oscillation and nonoscillation criteria for second order half-linear differential equations. In recent years these equations have attracted considerable attention. This is largely due to the fact that half-linear differential equations occur in a variety of real world problems; moreover, these are the natural generalizations of second order linear differential equations. In Section 3.1, we shall provide some preliminaries for the study of half-linear differential equations. In Sections 3.2 and 3.3, respectively, Sturm’s and Levin’s type comparison theorems are developed. In Section 3.4, we shall establish a Liapunov type inequality. Section 3.5 presents an oscillation criterion for almost periodic Sturm-Liouville equations. A systematic study on the zeros of solutions of singular half-linear equations is made in Section 3.6. Nonoscillation characterizations (necessary and sufficient conditions), comparison results as well as several sufficient criteria for the nonoscillation are presented in Section 3.7. Section 3.8 is devoted to the study of oscillation of half-linear equations. In Section 3.9, we shall establish oscillation criteria by employing integral and weighted averaging techniques. Here, interval criteria for the oscillation of half-linear equations are also provided. Section 3.10 deals with the oscillation of half-linear equations with integrable coefficients. Section 3.11 addresses the oscillation of damped and forced equations. In Section 3.12, we shall derive lower bounds for the distance between consecutive zeros of an oscillatory solution. Finally, in Section 3.13, we shall present a systematic study of the oscillation and nonoscillation of half-linear equations with a deviating argument. Here, classifications of the nonoscillatory solutions, and the existence results which guarantee that the solutions have prescribed asymptotic behavior are also presented.

The oscillation and nonoscillation property of solutions of second order linear differential equations is of special interest, and therefore, it has been the subject of many investigations. The interest in second order linear oscillations is due, in a large part, to the fact that many physical systems are modelled by such equations. In this chapter we shall discuss some of the most basic results in the theory of oscillations of linear ordinary differential equations of second order. In Section 2.1, we shall present Sturm and Sturm-Picone comparison theorems which are useful in oscillation theory. In Section 2.2, we shall provide some necessary and sufficient conditions for the nonoscillation as well as some comparison theorems of Sturm’s type. Sufficiency criteria for the nonoscillation are given in Section 2.3. In Section 2.4, we shall establish sufficient conditions for the oscillation of second order differential equations with alternating coefficients. Integral averaging techniques as well as interval criteria for the oscillation are discussed in Section 2.5. In Section 2.6, several criteria for oscillation of linear second order differential equations with integrable coefficients are established. Finally, in Section 2.7 we shall discuss the problems of forced oscillations.

Sun–Wong type theorems for second order damped elliptic equations

Oscillation criteria for second order forced elliptic differential equations with mixed nonlinearities

