The existence of solutions to certain type of nonlinear difference-differential equations

Weiran Lü; Linlin Wu; Dandan Wang; Chungchun Yang

doi:10.1515/math-2018-0071

Article Open Access

The existence of solutions to certain type of nonlinear difference-differential equations

Weiran Lü , Linlin Wu , Dandan Wang and Chungchun Yang

Published/Copyright: July 17, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

In this paper we study the entire solutions to a certain type of difference-differential equations. We also give an affirmative answer to the conjecture of Zhang et al. In addition, our results improve and complement earlier ones due to Yang-Laine, Latreuch, Liu-Lü et al. and references therein.

Keywords: Nevanlinna theory; Entire solution; Difference equation; Differential polynomial

MSC 2010: 34M05; 30D35; 39A10; 39B32

1 Introduction and main results

In studying difference-differential equations in the complex plane ℂ, it is always an interesting and quite difficult problem to prove the existence or uniqueness of the entire or meromorphic solutions to a given difference-differential equation. There have been many studies and results obtained lately that relate to the existence or growth of the entire or meromorphic solutions of various types of difference or differential equations, see, e.g., [1, 2, 3, 4, 5, 6, 7, 8, 9] and references therein.

Herein let f denote a non-constant meromorphic function and we assume that the reader is familiar with the standard terminology and results of Nevanlinna theory such as the characteristic function T(r, f), the proximity function m(r, f) and the counting function N(r, f) ( see, e.g., [10, 11, 12] ). However, for the convenience of the reader, we shall repeat some notations needed below.

We call a meromorphic function α ≢ 0, ∞ a small function with respect to f, if T(r, α) = S(r, f), where S(r, f) denotes any quantity satisfying S(r, f) = o{T(r, f)} as r → ∞, possibly outside a set of r of finite linear measure. The order of f is

ρ(f)=lim supr→∞log⁡T(r,f)log⁡r,

and the hyper-order ρ₂(f) is defined as

ρ2(f)=lim supr→∞loglog⁡T(r,f)log⁡r.

Definition 1.1

A difference polynomial, respectively, a difference-differential polynomial, infis a finite sum of difference products offand its shifts, respectively, of products of f, derivatives offand of their shifts, with all the coefficients of these monomials being small functions of f.

Definition 1.2

Given a nonzero constant c, we define the difference operators by

Δcf(z)=f(z+c)−f(z)andΔcnf(z)=Δc(Δcn−1f(z))(n≥2).

For the sake of simplicity, we letΔf(z) = f(z + 1) – f(z) andΔⁿf(z) = Δ(Δ^n–1f(z)) (n ≥ 2) for the casec = 1 (see, e. g., [2, 13] and [14]).

For the benefit of the readers, we shall give some related results. Yang and Laine considered the following difference equation and proved:

Theorem A

([7]). A nonlinear difference equation

f3(z)+q(z)f(z+1)=csin⁡bz,

whereqis a non-constant polynomial and b, c ∈ ℂ ∖ {0}, does not admit entire solutions of finite order. Ifqis a nonzero constant, then the above equation possesses three distinct entire solutions of finite order, provided thatb = 3nπand q³ = (–1)ⁿ⁺¹c²27/4 for a nonzero integer n.

In 2014, Liu and Lü et al. proved the following result.

Theorem B

([15]). Let n ≥ 4 be an integer, qbe a polynomial, and p₁, p₂, α₁, α₂be nonzero constants such thatα₁ ≠ α₂. If there exists some entire solution f of finite order to the following equation

fn(z)+q(z)Δf(z)=p1eα1z+p2eα2z,

thenqis a constant, and one of the following relations holds:

f(z)=c1eα1nz,and c₁(e^α₁/n – 1)q = p₂, α₁ = nα₂,
f(z)=c2eα2nz,and c₂(e^α₂/n – 1)q = p₁, α₂ = nα₁,

wherec₁, c₂are constants satisfyingc1n=p1,c2n=p2.

