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Entire solutions of certain class of differentialdifference equations
Advances in Difference Equationsvolume 2015, Article number: 150 (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 differentialdifference 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 \(\alpha_{1}\neq\alpha_{2}\).
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 \(\Delta f(z)=f(z+1)f(z)\), \(\Delta^{n}f(z)=\Delta(\Delta^{n1}f(z)) \) (\(n\geq 2\)). 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 differentialdifference 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.
Consider a transcendental meromorphic function f and let
where \(a_{l} \) (\(l=1,2,\ldots, n\)) are small functions of f, and \(c_{lj} \) (\(l=1,2,\ldots, n\); \(j=0,1,\ldots, k\)) are complex constants, and \(n_{lj} \) (\(l=1,2,\ldots, n\); \(j=0,1,\ldots, k\)) are natural numbers.
Definition 1.2
We define the total degree d of \(P(z,f) \) by
Moreover, a functional equation involving f and its shifts as well as their derivatives is called a differentialdifference 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
A nonlinear difference equation
where q is a nonconstant polynomial and \(b,c\in\mathbb{C}\) are nonzero constants, does not admit entire solutions of finite order. If q is a nonzero constant, then the above equation possesses three distinct entire solutions of finite order, provided that \(b=3n\pi\) and \(q^{3}=(1)^{n+1}c^{2}27/4\) for a nonzero integer n.
Very recently, Liu et al. [9] proved the following.
Theorem B
Let \(n\geq4\) be an integer, q be a polynomial, and \(p_{1}\), \(p_{2}\), \(\alpha_{1}\), \(\alpha_{2}\) be nonzero constants such that \(\alpha_{1}\neq\alpha_{2}\). If f is an entire solution of finite order to the following equation:
then q is a constant, and one of the following relations holds:

(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 \(c_{2}(\exp (\alpha_{2}/n)1)q=p_{1}\), \(\alpha_{2}=n\alpha_{1}\),
where \(c_{1}\), \(c_{2}\) are constants satisfying \(c_{1}^{n}=p_{1}\), \(c_{2}^{n}=p_{2}\).
In this paper, we are going to discuss the case when \(n=3\) in the above theorem and prove some results in Section 3.
Preliminaries
In order to prove our conclusions, we need some lemmas.
Lemma 2.1
([8])
Let f be a transcendental meromorphic solution of finite order ρ of a difference equation of the form
where \(H(z,f)\), \(P(z,f)\), \(Q(z,f)\) are difference polynomials in f such that the total degree of \(H(z,f)\) in f and its shifts is n, and that the corresponding total degree of \(Q(z,f)\) is less or equal to n. If \(H(z,f)\) contains just one term of maximal total degree, then for any small enough \(\varepsilon>0\),
The following result is a Clunie [10] type lemma for the differencedifferential 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 differentialdifference 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])
Let f be a nonconstant meromorphic solution with finite order of
where \(P (z, f )\) is difference polynomial in f, and let \(\delta< 1\) and \(\varepsilon> 0\). If \(P (z, a) \not\equiv0\) for a slowly moving target function a, that is, \(T(r,a)=S(r,f)\), then
for all r outside of a possible exceptional set with finite logarithmic measure.
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 differentialdifference polynomial in f.
Lemma 2.3
([13])
Assume that c is a nonzero constant, α is a nonconstant meromorphic function. Then the differential equation \(f^{2}+(cf^{(n)})^{2}=\alpha\) has no transcendental meromorphic solutions satisfying \(T(r,\alpha)=S(r,f)\).
Main results
Now, we shall derive the following result similar to that of Theorems A and B, for extended differentialdifference equations. First, we will prove the following result.
Theorem 3.1
Let q be a polynomial, and \(p_{1}\), \(p_{2}\), \(\alpha_{1}\), \(\alpha_{2}\) be nonzero constants such that \(\alpha_{1}\neq\alpha_{2}\). If f is an entire solution of finite order to the following equation:
then q is a constant, and one of the following relations holds:

(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 \(c_{2}(\exp(\alpha_{2}/3)1)q=p_{1}\), \(\alpha_{2}=3\alpha_{1}\),
where \(N_{1)}(r,\frac{1}{f})\) denotes the counting function corresponding to simple zeros of f, and \(c_{1}\), \(c_{2}\) are constants satisfying \(c_{1}^{3}=p_{1}\), \(c_{2}^{3}=p_{2}\).
Proof of Theorem 3.1
Suppose that f is a transcendental entire solution of finite order of (3.1). For the simplicity, we replace \(f(z)\), \(f'(z)\), \(\Delta f(z)\), \(q(z)\) by f, \(f'\), Δf, q. By differentiating both sides of (3.1), we have
From (3.1) and (3.2), we obtain
Differentiating (3.3) yields
It follows from (3.3) and (3.5) that
where
Since \(T(z,f)\) is a differentialdifference polynomial in f of degree 1, it follows from (3.6) and Proposition 2.1 that \(m(r,\varphi)=S(r,f)\), and \(T(r,\varphi)=S(r,f)\). If \(\varphi\not\equiv0\), Lemma 2.2 gives \(m(r,\frac{1}{f})=S(r,f)\). This together with the first theorem will result in
where \(N_{1)}(r,\frac{1}{f})\) denotes the counting function corresponding to simple zeros of f, which is our desired result.
