On Solutions to Some Nonlinear Difference and Differential Equations

Chen, Wei; Hu, Pei-Chu; Zhang, Ying-Ying

doi:10.4134/jkms.j150296

Cited by 5 publications

(9 citation statements)

References 15 publications

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“…Take f (z) = e −z + z + 1. By an elementary computation, we see that ( f (z) 2 ) + ( f (z) 2 ) = −4e −2z + 2e −z + 2 is a periodic function, but f (z) is not periodic. The case of k = l is [16, Theorem 1.1].…”

Section: Introduction and Main Resultsmentioning

confidence: 97%

“…Some results on the above equation can be found in [2,11], where h(z) is an entire function with finitely many zeros or a nonzero constant. Subcase 2.1.…”

Section: Lemmasmentioning

confidence: 99%

“…If e −c/2 i e i(h(z+c)+ 1 2 π+2kπ−h(z)) ≡ 1, − e −c/2 i e −i(h(z+c)+ 1 2 π+2kπ+h(z)) + e −2ih(z) ≡ 0, then e −c = −1. By shifting the equation (23) forward, f (z + 2c) 2 − f (z + c) 2 = −e −z and so f (z + 2c) 2 = f (z) 2 , which implies that f (z) is a periodic function with period 2c or 4c.…”

Section: Lemmasmentioning

confidence: 99%

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On the Periodicity of Transcendental Entire functions

Liu

Korhonen

2020

Bull. Aust. Math. Soc.

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According to a conjecture by Yang, if $f(z)f^{(k)}(z)$ is a periodic function, where $f(z)$ is a transcendental entire function and $k$ is a positive integer, then $f(z)$ is also a periodic function. We propose related questions, which can be viewed as difference or differential-difference versions of Yang’s conjecture. We consider the periodicity of a transcendental entire function $f(z)$ when differential, difference or differential-difference polynomials in $f(z)$ are periodic. For instance, we show that if $f(z)^{n}f(z+\unicode[STIX]{x1D702})$ is a periodic function with period $c$, then $f(z)$ is also a periodic function with period $(n+1)c$, where $f(z)$ is a transcendental entire function of hyper-order $\unicode[STIX]{x1D70C}_{2}(f)<1$ and $n\geq 2$ is an integer.

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Section: Introduction and Main Resultsmentioning

confidence: 97%

“…Some results on the above equation can be found in [2,11], where h(z) is an entire function with finitely many zeros or a nonzero constant. Subcase 2.1.…”

Section: Lemmasmentioning

confidence: 99%

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On the Periodicity of Transcendental Entire functions

Liu

Korhonen

2020

Bull. Aust. Math. Soc.

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show abstract

“…For the lemma of logarithmic derivative for functions on

C^{M}

, an important and interesting new result was observed by Li that has applications, for instance, to problems related to normal families and solutions of partial differential equations. It could be nice to see how these results obtained in Nevanlinna theory may be used to solve problems in mixed differential‐difference equations – some results have already been described, with several problems suggested, in , , , , , , , , , , and the references therein.…”

Section: Resultsmentioning

confidence: 98%

“…. , n. As a consequence, following [8,Section 9] or the similar argument in [16, Section 5], estimate (2.13) becomes, with N c (r, f) denoting the counting function of the zero divisor corresponding to W c (f) that depends only on f (but not necessarily on a particular f), Nevanlinna theory may be used to solve problems in mixed differential-difference equations -some results have already been described, with several problems suggested, in [3], [6], [7], [8], [11], [13], [15], [16], [18], [19], [23] and the references therein.…”

Section: Denote Byνmentioning

confidence: 99%