Abstract

In this paper, we study finite-order entire solutions of nonlinear differential-difference equations and solve a conjecture proposed by Chen, Gao, and Zhang when the solution is an exponential polynomial. We also find that any exponential polynomial solution of a nonlinear difference equation should have special forms.

1. Introduction and Main Result

Extensive application of Nevanlinna theory has prompted scholars to acquire a number of results on differential equations, difference equations, and differential-difference equations. In this paper, we assume readers are familiar with the standard notations and fundamental results, see [1–4].

Given a meromorphic function f and a constant c. We take for simplicity. and are the first-order difference operator and n-th order difference operator of f, respectively. We adopt the notations and to denote the order and the exponent of convergence of zeros of f, respectively.

Recall the definition of exponential polynomial of the formwhere and are polynomials in z. Denote

Many papers recently have focused on solvability and existence of solutions of nonlinear differential-difference equations, see [5–16].

In 2012, Wen et al. [17] classified finite-order entire solutions of the following nonlinear difference equation:where is an integer and , , and are polynomials such that is not identically zero and is not a constant. They obtained the following result.

Theorem 1 (see [17]). Let be an integer, , and , , and be polynomials such that is not identically zero and is not a constant. Then, the finite-order entire solutions f of equation (3) should satisfy the following:(a)Every solution f satisfies and is of mean type.(b)Every solution f satisfies if and only if .(c)A solution f belongs to if and only if . In particular, this is the case .(d)If a solution f belongs to and is any other finite-order entire solution to equation (3), then , where .(e)If f is an exponential polynomial solution of form (1), then . Moreover, if , then .

In 2016, Liu [9] investigated finite-order transcendental entire solutions of the following nonlinear differential-difference equation:where and are integers and , , and are polynomials such that is not identically zero and is not a constant. He obtained a result which is similar to Theorem 1.

In 2019, Chen et al. [6] considered solutions of equation (3), where is replaced by . They obtained the following result.

Theorem 2 (see [6]). Let be an integer and c, λ, , and be nonzero constants. Suppose and are polynomials such that is nonvanishing and is not a constant. If f is an entire solution of finite order ofthen the following conclusions hold:(1)Every solution f satisfies .(2)If a solution f belongs to , then must be a constant and one of the following two relation groups holds:(a) and (b) and , where both b and B are constants.

Remark 1. Chen et al. [6] gave an example: is an entire solution of finite order of the following difference equation:From the example, they conjectured that the conclusions of Theorem 2 are still valid if .
We consider the conjecture and prove a more generalized result. Moreover, we solve Chen, Gao, and Zhang’s conjecture when is an exponential polynomial of form (1).

Theorem 3. Let be an integer and c, , , , and be nonzero constants such that . Suppose is a nonvanishing polynomial and is a nonconstant polynomial. If the differential-difference equationhas a transcendental entire solution f, then(1)Every solution f satisfies .(2)If f is an exponential polynomial of form (1), then .(3)If f belongs to , then one of the following two relation groups holds:(a), , , and (b), , , and , where both b and B are constants and is a polynomial.

Corollary 1. Let be an integer and c, , , , and be nonzero constants such that . Suppose is a nonvanishing polynomial and is a nonconstant polynomial. If the differential-difference equation (7) has solutions f satisfying , then and must be a constant and one of the following two relation groups holds:(1), , , and (2), , , and , where both b and B are constants.Next, we give two examples to illustrate equation (7).

Example 1. is an entire solution of finite order of the following differential-difference equation:where , , , and . Thus, case (1) occurs.

Example 2. is an entire solution of finite order of the following difference equation:where , , , and . Thus, case (2) occurs.
In 2015, Zhang et al. [18] studied the existence of entire solutions of the following nonlinear difference equation:They obtained the following result.

Theorem 4 (see [18]). Let , , and α be nonzero constants. Suppose and are polynomials. Then, the nonlinear difference equation (10) possesses solutions of finite order of the form with , , and . and , , and satisfy the following condition:Moreover, one of the following conclusions holds:(1)If , then (2)If , then and , where n is an integer(3)If and , then , , and satisfy the following equation:

In the following, we consider a difference equation which is similar to (10) and obtain the following result.