Previous article Next article Full AccessConditions for the Existence of Conjugate Points for a Fourth Order Linear Differential EquationJohn S. BradleyJohn S. Bradleyhttps://doi.org/10.1137/0117086PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Dallas Banks, Bounds for the eigenvalues of some vibrating systems, Pacific J. Math., 10 (1960), 439–474 MR0117378 0097.16501 CrossrefGoogle Scholar[2] John H. Barrett, Two-point boundary problems for linear self-adjoint differential equations of the fourth order with middle term, Duke Math. J., 29 (1962), 543–554 10.1215/S0012-7094-62-02955-1 MR0148981 0108.08204 CrossrefISIGoogle Scholar[3] E. S. Čičkin, A non-oscillation theorem for a linear self-adjoint differential equation of fourth order, Izv. Vysš. Učebn. Zaved. Matematika, 1960 (1960), 206–209 MR0137900 Google Scholar[4] A. M. Fink, The functional T$\int \sb{0}\sp{T}$R and the zeroes of a second order linear differential equation, J. Math. Pures Appl. (9), 45 (1966), 387–394 MR0208053 0144.09402 Google Scholar[5] A. M. Fink, On the zeros of $y\sp{\prime\prime}+py=0$ with linear, convex and concave p, J. Math. Pures Appl. (9), 46 (1967), 1–10 MR0213651 0153.11401 Google Scholar[6] Don B. Hinton, Clamped end boundary conditions for fourth-order selfadjoint differential equations, Duke Math. J., 34 (1967), 131–138 10.1215/S0012-7094-67-03415-1 MR0208054 0148.06402 CrossrefISIGoogle Scholar[7] Walter Leighton, On the zeros of solutions of a second-order linear differential equation, J. Math. Pures Appl. (9), 44 (1965), 297–310 MR0186868 0128.30803 Google Scholar[8] Walter Leighton, Erratum: “On the zero of solutions of a second-order linear differential equation”, J. Math. Pures Appl. (9), 46 (1967), 10– MR0214856 0189.37401 Google Scholar[9] Walter Leighton and , Zeev Nehari, On the oscillation of solutions of self-adjoint linear differential equations of the fourth order, Trans. Amer. Math. Soc., 89 (1958), 325–377 MR0102639 0084.08104 CrossrefGoogle Scholar[10] A. Ju. Levin, Distribution of the zeros of solutions of a linear differential equation, Soviet Math. Dokl., 5 (1964), 818–821 0117.05003 Google Scholar[11] L. D. Nikolenko, Some criteria for non-oscillation of a fourth order differential equation, Dokl. Akad. Nauk SSSR (N.S.), 114 (1957), 483–485 MR0091394 0079.11101 Google Scholar[12] William T. Reid, Riccati matrix differential equations and non-oscillation criteria for associated linear differential systems, Pacific J. Math., 13 (1963), 665–685 MR0155049 0119.07401 CrossrefISIGoogle Scholar[13] Thomas L. Sherman, Properties of solutions of $n{\rm th}$ order linear differential equations, Pacific J. Math., 15 (1965), 1045–1060 MR0185185 0132.31204 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Lyapunov-Type Inequalities for Higher-Order Linear Differential EquationsLyapunov Inequalities and Applications | 28 January 2021 Cross Ref Existence of conjugate points for second and fourth order differential equationsProceedings of the Royal Society of Edinburgh: Section A Mathematics, Vol. 89, No. 3-4 | 14 November 2011 Cross Ref The existence of conjugate points for selfadjoint differential equations of even orderProceedings of the American Mathematical Society, Vol. 56, No. 1 | 1 January 1976 Cross Ref Green's function for n-n boundary value problem and an analogue of Hartman's resultJournal of Mathematical Analysis and Applications, Vol. 51, No. 3 | 1 Sep 1975 Cross Ref Separation Theorems for Self-Adjoint Linear Differential Equations of the Fourth OrderWilliam Brunner MillerSIAM Journal on Mathematical Analysis, Vol. 6, No. 4 | 17 February 2012AbstractPDF (1582 KB)Bounds for Eigenvalues and Conditions for Existence of Conjugate PointsD. O. BanksSIAM Journal on Applied Mathematics, Vol. 27, No. 3 | 12 July 2006AbstractPDF (1060 KB)Oscillation Criteria for a Class of Fourth Order Differential EquationsKurt KreithSIAM Journal on Applied Mathematics, Vol. 22, No. 1 | 12 July 2006AbstractPDF (189 KB) Volume 17, Issue 5| 1969SIAM Journal on Applied Mathematics835-1015 History Submitted:05 January 1968Published online:12 July 2006 InformationCopyright © 1969 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0117086Article page range:pp. 984-991ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics

I. Third Order Linear Homogeneous Differential Equations in Normal Form.- 1. Fundamental Properties of Solutions of the Third Order Linear Homogeneous Differential Equation.- 1. The Normal Form of a Third Order Linear Homogeneous Differential Equation.- 2. Adjoint and Self-adjoint Third Order Linear Differential Equations.- 3. Fundamental Properties of Solutions.- 4. Relationship between Solutions of the Differential Equations (a) and (b).- 5. Integral Identities.- 6. Notion of a Band of Solutions of the First, Second and Third Kinds.- 7. Further Properties of Solutions of the Differential Equation (a) Implied by Properties of Bands.- 8. Weakening of Property (v) for the Laguerre Invariant.- 2. Oscillatory Properties of Solutions of the Differential Equation (a).- 1. Basic Definitions.- 2. Sufficient Conditions for the Differential Equation (a) to Be Disconjugate.- 3. Sufficient Conditions for Oscillatoricity of Solutions of the Differencial Equation (a).- 4. Further Conditions Concerning Oscillatoricity or Non-oscillatoricity of Solutions of the Differential Equation (a).- 5. Relation between Solutions without Zeros and Oscillatoricity of the Differential Equation (a).- 6. Sufficient Conditions for Oscillatoricity of Solutions of the Differential Equation (a) in the Case A(x) ? 0, x ? (a, ?).- 7. Conjugate Points, Principal Solutions and the Relationship between the Adjoint Differential Equations (a) and (b).- 8. Criteria for Oscillatoricity of the Differential Equations (a) and (b) Implied by Properties of Conjugate Points.- 9. Further Criteria for Oscillatoricity of the Differential Equation (b).- 10. The Number of Oscillatory Solutions in a Fundamental System of Solutions of the Differential Equation (a).- 11. Criteria for Oscillatoricity of Solutions of the Differential Equation (a) in the Case that the Laguerre Invariant Does Not Satisfy Condition (v).- 12. The Case, When the Laguerre Invariant Is an Oscillatory Function of x.- 13. The Differential Equation (a) Having All Solutions Oscillatory in a Given Interval.- 3. Asymptotic Properties of Solutions of the Differential Equations (a) and (b).- 1. Asymptotic Properties of Solutions without Zeros of the Differential Equations (a) and (b).- 2. Asymptotic Properties of Oscillatory Solutions of the Differential Equation (b).- 3. Asymptotic Properties of All Solutions of the Differential Equation (a).- 4. Boundary Value Problems.- 1. The Green Function and Its Applications.- 2. Further Applications of Integral Equations to the Solution of Boundary-value Problems.- 3. Generalized Sturm Theory for Third Order Boundary-value Problems.- 4. Special Boundary-value Problems.- II. Third Order Linear Homogeneous Differential Equations with Continuous Coefficients.- 5. Principal Properties of Solutions of Linear Homogeneous Third Order Differential Equations with Continuous Coefficients.- 1. Principal Properties of Solutions of the Differential Equation (A).- 2. Bands of Solutions of the Differential Equation (A).- 3. Application of Bands to Solving a Three-point Boundary-value Problem.- 6. Conditions for Disconjugateness, Non-oscillatoricity and Oscillatoricity of Solutions of the Differential Equation (A).- 1. Conditions for Disconjugateness of Solutions of the Differential Equation (A).- 2. Solutions without Zeros and Their Relation to Oscillatoricity of Solutions of the Differential Equation (A).- 3. Conditions for the Existence of Oscillatory Solutions of the Differential Equation (A).- 4. On Uniqueness of Solutions without Zeros of the Differential Equation (A).- 5. Some Properties of Solutions of the Differential Equation (A) with r(x) ? 0.- 7. Comparison Theorems for Differential Equations of Type (A) and Their Applications.- 1. Comparison Theorems.- 2. A Simple Application of Comparison Theorems.- 3. Remark on Asymptotic Properties of Solutions of the Differential Equation (A).- III. Concluding Remarks.- 1. Special Forms of Third Order Differential Equations.- 2. Remark on Mutual Transformation of Solutions of Third Order Differential Equations.- IV. Applications of Third Order Linear Differential Equation Theory.- 8. Some Applications of Linear Third Order Differential Equation Theory to Non-linear Third Order Problems.- 1. Application of Quasi-linearization to Certain Problems Involving Ordinary Third Order Differential Equations.- 2. Three-point Boundary-value Problems for Third Order Non-linear Ordinary Differential Equations.- 3. On Properties of Solutions of a Certain Non-linear Third Order Differential Equation.- 9. Physical and Engineering Applications of Third Order Differential Equations.- 1. On Deflection of a Curved Beam.- 2. Three-layer Beam.- 3. Survey of Some Other Applications of Third Order Differential Equations.- References.

One of types of stochastic retarded systems is under consideration. Our scheme of analysis is applicable for investigation of linear and nonlinear differential difference equations with single and multiple constant delays, linear differential equations with variable delays, linear neutral delay differential equations, and separate linear differential difference equations. In addition, a problem of sensitivity estimation for linear dynamic systems described by stochastic differential difference equations can be explored too. All these schemes are based on extensions of phase spaces.

One of types of stochastic retarded systems is under consideration. Our scheme of analysis is applicable for investigation of linear and nonlinear differential difference equations with single and multiple constant delays, linear differential equations with variable delays, linear neutral delay differential equations, and separate linear differential difference equations. In addition, a problem of sensitivity estimation for linear dynamic systems described by stochastic differential difference equations can be explored too. All these schemes are based on extensions of phase spaces.

