On the oscillation of certain second-order linear differential equations

نویسندگان

چکیده

Abstract This paper consists of three parts: First, letting $b_1(z)$ , $b_2(z)$ $p_1(z)$ and $p_2(z)$ be nonzero polynomials such that have the same degree $k\geq 1$ distinct leading coefficients $1$ $\alpha$ respectively, we solve entire solutions Tumura–Clunie type differential equation $f^{n}+P(z,\,f)=b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}$ where $n\geq 2$ is an integer, $P(z,\,f)$ a polynomial in $f$ $\leq n-1$ with having growth. Second, study oscillation second-order $f''-[b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}]f=0$ prove $\alpha =[2(m+1)-1]/[2(m+1)]$ for some integer $m\geq 0$ if this admits nontrivial solution $\lambda (f)<\infty$ . partially answers question Ishizaki. Finally, $b_2\not =0$ $b_3$ constants $l$ $s$ relatively prime integers $l> s\geq $l=2$ $f''-(e^{lz}+b_2e^{sz}+b_3)f=0$ two linearly independent $f_1$ $f_2$ $\max \{\lambda (f_1),\,\lambda (f_2)\}<\infty$ In particular, precisely characterize all when $l=4$

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

On the stability of linear differential equations of second order

The aim of this paper is to investigate the Hyers-Ulam stability of the  linear differential equation$$y''(x)+alpha y'(x)+beta y(x)=f(x)$$in general case, where $yin C^2[a,b],$  $fin C[a,b]$ and $-infty

متن کامل

On the Oscillation of Second Order Linear Impulsive Differential Equations

For the second order linear impulsive differential equation with oscillatory coefficient ⎧⎨ ⎩ (r(t)x′(t))′ +h(t)x(t) = 0, t = tk, tk t0, k = 1,2, · · · , x(t+ k ) = akx(tk), x ′(t+ k ) = bkx ′(tk), k = 1,2, · · · , x(t+ 0 ) = x0, x ′(t+ 0 ) = x ′ 0, (E) where h can be changed sign on [t0,∞) , by using the equivalence transformation, we establish an associated impulsive differential equation wit...

متن کامل

Oscillation of Second Order Half-linear Differential Equations with Damping

This paper is concerned with a class of second order half-linear damped differential equations. Using the generalized Riccati transformation and the averaging technique, new oscillation criteria are obtained which are either extensions of or complementary to a number of the existing results. 2000 Mathematics Subject Classification: 34A30, 34C10.

متن کامل

Oscillation of Certain Second-Order Sub-Half-Linear Neutral Impulsive Differential Equations

By introducing auxiliary functions, we investigate the oscillation of a class of second-order subhalf-linear neutral impulsive differential equations of the form r t φβ z′ t ′ p t φα x σ t 0, t / θk,Δφβ z′ t |t θk qkφα x σ θk 0,Δx t |t θk 0, where β > α > 0, z t x t λ t x τ t . Several oscillation criteria for the above equation are established in both the case 0 ≤ λ t ≤ 1 and the case −1 < −μ ...

متن کامل

Oscillation Criteria for Second-order Linear Differential Equations^)

where p(x) is a continuous positive function for 0<x< oo. Equation (1) is said to be nonoscillatory in (a, oo) if no solution of (1) vanishes more than once in this interval. Because of the Sturm separation theorem, this is equivalent to the existence of a solution which does not vanish at all in (a, oo). The equation will be called nonoscillatory—without the interval being mentioned —if there ...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

ژورنال

عنوان ژورنال: Proceedings

سال: 2022

ISSN: ['0890-1740']

DOI: https://doi.org/10.1017/prm.2022.80