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Entire solutions of certain class of differential-difference equations
- Fengrong Zhang^{1},
- Nana Liu^{1},
- Weiran Lü^{1}Email author and
- Chungchun Yang^{2}
https://doi.org/10.1186/s13662-015-0488-5
© Zhang et al.; licensee Springer. 2015
Received: 30 December 2014
Accepted: 27 April 2015
Published: 9 May 2015
Abstract
As a continuation of our previous studies Liu et al. (J. Inequal. Appl. 2014:63, 2014), we will discuss the transcendental entire solutions of the following type of differential-difference equation: \(f^{3}(z)+P_{1}(z, \Delta f ,\ldots, f',\ldots, f^{(k)} ) =\lambda_{1}e^{\alpha_{1} z}+\lambda_{2}e^{\alpha_{2} z}\), where \(P_{1}\) is a linear polynomial in \(f, \Delta f,\ldots, f^{(k)} \), with polynomials as its coefficients, and \(\lambda_{1},\lambda_{2},\alpha_{1},\alpha_{2}\in\mathbb{C}\) are nonzero constants such that \(\alpha_{1}\neq\alpha_{2}\).
Keywords
- transcendental entire solution
- differential-difference equation
- Nevanlinna theory
MSC
- 30D35
- 30A06
1 Introduction
In this paper, we shall adopt the standard notations in Nevanlinna’s value distribution theory of meromorphic functions. For example, the characteristic function \(T(r,f)\), the counting function of the poles \(N(r,f)\), and the proximity function \(m(r,f)\) (see, e.g., [1, 2]).
Let f be a meromorphic function. Recall that \(\alpha\not\equiv0, \infty\) is a small function with respect to f, if \(T (r,\alpha) = S(r, f )\), where \(S(r,f)\) denotes any quantity satisfying \(S(r,f)=o\{T(r,f)\}\) as \(r\to+\infty\), possibly outside a set of r of finite linear measure.
For completeness, we recall basic notions to this end: Given a meromorphic function f and a constant c, \(f(z+c)\) is called a shift of \(f(z)\). For the sake of simplicity, we let \(\Delta f(z)=f(z+1)-f(z)\), \(\Delta^{n}f(z)=\Delta(\Delta^{n-1}f(z)) \) (\(n\geq 2\)). As for a difference product, we mean a difference monomial of type \(\prod^{k}_{j=1}f(z+c_{j})^{n_{j}}\), where \(c_{1},\ldots, c_{k}\) are complex constants, and \(n_{1},\ldots, n_{k}\) are natural numbers.
Definition 1.1
A difference polynomial, resp. a differential-difference polynomial, in f is a finite sum of difference products of f and its shifts, resp. of products of f, derivatives of f and of their shifts, with all the coefficients of these monomials being small functions of f.
Definition 1.2
Moreover, a functional equation involving f and its shifts as well as their derivatives is called a differential-difference equation.
Recently, many papers (see, e.g., [3] and [4]) are focused on complex difference polynomials and difference equations. Lots of results have been obtained by using value distribution theory (see, e.g., [5] and [6, 7]).
Yang and Laine [8] considered the following difference equation and proved the following.
Theorem A
Very recently, Liu et al. [9] proved the following.
Theorem B
- (1)
\(f(z)=c_{1}\exp(\frac{\alpha_{1}}{n}z)\), and \(c_{1}(\exp (\alpha_{1}/n)-1)q=p_{2}\), \(\alpha_{1}=n\alpha_{2}\);
- (2)
\(f(z)=c_{2}\exp(\frac{\alpha_{2}}{n}z)\), and \(c_{2}(\exp (\alpha_{2}/n)-1)q=p_{1}\), \(\alpha_{2}=n\alpha_{1}\),
In this paper, we are going to discuss the case when \(n=3\) in the above theorem and prove some results in Section 3.
2 Preliminaries
In order to prove our conclusions, we need some lemmas.
Lemma 2.1
([8])
The following result is a Clunie [10] type lemma for the difference-differential polynomials of a meromorphic function f. It can be proved by applying Lemma 2.1 and stated as follows.