Recently, Zhang et al. obtained the following result.

Theorem C

([16]). Letqbe a polynomial, andp₁, p₂, α₁, α₂be nonzero constants such thatα₁ ≠ α₂. Iffis an entire solution of finite order to the following equation:

f3(z)+q(z)Δf(z)=p1eα1z+p2eα2z,(1)

thenqis a constant, and one of the following relations holds:

T(r, f) = N₁₎(r, 1f) + S(r, f),
f(z) = c₁exp( α1z3), andc₁(exp( α13) – 1)q = p₂, α₁ = 3α₂,
f(z) = c₂ exp( α2z3), andc₂(exp( α23) – 1)q = p₁, α₂ = 3α₁,

where N₁₎(r, 1f) denotes the counting function corresponding to simple zeros of f, andc₁, c₂are constants satisfyingc13=p1, c23=p2.

Remark 1.3

In [16], the authors also gave an example to show that the case (1) occurs indeed.

Example 1.4

Letf(z) = e^πiz + e^–πiz = 2i sin(πiz). Thenfis a solution of the following equation:

f3(z)+3Δf(z)=e3πiz+e−3πiz.

Obviously, T(r, f) = N₁₎(r, 1f) + S(r, f). Thus, the case (1) occurs indeed.

Since in the above example α₁ + α₂ = 3πi + (–3πi) = 0, consequently, Zhang et al. posed the following conjecture.

Conjecture 1.5

([16]). Ifα₁ ≠ α₂, α₁ + α₂ ≠ 0, then the conclusion (1) of Theorem A is impossible. In fact, any entire solution f of (1) must have 0 as its Picard exceptional value.

In 2017, Latreuch gave an affirmative answer to Conjecture 1.5. In fact, he obtained the following result.

Theorem D

([17]). Letqbe a polynomial, andp₁, p₂, α₁, α₂be nonzero constants such thatα₁ ≠ α₂andα₁ + α₂ ≠ 0. Iffis an entire solution of finite order of (1), thenqis a constant, and one of the following relations holds:

f(z) = c₁ exp( α1z3), andc₁(exp( α13) – 1)q = p₂, α₁ = 3α₂;
f(z) = c₂ exp( α2z3), andc₂(exp( α23) – 1)q = p₁, α₂ = 3α₁,
wherec₁, c₂are constants satisfyingc13=p1,c23=p2.Furthermore, (1) does not have any entire solution of infinite order satisfies any one of the following conditions:
ρ₂(f) < 1;
λ(f) < ρ(f) = ∞ andρ₂(f) < ∞,

here λ(f) denotes the exponent of convergence of zeros sequence of f.

In the present paper we continue discussing Conjecture 1.5. Moreover, our result will include several known results for difference or differential equations obtained earlier as its special case. In fact, we consider a slightly more general form of (1) and obtain the following result.

Theorem 1.6

LetL(z, f) denote a difference-differential polynomial infof degree one with small functions as its coefficients such that L(z, 0) ≡ 0, and letp₁, p₂, α₁, α₂be nonzero constants such thatα₁ ≠ α₂. Iffis an entire solution withρ₂(f) < 1 to the following equation:

f3+L(z,f)=p1eα1z+p2eα2z,(2)

then one of the following relations holds:

f(z) = c₁exp( α1z3) + c₂exp( α2z3), wherec₁andc₂are two nonzero constants satisfyingc13=p1,c23=p2andα₁ + α₂ = 0;
f³(z) = (p₁ – c₁) exp(α₁z), andL(z, f) = c₁ exp(α₁z) + p₂ exp(α₂z), wherec₁is a constant;
f³(z) = (p₂ – c₂) exp(α₂z), andL(z, f) = p₁ exp(α₁z) + c₂ exp(α₂z), wherec₂is a constant.

From Theorem 1.6, we have

Corollary 1.7

Equation(2)does not have any entire solutionfwithρ(f) = ∞ andρ₂(f) < 1.

