Startseite Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
Artikel Open Access

Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#

  • Hong Yan Xu und Goutam Haldar EMAIL logo
Veröffentlicht/Copyright: 21. November 2024
Artikel downloaden (PDF)
Veröffentlichen auch Sie bei De Gruyter Brill
Informationen für Autor*innen Erkunden Sie dieses Fachgebiet
Demonstratio Mathematica
Aus der Zeitschrift Demonstratio Mathematica Band 57 Heft 1

Abstract

This study is devoted to exploring the existence and the precise form of finite-order transcendental entire solutions of second-order trinomial partial differential-difference equations

L ( f ) 2 + 2 h L ( f ) f ( z 1 + c 1 , z 2 + c 2 ) + f ( z 1 + c 1 , z 2 + c 2 ) 2 = e g ( z 1 , z 2 )

and

L ˜ ( f ) 2 + 2 h L ˜ ( f ) ( f ( z 1 + c 1 , z 2 + c 2 ) − f ( z 1 , z 2 ) ) + ( f ( z 1 + c 1 , z 2 + c 2 ) − f ( z 1 , z 2 ) ) 2 = e g ( z 1 , z 2 ) ,

where L ( f ) and L ˜ ( f ) are defined in (2.1) and (2.2), respectively, and g ( z ) is a polynomial in C 2 . Our results are the extensions of some of the previous results of Liu et al. Also, we exhibit a series of examples to explain that the forms of transcendental entire solutions of finite-order in our results are precise.

Keywords: functions of several complex variables; Fermat-type equations; entire solutions; Nevanlinna theory
MSC 2010: 30D35; 35M30; 32W50; 39A45

1 Introduction

Fermat’s last theorem [1] says that the Fermat equation x m + y m = 1 does not admit nontrivial solutions in rational numbers when m ⩾ 3 and does admit nontrivial rational solutions when m = 2 . For a positive integer m , the equation

(1.1) f m + g m = 1

is known as Fermat-type equation over function fields. With the help of the Nevanlinna theory [2,3], Montel [4], Iyer [5], and Gross [6] studied the existence and form of the solutions of the functional equation (1.1) and pointed out the following:

  1. For m = 2 , the entire solutions of (1.1) are f ( z ) = cos ( ξ ( z ) ) and g ( z ) = sin ( ξ ( z ) ) , where ξ is a non-constant entire function.

  2. For m > 2 , there are no non-constant entire solutions of (1.1).

  3. For m = 2 , the meromorphic solutions of (1.1) are of the form

    f ( z ) = 2 ϕ ( z ) 1 − ϕ 2 ( z ) and g ( z ) = 1 − ϕ 2 ( z ) 1 + ϕ 2 ( z ) ,

    where ϕ is a non-constant meromorphic function.

  4. For m = 3 , the meromorphic solutions of (1.1) are of the form

    f ( z ) = 1 2 ℘ ( h ) 1 + ℘ ′ ( h ) 3 and g ( z ) = ω 2 ℘ ( h ) 1 − ℘ ′ ( h ) 3 ,

    where ω 3 = 1 and ℘ satisfies ( ℘ ′ ) 2 = 4 ℘ 3 − 1 .

  5. For m > 3 , there are no non-constant meromorphic solutions.

In 2004, Yang and Li [7] investigated (1.1) by replacing g with f ′ when m = 2 , and proved that the transcendental entire solution of f ( z ) 2 + f ′ ( z ) 2 = 1 has the form f ( z ) = A e α z ⁄ 2 + e − α z ⁄ 2 A , where A and α are non-zero complex constants.

The advent of the difference analog lemma of the logarithmic derivative [8,9] expedites the research activity to characterize the entire or meromorphic solutions of Fermat-type difference and differential-difference equations [10–13]. In 2012, Liu et al. [14] proved that the transcendental entire solutions with finite-order of the Fermat-type difference equation f ( z ) 2 + f ( z + c ) 2 = 1 must satisfy f ( z ) = sin ( A z + B ) , where B is a constant and A = ( 4 k + 1 ) π ⁄ 2 c , where k is an integer. In 2016, Liu and Yang [15] studied the existence and the form of solutions of some quadratic trinomial functional equations and obtained some important results as follows. The equation f ( z ) 2 + 2 α f ( z ) f ′ ( z ) + f ′ 2 ( z ) = 1 has no transcendental meromorphic solutions, and the finite-order transcendental entire solution of f ( z ) 2 + 2 α f ( z ) f ( z + c ) + f ( z + c ) 2 = 1 must be of order equal to one, where in both the equations, α ≠ 0 , ± 1 .

The study of several characteristics of the solutions to partial differential equations in several complex variables [16–24] is an important topic. The Fermat-type equation like appears in particle mechanics; in particular, it appears as the Lagrange function in the Lagrangian functional describing the “action” of a system. As far as the author’s knowledge is concerned, Saleebly, in 1999, first started investigation about the existence and form of entire and meromorphic solutions of Fermat-type partial differential equations [25,26]. Most noticeably, Khavinson [27] proved that any entire solution of the partial differential equation f z 1 2 + f z 2 2 = 1 must be linear, i.e., f ( z 1 , z 2 ) = a z 1 + b z 2 + c , where a , b , c ∈ C , and a 2 + b 2 = 1 . Later, Li [28,29] investigated the partial differential equations with more general forms such as f z 1 2 + f z 2 2 = p and f z 1 2 + f z 2 2 = e q , where p and q are the polynomials in C 2 .

Hereinafter, we denote by z + w = ( z 1 + w 1 , z 2 + w 2 ) for any z = ( z 1 , z 2 ) , w = ( w 1 , w 2 ) ∈ C 2 .

Recently, Xu and Cao [30] extended several results from one complex variable to several complex variables. In fact, they considered two Fermat-type equations f ( z ) 2 + f ( z + c ) 2 = 1 and f ( z + c ) 2 + f z 1 2 = 1 and proved that any finite-order transcendental entire solution f : C n → P 1 ( C ) has the form f ( z ) = cos ( L ( z ) + B ) , where L is a linear function of the form L ( z ) = a 1 z 1 + … + a n z n on C n such that L ( c ) = − π ⁄ 2 − 2 k π ( k ∈ Z ), and B ∈ C and f ( z ) = cos ( A 1 z 1 + A 2 z 2 + Constant ) , where A 1 c 1 + A 2 c 2 = − π ⁄ 2 − 2 k π , k ∈ Z , respectively.

In 2022, Xu et al. [31] explored the precise form of entire and meromorphic solutions of the following partial differential difference equations:

(1.2) α ∂ f ∂ z 1 + β ∂ f ∂ z 2 2 + f ( z 1 + c 1 , z 2 + c 2 ) 2 = e g ( z 1 , z 2 )

and

(1.3) α ∂ f ∂ z 1 + β ∂ f ∂ z 2 2 + ( f ( z 1 + c 1 , z 2 + c 2 ) − f ( z 1 , z 2 ) ) 2 = e g ( z 1 , z 2 ) ,

where g ( z ) is a polynomial in C 2 and α and β are the constants in C . For detail study, we insist the readers to go through [31].

As far as we know, it was Saleeby [32], who first considered another quadratic functional equation

(1.4) f 2 + 2 α f g + g 2 = 1 ,

and investigated the existence and form of transcendental entire and meromorphic solutions in C n . We recall the result.

Theorem A

[32] The transcendental entire solutions of (1.4) must be of the form f ( z ) = 1 2 cos ( h ( z ) ) 1 + α + sin ( h ( z ) ) 1 − α and 1 2 cos ( h ( z ) ) 1 + α − sin ( h ( z ) ) 1 − α , where h is entire C n . The meromorphic solutions of (1.4) must be of the form f ( z ) = α 1 − α 2 β ( z ) 2 ( α 1 − α 2 ) β ( z ) and g ( z ) = 1 − β ( z ) 2 ( α 1 − α 2 ) β ( z ) , where β ( z ) is meromorphic in C n and α 1 = − α + α 2 − 1 , α 2 = − α − α 2 − 1 , and α 2 ≠ 0 , 1 .

In 2021, Li and Xu [33] considered the quadratic trinomial partial differential difference equations

(1.5) f ( z 1 + c 1 , z 2 + c 2 ) 2 + 2 α f ( z 1 + c 1 , z 2 + c 2 ) ∂ f ∂ z 1 + ∂ f ∂ z 1 2 = e g ( z 1 + c 1 , z 2 + c 2 )

and obtained the following result.

