On entire solutions of certain type of nonlinear differential equations

1.   Introduction and main results

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 differential equation. Note that for the past five or more decades, Nevanlinna theory of meromorphic functions has been used extensively to tackle problems and derive many interesting results regarding existence and growth of meromorphic solutions of differential equations in complex plane(see, e.g., ^[13,16,20] and ^[27]). Herein, we assume that the reader is familiar with the fundamental results of Nevanlinna theory and its standard notations such as $m(r, f), \, N(r, f), \, T(r, f), \, S(r, f)$ and etc, see e.g., ^{[11, 24]}. However, for the convenience of the reader, we shall repeat some notations needed below.

Given a meromorphic function $f,$ recall that $\alpha\not\equiv 0, \infty$ 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 $E$ of $r$ of finite linear measure.

It will be useful to recall the definition of the order $\rho(f)$ and the hyper-order $\rho_2(f)$ of a meromorphic function $f.$ The order is defined as

In addition, the hyper-order $\rho_2(f)$ is defined as

Moreover, let $k\geq 1$ be an integer, we need the following facts, which can be found, e.g., in ^{[11, 24]} and references therein.

A differential polynomial $Q(z, f)$ in $f$ is a finite sum of products of $f,$ derivatives of $f,$ with all the coefficients being small functions of $f.$ Namely

where $I$ is a finite index set. The degree of a single term in $Q(z, f)$ will now be defined as $|\lambda|: = \lambda_{0}+\lambda_{1}+\cdots+\lambda_{n}.$ Of course, the maximal degree of $Q(z, f)$ will then be defined as $|q|: = \max_{\lambda\in I}|\lambda|.$

Before stating our main results, we recall some previous results concerning algebraic differential equations. An algebraic differential equation is of the form

where $P$ is a polynomial in each of its variables. The equation is of order $k,$ if $w^{(k)}$is the highest derivative appearing in $P.$ An important part of the theory of algebraic differential equations is to investigate the order $\rho(w)$ of solutions $w$ meromorphic in $\mathbb{C},$ preferably in terms of $P$ only. For $k = 1,$ Gol'dberg ^[9] proved that $w(z)$ must be of finite order.

In 1933, Yosida ^[25] applied the Nevanlinna theory of meromorphic functions to differential equations in the complex plane for the first time and generalized a Malmquists theorem.

The following result was shown by Wittich (^[22], pp. 64–65), which can be stated as follows:

Theorem A. Let $P(z, w, w', \cdots, w^{(l)}) = \sum_{n_{0}\cdots n_{l}}a_{n_0\cdots n_l}(z)w^{n_0}(w')^{n_1}\cdots (w^{(l)})^{n_l} = 0$ be an algebraic differential equation, where $a_{n_0n_1\cdots n_l}(z)$ are polynomials. Then the above equation has no transcendental entire solution if only one term appears in the equation with a maximal degree.

Since then many interesting results related to the growth or existence of meromorphic solutions of certain types of generalized differential equations were derived or obtained, see, e.g., ^{[1,2,6,8,18,19]}. Since 1970's, Nevanlinna's value distribution theory (particularly Clunie type of lemmas relating equations involving differential polynomials) have been used or utilized by some authors (see, e.g., ^[15,16]) to tackle the nonlinear differential equations of the form

where $P_d(z, f)$ denotes a polynomial in $f$ and its derivatives with a total degree $d\leq n-1,$ with small functions of $f$ as the coefficients, and $h$ is a given entire or meromorphic function.

Zhang and Liao ^[26] obtained the following result.

Theorem B. Let $a, p_1, p_2$ and $\lambda$ be nonzero constants. Then the equation

does not have any transcendental entire solution.

In the paper, we first shall study the relevant problems for the following type of differential equation:

where $Q(z, f)$ denotes a differential polynomial in $f.$

Theorem 1.1. Assume that a differential polynomial $Q(z, f)$ (with no constant term, i.e., $Q(z, 0)\equiv 0$) contains just one term of maximal degree, $p\, (\not\equiv 0)$ is a small function of the exponential function ${\rm e}^z,$ and $a_i, \, b_i\, (i = 1, 2)$ are nonzero polynomials with $b_1\not\equiv b_2$. If $f$ is an admissible entire solution to Eq (1.3), then we have $\rho(f) = \max\{ \deg(b_1), \deg(b_2) \}.$

For example, the differential equation

has a transcendental entire solution $f(z) = ({\rm e}^z+{\rm e}^{-z})/4$ and $\rho(f) = 1.$

It is not difficult to verify that $f(z) = \sin z +1$ is a solution to the equation

and $\rho(f) = 1.$

The equation $f^2+8f''-8f = 16{\rm e}^{2z}+4{\rm e}^{-2z}-16$ has exactly two entire solutions, namely $f_1(z) = 4{\rm e}^{z}-2{\rm e}^{-z}$ and $f_2(z) = -4{\rm e}^{z}+2{\rm e}^{-z}.$ Obviously, $\rho(f_{1}) = \rho(f_{2}) = 1.$ In fact, this equation has no other meromorphic solutions satisfying $N(r, f) = S(r, f)$ (see, e.g., ^[14]).

