[Paper Review] On the oscillation of certain second-order linear differential equations

Yueyang Zhang|arXiv (Cornell University)|Oct 12, 2021

Meromorphic and Entire Functions22 references4 citations

TL;DR

This paper investigates the oscillation behavior of second-order linear differential equations with coefficients involving sums of exponential functions. It provides a complete characterization of entire solutions with finite exponent of convergence of zeros, proving that for equations of the form $ f'' - (e^{lz} + b_2e^{sz} + b_3)f = 0 $, two linearly independent solutions with $ \lambda(f) < \infty $ exist only if $ l = 2 $, and fully describes such solutions for $ l = 2 $ and $ l = 4 $.

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 $p_1(z)$ and $p_2(z)$ have the same degree $k\geq 1$ and distinct leading coefficients $1$ and $α$, respectively, we solve entire solutions of the 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, $P(z,f)$ is a differential polynomial in $f$ of degree $\leq n-1$ with coefficients having polynomial growth. Second, we study the oscillation of the second-order differential equation $f''-[b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}]f=0$ and prove that $α=[2(m+1)-1]/[2(m+1)]$ for some integer $m\geq 0$ if this equation admits a nontrivial solution such that $λ(f)&lt;\infty$. This partially answers a question of Ishizaki. Finally, letting $b_2 ot=0$ and $b_3$ be constants and $l$ and $s$ be relatively prime integers such that $l&gt; s\geq 1$, we prove that $l=2$ if the equation $f''-(e^{lz}+b_2e^{sz}+b_3)f=0$ admits two linearly independent solutions $f_1$ and $f_2$ such that $\max\{λ(f_1),λ(f_2)\}&lt;\infty$. In particular, we precisely characterize all solutions such that $λ(f)&lt;\infty$ when $l=2$ and $l=4$.

Motivation & Objective

• To characterize all entire solutions of second-order linear differential equations with coefficients that are sums of two exponential-type functions and polynomial growth.
• To resolve a question posed by Ishizaki regarding the oscillation of equations with coefficients $ b_1(z)e^{p_1(z)} + b_2(z)e^{p_2(z)} $, particularly when $ \lambda(f) < \infty $.
• To determine the necessary conditions on the exponents $ l $ and $ s $ for the equation $ f'' - (e^{lz} + b_2e^{sz} + b_3)f = 0 $ to admit two linearly independent solutions with finite exponent of convergence of zeros.
• To provide explicit recursive formulas and structural descriptions of such solutions for $ l = 2 $ and $ l = 4 $, and to conjecture that these are the only possible cases.

Proposed method

• Analyzes Tumura–Clunie type differential equations of the form $ f^n + P(z,f) = b_1(z)e^{p_1(z)} + b_2(z)e^{p_2(z)} $, where $ P(z,f) $ is a differential polynomial of degree $ \leq n-1 $ with polynomial coefficients.
• Applies Nevanlinna theory and Hadamard’s factorization theorem to derive the structure of entire solutions with $ \lambda(f) < \infty $, linking them to exponential polynomials.
• Uses recursive coefficient relations derived from substitution into the differential equation to determine conditions under which solutions with finite zero convergence exist.
• Employs the method of undetermined coefficients and structural analysis of exponential polynomials to classify solutions for $ l = 2 $ and $ l = 4 $, showing that $ b_2 $ and $ b_3 $ must satisfy specific algebraic constraints.
• Applies theorems on the growth and zero distribution of solutions to constrain possible values of $ l $, proving that $ l = 2 $ is the only possibility when two independent solutions with $ \lambda(f) < \infty $ exist.
• Uses symmetry and sign analysis of the exponents in the solution ansatz to eliminate other values of $ l $, particularly ruling out $ l \geq 3 $ when $ l $ and $ s $ are coprime integers.

Experimental results

Research questions

• RQ1 Under what conditions does the differential equation $ f'' - (b_1(z)e^{p_1(z)} + b_2(z)e^{p_2(z)})f = 0 $ admit a nontrivial solution with $ \lambda(f) < \infty $, given that $ p_1(z) $ and $ p_2(z) $ are polynomials of the same degree with distinct leading coefficients?
• RQ2 What is the necessary condition on the ratio of the leading coefficients of $ p_1(z) $ and $ p_2(z) $ for such solutions to exist, as posed by Ishizaki?
• RQ3 For the equation $ f'' - (e^{lz} + b_2e^{sz} + b_3)f = 0 $ with $ l $ and $ s $ relatively prime integers, what values of $ l $ allow two linearly independent solutions with $ \lambda(f) < \infty $?
• RQ4 Can a complete and explicit characterization of all solutions with $ \lambda(f) < \infty $ be given for $ l = 2 $ and $ l = 4 $, and what algebraic constraints must $ b_2 $ and $ b_3 $ satisfy?
• RQ5 Is it possible for such equations to admit solutions with $ \lambda(f) < \infty $ when $ l \neq 2,4 $, or is $ l = 2 $ the only possible case?

Key findings

• For the equation $ f'' - (b_1(z)e^{p_1(z)} + b_2(z)e^{p_2(z)})f = 0 $, if a nontrivial solution satisfies $ \lambda(f) < \infty $, then the ratio of the leading coefficients of $ p_1(z) $ and $ p_2(z) $ must be $ \alpha = \frac{2(m+1)-1}{2(m+1)} $ for some integer $ m \geq 0 $, partially answering a question of Ishizaki.
• When the coefficient is $ A(z) = e^{lz} + b_2e^{sz} + b_3 $ with $ l $ and $ s $ relatively prime integers and $ b_2, b_3 $ constants, two linearly independent solutions with $ \lambda(f) < \infty $ exist only if $ l = 2 $.
• For $ l = 2 $, all solutions with $ \lambda(f) < \infty $ are explicitly characterized: they are of the form $ f = \kappa e^{h} $, where $ \kappa $ is a polynomial in $ e^z $, and $ h $ is a linear combination of $ e^z $ and $ z $, with coefficients satisfying specific recursive relations.
• For $ l = 4 $, a similar explicit characterization is provided, showing that solutions exist only when $ b_2 $ and $ b_3 $ satisfy $ b_2 = c_0(k_1 - k_2) $ and $ 4b_3 = (k_1 + k_2 + 1)^2 $, where $ k_1, k_2 $ are nonnegative integers.
• The paper constructs two linearly independent solutions $ f_1 $ and $ f_2 $ with $ \lambda(f_1), \lambda(f_2) < \infty $, and shows that $ \min\{\lambda(f_1), \lambda(f_2)\} = 0 $ is possible by choosing one of the degrees $ k_1 $ or $ k_2 $ to be zero.
• The authors conjecture that no such solutions exist for $ l \neq 2,4 $, and provide recursive formulas for the coefficients of the solution polynomials, though existence of nontrivial solutions satisfying these for $ l \geq 3 $ remains unproven.