Theorem B

[33] Let α 2 ≠ 0 , 1 , c 2 ≠ 0 , and g be a non-constant polynomial in C 2 , and not the form of ϕ ( z 2 ) . If (1.5) admits a finite-order transcendental entire solution f, then g must be of the form g ( z 1 , z 2 ) = a 1 z 1 + a 2 z 2 + b , where a 1 ( ≠ 0 ) , a 2 , b 2 ∈ c . Furthermore, f ( z ) must satisfy one of the following cases:

  1. f ( z 1 , z 2 ) = 2 a 1 ( A 2 ξ + A 1 ξ − 1 ) e 1 2 ( a 1 z 1 + a 2 z 2 + b ) ,

    where ξ ( ≠ 0 ) , a 1 , a 2 , b , c 1 , c 2 , A 1 , A 2 ∈ C satisfying

    e 1 2 ( a 1 c 1 + a 2 c 2 ) = a 1 ( A 1 ξ + A 2 ξ − 1 ) 2 ( A 2 ξ + A 1 ξ − 1 ) .

  2. f ( z 1 , z 2 ) = 1 2 A 2 a 11 e a 11 z 1 + a 12 z 2 + b 1 + A 1 a 21 e a 21 z 1 + a 22 z 2 + b 2 ,

    where a j ( ≠ 0 ) , b j ∈ C , ( j = 1 , 2 ) satisfy a 11 z 1 + a 12 z 2 ≠ a 21 z 1 + a 22 z 2 ,

    g ( z 1 , z 2 ) = ( a 11 + a 21 ) z 1 + ( a 12 + a 22 ) z 2 + b 1 + b 2 , and

    e a 11 c 1 + a 12 c 2 = A 2 A 1 a 11 , e a 21 c 1 + a 2 c 2 = A 1 A 2 a 21 , e a 1 c 1 + a 2 c 2 = a 11 a 21 .

2 Main results

Inspired by Theorems A and B, and utilizing difference analogs of the Nevanlinna theory of several complex variables [34,35], we investigate the existence and forms of transcendental entire solutions of the following trinomial second-order partial differential-difference equations:

(2.1) L ( f ) 2 + 2 h L ( f ) f ( z 1 + c 1 , z 2 + c 2 ) + f ( z 1 + c 1 , z 2 + c 2 ) 2 = e g ( z 1 , z 1 )

and

(2.2) L ˜ ( f ) 2 + 2 h L ˜ ( f ) f ( z 1 + c 1 , z 2 + c 2 ) + ( f ( z 1 + c 1 , z 2 + c 2 ) − f ( z 1 , z 2 ) ) 2 = e g ( z 1 , z 2 ) ,

where L ˜ ( f ) = γ ∂ 2 f ( z 1 , z 2 ) ∂ z 1 2 + δ ∂ 2 f ( z 1 , z 2 ) ∂ z 2 2 + η ∂ 2 f ( z 1 , z 2 ) ∂ z 1 ∂ z 2 , and L ( f ) = L ˜ + α ∂ f ( z 1 , z 2 ) ∂ z 1 + β ∂ f ( z 1 , z 2 ) ∂ z 2 , α , β , γ , δ , and η are the constants in C , c = ( c 1 , c 2 ) ∈ C 2 , and g ( z ) is a polynomial in C 2 .

Before we state our main results, let us first set that

(2.3) A 1 ≔ 1 2 1 + h − 1 2 i 1 − h and A 2 ≔ 1 2 1 + h + 1 2 i 1 − h ,

where h ≠ 0 , ± 1 , 2 is a constant in C ,

(2.4) R 1 = α + γ a 1 + 1 2 η a 2 , R 2 = β + δ a 2 + 1 2 η a 1 , R 3 = γ c 2 2 + δ c 1 2 − η c 1 c 2 , R 4 = α a 1 + β a 2 + 1 2 ( γ a 1 2 + δ a 2 2 + η a 1 a 2 ) , R j 5 = α + 2 γ a j 1 + η a j 2 , R j 6 = β + 2 δ a j 2 + η a j 1 , R j 7 = α a j 1 + β a j 2 + γ a j 1 2 + δ a j 2 2 + η a j 1 a j 2 , j = 1 , 2 .

Now, we state our results as follows.

Theorem 2.1

Let c = ( c 1 , c 2 ) ∈ C 2 , h ∈ C such that h ≠ 0 , ± 1 and g ( z 1 , z 2 ) is a polynomial in C 2 . If f ( z ) be a finite-order transcendental entire solution of equation (2.1), then one of the following cases occurs.

  1. The form of the solution f is of the form f ( z 1 , z 2 ) = ∑ A e l 1 z 1 + l 2 z 2 , where A , l 1 , and l 2 are the arbitrary constants in C with α l 1 + β l 2 + γ l 1 2 + δ l 2 2 + η l 1 l 2 = 0 and

    g ( z 1 , z 2 ) = 2 L n ± ∑ A ′ e l 1 z 1 + l 2 z 2 , where A ′ = A e ( l 1 c 1 + l 2 c 2 ) .

  2. g must be of the form g ( z 1 , z 2 ) = L ( z 1 , z 2 ) + H ( s 1 ) + B , where L ( z 1 , z 2 ) = a 1 z 1 + a 2 z 2 , H ( s 1 ) is a polynomial in s 1 = c 2 z 1 − c 1 z 2 , a 1 , a 2 , B are constants in C , and the form of the solution is

    f ( z 1 , z 2 ) = 1 2 ( A 2 ξ + A 1 ξ − 1 ) e 1 2 [ L ( z 1 , z 2 ) + H ( s 1 ) − L ( c ) + B ] .

    L ( z 1 , z 2 ) satisfies relation

    e 1 2 L ( c 1 , c 2 ) = A 2 ξ + A 1 ξ − 1 2 ( A 1 ξ + A 2 ξ − 1 ) R 4 + ( R 1 c 2 − R 2 c 1 ) a 0 + 1 2 R 3 a 0 2 ,

    where a 0 is the coefficient of s 1 of the polynomial H ( s 1 ) and R j ’s are defined in (2.4). In particular, if R 1 c 2 − R 2 c 1 ≠ 0 or R 3 ≠ 0 , then H ( s 1 ) becomes linear in s 1 .

  3. g ( z 1 , z 2 ) must be of the form g ( z 1 , z 2 ) = L ( z 1 , z 2 ) + H ( s 1 ) + B , where L ( z 1 , z 2 ) = L 1 ( z 1 , z 2 ) + L 2 ( z 1 , z 2 ) and H ( s 1 ) = H 1 ( s 1 ) + H 2 ( s 1 ) with L 1 ( z 1 , z 2 ) + H 1 ( s 1 ) ≠ L 2 ( z 1 , z 2 ) + H 2 ( s 1 ) and L j ( z 1 , z 2 ) = a j 1 z 1 + a j 2 z 2 , B = B 1 + B 2 ; H j ( s 1 ) is a polynomial in s 1 = c 2 z 1 − c 1 z 2 for j = 1 , 2 , and B 1 , B 2 , and a j i are the constants in C , and the form of the solution is

    f ( z 1 , z 2 ) = 1 2 [ A 2 e ( L 1 ( z 1 , z 2 ) + H 1 ( s 1 ) − L 1 ( c ) + B 1 ) + A 1 e ( L 2 ( z 1 , z 2 ) + H 2 ( s 1 ) − L 2 ( c ) + B 2 ) ] ,

    where L 1 ( z 1 , z 2 ) and L 2 ( z 1 , z 2 ) , respectively, satisfy the relations

    e L 1 ( c ) = A 2 A 1 [ R 17 + ( R 15 c 2 − R 16 c 1 ) a 0 + R 3 a 0 2 ] and e L 2 ( c ) = A 1 A 2 [ R 27 + ( R 25 c 2 − R 26 c 1 ) a 00 + R 3 a 00 2 ] ,

    a 0 and a 00 , respectively, the coefficients of the linear term of the polynomials H 1 ( s 1 ) and H 2 ( s 1 ) , and R i j ’s are defined in (2.4). In particular, if R 15 c 2 − R 16 c 1 ≠ 0 or R 3 ≠ 0 , then H 1 becomes linear in s 1 . Similarly, if R 25 c 2 − R 26 c 1 ≠ 0 or A 3 ≠ 0 , then H 2 becomes linear in s 1 .

Next, we exhibit some examples in support of Theorem 2.1.