Theorem 1.1 fails if $Q(z, f)$ contains more than one term of the maximal degree. Indeed, the differential equation

is solved by $f(z) = {\rm e}^{\sin z}+z,$ and we would get $\rho(f) = +\infty.$

Next, we will deal with the growth of entire transcendental solutions of nonlinear differential equations. Before stating our next result, we recall some previous results concerning the second order homogeneous linear differential equation of type

where $P(e^z$) and $Q(e^z)$ are polynomials in $e^z$ and they are not both constants. It is well known that every solution $f$ of (1.4) is an entire function.

In fact, this type differential equation was discussed by Mathieu in 1868 in connection with the problem of vibrations of an elliptic membrane in the following manner:

Assume that the membrane, which is in the plane $\it {XOY}$ when it is in equilibrium, is vibrating with frequency $p.$ Then, if we write

the equation of two dimensional wave motion

becomes

By a slight transformation, Eq (1.5) will become a linear differential equation, of the second order, of the form

where $a, \, q$ are constants.

Mathieu Eq (1.6) has some important applications, which can be found, e.g., in ^[21] and references therein for interested readers.

Thereafter up to the present, in this direction, many good results were obtained, see, e.g., ^[5,12]. It is also worth mentioned that when they studied the growth on (1.4), the so called central index method was used. The purpose of the present paper is to extend these works (see, e.g., ^{[3,4,7,10,17]}) concerning the nature of solutions of linear differential equations to nonlinear differentials. Moreover, our results will include several known results for linear differential equations obtained earlier as special cases.

Before proceeding further, we give an idea of the problem we are dealing with the growth of the solution to (1.6) by the following assertion.

According to (1.6) and the fact (see, e.g., ^[11]):

where $P(z) = a_{k}z^k+a_{k-1}z^{k-1}+\cdots+a_0$ is a polynomial with $a_{k} (\neq 0),$ then, any solution $u\not\equiv 0$ of (1.6) satisfies

which shows that $\rho(u) = \infty.$

Motivated by this, our next aim is to discuss the growth of solutions to some more general forms of Eq (1.6). To bring about our results from the more general hypotheses without complicated calculations will probably be the most interesting feature of this note. For convenience, in what follows, we set

where $a_{0i}, a_{1i}, \cdots, a_{ki}$ are constants with $a_{0i}\neq 0 \, (i = 1, 2, \cdots, m).$

where $b_{0j}, b_{1j}, \cdots, b_{lj}$ are constants with $b_{0j}\neq 0 \, (j = 1, 2, \cdots, n).$

where $\lambda_{0j}, \lambda_{1j}, \cdots, \lambda_{qj} \, (j = 1, 2, \cdots, n)$ are nature numbers.

Theorem 1.2 Suppose that $\alpha_{i}(z)(i = 1, \cdots, m)$, $\beta_{j}(z)(j = 1, \cdots, n)$ are nonzero polynomials and $k^{*} > l^{*}.$ If $d_1 = d_2 = \cdots = d_n = d\geq 1,$ then every entire solution $f(\not\equiv 0)$ of the equation

satisfies $\rho(f) = \infty$ and $\rho_2(f) = k^{*}.$

The following examples show that the conclusion of Theorem 1.2 can occur.

Example 1.1 The equation

is satisfied by $f(z) = {\rm e}^{\sin z}.$ It is easy to see that $\rho(f) = \infty$ and $\rho_2(f) = 1.$

Example 1.2 Consider the nonlinear equation

Then $f(z) = {\rm e}^{{\rm e}^z}$ solves the above equation and $\rho(f) = \infty,$ $\rho_2(f) = 1.$

Example 1.3 The nonlinear equation

has an entire solution $f(z) = {\rm e}^{{\rm e}^{z^{2}}}$ with $\rho(f) = \infty$ and $\rho_2(f) = 2.$

Example 1.4 The linear differential equation

has three entire solutions $f_j(z) = {\rm e}^{c_j{\rm e}^{z}-z},$ in which $c_j^3 = 1, \, j = 1, 2, 3.$ Obviously, $\rho(f_j) = \infty, \, \rho_2(f_j) = 1\, \, (j = 1, 2, 3).$

The following examples show that the condition $k^{*} > l^{*}$ is necessary.

