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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 $\alpha$
, 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 $\alpha =[2(m+1)-1]/[2(m+1)]$
for some integer $m\geq 0$
if this equation admits a nontrivial solution such that $\lambda (f)<\infty$
. This partially answers a question of Ishizaki. Finally, letting $b_2\not =0$
and $b_3$
be constants and $l$
and $s$
be relatively prime integers such that $l> s\geq 1$
, 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 $\max \{\lambda (f_1),\,\lambda (f_2)\}<\infty$
. In particular, we precisely characterize all solutions such that $\lambda (f)<\infty$
when $l=2$
and $l=4$
.
In this work, we consider the first-order difference equation with general argument\[ \varDelta x(n)+p(n)x\left( \tau (n)\right) =0,\quad n\geq 0, \]where $(p(n))$
is a sequence of non-negative real numbers, $(\tau (n))$
is a sequence of integers such that $\tau (n)< n$
for$\ n\in \mathbf {N},\,$
and$\ \lim _{n\rightarrow \infty }\tau (n)=\infty.$
Under the assumption that the deviating argument is not necessarily monotone, we obtain some new oscillation conditions and improve the all known results for the above equation in the literature, involving only upper and only lower limit conditions. Two examples illustrating the results are also given.
Let λ > 0 and let
be the Bessel operator on ℝ+ := (0,∞). We show that the oscillation operator 𝒪(RΔλ,∗) and variation operator 𝒱ρ(RΔλ,∗) of the Riesz transform RΔλ associated with Δλ are both bounded on Lp(ℝ+, dmλ) for p ∈ (1,∞), from L1(ℝ+, dmλ) to L1,∞(ℝ+, dmλ), and from L∞(ℝ+, dmλ) to BMO(ℝ+, dmλ), where ρ ∈ (2,∞) and dmλ(x) := x2λ dx. As an application, we give the corresponding Lp-estimates for β-jump operators and the number of up-crossings.