Example 2.2

Let α = β = 0 , γ = δ = 1 , η = − 2 , h = − 5 ⁄ 4 , c 1 = c 2 = π i , and g ( z ) = 2 ( z 1 + z 2 ) . Then, it can be easily seen that f ( z 1 , z 2 ) = e z 1 + z 2 is a solution of (2.1), which is the conclusion (i) of Theorem 2.1.

Example 2.3

α = β = 0 , ξ = c 1 = c 2 = γ = δ = 1 , η = 2 , and g ( z 1 , z 2 ) = z 1 + z 2 + ( z 1 − z 2 ) 2 + 1 . Then, in view of conclusion (ii) of Theorem 2.1, one can easily verify that f ( z 1 , z 2 ) = A 1 + A 2 2 e [ z 1 + z 2 + ( z 1 − z 2 ) 2 + 1 ] ⁄ 2 is a solution of

∂ 2 f ( z 1 , z 2 ) ∂ z 1 2 + ∂ 2 f ( z 1 , z 2 ) ∂ z 2 2 + 2 ∂ 2 f ( z 1 , z 2 ) ∂ z 1 ∂ z 2 2 + 2 h ∂ 2 f ( z 1 , z 2 ) ∂ z 1 2 + ∂ 2 f ( z 1 , z 2 ) ∂ z 2 2 + 2 ∂ 2 f ( z 1 , z 2 ) ∂ z 1 ∂ z 2 f ( z 1 + c 1 , z 2 + c 2 ) + f ( z 1 + c 1 , z 2 + c 2 ) 2 = e [ z 1 + z 2 + ( z 1 − z 2 ) 2 + 1 ] ,

where A 1 and A 2 are given by (2.3).

Example 2.4

Let α = β = 0 , γ = δ = 1 , η = − 2 , L 1 ( z ) = z 1 + z 2 , L 2 ( z ) = 2 z 1 + z 2 , H 1 ( s 1 ) = H 2 ( s 2 ) = 0 , and B 1 = B 2 = 1 . Then, in view of conclusion (iii) of Theorem 2.1, it can be easily verified that f ( z 1 , z 2 ) = 1 2 [ A 1 e z 1 + 2 z 2 + 1 + A 2 e 2 z 1 + z 2 + 1 ] is a solution of

∂ 2 f ( z 1 , z 2 ) ∂ z 1 2 + ∂ 2 f ( z 1 , z 2 ) ∂ z 2 2 − 2 ∂ 2 f ( z 1 , z 2 ) ∂ z 1 ∂ z 2 2 + 2 h ∂ 2 f ( z 1 , z 2 ) ∂ z 1 2 + ∂ 2 f ( z 1 , z 2 ) ∂ z 2 2 − 2 ∂ 2 f ( z 1 , z 2 ) ∂ z 1 ∂ z 2 f ( z 1 + c 1 , z 2 + c 2 ) + f ( z 1 + c 1 , z 2 + c 2 ) 2 = e [ 3 z 1 + 3 z 2 + 2 ] , where  A 1  and  A 2  are given by (2.3) .

Theorem 2.5

Let c = ( c 1 , c 2 ) ∈ C 2 , γ , δ , η , h ∈ C such that γ ≠ 0 , h ≠ 0 , ± 1 and g ( z 1 , z 2 ) is a polynomial in C 2 . Let f ( z ) be a finite-order transcendental entire solution of (2.2). Then, one of the following cases must occur.

  1. f ( z 1 , z 2 ) = ϕ 1 ( β 1 z 1 − α 1 z 2 ) + ϕ 2 ( β 2 z 1 − α 2 z 2 ) , where ϕ 1 and ϕ 2 are finite-order transcendental entire functions in C 2 satisfying

    ϕ 1 ( β 1 z 1 − α 1 z 2 + β 1 c 1 − α 1 c 2 ) + ϕ 2 ( β 2 z 1 − α 2 z 2 + β 2 c 1 − α 2 c 2 ) − ϕ 1 ( β 1 z 1 − α 1 z 2 ) − ϕ 2 ( β 2 z 1 − α 2 z 2 ) = ± e g ( z ) ⁄ 2 ,

    where α 1 α 2 = γ , β 1 β 2 = δ and α 1 β 2 + β 1 α 2 = η .

  2. g ( z 1 , z 2 ) = a 1 z 1 + a 2 z 2 + H ( c 2 z 1 − c 1 z 2 ) + B , where H is a polynomial in c 2 z 1 − c 1 z 2 and a 1 c 1 + a 2 c 2 = 4 k π i , k ∈ Z ,

    f ( z 1 , z 2 ) = ± 1 γ ∫ 0 z 1 ⁄ α 1 ∫ 0 z 1 ⁄ α 2 e 1 2 [ a 1 z 1 + a 2 z 2 + H ( c 2 z 1 − c 1 z 2 ) + B ] d z 1 d z 1 + 1 α 2 ∫ 0 z 1 ⁄ α 2 G 0 α 1 z 2 − β 1 z 1 α 1 d z 1 + G 1 α 2 z 2 − β 2 z 1 α 2 ,

    where α 1 , α 2 , β 1 , and β 2 are defined in (i) and G 0 and G 1 are finite-order transcendental entire functions in C 2 satisfying

    1 α 2 ∫ 0 z 1 ⁄ α 2 G 0 α 1 z 2 − β 1 z 1 α 1 + α 1 c 2 − β 1 c 1 α 1 − G 0 α 1 z 2 − β 1 z 1 α 1 d z 1 + G 1 α 2 z 2 − β 2 z 1 α 2 + α 2 c 2 − β 2 c 1 α 2 − G 1 α 2 z 2 − β 2 z 1 α 2 = 0 .

  3. If γ c 2 2 + δ c 1 2 ≠ η c 1 c 2 , then g must be of the form g ( z 1 , z 2 ) = a 1 z 1 + a 2 z 2 + B , a 1 , a 2 , B ∈ C , and the form of the solution is

    f ( z 1 , z 2 ) = ϕ 1 ( β 1 z 1 − α 1 z 2 ) + ϕ 2 ( β 2 z 1 − α 2 z 2 ) + 2 2 ( A 1 ξ + A 2 ξ − 1 ) γ a 1 2 + δ a 2 2 + η a 1 a 2 e 1 2 [ a 1 z 1 + a 2 z 2 + B ] ,

    where ξ ( ≠ 0 ) ∈ C , γ a 1 2 + δ a 2 2 + η a 1 a 2 ≠ 0 , α 1 , α 2 , β 1 , and β 2 are the same as in (i), A 1 and A 2 are defined in (2.3), ϕ 1 and ϕ 2 are finite-order transcendental entire functions in C 2 such that

    ϕ 1 ( β 1 z 1 − α 1 z 2 + β 1 c 1 − α 1 c 2 ) + ϕ 2 ( β 2 z 1 − α 2 z 2 + β 2 c 1 − α 2 c 2 ) = ϕ 1 ( β 1 z 1 − α 1 z 2 ) + ϕ 2 ( β 2 z 1 − α 2 z 2 ) and e 1 2 [ a 1 c 1 + a 2 c 2 ] = ( A 2 ξ + A 1 ξ − 1 ) ( γ a 1 2 + δ a 2 2 + η a 1 a 2 ) 4 ( A 1 ξ + A 2 ξ − 1 ) + 1 .

  4. If γ c 2 2 + δ c 1 2 ≠ η c 1 c 2 , then g must be of the form g ( z 1 , z 2 ) = L 1 ( z 1 , z 2 ) + L 2 ( z 1 , z 2 ) + B 1 + B 2 , where L j ( z 1 , z 2 ) = a j 1 z 1 + a j 2 z 2 with L 1 ( z 1 , z 2 ) ≠ L 2 ( z 1 , z 2 ) , a i j , B j ∈ C and the form of the solution is

    f ( z 1 , z 2 ) = ϕ 1 ( β 1 z 1 − α 1 z 2 ) + ϕ 2 ( β 2 z 1 − α 2 z 2 ) + A 1 e L 1 ( z 1 , z 2 ) + B 1 2 ( γ a 11 2 + δ a 12 2 + η a 11 a 12 ) + A 2 e L 2 ( z 1 , z 2 ) + B 2 2 ( γ a 21 2 + δ a 22 2 + η a 21 a 22 ) ,

    where γ a 21 2 + δ a 22 2 + η a 21 a 22 ≠ 0 , γ a 11 2 + δ a 12 2 + η a 11 a 12 ≠ 0 , α 1 , α 2 , β 1 , β 2 are the same as in (i), A 1 and A 2 are defined in (2.3), ϕ 1 and ϕ 2 are finite-order transcendental entire functions in C 2 such that

    ϕ 1 ( β 1 z 1 − α 1 z 2 + β 1 c 1 − α 1 c 2 ) + ϕ 2 ( β 2 z 1 − α 2 z 2 + β 2 c 1 − α 2 c 2 ) = ϕ 1 ( β 1 z 1 − α 1 z 2 ) + ϕ 2 ( β 2 z 1 − α 2 z 2 ) and e L 1 ( c ) = A 2 A 1 ( γ a 11 2 + δ a 12 2 + η a 11 a 12 ) + 1 , e L 2 ( c ) = A 1 A 2 ( γ a 21 2 + δ a 22 2 + η a 21 a 22 ) + 1 .