Next, we can assume that φ in (3.6) vanishes identically, dividing with \(f^{2}\), and recalling \(f''/f = (f'/f)' + (f'/f)^{2}\), we get the Riccati equation
where \(t := f'/f\). This equation has two constant solutions \(t = \alpha _{1}/3\), \(t = \alpha_{2}/3\). By Corollary 5.2 in the paper by Bank et al. [14], all other meromorphic solutions are of infinite order. Consider \(t =\alpha_{1}/3\), hence \(f(z) = A \exp(\alpha_{1}z/3)\) with a constant A and look at (3.6). Denote \(s(z) := q(z)\Delta f(z)\) and \(u(z) := s'(z)/s(z)\). Then, by (3.6), we get
As above, the only solutions of finite order are \(u =\alpha_{1}\), \(u =\alpha _{2}\), hence \(s(z) = D_{1} \exp(\alpha_{1}z)\) or \(s(z) = D_{2} \exp(\alpha_{2}z)\). Now, it is easy to see that the case \(s(z) = D_{1} \exp(\alpha_{1}z)\) results in a contradiction. Therefore, we have \(f(z) = A \exp(\alpha_{1}z/3)\) and \(q(z)\Delta f (z)= D_{2} \exp(\alpha_{2}z)\). The assertions \(A(\exp(\alpha_{1}/3)1)q = p_{2}\), \(A^{3} = p_{1}\) and \(\alpha_{1} = 3\alpha_{2}\) now immediately follow, and the claim (2) is done. As to the claim (2), the Riccati equations for t and u are symmetric with respect to \(\alpha_{1}\), \(\alpha_{2}\), therefore the claim (3) follows.
This completes the proof of Theorem 3.1. □
Remark 3.1
It is easy to see that \(f(z)=e^{\pi i z}+e^{\pi i z}=2i \sin(\pi i z)\) is a solution of the following equation:
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
Let \(q_{j} \) (\(j=1,2,3\)) and p be polynomials, and \(\lambda_{1}\), \(\lambda_{2}\), α be nonzero constants. Then nonlinear difference equation
possesses solutions of finite order of the form \(f(z)=c_{1}e^{\alpha z/3}+c_{2}e^{\alpha z/3}\) with \(c_{1}^{3}=\lambda_{1}\), \(c_{2}^{3}=\lambda_{2}\), and \(p\equiv0\), \(v=e^{\alpha/3}\), \(q_{1}\), \(q_{2}\), \(q_{3}\) satisfy the condition:
Moreover, if \(\alpha=(6n\pi\pm3\pi) i\), then \((8q_{3}4q_{2}+2q_{1})^{3}=27\lambda_{1}\lambda_{2}\); or if \(\alpha=(6n\pi\pm\pi)i\), then \(q_{2}+q_{1}=0\) and \((\pm\sqrt {3}i q_{1}q_{3})^{3}=27\lambda_{1}\lambda_{2}\), where n is an integer; if \(v\neq1\), \(\frac{1\pm\sqrt{3}i}{2}\), then \(q_{1}\), \(q_{2}\), \(q_{3}\) satisfy the equation
Proof of Theorem 3.2
For the simplicity the argument z will be omitted in \(f(z)\), \(f'(z)\), \(\Delta f(z)\) etc. Suppose that f is a transcendental entire solution of finite order of (3.8), and by differentiating (3.8) we get
Combining (3.8) and (3.9), we obtain
and
thus through the above two equations, we find
where \(T_{4}(z,f)\) is a differentialdifference polynomial of f, and its total degree is at most 4.
If \(T_{4}(z,f)\equiv0\), it follows from (3.10) that \(9(f')^{2}+\alpha ^{2}f^{2}\equiv0\), then \(f'=\pm\frac{\alpha}{3}f\), and there exists a nonzero constant c such that \(f=ce^{\pm(\alpha z)/3}\). Substituting the above expression of f into (3.8), we obtain a contradiction. Therefore, \(T_{4}(f)\not\equiv0\). Set \(\beta=9(f')^{2}+\alpha^{2}f^{2}\), then Proposition 2.1 shows that \(m(r,\beta)=S(r,f)\), i.e. β is a small function of f. Moreover, by Lemma 2.3, it is easy to see that β is a constant. By differentiating both sides of \(\beta=9(f')^{2}+\alpha^{2}f^{2}\), we get
It follows from (3.11) that
where \(c_{1}\), \(c_{2}\) are nonzero constants. Substituting (3.12) into (3.8), we have
where \(w=e^{(\alpha z)/3}\), \(a_{1}=c_{1}^{3}\lambda_{1}\), \(a_{2}=3c_{1}^{2}c_{2}+q_{3}c_{1}e^{\alpha}3q_{3}c_{1}e^{2\alpha /3}+3q_{3}c_{1}e^{\alpha/3}c_{1}q_{3} +q_{2}c_{1}e^{2\alpha/3}2q_{2}c_{1}e^{\alpha /3}+q_{2}c_{1}+q_{1}c_{1}e^{\alpha/3}q_{1}c_{1}\), \(a_{3}=3c_{1}c_{2}^{2}+c_{2}q_{3}e^{\alpha}3q_{3}c_{2}e^{2\alpha /3}+3q_{3}c_{2}e^{\alpha/3} c_{2}q_{3}+c_{2}q_{2}e^{2\alpha/3}2c_{2}q_{2}e^{\alpha/3} +c_{2}q_{2}+c_{2}q_{1}e^{\alpha/3}c_{2}q_{1}\), \(a_{4}=c_{2}^{3}\lambda_{2}\).
