Asymptotic Behaviour of Disconjugate nth Order Differential Equations

1971 ◽  
Vol 23 (2) ◽  
pp. 293-314 ◽  
Author(s):  
D. Willett
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Linear Differential Equation ◽  
Ordered Set ◽  
Minimal Solution ◽  
Linear Differential ◽  
Image Position

An ordered set (u1, …, un) of positive Cn(a, b)-solutions of the linear differential equation1.1will be called fundamental principal system on [a, b) provided that1.2and1.3A system (u1, …, un) satisfying just (1.2) will be called a principal system on [a, b). In any principal system (u1, …, un) the solution u1 will be called a minimal solution.

Generalized de la Vallée Poussin Disconjugacy Tests for Linear Differential Equations(1)

1971 ◽  
Vol 14 (3) ◽  
pp. 419-428 ◽  
Author(s):  
D. Willett
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Linear Equation ◽  
Nontrivial Solution ◽  
Oscillatory Behavior ◽  
Bounded Interval ◽  
Linear Differential ◽  
Image Position ◽  
De La Vallée Poussin

In this paper, we study the oscillatory behavior of the solutions of the linear differential equation(1.1)where(1.2)and all functions are assumed to be continuous on a bounded interval [a, b). An «th-order linear equation is said to be disconjugate on an interval I provided it has no nontrivial solution with more than n — 1 zeros, counting multiplicities, in I.

A non-oscillation theorem for nonlinear differential equations with p-Laplacian

2006 ◽  
Vol 136 (3) ◽  
pp. 633-647 ◽  
Author(s):  
Jitsuro Sugie ◽  
Masakazu Onitsuka
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Linear Differential Equation ◽  
Phase Plane ◽  
Nonlinear Differential Equations ◽  
Phase Plane Analysis ◽  
Oscillation Theorem ◽  
Plane Analysis ◽  
Linear Differential

The equation considered in this paper is tp(φp(x′))′ + g(x) = 0, where φp(x′) = |x′|p−2x′ with p > 1, and g(x) satisfies the signum condition xg(x) > 0 if x ≠ 0 but is not assumed to be monotone. Our main objective is to establish a criterion on g(x) for all non-trivial solutions to be non-oscillatory. The criterion is the best possible. The method used here is the phase-plane analysis of a system equivalent to this differential equation. The asymptotic behaviour is also examined in detail for eventually positive solutions of a certain half-linear differential equation.

Sturmian comparison theorem for half-linear second-order differential equations

1995 ◽  
Vol 125 (6) ◽  
pp. 1193-1204 ◽  
Author(s):  
Horng Jaan Li ◽  
Cheh Chih Yeh
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Comparison Theorem ◽  
Fixed Number ◽  
Lyapunov Inequality ◽  
Second Order Differential Equations ◽  
Wirtinger Inequality ◽  
Linear Differential ◽  
Image Position

Let φ:ℝ→ℝ be defined by φ(s) = |s|p−2s, with p > 1 a fixed number. We extend Sturm Comparison Theorem of the linear differential equationto the nonlinear differential equationby using the Wirtinger inequality. A Lyapunov inequality and some oscillation criteria of (E) are also given.

scholarly journals On the Solution of Differential Equations by Definite Integrals

1931 ◽  
Vol 2 (4) ◽  
pp. 189-204 ◽  
Author(s):  
E. T. Whittaker
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Positive Direction ◽  
Negative Direction ◽  
Definite Integrals ◽  
Particular Solution ◽  
Linear Differential ◽  
Image Position ◽  
Path Of Integration

It is well known that in many cases the solutions of a linear differential equation can be expressed as definite integrals, different solutions of the same equation being represented by integrals which have the same integrand, but different paths of integration. Thus, the various solutions of the hypergeometric differential equationcan be represented by integrals of the typethe path of integration being (for one particular solution) a closed circuit encircling the point t = 0 in the positive direction, then the point t = 1 in the positive direction, then the point t = 0 in the negative direction, and lastly the point t = 1 in the negative direction; or (for another particular solution) an arc in the t-plane joining the points t = 1 and t = ∞.

On a Relation Between a Theorem of Hartman and a Theorem of Sherman

1973 ◽  
Vol 16 (2) ◽  
pp. 275-281 ◽  
Author(s):  
A. C. Peterson
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
General Result ◽  
Nonlinear Differential Equations ◽  
Conjugate Point ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
Linear Differential ◽  
Image Position

We are concerned with the nth-order linear differential equation1where the coefficients are assumed to be continuous. Hartman [1] proved that (see Definition 2) the first conjugate point η1(t) of t satisfies2Hartman actually proved a more general result which has very important applications in nonlinear differential equations.

Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kusano Takaŝi ◽  
Jelena V. Manojlović
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Continuous Function ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Regular Variation ◽  
Asymptotic Formulas ◽  
Asymptotic Behavior Of Solutions ◽  
Positive Continuous Function ◽  
Linear Differential

AbstractWe study the asymptotic behavior of eventually positive solutions of the second-order half-linear differential equation(p(t)\lvert x^{\prime}\rvert^{\alpha}\operatorname{sgn}x^{\prime})^{\prime}+q(% t)\lvert x\rvert^{\alpha}\operatorname{sgn}x=0,where q is a continuous function which may take both positive and negative values in any neighborhood of infinity and p is a positive continuous function satisfying one of the conditions\int_{a}^{\infty}\frac{ds}{p(s)^{1/\alpha}}=\infty\quad\text{or}\quad\int_{a}^% {\infty}\frac{ds}{p(s)^{1/\alpha}}<\infty.The asymptotic formulas for generalized regularly varying solutions are established using the Karamata theory of regular variation.

A property of the asymptotic series for a class of Titchmarsh–Weyl m-functions

1986 ◽  
Vol 102 (3-4) ◽  
pp. 253-257 ◽  
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Linear Differential Equation ◽  
Asymptotic Series ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
The Real ◽  
Linear Differential ◽  
Image Position

SynopsisIn an earlier paper [6] we showed that if q ϵ CN[0, ε) for some ε > 0, then the Titchmarsh–Weyl m(λ) function associated with the second order linear differential equationhas the asymptotic expansionas |A| →∞ in a sector of the form 0 < δ < arg λ < π – δ.We show that if the real valued function q admits the expansionin a neighbourhood of 0, then

scholarly journals On [p,q]-order of growth of solutions of complex linear differential equations near a singular point

Filomat ◽  
2019 ◽  
Vol 33 (13) ◽  
pp. 4013-4020
Author(s):  
Jianren Long ◽  
Sangui Zeng
Keyword(s):  
Differential Equation ◽  
Singular Point ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Linear Differential Equations ◽  
Order Of Growth ◽  
Complex Linear ◽  
Linear Differential ◽  
Growth Of Solutions

We investigate the [p,q]-order of growth of solutions of the following complex linear differential equation f(k)+Ak-1(z) f(k-1) + ...+ A1(z) f? + A0(z) f = 0, where Aj(z) are analytic in C? - {z0}, z0 ? C. Some estimations of [p,q]-order of growth of solutions of the equation are obtained, which is generalization of previous results from Fettouch-Hamouda.

scholarly journals On the Growth of Solutions of Some Second Order Linear Differential Equations with Entire Coefficients

2013 ◽  
Vol 21 (2) ◽  
pp. 35-52
Author(s):  
Benharrat Belaïdi ◽  
Habib Habib
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Infinite Order ◽  
Entire Functions ◽  
Complex Numbers ◽  
Order Of Growth ◽  
Hyper Order ◽  
Linear Differential ◽  
Growth Of Solutions

Abstract In this paper, we investigate the order and the hyper-order of growth of solutions of the linear differential equation where n≥2 is an integer, Aj (z) (≢0) (j = 1,2) are entire functions with max {σ A(j) : (j = 1,2} < 1, Q (z) = qmzm + ... + q1z + q0 is a nonoonstant polynomial and a1, a2 are complex numbers. Under some conditions, we prove that every solution f (z) ≢ 0 of the above equation is of infinite order and hyper-order 1.

Solution Space Decompositions for nth Order Linear Differential Equations

1975 ◽  
Vol 27 (3) ◽  
pp. 508-512
Author(s):  
G. B. Gustafson ◽  
S. Sedziwy
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Solution Space ◽  
Decomposition Theorem ◽  
Linear Differential Equations ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Linear Differential ◽  
Image Position

Consider the wth order scalar ordinary differential equationwith pr ∈ C([0, ∞) → R ) . The purpose of this paper is to establish the following:DECOMPOSITION THEOREM. The solution space X of (1.1) has a direct sum Decompositionwhere M1 and M2 are subspaces of X such that(1) each solution in M1\﹛0﹜ is nonzero for sufficiently large t ﹛nono sdilatory) ;(2) each solution in M2 has infinitely many zeros ﹛oscillatory).

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