The following examples show that the forms of solutions are precise.

Example 2.6

Let α 1 = α 2 = β 1 = β 2 = 1 . Choose c = ( c 1 , c 2 ) ∈ C 2 such that c 1 − c 2 = 2 log 2 . Then, in view of conclusion (i) of Theorem 2.5, we can easily deduce that f ( z 1 , z 2 ) = 2 e ( z 1 + z 2 ) ⁄ 2 + e z 2 ⁄ 2 is a solution of (2.2) with g ( z 1 , z 2 ) = z 1 − z 2 .

Example 2.7

Let α 1 = α 2 = 1 , β 1 = β 2 = 0 , a 1 = a 2 = 1 , and c 1 = c 2 = 4 π i . Let g ( z 1 , z 2 ) = z 1 + z 2 , G 0 ( z 2 ) = 2 e z 2 ⁄ 2 and G 1 ( z 2 ) = 4 e z 2 ⁄ 2 . Then, in view of conclusion (ii) of Theorem 2.5, we can easily deduce that f ( z 1 , z 2 ) = 4 e ( z 1 + z 2 ) ⁄ 2 is a solution of (2.2).

Example 2.8

Let ξ = α 1 = α 2 = β 1 = β 2 = 1 , c 1 = log 2 + π i , c 2 = log 2 − π i , and ψ 1 ( z 1 − z 2 ) = ϕ 1 ( z 1 − z 2 ) + ϕ 2 ( z 1 − z 2 ) = e z 1 − z 2 . Then, in view of conclusion (iii) of Theorem 2.5, we can easily see that f ( z 1 , z 2 ) = e z 1 − z 2 + A 1 + A 2 2 e ( z 1 + z 2 + 1 ) ⁄ 2 is a solution of (2.2), where A 1 and A 2 are given by (2.3).

Example 2.9

Let α 1 = α 2 = β 1 = β 2 = 1 , ϕ 1 ( z 1 − z 2 ) = e z 1 − z 2 , and ϕ 2 ( z 1 − z 2 ) = 0 . Choose c = ( c 1 , c 2 ) ∈ C 2 such that c 1 − c 2 = 4 π i . Let L 1 ( z ) = z 1 + 2 z 2 and L 2 ( z ) = 2 z 1 + z 2 such that e L 1 ( c ) = ( 9 A 2 + A 1 ) ⁄ A 1 and e L 2 ( c ) = ( 9 A 1 + A 2 ) ⁄ A 2 , where A 1 and A 2 are given by (2.3). Then, we can easily verify that f ( z 1 , z 2 ) = e z 1 − z 2 + 1 9 2 ( A 1 e z 1 + 2 z 2 + 5 + A 2 e 2 z 1 + z 2 + 6 ) is a solution of (2.2) with g ( z 1 , z 2 ) = 2 z 1 + 2 z 2 + 11 .

3 Proofs of the main results

Before proving the main results, we present here some necessary lemmas that will play a key role to prove the main results of this article.

Lemma 3.1

[36] Let f j ≢ 0   ( j = 1 , 2 , 3 ) be meromorphic functions on C n such that f 1 is not constant, f 1 + f 2 + f 3 = 1 , and such that

∑ j = 1 3 N 2 r , 1 f j + 2 N ¯ ( r , f j ) < λ T ( r , f j ) + O ( log + T ( r , f j ) )

holds for all r outside possibly a set with finite logarithmic measure, where λ < 1 is a positive number. Then, either f 2 = 1 or f 3 = 1 .

Lemma 3.2

[36] Let a 0 ( z ) , a 1 ( z ) , … , a n ( z )   ( n ⩾ 1 ) be meromorphic functions on C m and g 0 ( z ) , g 1 ( z ) , … , g n ( z ) are entire functions on C m such that g j ( z ) − g k ( z ) are not constants for 0 ⩽ j < k ⩽ n . If ∑ j = 0 n a j ( z ) e g j ( z ) ≡ 0 , and ∣ ∣ T ( r , a j ) = o ( T ( r ) ) , where T ( r ) = min 0 ⩽ j < k ⩽ n T ( r , e g j − g k ) for j = 0 , 1 , … , n , then a j ( z ) ≡ 0 for each j = 0 , 1 , … , n .

Lemma 3.3

[37–39] For an entire function F on C n , F ( 0 ) ≢ 0 and put ρ ( n F ) = ρ < ∞ . Then, there exist a canonical function f F and a function g F ∈ C n such that F ( z ) = f F ( z ) e g F ( z ) . For the special case n = 1 , f F is the canonical product of Weierstrass.

Lemma 3.4

[40] If g and h are entire functions on the complex plane C and g ( h ) is an entire function of finite-order, then there are only two possible cases: either

  1. the internal function h is a polynomial and the external function g is of finite-order; or else

  2. the internal function h is not a polynomial but a function of finite-order, and the external function g is of zero order.

Proof of Theorem 2.1

Let f ( z 1 , z 2 ) be a transcendental entire solution of equation (2.1). Let us make a transformation

(3.1) L ( f ) = 1 2 ( U − V ) , f ( z 1 + c 1 , z 2 + c 2 ) = 1 2 ( U + V ) .

Then, equation (2.1) becomes

( 1 + h ) U 2 + ( 1 − h ) V 2 = e g ( z 1 , z 2 ) ,

which can be rewritten as

(3.2) 1 + h U e g ( z 1 , z 2 ) ⁄ 2 + i 1 − h V e g ( z 1 , z 2 ) ⁄ 2 1 + h U e g ( z 1 , z 2 ) ⁄ 2 − i 1 − h V e g ( z 1 , z 2 ) ⁄ 2 = 1 .

Since f is finite-order transcendental entire, in view of Lemmas 3.3 and 3.4, it follows from (3.2) that

1 + h U e g ( z 1 , z 2 ) ⁄ 2 + i 1 − h V e g ( z 1 , z 2 ) ⁄ 2 = e h 1 ( z 1 , z 2 ) 1 + h U e g ( z 1 , z 2 ) ⁄ 2 − i 1 − h V e g ( z 1 , z 2 ) ⁄ 2 = e − h 1 ( z 1 , z 2 ) ,

where h 1 ( z 1 , z 2 ) is a polynomial in C 2 , from which we obtain

(3.3) U = e h 1 ( z 1 , z 2 ) + e − h 1 ( z 1 , z 2 ) 2 1 + h e g ( z 1 , z 2 ) ⁄ 2 and V = e h 1 ( z 1 , z 2 ) − e − h 1 ( z 1 , z 2 ) 2 i 1 − h e g ( z 1 , z 2 ) ⁄ 2 .

Set

(3.4) γ 1 ( z 1 , z 2 ) = g ( z 1 , z 2 ) 2 + h 1 ( z 1 , z 2 ) and γ 2 ( z 1 , z 2 ) = g ( z 1 , z 2 ) 2 − h 1 ( z 1 , z 2 ) .

Therefore, it follows from (3.3) and (3.4) that

(3.5) U = e γ 1 ( z 1 , z 2 ) + e γ 2 ( z 1 , z 2 ) 2 1 + h and V = e γ 1 ( z 1 , z 2 ) − e γ 2 ( z 1 , z 2 ) 2 i 1 − h .

In view of (3.1) and (3.5), we obtain that

(3.6) L ( f ) = 1 2 [ A 1 e γ 1 ( z 1 , z 2 ) + A 2 e γ 2 ( z 1 , z 2 ) ] , f ( z 1 + c 1 , z 2 + c 2 ) = 1 2 [ A 2 e γ 1 ( z 1 , z 2 ) + A 1 e γ 2 ( z 1 , z 2 ) ] .