Proposition 2.1
If in the above lemma, \(H(z,f)=f^{n}\), \(P(z,f)\), and \(Q(z,f)\) are differential-difference polynomials in f, then \(m(r,P(z,f))=S(r,f)\).
The following result is an analogue of a result due to Mohon’ko and Mohon’ko [12] for differential equations.
Lemma 2.2
([4])
Remark 2.1
By some further analysis on the proof of Theorem 2.4 in [6], we can obtain the conclusion of Lemma 2.2 when \(P (z, f )\) is a differential-difference polynomial in f.
Lemma 2.3
([13])
Assume that c is a nonzero constant, α is a non-constant meromorphic function. Then the differential equation \(f^{2}+(cf^{(n)})^{2}=\alpha\) has no transcendental meromorphic solutions satisfying \(T(r,\alpha)=S(r,f)\).
3 Main results
Now, we shall derive the following result similar to that of Theorems A and B, for extended differential-difference equations. First, we will prove the following result.
Theorem 3.1
- (1)
\(T(r,f)=N_{1)}(r,\frac{1}{f})+S(r,f)\);
- (2)
\(f(z)=c_{1}\exp(\frac{\alpha_{1}}{3}z)\), and \(c_{1}(\exp(\alpha_{1}/3)-1)q=p_{2}\), \(\alpha_{1}=3\alpha_{2}\);
- (3)
\(f(z)=c_{2}\exp(\frac{\alpha_{2}}{3}z)\), and \(c_{2}(\exp(\alpha_{2}/3)-1)q=p_{1}\), \(\alpha_{2}=3\alpha_{1}\),
Proof of Theorem 3.1
This completes the proof of Theorem 3.1. □
Remark 3.1
Note \(f(z)=e^{\pi i z}+e^{-\pi i z}=2i \sin(\pi i z)\) has infinitely many zeros, and \(T(r,f)=N_{1)}(r,\frac{1}{f})+S(r,f)\). Thus, we would like to pose the following conjecture.
Conjecture 3.1
If \(\alpha_{1}\neq \alpha_{2}\), \(\alpha_{1}+\alpha_{2}\neq0\), then the conclusion (1) of Theorem 3.1 is impossible, in fact, any entire solution f of (3.1) must have 0 as its Picard exceptional value.
For \(\alpha_{1}+\alpha_{2}=0\), we have the following.
Theorem 3.2
Proof of Theorem 3.2
Now we distinguish four cases.
Case 1. Assume that \(v=1\), then \(\alpha=6n\pi i\), substituting \(v=1\) into \(a_{2}\) gives \(c_{1}c_{2}=0\), this is a contradiction, thus \(v\neq1\).
Case 2. If \(v=-1\), then \(\alpha=(6n\pi\pm3\pi)i\). By substituting \(v=-1\) into \(a_{2}\), we deduce \((8q_{3}-4q_{2}+2q_{1})^{3}=27\lambda_{1}\lambda_{2}\).
Case 3. While \(v=\frac{1\pm\sqrt{3}i}{2}\), in a similar way to above, we get \(\alpha=(6n\pi\pm\pi)i\) and \(q_{1}+q_{2}=0\). Moreover, \((\pm\sqrt{3} i q_{1}-q_{3})^{3}=27\lambda_{1}\lambda_{2}\) and \(\deg q_{1}=\deg q_{2}=\deg q_{3}\).
This also completes the proof of Theorem 3.2. □
Theorem 3.3
Proof of Theorem 3.3
It is not difficult to see that a similar argument can be used to obtain the following result.
Theorem 3.4
4 Conclusion
Declarations
Acknowledgements
The authors would like to thank the referee for his/her several important suggestions and for pointing out some errors in our original manuscript. These comments greatly improved the readability of the paper. This work was supported by the research funds of China University of Petroleum (No. 14CX06085A), the Special Funds of the National Natural Science Foundation of China (No. 11426217) and the Fundamental Research Funds for the Central Universities (No. 14CX02012A).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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