Remark 1.8

It is obvious from Theorem 1.6 that the above conjecture is true. We also point out that if an entire function solutionfin Theorem 1.6 is replaced by a meromorphic solution withN(r, f) = S(r, f), the conclusion of Theorem 1.6 still holds.

Remark 1.9

Lemma 2.2 (in section 2) is crucial to the proofs of our main results. However, it may be false if the condition “ ρ₂(f) < 1 ” is violated. There is no difficulty in showing thatf(z) = exp(exp(z)) is a counterexample. Now one may raise the questions: what will happen if we delete the conditionρ₂(f) < 1 in Theorem 1.6, Corollary 1.7 and so on?

2 Some lemmas

In order to prove Theorem 1.6, we need the following results.

Lemma 2.1

([18]). Let m, nbe positive integers satisfying1m+1n<1.Then there are no transcendental entire solutions offandgsatisfy the following equation

a(z)f(z)n+b(z)g(z)m=1,(3)

with a, b being small functions of f, and g, respectively.

Lemma 2.2

([19]). Letfbe a transcendental meromorphic function of hyper-orderρ₂(f) < 1. Then forc ∈ ℂ, we have

m(r,f(z+c)f(z))=S(r,f),(4)

outside of a possible exceptional set with finite logarithmic measure.

Remark 2.3

The following result is the analogue of the logarithmic derivatives lemma [10, 11] for the difference-differential polynomials of a meromorphic function f. It can be proved by applying Lemma 2.2 and the logarithmic derivatives lemma with a similar reasoning as in [19, 20, 21, 22] and stated as follows.

Lemma 2.4

Letfbe a transcendental meromorphic function withρ₂(f) < 1. GivenL(z, f) as to Theorem 1.6, then for any positive integer k, we have

m(r,L(z,f)f(z))+m(r,L(k)(z,f)f(z))=S(r,f),(5)

outside of a possible exceptional set with finite logarithmic measure.

Lemma 2.5

([12], Theorem 1.55). Letg₁, g₂,⋯,g_pbe transcendental meromorphic functions satisfyingΘ(∞, g_j) = 1 (j = 1, 2, ⋯, p). If∑j=1pajgj=1,then fora_j ∈ ℂ ∖ {0} (j = 1, 2, ⋯, p), we have∑j=1pδ(0, g_j) ≤ p – 1.

Remark 2.6

By the same methods as in the proof of Theorem 1.55 used in [12], we also point out that if nonzero constantsa₁, a₂, ⋯, a_p are replaced by small functions ofg₁, g₂, ⋯, g_p, the conclusion of Lemma 2.5 still holds.

Lemma 2.7

([22]). Suppose thatfis a transcendental meromorphic function, a, b, c anddare small functions offsuch that acd ≢ 0. If

af2+bff′+c(f′)2=d,

then

c(b2−4ac)d′d+b(b2−4ac)−c(b2−4ac)′+(b2−4ac)c′=0.

Lemma 2.8

([23]). Assume that c ∈ ℂ is a nonzero constant, αis a non-constant meromorphic function. Then the differential equationf² + (cf⁽ⁿ⁾)² = αhas no transcendental meromorphic solutions satisfyingT(r, α) = S(r, f).

Lemma 2.9

([12]). Assume thatfis a meromorphic function. Then for all irreducible rational functions in f,

R(z,f)=Σi=0paifiΣj=0qbjfj,

with meromorphic coefficientsa_i, b_j satisfying

T(r,ai)=S(r,f),i=0,⋯,p,T(r,bj)=S(r,f),j=0,⋯,q,

the characteristic function ofR(z, f) satisfies

T(r,R(z,f))=dT(r,f)+S(r,f),

whered = max(p, q).

3 Proof of Theorem 1.6

Suppose that f is an entire solution with ρ₂(f) < 1 to (2). Obviously, f is a transcendental function. For the simplicity, we replace f(z), f′(z) and L(z, f) by f, f′ and L, respectively.