After simple calculation, it follows from the two equations of (3.6) that

(3.7) A 2 A 1 Q 1 ( z 1 , z 2 ) e γ 1 ( z 1 , z 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) + Q 2 ( z 1 , z 2 ) e γ 2 ( z 1 , z 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) − A 2 A 1 e γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) = 1 ,

where

(3.8) Q j ( z 1 , z 2 ) = α ∂ γ j ∂ z 1 + β ∂ γ j ∂ z 2 + γ ∂ γ j ∂ z 1 2 + ∂ 2 γ j ∂ z 1 2 + δ ∂ γ j ∂ z 2 2 + ∂ 2 γ j ∂ z 2 2 + η ∂ γ j ∂ z 1 ∂ γ j ∂ z 2 + ∂ 2 γ j ∂ z 1 ∂ z 2 , j = 1 , 2 .

Now, we discuss two possible cases.

Case 1. Let γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) = k , a constant in C . In view of (3.4), we conclude that h 1 ( z 1 , z 2 ) is constant. Let ξ = e h 1 ( z 1 , z 2 ) ∈ C . Then, (3.6) yields that

(3.9) L ( f ) = C 1 e g ( z 1 , z 2 ) ⁄ 2 , f ( z 1 + c 1 , z 2 + c 2 ) = C 2 e g ( z 1 , z 2 ) ⁄ 2 ,

where C 1 = 1 2 ( A 1 ξ + A 2 ξ − 1 ) , C 2 = 1 2 ( A 2 ξ + A 1 ξ − 1 ) .

Note that

(3.10) C 2 ≠ 0 and C 1 2 + C 2 2 = 1 2 [ ( A 1 2 + A 2 2 ) ( ξ 2 + ξ − 2 ) + 4 A 1 A 2 ] .

If C 1 = 0 , in view of (3.9) and (3.10), it follows that

(3.11) α ∂ f ∂ z 1 + β ∂ f ∂ z 2 + γ ∂ 2 f ∂ z 1 2 + δ ∂ 2 f ∂ z 2 2 + η ∂ 2 f ∂ z 1 ∂ z 2 = 0 , f ( z 1 + c 1 , z 2 + c 2 ) = ± e g ( z 1 , z 2 ) ⁄ 2 .

From the first equation of (3.11), we obtain that

f ( z 1 , z 2 ) = ∑ A e l 1 z 1 + l 2 z 2 ,

where A , l 1 , and l 2 are the arbitrary constants in C satisfying the relation α l 1 + β l 2 + γ l 1 2 + δ l 2 2 + η l 1 l 2 = 0 . From the second equation of (3.11), we obtain that

g ( z 1 , z 2 ) = 2 L n ± ∑ A ′ e l 1 z 1 + l 2 z 2 ,

where A ′ = A e l 1 c 1 + l 2 c 2 . This is conclusion (i).

If C 1 ≠ 0 , then from (3.9), we obtain that

(3.12) α ∂ g ∂ z 1 + β ∂ g ∂ z 2 + γ 1 2 ∂ g ∂ z 1 2 + ∂ 2 g ∂ z 1 2 + δ 1 2 ∂ g ∂ z 2 2 + ∂ 2 g ∂ z 2 2 + η 1 2 ∂ g ∂ z 1 ∂ g ∂ z 2 + ∂ 2 g ∂ z 1 ∂ z 2 = 2 C 1 C 2 e 1 2 [ g ( z 1 + c 1 , z 2 + c 2 ) − g ( z 1 , z 2 ) ] .

Since g is a polynomial in C 2 , it follows from (3.12) that g ( z 1 + c 1 , z 2 + c 2 ) − g ( z 1 , z 2 ) must be constant, say ζ ∈ C .

Then, g can be written as g ( z 1 , z 2 ) = L ( z 1 , z 2 ) + H ( s 1 ) + B , where L ( z 1 , z 2 ) = a 1 z 1 + a 2 z 2 , H ( s 1 ) is a polynomial in s 1 = c 2 z 1 − c 1 z 2 , a 1 , a 2 , and B are the constants in C . Hence, it follows from (3.12) that

(3.13) ( R 1 c 2 − R 2 c 1 ) H ′ + R 3 1 2 H ′ 2 + H ′ ′ = 2 ( A 1 ξ + A 2 ξ − 1 ) A 2 ξ + A 1 ξ − 1 e 1 2 L ( c ) − R 4 ,

where R ′ s are defined in (2.4).

If R 1 c 2 − R 2 c 1 = 0 = R 3 , then from (3.13), we obtain

e 1 2 L ( c ) = A 2 ξ + A 1 ξ − 1 2 ( A 1 ξ + A 2 ξ − 1 ) R 4 .

If R 1 c 2 − R 2 c 1 ≠ 0 or (here “or” is in inclusive sense) R 3 ≠ 0 , then it follows from (3.13) that H ′ must be constant, say a 0 , which is the coefficient of s 1 in the polynomial H ( s 1 ) .

Therefore, from (3.13), we obtain that

(3.14) e 1 2 L ( c ) = A 2 ξ + A 1 ξ − 1 2 ( A 1 ξ + A 2 ξ − 1 ) R 4 + ( R 1 c 2 − R 2 c 1 ) a 0 + 1 2 R 3 a 0 2 .

Hence, in either case L ( z ) satisfies relation (3.14).

Therefore, in view of the second equation of (3.9), we obtain the form of the solution as

f ( z 1 , z 2 ) = 1 2 ( A 2 ξ + A 1 ξ − 1 ) e 1 2 [ L ( z 1 , z 2 ) + H ( s 1 ) − L ( c ) + B ] .

This is conclusion (ii).

Case 2 Let γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) be non-constant. Then, in view of (3.7), it follows that Q 1 ( z 1 , z 2 ) and Q 2 ( z 1 , z 2 ) both cannot be zero at the same time.

If Q 1 ( z 1 , z 2 ) ≡ 0 and Q 2 ( z 1 , z 2 ) ≢ 0 , then (3.7) yields that

Q 2 ( z 1 , z 2 ) e γ 2 ( z 1 , z 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) − A 2 A 1 e γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) = 1 .

In view of the aforementioned equation, it follows that

N r , 1 A 2 e γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) + A 1 = N r , 1 Q 2 ( z 1 , z 2 ) e γ 2 ( z 1 , z 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) = S ( r , e γ 2 ( z 1 , z 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) ) .

Also, note that

N ( r , e γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) ) = S ( r , e γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) ) ,

N r , 1 e γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) = S ( r , e γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) ) .

By the second main theorem of Nevanlinna for several complex variables, we obtain

T ( r , e γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) ) ⩽ N ¯ ( r , e γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) ) + N ¯ r , 1 e γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) + N ¯ r , 1 e γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) − A 1 ⁄ A 2 + S ( r , e γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) ) ⩽ S ( r , e γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) ) + S ( r , e γ 2 ( z 1 , z 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) ) .

This implies that γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) is constant, which is a contradiction. Similarly, we can obtain a contradiction for the case Q 1 ( z 1 , z 2 ) ≢ 0 and Q 1 ( z 1 , z 2 ) ≡ 0 . Hence, Q 1 ( z 1 , z 2 ) ≢ 0 and Q 2 ( z 1 , z 2 ) ≢ 0 .

Since γ 1 ( z 1 , z 2 ) and γ 2 ( z 1 , z 2 ) are the polynomials in C 2 and γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) is non-constant, applying Lemma 3.1 to equation (3.7), we obtain that either

A 2 A 1 Q 1 ( z ) e γ 1 ( z ) − γ 1 ( z + c ) = 1 , or Q 2 ( z 1 , z 2 ) e γ 2 ( z 1 , z 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) = 1 .

If

(3.15) A 2 A 1 Q 1 ( z 1 , z 2 ) e γ 1 ( z 1 , z 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) = 1 ,

then from (3.7), it follows that

(3.16) Q 2 ( z 1 , z 2 ) e γ 2 ( z 1 , z 2 ) − γ 2 ( z 1 + c 1 , z 2 + c 2 ) = A 1 A 2 .