Preliminaries. Introduction. Initial Value Problem, Oscillation and Nonoscillation. Continuability and Boundedness. Some Basic Results for Second Order Linear Ordinary Differential Equations. Some Useful Criteria for First Order. Some Useful Results from Analysis and Fixed Point Theorems. Notes and General Discussions. References. Oscillations of Differential Equations with Deviating Arguments. Oscillation Theorems (I). Oscillation Theorems (II). Comparison Theorems for Second Order Functional Differential Equations. Oscillation of Functional Equations with a Damping Term. Oscillation of Second Order Linear Delay Differential Equations. Oscillation of Forced Functional Differential Equations. Oscillation of Functional Equations with Damping and Forcing Terms. Necessary and Sufficient Conditions for the Oscillation of Forced Equations. Oscillation for Perturbed Differential Equations. Asymptotic Behavior of Oscillatory Solutions of Functional Equations. Notes and General Discussions. References. Oscillation of Neutral Functional Differential Equations. Oscillation of Nonlinear Neutral Equations. Oscillation of Neutral Equations with Damping. Oscillation of Forced Neutral Equations. Oscillation of Neutral Equations with Mixed Type. Necessary and Sufficient Conditions for Oscillations of Neutral Equations with Deviating Arguments. Comparison and Linearized Oscillation Theorems for Neutral Equations. Existence of Nonoscillatory Solutions of Neutral Delay Differential Equations. Asymptotic Behavior of Nonoscillatory Solutions of Neutral Nonlinear Delay Differential Equations. Notes and General Discussions. References. Conjugacy and Nonoscillation for Second Order Differential Equations. Conjugacy of Linear Second Order Ordinary Differential Equations. Nonoscillation Theorems. Integral Conditions and Nonoscillations. Notes and General Discussions. References. Oscillation of Impulsive Differential Equations. Oscillation Criteria for Impulsive Delay Differential Equations. Oscillation of Second Order Linear Differential Equations with Impulses. Notes and General Discussions. References. Subject Index with Deviating Arguments. Oscillation of Neutral Functional Differential Equations. Conjugacy and Nonoscillation for Second Order Differential Equations. Oscillation of Impulsive Differential Equations.

We obtain new sufficient criteria for the oscillation of all solutions of linear delay difference equations with several (variable) finite delays. Our results relax numerous well-known limes inferior-type oscillation criteria from the literature by letting the limes inferior be replaced by the limes superior under some additional assumptions related to slow variation. On the other hand, our findings generalize an oscillation criterion recently given for the case of a constant, single delay.

New oscillation criteria are given for second order nonlinear ordinary differential equations with alternating coefficients. The results involve a condition obtained by Kamenev for linear differential equations. The obtained criterion for superlinear differential equations is a complement of the work established by Kwong and Wong, and Philos, for sublinear differential equations and by Yan for linear differential equations.

We consider systems of linear differential and difference equations \delta Y(x)/dx = A(x) Y(x), \: \sigma s(Y(x)) = B(x)Y(x) with \delta = \frac{d}{dx} , \sigma a shift operator \sigma(x) = x+a , q -dilation operator \sigma(x) = qx or Mahler operator \sigma(x) = x^p and systems of two linear difference equations \sigma_1 Y(x) =A(x)Y(x), \: \sigma_2 Y(x) =B(x)Y(x) with (\sigma_1,\sigma_2) a sufficiently independent pair of shift operators, pair of q -dilation operators or pair of Mahler operators. Here A(x) and B(x) are n\times n matrices with rational function entries. Assuming a consistency hypothesis, we show that such systems can be reduced to a system of a very simple form. Using this we characterize functions satisfying two linear scalar differential or difference equations with respect to these operators. We also indicate how these results have consequences both in the theory of automatic sets, leading to a new proof of Cobham's Theorem, and in the Galois theories of linear difference and differential equations, leading to hypertranscendence results.

This chapter presents oscillation and nonoscillation theory for solutions of second order nonlinear differential equations of superlinear type. Section 4.1 deals with the oscillation of superlinear equations with sign changing coefficients. Here, first we shall discuss some results which involve integrals and weighted integrals of the alternating coefficients, and then provide several criteria which use average behaviors of these integrals. More general averages such as ‘weighted averages’ and ‘iterated averages’ are also employed. In Section 4.2, we impose some additional conditions on the superlinear terms which allow us to proceed further and extend and improve some of the results established in Section 4.1. In fact, an asymptotic study has been made and interesting oscillation criteria have been proved. In Section 4.3, first we shall provide sufficient conditions which guarantee the existence of nonoscillatory solutions, and then present necessary and sufficient conditions for the oscillation of superlinear equations. Oscillation results via comparison of nonlinear equations of the same form as well as with linear ones of the same order are also established. In Section 4.4, we shall extend some of the results of the previous sections and establish several new oscillation criteria for more general superlinear equations. Necessary and sufficient conditions for such equations to be oscillatory are also given. Section 4.5 deals with the oscillation of forced-superlinear differential equations with alternating coefficients. Finally, Section 4.6 presents the oscillation and nonoscillation criteria for second order superlinear equations with nonlinear damping terms.

Hille and Nehari type criteria for third-order dynamic equations

In this note we give a simple geometrical picture that explains why an inverted pendulum is stabilized by high frequency vibrations.