Theorem 5. Suppose that , , and λ are nonzero constants and that and are nonzero polynomials. If f is a nontrivial exponential polynomial solution ofthen f has solutions of finite order of the following form:where , , , and . is a nonzero polynomial, , and and satisfyMoreover, one of the following conclusions holds:(1)If , then , , and and satisfy Especially, if and are constants, then .(2)If and is the solution of , then and and satisfy Especially, if is a constant, then .The following examples show the existences of solution of equation (13).

Example 3. An entire solution solves the following difference equation:where , , . The case occurs.

Example 4. An entire solution solves the following difference equation:where , . The case satisfies .

Example 5. An entire solution solves the following difference equation:where , . The case satisfies .
This paper is organized as follows. In Section 2, we introduce the background of exponential polynomials and some indispensable lemmas. Sections 3 and 4 contain the detailed proofs on Theorems 3 and 5. In Section 5, we will discuss the methods of the main results obtained in the paper.

2. Preliminaries

We recollect a basic result on exponential polynomials. Let , where . We know ([1], p.7) that

For exponential polynomials of form (1), Wen et al. [17] followed the reasoning in [19] and acquired some instrumental tools.

Suppose the polynomials in (1) are pairwise different and normalized by . Then, representation (1) is uniquely determined and the functions are linearly independent. Letand let be pairwise different leading coefficients of the polynomials of maximum degree q. Thus, (1) can be written in the following normalized form:where are either exponential polynomials of degree less than q or ordinary polynomials in z. hold for .

A convex hull of a set , denoted by , is the intersection of all convex sets containing . If contains only finitely many elements, then is obtained as an intersection of finitely many closed half-planes. Hence, is either a compact polygon (with a nonempty interior) or a line segment. We denote the perimeter of by . If is a line segment, then equals to twice the length of this line segment. We fix the notation for , , and .

Theorem 6 (see [19], Satz 1]). Let f be given by (23). Then,

Next, we can find the following consequence from the result of Steinmetz ([20], Satz 1), i.e.,holds for an exponential polynomial in form (23) (also see [21], Section 3).

Some auxiliary results are necessary. The first one is a difference analogue of logarithmic derivative lemma given by Chiang and Feng.

Lemma 1 (see [22], Corollary 2.5). Let be a meromorphic function with finite order . Suppose c is a fixed nonzero complex constant. Then, for each , we haveThe following lemma is a useful tool to solve differential-difference equations and difference equations.

Lemma 2 (see [3]). Suppose that are meromorphic functions and that are entire functions. They satisfy the following conditions:(1)(2) are not constants for (3) holds, for and , where is finite linear measure or finite logarithmic measure.Then, .

Halburd and Korhonen proved a difference analogue of Clunie lemma under the condition finite order.

Lemma 3 (see [23]). Let be a nonconstant finite-order meromorphic solution ofwhere and are difference polynomials in f with small meromorphic coefficients. Suppose and . If the total degree of is a polynomial in f and its shifts are less than or equal to n, thenfor all r outside of a possible exceptional set with finite logarithmic measure.

Remark 2. Similar to Lemma 3, if f is a transcendental exponential polynomial in form (23), and are differential-difference polynomials in f and the coefficients of and are polynomials , for each , then an obtained result iswhere r is sufficiently large.
Chen and Yang proved the following lemma.

Lemma 4 (see [24]). Let λ be a nonzero constant and be a nonvanishing polynomial. Then, the differential equationhas a special solution which is a nonzero polynomial.

In addition, the following lemma is similar to Lemma 5.3 of [17] and Lemma 2.7 of [9]. The proof can be given word by word.

Lemma 5. Let f be given by (23), where . If f is a solution of equation (7), then .

3. Proof of Theorem 3

Proof of Conclusion 1. Suppose that is a finite-order entire solution of equation (7). Applying the lemma on the logarithmic derivative and Lemma 1 to equation (7), we obtainThus,which implies that .
If , then the order of left side of equation (7) is equal to . Since the order of right side of equation (7) is equal to 1, we have and . Let , where . Equation (7) can be written aswhere and satisfy and , respectively. Next, we consider the following three cases: Case 1. and . By equation (33) and Lemma 2, we have , which is a contradiction. Case 2. and . Equation (33) can be rewritten as Using Lemma 2, we have , which is a contradiction. Case 3. and . Similar to the proof of Case 2, we can get a contradiction. Thus, we have . Noting , we obtain .