As γ 1 ( z 1 , z 2 ) and γ 2 ( z 1 , z 2 ) are polynomials, in view of (3.15) and (3.16), we conclude that γ 1 ( z 1 , z 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) and γ 2 ( z 1 , z 2 ) − γ 2 ( z 1 + c 1 , z 2 + c 2 ) both are constants in C , and hence, we can obtain that γ 1 ( z 1 , z 2 ) = L 1 ( z 1 , z 2 ) + H 1 ( s 1 ) + B 1 and γ 2 ( z 1 , z 2 ) = L 2 ( z 1 , z 2 ) + H 2 ( s 1 ) + B 2 , where L j ( z 1 , z 2 ) = a j 1 z 1 + a j 2 z 2 , H j ( s 1 ) is a polynomial in s 1 = c 2 z 1 − c 1 z 2 , and a j 1 , a j 2 , B 1 , and B 2 are the constants in C for j = 1 , 2 . Note that L 1 ( z 1 , z 2 ) + H 1 ( s 1 ) ≠ L 2 ( z 1 , z 2 ) + H 2 ( s 1 ) . Otherwise, γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) would become constant, a contradiction to our assumption. Hence, the form of the polynomial g is g ( z 1 , z 2 ) = L ( z 1 , z 2 ) + H ( s 1 ) + B , where L ( z 1 , z 2 ) = L 1 ( z 1 , z 2 ) + L 2 ( z 1 , z 2 ) , H ( s 1 ) = H 1 ( s 1 ) + H 2 ( s 1 ) , and B = B 1 + B 2 .

Therefore, in view of (3.15) and (3.16), we obtain that

(3.17) ( R 15 c 2 − R 16 c 1 ) H 1 ′ + R 3 ( H 1 ′ 2 + H 1 ″ ) = A 1 A 2 e L 1 ( c ) − R 17 , ( R 25 c 2 − R 26 c 1 ) H 2 ′ + R 3 ( H 2 ′ 2 + H 2 ″ ) = A 2 A 1 e L 2 ( c ) − R 27 ,

where R ′ s are defined in (2.4).

Then, by similar arguments as in Case 1, we obtain from (3.17) that

(3.18) e L 1 ( c ) = A 2 A 1 ( R 17 + ( R 15 c 2 − R 16 c 1 ) a 0 + R 3 a 0 2 ) , e L 2 ( c ) = A 1 A 2 ( R 27 + ( R 25 c 2 − R 26 c 1 ) a 00 + R 3 a 00 2 ) ,

where a 0 and a 00 , respectively, the coefficients of the linear term of the polynomials H 1 ( s 1 ) and H 2 ( s 1 ) .

Therefore, in view of the second equation of (3.6), we obtain

f ( z 1 , z 2 ) = 1 2 ( A 2 e L 1 ( z 1 , z 2 ) + H 1 ( s 1 ) − L 1 ( c ) + B 1 + A 1 e L 2 ( z 1 , z 2 ) + H 2 ( s 1 ) − L 2 ( c ) + B 2 ) ,

where L 1 ( c ) and L 2 ( c ) can be found from (3.18).

This is conclusion (iii).

If Q 2 ( z 1 , z 2 ) e γ 2 ( z 1 , z 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) = 1 , then it follows from equation (3.7) that Q 1 ( z 1 , z 2 ) e γ 1 ( z 1 , z 2 ) − γ 2 ( z 1 + c 1 , z 2 + c 2 ) = 1 . Since γ 1 ( z 1 , z 2 ) and γ 2 ( z 1 , z 2 ) are both polynomials in C 2 , it follows that γ 2 ( z 1 , z 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) = η 1 and γ 1 ( z 1 , z 2 ) − γ 2 ( z 1 + c 1 , z 2 + c 2 ) = η 2 , where η 1 , η 2 ∈ C . This implies that γ 1 ( z 1 , z 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) = γ 2 ( z 1 , z 2 ) − γ 2 ( z 1 + c 1 , z 2 + c 2 ) = η 1 + η 2 . Therefore, we can write γ 1 ( z 1 , z 2 ) = L ( z 1 , z 2 ) + H ( s 1 ) + ζ 1 and γ 2 ( z 1 , z 2 ) = L ( z 1 , z 2 ) + H ( s 1 ) + ζ 2 . But, then we obtain γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) = ζ 2 − ζ 1 , constants, which is a contradiction.□

Proof of Theorem 2.5

Let f ( z 1 , z 2 ) be a finite-order transcendental entire solution of equation (2.2). Let us make a transformation

(3.19) L ˜ ( f ) = 1 2 ( U − V ) , f ( z + c ) − f ( z ) = 1 2 ( U + V ) .

Then, by similar arguments as in Theorem 2.1, we can obtain that

(3.20) L ˜ ( f ) = 1 2 [ A 1 e γ 1 ( z 1 , z 2 ) + A 2 e γ 2 ( z 1 , z 2 ) ] f ( z 1 + c 1 , z 2 + c 2 ) − f ( z 1 , z 2 ) = 1 2 [ A 2 e γ 1 ( z 1 , z 2 ) + A 1 e γ 2 ( z 1 , z 2 ) ] ,

where γ 1 and γ 2 are defined in (3.4) and A 1 and A 2 are defined in (2.3).

In view of equations in (3.20), we obtain that

(3.21) A 2 Q 1 ( z 1 , z 2 ) + A 1 A 1 e γ 1 ( z 1 , z 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) + A 1 Q 2 ( z 1 , z 2 ) + A 2 A 1 e γ 2 ( z 1 , z 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) − A 2 A 1 e γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) = 1 ,

where

(3.22) Q j ( z 1 , z 2 ) = γ ∂ γ j ∂ z 1 2 + ∂ 2 γ j ∂ z 1 2 + δ ∂ γ j ∂ z 2 2 + ∂ 2 γ j ∂ z 2 2 + η ∂ γ j ∂ z 1 ∂ γ j ∂ z 2 + ∂ 2 γ j ∂ z 1 ∂ z 2 , j = 1 , 2 .

Now, we consider two possible cases.

Case 1. Let γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) = k ∈ C . In view of (3.4), we conclude that h 1 ( z 1 , z 2 ) is constant. Set e h 1 = ξ ∈ C . Then, (3.20) yields that

(3.23) L ˜ ( f ) = C 1 e g ( z 1 , z 2 ) ⁄ 2 , f ( z 1 + c 1 , z 2 + c 2 ) − f ( z 1 , z 2 ) = C 2 e g ( z 1 , z 2 ) ⁄ 2 ,

where C 1 = 1 2 ( A 1 ξ + A 2 ξ − 1 ) , C 2 = 1 2 ( A 2 ξ + A 1 ξ − 1 ) .

Note that

(3.24) C 1 2 + C 2 2 = 1 2 [ ( A 1 2 + A 2 2 ) ( ξ 2 + ξ − 2 ) + 4 A 1 A 2 ] .

Subcase 1.1. Let C 1 = 0 . Then, in view of (3.24), we obtain that C 2 = ± 1 . Therefore, it follows from (3.23) that

(3.25) γ ∂ 2 f ( z 1 , z 2 ) ∂ z 1 2 + δ ∂ 2 f ( z 1 , z 2 ) ∂ z 2 2 + η ∂ 2 f ( z 1 , z 2 ) ∂ z 1 ∂ z 2 = 0 , f ( z 1 + c 1 , z 2 + c 2 ) − f ( z 1 , z 2 ) = ± e g ( z 1 , z 2 ) ⁄ 2 .

Now, in view of the results in [35, page 2178, line 21], the first equation of (3.25) can be written as

(3.26) ( α 1 D + β 1 D ′ ) ( α 2 D + β 2 D ′ ) f ( z ) = 0 ,

where D ≡ ∂ ∂ z 1 , D ′ ≡ ∂ ∂ z 2 , α 1 α 2 = γ , β 1 β 2 = δ , and α 1 β 2 + α 2 β 1 = η .

Solving (3.26), we obtain that

f ( z 1 , z 2 ) = ϕ 1 ( β 1 z 1 − α 1 z 2 ) + ϕ 2 ( β 2 z 1 − α 2 z 2 ) ,

where ϕ 1 and ϕ 2 are finite-order transcendental entire functions in C 2 .

In view of the second equation of (3.25), we obtain that

ϕ 1 ( β 1 z 1 − α 1 z 2 + β 1 c 1 − α 1 c 2 ) + ϕ 2 ( β 2 z 1 − α 2 z 2 + β 2 c 1 − α 2 c 2 ) − ϕ 1 ( β 1 z 1 − α 1 z 2 ) − ϕ 2 ( β 2 z 1 − α 2 z 2 ) = ± e 1 2 g ( z 1 , z 2 ) .

