The existence of solutions of q-difference-differential equations

By using the Nevanlinna theory of value distribution, we investigate the existence of solutions of some types of non-linear q-difference differential equations. In particular, we generalize the Rellich–Wittich-type theorem and Malmquist-type theorem about differential equations to the case of q-difference differential equations (system).

have focused on the existence and growth of solutions of difference equation.
The following two results had been proved by F. Rellich and H. Wittich, respectively.
Theorem 1 (see He 1981, Rellich). Let the differential equation be the following form If f(w) is transcendental meromorphic function of w, then Eq.
(1) has no non-constant entire solution. Wittich (1955) studied the more general differential equation than Eq.
(1) and obtained the following result.
Theorem 2 (see Wittich 1955). Let be differential polynomial, with coefficients a (i) (z) are polynomial of z. If the right-hand side of the differential equation f(w) is the transcendental meromorphic function of w, then the Eq. (2) has no non-constant entire solution.
In the 1980s, Yanagihara and Shimomura extended the above type theorem to the case of difference equations (see Yanagihara 1980Yanagihara , 1983Shimomura 1981), and obtained the following two results Theorem 3 (see Shimomura 1981). For any non-constant polynomial P(w), the difference equation has a non-trivial entire solution.
Theorem 4 (see Yanagihara 1980). For any non-constant rational function R(w), the difference equation has a non-trivial meromorphic solution in the complex plane.

Conclusions and our main results
In the present paper, we mainly study the above Rellich-Wittich-type theorem of q-difference differential equation (system).

Definition 5
We call the equation a q-difference differential equation (system) if a equation (system) contains the q-difference term f(qz) and differential term f ′ (z) of one function f(z) at the same time.
We consider the system of q-difference differential equation of the form where a J 1 (z), b J 2 (z) are polynomials of z and q ∈ C\{0}, P m [f ] is a polynomial of f of degree m, and d m (z), . . . , d 0 (z) are polynomials of z, and obtain the following results. (3), if s ≥ 1, t ≥ 1 and f is a transcendental meromorphic function, then the system (3) has no non-constant transcendental entire solutions (w 1 , w 2 ) with zero order.

Theorem 6 For system
Remark 7 Under the assumptions of Theorem 6, the system of q-difference differential equation has no non-constant transcendental entire solutions (w 1 , w 2 ) with zero order, where s 1 , s 2 ≥ 1 and P s i [f ] and Q t i [f ] are irreducible polynomials in f.
If s = t and w 1 = w 2 , we can get the following theorem easily has no non-constant transcendental entire solution with zero order, where P s [f ] and Q t [f ] are irreducible polynomials in f. As we know, it is very interest problem about the Malmquist theorem of differential equations, Laine (1993) gave the following results Theorem 9 (see Laine 1993). Let where R(z, w) is defined as If Eq. (5) has transcendental meromorphic solution, then there will be l = 0 and k ≤ 2n.
Theorem 10 (see Laine 1993). Let where R(z, w) is defined as in Theorem 9. If Eq. (6) has transcendental meromorphic solution, then there will be l = 0 and k ≤ min{�, Recently, there were a number of papers focused on the Malmquist-type theorem of the complex difference equations. Ablowitz et al. (2000) proved some results on the classical Malmquist-type theorem of the complex difference equations by applying Nevanlinna theory. Besides, Gao, Xu and Li also studied some systems of complex difference equation and obtained some more precise results related to Malmquist-type theorem (see Gao 2012a, b, c;Li and Gao 2015;Xu and Xuan 2015). In this paper, we mainly study the q-difference differential equation about the Maimquist-type theorem, and obtain the following theorem.

Theorem 11 Let
where R(z, w) is defined as (7) (w ′ (qz)) n = R(z, w), P(z, w) and Q(z, w) are irreducible polynomials in w, coefficients a i (z), b j (z) are rational functions of z. If Eq. (7) exists transcendental meromorphic solutions with zero order, then we also think that l = 0 and k ≤ 2n. Similar to the proof of Theorem 11, we can get the following corollary easily.

. Let f(z) be a meromorphic function. Then for all irreducible rational functions in f, with meromorphic coefficients a i (z), b j (z), the characteristic function of R(z, f(z)) satisfies
where d = max{m, n} and �(r) = max i,j {T (r, a i ), T (r, b j )}.

Lemma 14 (Zhang and Korhonen 2010, Theorem 1 and Theorem 3) Let f(z) be a transcendental meromorphic function of zero order and q be a nonzero complex constant. Then
and on a set of logarithmic density 1.
Lemma 15 (see Barnett et al. 2007). Let f(z) be a nonconstant zero-order meromorphic function and q ∈ C\{0}. Then on a set of logarithmic density 1 for all r outside a possible exceptional set of logarithmic density 0.
Lemma 16 (see Yi andYang 1995, p. 37 or Yang 1993). Let f(z) be a nonconstant meromorphic function in the complex plane and l be a positive integer. Then
Suppose that w(z) is a transcendental meromorphic solution of equation (7) with zero order, then ϕ(z) = 1 w(z)−a is also a transcendental meromorphic solution of Eq. (17). We will discuss two cases as follows.
If 2n + l − k ≥ 0, then then deg ϕ P(z, ϕ) = 2n + l and deg ϕ Q(z, ϕ) = l. It follows by Lemma 13 that Similar to the argument as in above, we can get l = 0 and k ≤ 2n.
This completes the proof of Theorem 11.