Example 1.5 The equation

is satisfied by $f(z) = z^3,$ and $\rho(f) = 0.$

Example 1.6 It is clear that the differential equation

has a solution $f = {\rm e}^{z}$, but $\rho(f) = 1,$ and $\rho_2(f) = 0.$

2.   Some Lemmas

In order to prove our results, we also need the following results.

The determinant $\omega(f_1, \cdots, f_n)$ is called the Wronskian of $f_{1}, \cdots, f_{n},$ and which is given by

Lemma 2.1 (see, e.g., Proposition 1.4.2 in ^[13]) Let $f_1, \cdots, f_n$ be meromorphic functions. Then $\omega(f_1, \cdots, f_n)$ vanishes identically if and only if $f_1, \cdots, f_n$ are linearly dependent.

Lemma 2.2 (see, e.g., Lemma 1.9 in ^[24]) Suppose that $g_1, g_2, \cdots, g_n$ are entire functions and $a_0, a_1, a_2, \cdots, a_n$ are meromorphic functions such that

where $E$ is a set whose linear measure is finite. If

then there exist constants $c_1, c_2, \cdots, c_n,$ at least one of them is not zero, such that

Lemma 2.3 (see, e.g., pp.35, Corollary in ^[24]) Let $f$ and $g$ be meromorphic functions such that $T(r, f) = O(T(r, g))\, (r\to\infty, r\not\in E),$ where $E$ is a set of finite measure. Then $\rho(f)\leq \rho(g).$

The following Lemma is crucial to the proofs of our main results.

Lemma 2.4 Let $F(z) = \sum_{i = 1}^{n}a_i(z){\rm e}^{b_i (z)},$ and $a_i, \, b_i \, (i = 1, 2, \cdots, n)$ be nonzero polynomials with $b_i\not\equiv b_j, i\neq j$. Then

in which $\deg b_s: = \max\{ \deg b_i, \, i = 1, 2, \cdots, n\}.$

Proof: First, by Lemma 2.1 and Lemma 2.2, we assume without loss of generality, that $a_{1}(z){\rm e}^{b_1(z)},$ $a_{2}(z){\rm e}^{b_2(z)}, \cdots,$ $a_{n}(z){\rm e}^{b_n(z)}$ are linear independence. Thus we see that

and which is normally written as $\omega = \omega(a_{1}{\rm e}^{b_1}, a_{2}{\rm e}^{b_2}, \cdots, a_{n}{\rm e}^{b_n}).$

Again, by repeated differentiation in both sides of $F(z) = \sum_{i = 1}^{n}a_{i}(z){\rm e}^{b_i(z)},$ we have

with $a_{1j} = a_j, \, a_{kj} = a'_{k-1\, j}+a_{k-1\, j}b'_{j}, \, k = 2, 3, \cdots, n, \, \, j = 1, 2, \cdots, n.$

On the other hand, one may show that

Further, we set $D = \left|\begin{array}{cccc} a_{1\, 1} & a_{1\, 2} & \cdots & a_{1\, n} \\ a_{2\, 1} & a_{2\, 2} & \cdots & a_{2\, n} \\ \vdots & \vdots & \cdots & \vdots \\ a_{n\, 1} & a_{n\, 2} & \cdots & a_{n\, n} \end{array}\right|.$

It follows from the properties of determinant and (2.2) that $D\not\equiv 0.$ For sake of brevity and simplicity, we, as a rule, now make the following reasonable assumption: $\deg b_1 = \max\{ \deg { b_i}, \, i = 1, 2, \cdots, n\}.$ Thus, by Cramer's ruler, (2.1) gives

with $D_1 = \left|\begin{array}{cccc} F & a_{1\, 2} & \cdots & a_{1\, n} \\ F' & a_{2\, 2} & \cdots & a_{2\, n} \\ \vdots & \vdots & \cdots & \vdots \\ F^{(n-1)} & a_{n\, 2} & \cdots & a_{n\, n} \end{array}\right|.$