This is conclusion (i).

Subcase 1.2. Let C 2 = 0 . Then, in view of (3.24), it yields that C 1 = ± 1 .

Therefore, it follows from (3.23) that

(3.27) γ ∂ 2 f ( z 1 , z 2 ) ∂ z 1 2 + δ ∂ 2 f ( z 1 , z 2 ) ∂ z 2 2 + η ∂ 2 f ( z 1 , z 2 ) ∂ z 1 ∂ z 2 = ± e 1 2 g ( z 1 , z 2 ) , f ( z 1 + c 1 , z 2 + c 2 ) − f ( z 1 , z 2 ) = 0 .

Clearly, second equation of (3.27) shows that f is a periodic function of period c . In view of the two equations in (3.27), it follows that e 1 2 ( g ( z 1 + c 1 , z 2 + c 2 ) − g ( z 1 , z 2 ) ) = 1 . This implies that g ( z 1 , z 2 ) = a 1 z 1 + a 2 z 2 + H ( c 2 z 1 − c 1 z 2 ) + B , where H is a polynomial in c 2 z 1 − c 1 z 2 and a 1 c 1 + a 2 c 2 = 4 k π i , k ∈ Z .

Now, in view of the results in [35, page 2178, line 21], the first equation of (3.27) can be written as

(3.28) ( α 1 D + β 1 D ′ ) ( α 2 D + β 2 D ′ ) f ( z 1 , z 2 ) = ± e 1 2 g ( z 1 , z 2 ) ,

where D , D ′ , α 1 , α 2 , β 1 , and β 2 are defined in (3.26).

Let ( α 2 D + β 2 D ′ ) f ( z 1 , z 2 ) = u ( z 1 , z 2 ) . Then, in view of the results in [[31], page 2182, line 22], we obtain

u ( z 1 , z 2 ) = ± 1 α 1 ∫ 0 z 1 ⁄ α 1 e 1 2 g ( z 1 , z 2 ) d z 1 + G 0 α 1 z 2 − β 1 z 1 α 1 ,

where G 0 is a finite-order transcendental entire function in C 2 .

Since we have assumed that ( α 2 D + β 2 D ′ ) f ( z 1 , z 2 ) = u ( z 1 , z 2 ) , we obtain that

(3.29) α 2 ∂ f ∂ z 1 + β 2 ∂ f ∂ z 2 = ± 1 α 1 ∫ 0 z 1 ⁄ α 1 e 1 2 g ( z 1 , z 2 ) d z 1 + G 0 α 1 z 2 − β 1 z 1 α 1 .

By similar argument, we obtain from (3.29) that

f ( z 1 , z 2 ) = ± 1 γ ∫ 0 z 1 ⁄ α 1 ∫ 0 z 1 ⁄ α 2 e 1 2 [ a 1 z 1 + a 2 z 2 + H ( c 2 z 1 − c 1 z 2 ) + B ] d z 1 d z 1 + 1 α 2 ∫ 0 z 1 ⁄ α 2 G 0 α 1 z 2 − β 1 z 1 α 1 d z 1 + G 1 α 2 z 2 − β 2 z 1 α 2 ,

where G 1 is a finite-order transcendental entire function in C 2 .

In view of the fact that a 1 c 1 + a 2 c 2 = 4 k π i , k ∈ Z , it follows from the second equation of (3.27) that

1 α 2 ∫ 0 z 1 ⁄ α 2 G 0 α 1 z 2 − β 1 z 1 α 1 + α 1 c 2 − β 1 c 1 α 1 − G 0 α 1 z 2 − β 1 z 1 α 1 d z 1 + G 1 α 2 z 2 − β 2 z 1 α 2 + α 2 c 2 − β 2 c 1 α 2 − G 1 α 2 z 2 − β 2 z 1 α 2 = 0 .

This is conclusion (ii).

Subcase 1.3. Let C 1 ≠ 0 and C 2 ≠ 0 . Then, after simple calculations, (3.23) yields that

(3.30) γ ∂ 2 g ∂ z 1 2 + 1 2 ∂ g ∂ z 1 2 + δ ∂ 2 g ∂ z 2 2 + 1 2 ∂ g ∂ z 2 2 + η ∂ 2 g ∂ z 1 ∂ z 2 + 1 2 ∂ g ∂ z 1 ∂ g ∂ z 2 = 2 C 1 C 2 e 1 2 [ g ( z 1 + c 1 , z 2 + c 2 ) − g ( z 1 , z 2 ) ] − 1 .

Since g is a polynomial in C 2 , in view of (3.30), we conclude that g ( z 1 + c 1 , z 2 + c 2 ) − g ( z 1 , z 2 ) = ξ , ξ ∈ C . This implies that g ( z 1 , z 2 ) = L 1 ( z 1 , z 2 ) + H ( s 1 ) + B 1 , where L 1 ( z 1 , z 2 ) = a 11 z 1 + a 12 z 2 , H ( s 1 ) is a polynomial in s 1 ≔ c 2 z 1 − c 1 z 2 , a 11 , a 12 , B 1 ∈ C . Hence, we obtain from (3.30) that

(3.31) γ a 11 + 1 2 η a 12 c 2 − δ a 12 + 1 2 η a 11 c 1 H ′ + ( γ c 2 2 + δ c 1 2 − η c 1 c 2 ) 1 2 H ′ 2 + H ″ = 2 C 1 C 2 e 1 2 L 1 ( c ) − 1 .

Since γ c 2 2 + δ c 1 2 ≠ η c 1 c 2 , in view of (3.31), we conclude that H ′ is constant. This implies that H ( s 1 ) = a 0 s 1 + b 0 . Hence, g ( z ) reduces to the form

(3.32) g ( z 1 , z 2 ) = L ( z 1 , z 2 ) + B = a 1 z 1 + a 2 z 2 + B ,

where a 1 = a 11 + a 0 c 2 , a 2 = a 12 − a 0 c 1 and B = B 1 + b 0 .

Therefore, in view of (3.30) and (3.32), we obtain that

(3.33) e 1 2 [ a 1 c 1 + a 2 c 2 ] = C 2 4 C 1 ( γ a 1 2 + δ a 2 2 − η a 1 a 2 ) + 1 .

Now, the first equation of (3.23) can be rewritten as

(3.34) ( γ D 2 + δ D ′ 2 + η D D ′ ) f ( z 1 , z 2 ) = C 1 e 1 2 [ a 1 z 1 + a 2 z 2 + B ] ,

where D ≡ ∂ ∂ z 1 and D ′ ≡ ∂ ∂ z 2 .

Therefore, complementary function of (3.34) is C.F. = ϕ 1 ( β 1 z 1 − α 1 z 2 ) + ϕ 2 ( β 2 z 1 − α 2 z 2 ) , where α 1 , α 2 , β 1 , and β 2 are same as in (i), ϕ 1 and ϕ 2 are finite-order transcendental entire functions in C 2 , and the particular integral is

P.I. = 4 C 1 e B ⁄ 2 γ a 1 2 + δ a 2 2 + η a 1 a 2 ∫ ∫ e v d v d v = 4 C 1 γ a 1 2 + δ a 2 2 + η a 1 a 2 e 1 2 [ a 1 z 1 + a 2 z 2 + B ] ,

where v = a 1 z 1 + a 2 z 2 .

Hence, from (3.23), we obtain

f ( z 1 , z 2 ) = ϕ 1 ( β 1 z 1 − α 1 z 2 ) + ϕ 2 ( β 2 z 1 − α 2 z 2 ) + 2 2 ( A 1 ξ + A 2 ξ − 1 ) γ a 1 2 + δ a 2 2 + η a 1 a 2 e 1 2 [ a 1 z 1 + a 2 z 2 + B ] .

Substituting f ( z 1 , z 2 ) into the second equation of (3.23) and combining with (3.33), we obtain that

ϕ 1 ( β 1 z 1 − α 1 z 2 + β 1 c 1 − α 1 c 2 ) + ϕ 2 ( β 2 z 1 − α 2 z 2 + β 2 c 1 − α 2 c 2 ) = ϕ 1 ( β 1 z 1 − α 1 z 2 ) + ϕ 2 ( β 2 z 1 − α 2 z 2 ) .

This is conclusion (iii).