By differentiating both sides of (2), we obtain

3f2f′+L′=α1p1eα1z+α2p2eα2z.(6)

Combining (2) and (6) yields

α2f3+α2L−3f2f′−L′=(α2−α1)p1eα1z.(7)

By differentiating (7) again, we derive that

3α2f2f′+α2L′−6f(f′)2−3f2f″−L″=α1(α2−α1)p1eα1z.(8)

It follows from (7) and (8) that

fφ=T(z,f),(9)

where

φ=α1α2f2−3(α1+α2)ff′+6(f′)2+3ff″,(10)

T(z,f)=−α1α2L+(α1+α2)L′−L″.

Two cases will now be considered below, depending on whether or not φ vanishes identically.

If φ ≡ 0, then (9) shows that T(z, f) ≡ 0, namely

L″−(α1+α2)L′+α1α2L=0.(11)

Further, the general solution of (11) is given by

L=c1eα1z+c2eα2z,(12)

where c₁ and c₂ are constants. Thus, (2) and (12) would give

f3=(p1−c1)eα1z+(p2−c2)eα2z.(13)

We claim that p₁ = c₁ or p₂ = c₂. Assume now, contrary on the assertion, that p₁ ≠ c₁ and p₂ ≠ c₂. We rewrite (13) as

(e−α2z/3fp2−c23)3−(p1−c1p2−c23e(α1−α2)z/3)3=1,

which contradicts Lemma 2.1. Hence, p₁ = c₁ or p₂ = c₂. In this case, we can derive the conclusions (2) and (3).

In the following, we will consider the case φ ≢ 0. In this case, (9) gives

φ(z)=T(z,f)f(z).(14)

Since T(z, f) is a difference-differential polynomial in f of degree 1, and L(z, 0) ≡ 0, it follows from (14), Lemma 2.4 and the lemma on the logarithmic derivatives that m(r, φ) = S(r, f). Note that φ is an entire function, so T(r,φ) = m(r, φ) = S(r, f), which means that φ is a small function of f.

Now, we rewrite (10) as

φf2=α1α2−3(α1+α2)f′f+6(f′f)2+3f″f.(15)

Applying the lemma on the logarithmic derivatives to (15), we find m(r, φf2) = S(r, f). Since φ is a small function of f, one can get m(r, 1f) = S(r, f). Thus, the first fundamental theorem implies T(r, f) = N(r, 1f) + S(r, f).

On the other hand, by (10) again, we have N₍₂(r, 1/f) = S(r, f), and

T(r,f)=N1)(r,1f)+S(r,f),(16)

where N₁₎(r, 1f) denotes the counting function corresponding to simple zeros of f. Differentiating (10) yields

φ′=2α1α2ff′−3(α1+α2)[ff″+(f′)2]+15f′f″+3ff‴.(17)

For brevity, in the following, we assume that z₀ is a simple zero of f, and we can assume, without loss of generality, by (16) that ψ(z₀) ≠ 0, ∞, where ψ is any non-vanishing small function of f. Thus, (10) enables us to deduce the following fact

φ(z0)=6(f′(z0))2.(18)

Now, we are ready to present φ′ ≢ 0. Suppose, contrary to our assertion, that φ′ ≡ 0, namely, φ is a constant, say A.

If z₀ is a zero of f′(z) – A/6, then we set

h(z)=f′(z)−A/6f(z).(19)

Trivially, h ≢ 0. Then by the lemma on the logarithmic derivatives, the facts m(r,1/f) = S(r, f), N₍₂(r,1/f) = S(r, f) and (18), we have T(r, h) = S(r, f).

By (19), we therefore have

f′=hf+A/6,f″=(h′+h2)f+hA6.(20)

Substituting (20) into (10) yields

[α1α2−3(α1+α2)h+3h′+9h2]f=3[(α1+α2)−5h]A6,

which implies

α1+α2≡5h,α1α2−3(α1+α2)h+9h2≡0.