In view of (2.3) and the expression of $D_1,$ we find

where $A_{11}, A_{21}, \cdots, A_{n1}$ are co-factors of the first column of $D_1.$ In view of $a_i, \, b_i \, (i = 1, 2, \cdots, n)$ are nonzero polynomials, we deduce that $\frac{A_{11}}{D}, \frac{A_{21}}{D}, \cdots, \frac{A_{n1}}{D}$ are small functions of ${\rm e}^{b_1}.$ In addition, it is easy to see by (2.4) that they are also small functions of $F.$

Consequently, it follows from the logarithmic derivative lemma and (2.4) that

Moreover, note that ${\rm e}^{b_1} = F\, \frac{{\rm e}^{b_1}}{F}$ and (2.5), we would get

Therefore, $T(r, {\rm e}^{b_s}) = m(r, {\rm e}^{b_s})\leq T(r, F)+S(r, F)$ is satisfies, and then Lemma 2.3 immediately implies $b_{s} = \rho({\rm e}^{b_s})\leq \rho(F).$ It is easy to see that

Consequently, on ground of Lemma 2.3, $\rho(F)\leq \rho({\rm e}^{b_s}) = b_{s}$ is correct.

This completes the proof of Lemma 2.4.

Lemma 2.5 (^[13]) Let $f$ be a meromorphic solution of an algebraic equation

where $P$ is a polynomial in $f, f', \cdots, f^{(n)}$ with meromorphic coefficients small with respect to $f.$ If a complex constant $c$ does not satisfy Eq (2.6), then

3.   Proofs of Theorems

Proof of Theorem 1.1.

Now, we no loss in generality in supposing $b_1$ and $b_2$ are not constants. If (1.3) has an entire solution $f,$ we then from (1.3) and Lemma 2.4 get

this shows that $p$ is a small function of $Q,$ and we see that $p$ is also a small function of $f.$

Again, by (1.3), we have

which, and (1.3), gives

and therefore

According to (3.1) and (3.2), we obtain

If $s_1'p_1-s_1p_1'+s_1p_1b_2'\equiv 0,$ then $Ap_1 = s_1{\rm e}^{b_2}$ for a constant $A,$ it is a contradiction. Now, we may assume that $s_1'p_1-s_1p_1'+s_1p_1b_2'\not\equiv 0.$ Thus applying Lemma 2.5 (where $c = 0$ is used) to (3.3), we have

Let $N$ denote the maximal degree of $Q(z, f).$ Now we claim that

To prove (3.5), we rewrite $Q(z, f)$ as

where $P_{j}\, (j = 1, 2, \cdots, N)$ denote a polynomial of $f^{(k)}/f, k\in \mathbb{N}.$ Obviously, it follows by (3.4) and (3.6) that

On the other hand, according to (3.6)

is correct. By the same methods as used in ^[23], we then by (3.8) have

and thereby (3.5) immediately follows from (3.7) and (3.9).

Therefore, we have $N\, m(r, f) = \max\{T(r, e^{b_1}), T(r, e^{b_2})\}+S(r, f),$ which implies $\rho(f) = \max\{ \deg(b_1), \deg(b_2) \},$ and Theorem 1.1 follows.

Proof of Theorem 1.2.

We first of all rewrite (1.7) as

Suppose that $f$ is of finite order. Since $\alpha_{i}(z)(i = 1, \ldots, m)$, $\beta_{j}(z)(j = 1, \ldots, n)$ are nonzero polynomials, it follows by Lemma 2.4 and (3.10) that

which is impossible. Thus the order of $f$ must be infinite and then it is easily shown in this case there exists an integer $l\, (1\leq l\leq n)$ such that

which gives

where $E$ is a set whose linear measure is not greater than $2.$ Thus, we find

or

which and Lemma 2.3 will lead to the conclusion that the order of $\log T(r, f)$ is $k^*,$ which will lead to the conclusion that hyper-order of $f$ must be $k^{*}.$ The assertion follows.

Finally, we would like to pose the following conjecture, for further studies.

Conjecture. The asserts of the present paper remain to be valid for meromorphic solutions $f$ satisfying $N(r, f) = S(r, f)$ or $N(r, f)$ is of finite order.

4.   Conclusions

Using the theory of meromorphic functions and the Cramer's rule, this paper extends some results on entire solutions of certain type of linear differential equations to that of nonlinear differential equations. Some examples show that the existence of solutions for such equations. Meanwhile, a conjecture is posed for further studies.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions, which led to considerable improvement of the article.

This work was supported by the National Natural Science Foundation of China (Nos. 11602305, 11601521) and the Fundamental Research Funds for the Central Universities (Nos. 18CX02045A, 17CX02048).

Conflict of interest

The authors declare no conflict of interest.