Since w is transcendental, we must have \(a_{1}=a_{4}=0\), and \(a_{2}=a_{3}=p\equiv0\). Therefore,
Set \(v=e^{\alpha/3}\), it follows by (3.13) that
or
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 \(\deg q_{1}=\deg q_{2}=\deg q_{3}\).
Case 4. When \(v\neq1\), \(\frac{1\pm\sqrt{3}i}{2}\), by (3.14), we could get the following equation:
This also completes the proof of Theorem 3.2. □
Theorem 3.3
Let \(p_{j}\) (\(j=1,2\)) be polynomials, \(\lambda_{1}\), \(\lambda_{2}\), α be nonzero constants, \(d_{1}\), \(d_{2}\) be constants, and k be a positive integer. If f is an entire solution of finite order to the following nonlinear differentialdifference equation:
then \(f(z)=c_{1}e^{\alpha z/3}+c_{2}e^{\alpha z/3}\), \(c_{1}^{3}=\lambda_{1}\), \(c_{2}^{3}=\lambda_{2}\),
and
Proof of Theorem 3.3
For simplicity the argument z will be omitted in \(f(z)\), \(f'(z)\), \(f'(z+d_{1})\) etc. Suppose that f is a transcendental entire solution of finite order of (3.15), by differentiating (3.15) we get
where \(Q(z,f)=p'_{1}(z)f^{(k)}(z+d_{1})+p_{1}(z)f^{(k+1)}(z+d_{1})+p'_{2}(z)f(z+d_{2})+p_{2}(z)f'(z+d_{2})\). Obviously, \(Q(z,f)\) is a differentialdifference polynomial of f, and its total degree is at most 1.
Set \(Q_{1}(z,f)=p_{1}(z)f^{(k)}(z+d_{1})+p_{2}(z)f(z+d_{2})\). It follows by (3.15) and (3.16) that
and
Thus, for the above two equations, we can also get (3.10) and by a similar method to the one above, we have (3.12). Consequently
and
In view of (3.15), (3.12), (3.17), and (3.18), we obtain \(f(z)=c_{1}e^{\alpha z/3}+c_{2}e^{\alpha z/3}\) with \(c_{1}^{3}=\lambda_{1}\), \(c_{2}^{3}=\lambda_{2}\). Moreover, we have
and
Thus, we obtain the desired result and finish the proof of this theorem. □
It is not difficult to see that a similar argument can be used to obtain the following result.
Theorem 3.4
Let f be an entire solution of finite order to the following nonlinear differentialdifference equation:
here \(P_{1}\) is a linear polynomial in \(\Delta f(z),\ldots, f'(z),\ldots,f^{(k)}(z)\), with polynomials as its coefficients, and \(\lambda_{1},\lambda_{2},\alpha\in\mathbb{C}\) are nonzero constants. Then \(f(z)=c_{1}e^{\alpha z/3}+c_{2}e^{\alpha z/3}\) with \(c_{1}^{3}=\lambda_{1}\), \(c_{2}^{3}=\lambda_{2}\).
Conclusion
In studying differential equations in the complex plane \(\mathbb{C}\), it is always an interesting and quite difficult problem to prove the existence or uniqueness of the entire or meromorphic solution of a given equation. There have been many studies and results obtained lately that relate to the existence or growth of entire or meromorphic solutions of various types of complex difference equations and differencedifferential equations. By our methods, one can discuss the entire solutions for the algebraic difference equation and differencedifferential equation of the form
here \(P_{l}\) is a differencedifferential polynomial in \(\Delta f(z),\ldots, f'(z),\ldots,f^{(k)}(z)\), with polynomials as its coefficients, and \(\deg(P_{l})\leq n2\), \(\lambda_{1}\), \(\lambda_{2}\), \(\alpha_{1}\), \(\alpha_{2}\) are nonzero polynomials.
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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).
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MSC
 30D35
 30A06
Keywords
 transcendental entire solution
 differentialdifference equation
 Nevanlinna theory