Thereby we have

h=α1+α25=α13,orh=α1+α25=α23.(21)

Thus, (19) and (21) would give

f(z)=Bhehz−1hA6,(22)

where B is a nonzero constant.

On the other hand, substituting (22) into (2), it follows by Lemma 2.5 that 1hA6=0. This, however, contradicts (16), and thus φ′ ≢ 0.

Using the same way as above, φ′ ≢ 0 is also obtained by setting

ℏ(z)=f′(z)+A/6f(z)

assuming that f′(z₀) + A/6=0.

Moreover, (17) gives

φ′(z0)=[−3(α1+α2)(f′)2+15f′f″](z0)=0.(23)

In order to prove Theorem 1.6, we discuss two cases below:

[2φ′ + (α₁ + α₂)φ]f′ – 5φf″ ≡ 0.
In this case, let us write it in the following form
f″=[25φ′φ+15(α1+α2)]f′:=sf′,(24)
and consequently
f‴=(s′+s2)f′.(25)
Substituting (24) and (25) into (17), we then immediately derive
α1α2φ′f=[2α1α2φ−3(α1+α2)sφ+3(s′+s2)φ+3(α1+α2)φ′−3sφ′]f′,
which is impossible by (16) and the facts that the coefficients φ′(≢ 0), 2α₁α₂φ – 3(α₁ + α₂)sφ + 3(s′ + s²)φ + 3(α₁ + α₂)φ′ – 3sφ′ are small functions of f. So, this case can not occur.
[2φ′ + (α₁ + α₂)φ]f′ – 5φf″ ≢ 0.
Obviously, in this case, by (18), (23) and f′(z₀) ≠ 0, we then see that z₀ is a zero of the function [2φ′ + (α₁ + α₂)φ]f′ – 5φf″.
Accordingly, we set
ϕ(z)=[2φ′(z)+(α1+α2)φ(z)]f′(z)−5φf″(z)f(z).(26)
Then by the lemma on the logarithmic derivatives, the facts N₍₂(r, 1/f) = S(r, f), m(r, 1/f) = S(r, f), and (17), we have T(r, ϕ) = S(r, f). Thereby, from (26), we obtain
f″=[25φ′φ+15(α1+α2)]f′−ϕ5φf:=sf′+tf.(27)
Trivially, s, t are small functions of f.
By (10) and (27), we have
af2+bff′+6(f′)2=φ,(28)
where a = α₁α₂ + 3t, b = 3[s – (α₁ + α₂)].
In the following, we consider two subcases.

