In this paper, we study the existence and nonexistence of entire solutions of finite and infinite order of certain nonlinear difference equations of the form 

$$\begin{aligned} f^{n}\left( z\right) +L\left( z,f\right) =p_{1}\left( z\right) e^{\alpha _{1}\left( z\right) }+p_{2}\left( z\right) e^{\alpha _{2}\left( z\right) }, \quad \text { }n\ge 3 \end{aligned}$$

where $p_{i}\left( z\right) $, $\alpha _{i}\left( z\right) $$\left( i=1,2\right) $ are polynomials and $L\left( z,f\right) $ is a nonzero linear difference polynomial in f, we give also an affirmative answer to the conjecture posed by Zhang et al. (Adv Differ Equ 2015:150, 2015).

On the Existence of Entire Solutions of Certain Class of Nonlinear Difference Equations

/pdf/value-distribution-of-difference-polynomials-23u61qpmeg.pdf

Value Distribution of Difference Polynomials

We mainly study the transcendental entire solutions of the differential equation f n ( z ) + P ( f ) = p 1 e α 1 z + p 2 e α 2 z , where p 1 , p 2 , α 1 and α 2 are nonzero constants satisfying α 1 ≠ α 2 and P ( f ) is a differential polynomial in f of degree n − 1 . We improve Chen and Gao’s results and partially answer a question proposed by Li (J. Math. Anal. Appl. 375 (2011), pp. 310–319).

https://www.mdpi.com/2227-7390/7/6/539/pdf

Three Results on the Nonlinear Differential Equations and Differential-Difference Equations

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.

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

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 $\alpha_{1}\neq\alpha_{2}$
 .

/pdf/entire-solutions-of-certain-class-of-differential-difference-gqvksdcysx.pdf

Entire solutions of certain class of differential-difference equations

In this paper, we shall study the conditions regarding the existence of transcendental entire solutions of certain type of difference equations. Our results are either supplements to some results obtained recently, or are relating to the conjecture raised in Yang and Laine (Proc. Jpn. Acad., Ser. A, Math. Sci. 86:10-14, 2010). Finally, two relevant conjectures are posed for further studies. MSC:39B32, 34M05, 30D35.

/pdf/entire-solutions-of-certain-type-of-difference-equations-2cqr0aw484.pdf

Entire solutions of certain type of difference equations

Let f denote a transcendental meromorphic function with N(r, f) = S(r, f) and k be an integer. By using methods different from others, we have been able to derive several new results and pose some new conjectures that relate to the yet to be resolved conjecture concerning the quantitative estimates on the zeros of ff(k)-b, for a non-vanishing small function b.

Notes on the value distribution of ff(k)-b

Let α be an entire function, $a_{n-1},\ldots,a_{1},a_{0}$
 , R be small functions of f, and let $n\geq2$
 be an integer. Then, for any positive integer k, the differential equation $f^{n}f^{(k)}+a_{n-1}f^{n-1}+\cdots+a_{1}f+a_{0}=R\mathrm{e}^{\alpha}$
 has transcendental meromorphic solutions under appropriate conditions on the coefficients. In addition, for $n=1$
 and $k=1$
 , we have extended some well-known and relevant results obtained by others, by using different arguments.

https://link.springer.com/content/pdf/10.1186%2Fs13660-015-0672-x.pdf

On the meromorphic solutions of certain class of nonlinear differential equations

Let f denote a transcendental meromorphic function of finite order, we have shown that the difference polynomial Pf of the form:fnΔfn1Δ2fn2⋯Δkfnk−αcannot have 0 as its Picard exceptional value, where α is a small function of f. As a result of the studies, we are able to provide a unified and simplified proof to several known results, particularly to those Hayman-type of problems and conjecture, with generalizations and extensions.

Weiran Lü

Papers

Entire solutions of certain type of difference equations

Entire solutions of certain class of differential-difference equations

Notes on the value distribution of ff(k)-b

On the meromorphic solutions of certain class of nonlinear differential equations

On Picard exceptional values of difference polynomials