Suppose that a ≡ 0. In this case, (28) becomes
(bf+6f′)f′=φ,
which gives
f′=φ1eβ,bf+6f′=φ2e−β,(29)
where β, φ₁ and φ₂ are entire functions such that φ₁φ₂ = φ.
Trivially, in this case, b ≢ 0, and it follows by (29) that
f=φ2e−β−6φ1eβb.(30)
Thus, by (29) and (30), we have
[φ1+6(φ1b)′+6φ1bβ′]e2β=(φ2b)′−φ2bβ′,
which shows that
(φ2b)′−φ2bβ′≡0.(31)
Obviously, (31) gives log φ2b=β+C, where C is a constant. Therefore, e^β is a small function of f, this shows that f′ is also a small function of f. The contradiction T(r, f) = S(r, f) now follows by (30).
a ≢ 0. In this case, applying Lemma 2.7 to (28), we immediately get the following equation
6(b2−24a)φ′φ+b(b2−24a)−6(b2−24a)′≡0.(32)
Now, we consider two cases.
Firstly, assume that b² – 24a ≢ 0.
Note that b=65φ′φ−125(α1+α2), and we then rewrite (32) as
115φ′φ−125(α1+α2)=(b2−24a)′b2−24a.
By integrating the above equation, we have
11log⁡φ−12(α1+α2)z=5log⁡(b2−24a)+log⁡D,(33)
where D is a constant. Obviously, (33) gives φ¹¹ = D(b² – 24a)⁵e^{12(α₁+α₂)z}. If α₁ + α₂ ≠ 0, then e^α₁z, e^α₂z are also two small functions of f because φ and b² – 24a are small functions of f. Rewrite (2) as
f2=−Lf+p1eα1z+p2eα2zf.
Then
2m(r,f)=m(r,f2)=m(r,−Lf+p1eα1z+p2eα2zf)≤m(r,1f)+S(r,f)=S(r,f),
which is a contradiction, since f is an entire function. Therefore, α₁ + α₂ = 0.
Combining (2) and (6) yields
α1f3+α1L−3f2f′−L′=(α1−α2)p2eα2z.(34)
It follows by (7), (34) and α₁ + α₂ = 0 that
f4[−α12f2+9(f′)2]=2α12Lf3−6L′f2f′+(α1L)2−(L′)2−4α12p1p2.(35)
Obviously, P₄(f) := 2 α12Lf³ – 6L′f²f′ + (α₁L)² – (L′)² – 4 α12p₁p₂ is a difference-differential polynomial of f, and its total degree at most 4.
If P₄(f) ≡ 0, it follows from (35) that 9(f′)² – α12f² ≡ 0, and then f′=±α13f. Substituting the above expression into (10), we arrive at φ ≡ 0, a contradiction. Therefore, P₄(f) ≢ 0. Set β = 9(f′)² – α12f². In this case, we rewrite (35) as
β=P4(f)f4,
which, Lemma 2.4 and the fact that we have proved m(r, 1f) = S(r, f) must show that m(r, β) = S(r, f), i.e. β is a small function of f. Moreover, from Lemma 2.8, it is easy to see that β is a constant. By differentiating both sides of β = 9(f′)² – α12f², we get
f″−(α13)2f=0.(36)
It follows from (36) that
f(z)=c1eα13z+c2e−α13z,(37)
where c₁, c₂ are constants. By (37) and (2), we have c13=p1,c23=p2. Conclusion (1) has consequently been proved.
Now, we assume that b² – 24a ≡ 0.
By making use of (28), we have 6(f′+b12f)2=φ, which shows that γ:=f′+b12f is a small function of f. Thus, f″=(b2144−b′12)f+γ′−b12γ.
Substituting two above expressions into (10), we obtain
[α1α2+b4(α1+α2)+b2144−b′4]f2+[−3(α1+α2)γ−bγ+3(γ′−b′12γ)]f=φ.(38)
Using (38) and Lemma 2.9, we therefore have
α1α2+b4(α1+α2)+b2144−b′4≡0,−3(α1+α2)γ−bγ+3(γ′−b′12γ)≡0andφ≡0.
This contradicts φ ≢ 0.
Thus, we finish the proof of Theorem 1.6.

Acknowledgement

The authors would like to thank the referees for their several important suggestions and for pointing out some errors in our original manuscript. These comments greatly improved the readability of the paper.

The research was supported by NNSF of China Project No. 11601521, and the Fundamental Research Fund for Central Universities in China Project Nos. 15CX05061A, 18CX02048A and 18CX02045A.

References

[1] Bergweiler W., Langley J. K., Zeros of differences of meromorphic functions, Math Proc Camb Phil Soc., 2007, 142, 133–14710.1017/S0305004106009777Search in Google Scholar

[2] Chen Z.X., Complex Differences and Difference Equations, Beijing: Science Press, 201410.1155/2014/124843Search in Google Scholar

[3] Chen Z. X., Yang C. C., On entire solutions of certain type of differential-difference equations, Taiwanese J. Math., 2014, 18, 677–68510.11650/tjm.18.2014.3745Search in Google Scholar

[4] Chiang Y. M., Feng S. J., On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane, Ramanujan J., 2008, 16, 105–12910.1007/s11139-007-9101-1Search in Google Scholar

[5] Laine I., Yang C.C., Clunie theorems for difference and q–difference polynomials, J. London Math. Soc., 2007, 76, 556–56610.1112/jlms/jdm073Search in Google Scholar

[6] Rieppo J., Malmquist-type Results for Difference Equations with Periodic Coefficients, Comput. Methods Funct. Theory, 2015, 15, 449–45710.1007/s40315-015-0109-zSearch in Google Scholar

[7] Yang C. C., Laine I., On analogies between nonlinear difference and differential equations, Proc. Japan. Acad., Ser. A, Math. Sci., 2010, 86, 10–1410.3792/pjaa.86.10Search in Google Scholar

[8] Zhang J. L., Some results on difference Painlevé IV equations, Journal of Difference Equations and Applications, 2016, 22, 1912–192910.1080/10236198.2016.1255207Search in Google Scholar

[9] Zhang J., Liao L. W., On entire solutions of a certain type of nonlinear differential and difference equations, Taiwanese J. Math., 2011, 15, 2145–215710.11650/twjm/1500406427Search in Google Scholar

[10] Hayman W. K., Meromorphic Functions, Oxford, Clarendon Press, 1964Search in Google Scholar

[11] Laine I., Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin/New York, 199310.1515/9783110863147Search in Google Scholar

[12] Yang C. C., Yi H. X., Uniqueness Theory of Meromorphic Functions, Science Press, Beijing/New York, 200310.1007/978-94-017-3626-8Search in Google Scholar

[13] Elaydi S., An introduction to difference equations, Springer-Verlag, 2005Search in Google Scholar

[14] Liu N. N., Lü W. R., Yang C. C., On Picard exceptional values of difference polynomials, Journal of Difference Equations and Applications, 2014, 20, 1437–144310.1080/10236198.2014.936318Search in Google Scholar

[15] Liu N. N., Lü W. R., Shen T. T., Yang C. C., Entire solutions of certain type of difference equations, J. Inequal. Appl., 2014,6310.1186/1029-242X-2014-63Search in Google Scholar

[16] Zhang F. R., Liu N. N., Lü W. R., Yang C. C., Entire solutions of certain class of differential-difference equations, Advances in Difference Equations, 2015,15010.1186/s13662-015-0488-5Search in Google Scholar

[17] Latreuch Z., On the Existence of Entire Solutions of Certain Class of Nonlinear Difference Equations, Mediterr. J. Math.,2017, 14:115 10.1007/s00009-017-0914-x.Search in Google Scholar

[18] Yang C. C., A generalization of a theorem of P. Montel on entire functions, Proc. Amer. Math. Sci.,1970, 26, 332–33410.1090/S0002-9939-1970-0264080-XSearch in Google Scholar

[19] Halburd R. G., Korhonen R. J., Tohge K., Holomorphic curves with shift-invariant hyperplane preimages, Trans. Amer. Math. Soc., 2014, 366, 4267–429810.1090/S0002-9947-2014-05949-7Search in Google Scholar

[20] Halburd R. G., Korhonen R. J., Nevanlinna theory for the difference operators, Ann. Acad. Sci. Fenn., Math., 2006, 31, 463–478Search in Google Scholar

[21] Halburd R. G., Korhonen R. J., Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl., 2006, 314, 477–48710.1016/j.jmaa.2005.04.010Search in Google Scholar

[22] Li P., Entire solutions of certain type of differential equations II, J. Math. Anal. Appl., 2011, 375, 310–31910.1016/j.jmaa.2010.09.026Search in Google Scholar

[23] Yang C. C., Li P., On the transcendental solutions of a certain type of nonlinear differential equations, Arch. Math., 2004, 82, 442–44810.1007/s00013-003-4796-8Search in Google Scholar

Received: 2017-08-17

Accepted: 2018-03-22

Published Online: 2018-07-17

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2018-0071

Keywords for this article

Nevanlinna theory; Entire solution; Difference equation; Differential polynomial

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