Case 2. Let γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) be non-constant. Then, obviously, A 2 Q 1 ( z 1 , z 2 ) + A 1 and A 1 Q 2 ( z 1 , z 2 ) + A 2 cannot be identically zero at the same time. Otherwise, in view of (3.21), it follows that e γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) is a constant, which implies that γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) is a constant. This is a contradiction to our assumption.

If A 2 Q 1 ( z 1 , z 2 ) + A 1 ≡ 0 and A 1 Q 2 ( z 1 , z 2 ) + A 2 ≢ 0 , then (3.21) it yields

(3.35) ( A 1 Q 2 ( z 1 , z 2 ) + A 2 ) e γ 2 ( z 1 , z 2 ) − A 2 e γ 2 ( z 1 + c 1 , z 2 + c 2 ) − A 1 e γ 1 ( z 1 + c 1 , z 2 + c 2 ) ≡ 0 .

In view of Lemma 3.2 and (3.35), we can easily obtain a contradiction. Similarly, we can obtain a contradiction for the case A 2 Q 1 ( z 1 , z 2 ) + A 1 ≢ 0 and A 1 Q 2 ( z 1 , z 2 ) + A 2 ≡ 0 . Hence, we must have A 2 Q 1 ( z 1 , z 2 ) + A 1 ≢ 0 and A 1 Q 2 ( z 1 , z 2 ) + A 2 ≢ 0 .

Now, in view of Lemma 3.1, we obtain from (3.21) that either

A 2 Q 1 ( z 1 , z 2 ) + A 1 A 1 e γ 1 ( z 1 , z 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) ≡ 1

or

A 1 Q 2 ( z 1 , z 2 ) + A 2 A 1 e γ 2 ( z 1 , z 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) ≡ 1 .

If A 1 Q 2 ( z 1 , z 2 ) + A 2 A 1 e γ 2 ( z 1 , z 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) ≡ 1 , then in view of (3.21), it follows that A 2 Q 1 ( z 1 , z 2 ) + A 1 A 2 e γ 1 ( z 1 , z 2 ) − γ 2 ( z 1 + c 1 , z 2 + c 2 ) ≡ 1 . Therefore, we must obtain that γ 2 ( z 1 , z 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) = ξ 1 and γ 1 ( z 1 , z 2 ) − γ 2 ( z 1 + c 1 , z 2 + c 2 ) = ξ 2 , ξ 1 , ξ 2 ∈ C . Thus, it follows that γ 1 ( z 1 , z 2 ) − γ 1 ( z 1 + 2 c 1 , z 2 + 2 c 2 ) = γ 2 ( z 1 , z 2 ) − γ 2 ( z 1 + 2 c 1 , z 2 + 2 c 2 ) = ξ 1 + ξ 2 . This implies that γ 1 ( z 1 , z 2 ) = L ( z 1 , z 2 ) + H ( s 1 ) + B 1 and γ 2 ( z 1 , z 2 ) = L ( z 1 , z 2 ) + H ( s 1 ) + B 2 , where L ( z 1 , z 2 ) = a 1 z 1 + a 2 z 2 and H ( s 1 ) is a polynomial in s 1 ≔ c 2 z 1 − c 1 z 2 , a 1 , a 2 , B 1 , B 2 ∈ C . Hence, we must obtain that γ 2 ( z 1 + c 1 , z 2 + c 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) = B 2 − B 1 , a constant in C , which is a contradiction to the assumption.

Therefore, we must have

(3.36) A 2 Q 1 ( z 1 , z 2 ) + A 1 A 1 e γ 1 ( z 1 , z 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) ≡ 1 .

In view of (3.21) and (3.36), we obtain that

(3.37) A 1 Q 2 ( z 1 , z 2 ) + A 2 A 2 e γ 2 ( z 1 , z 2 ) − γ 2 ( z 1 + c 1 , z 2 + c 2 ) ≡ 1 .

Since γ 1 ( z 1 , z 2 ) and γ 2 ( z 1 , z 2 ) are the polynomials in C 2 , from (3.36) and (3.37), we can conclude that γ 1 ( z 1 , z 2 ) − γ 1 ( z 1 + c 1 , z 2 + c 2 ) = η 1 and γ 2 ( z 1 , z 2 ) − γ 2 ( z 1 + c 1 , z 2 + c 2 ) = η 2 , η 1 , η 2 ∈ C . Thus, we have γ 1 ( z 1 , z 2 ) = L 1 ( z 1 , z 2 ) + H 1 ( s 1 ) + B 1 and γ 2 ( z 1 , z 2 ) = L 2 ( z 1 , z 2 ) + H 2 ( s 1 ) + B 2 , where L j ( z 1 , z 2 ) = a j 1 z 1 + a j 2 z 2 and H j ( s 1 ) is a polynomial in s 1 ≔ c 2 z 1 − c 1 z 2 , a j 1 , a j 2 , B j ∈ C for j = 1 , 2 .

Therefore, in view of (3.22) and (3.36), we obtain that

[ ( 2 γ a 11 + η a 12 ) c 2 − ( 2 δ a 12 + η a 11 ) c 1 ] H 1 ′ + ( γ c 2 2 + δ c 1 2 − η c 1 c 2 ) ( H 1 ′ 2 + H 1 ″ ) = A 1 A 1 [ e L 1 ( c ) − 1 ] − ( γ a 11 2 + δ a 12 2 + η a 11 a 12 ) .

Since γ c 2 2 + δ c 1 2 − η c 1 c 2 ≠ 0 , we conclude that H 1 ( s 1 ) is a linear polynomial in s 1 , and thus, L 1 ( z ) + H 1 ( s 1 ) becomes linear in C . For the sake of convenience, we still denote that γ 1 ( z ) = L 1 ( z ) + B 1 . In a similar manner, from (3.22) and (3.37), we can conclude that γ 2 ( z ) = L 2 ( z ) + B 2 .

Therefore, it follows from (3.36) and (3.37) that

(3.38) e L 1 ( c ) = A 2 A 1 ( γ a 11 2 + δ a 12 2 + η a 11 a 12 ) + 1 , e L 2 ( c ) = A 1 A 2 ( γ a 21 2 + δ a 22 2 + η a 21 a 22 ) + 1 .

Now, in view of the first equation of (3.20), it follows that

(3.39) ( γ D 2 + δ D ′ 2 + η D D ′ ) f ( z 1 , z 2 ) = 1 2 [ A 1 e L 1 ( z 1 , z 2 ) + B 1 + A 2 e L 2 ( z 1 , z 2 ) + B 2 ] ,

where D ≡ ∂ ∂ z 1 and D ′ ≡ ∂ ∂ z 2 .

Now, in view of the results in [[31], page 2178, line 21], the complementary function of (3.39) is ϕ 1 ( β 1 z 1 − α 1 z 2 ) + ϕ 2 ( β 2 z 1 − α 2 z 2 ) , where α 1 , α 2 , β 1 , and β 2 are same as in (i), ϕ 1 and ϕ 2 are finite-order transcendental entire functions in C 2 , and the particular integral is

P.I. = A 1 e L 1 ( z 1 , z 2 ) + B 1 2 ( γ a 11 2 + δ a 12 2 + η a 11 a 12 ) + A 2 e L 2 ( z 1 , z 2 ) + B 2 2 ( γ a 21 2 + δ a 22 2 + η a 21 a 22 ) .

Hence, the form of the solution of (3.39) is

(3.40) f ( z 1 , z 2 ) = ϕ 1 ( β 1 z 1 − α 1 z 2 ) + ϕ 2 ( β 2 z 1 − α 2 z 2 ) + A 1 e L 1 ( z 1 , z 2 ) + B 1 2 ( γ a 11 2 + δ a 12 2 + η a 11 a 12 ) + A 2 e L 2 ( z 1 , z 2 ) + B 2 2 ( γ a 21 2 + δ a 22 2 + η a 21 a 22 ) .

Substituting (3.40) into the second equation of (3.20) and combining with (3.38), we obtain that

ϕ 1 ( β 1 z 1 − α 1 z 2 + β 1 c 1 − α 1 c 2 ) + ϕ 2 ( β 2 z 1 − α 2 z 2 + β 2 c 1 − α 2 c 2 ) = ϕ 1 ( β 1 z 1 − α 1 z 2 ) + ϕ 2 ( β 2 z 1 − α 2 z 2 ) .

This is conclusion (iv).□

# This work was completed with the support of our TE X-pert.

Acknowledgment

Authors would like to thank all the Referees for their valuable suggestions towards the clarity of the paper.

  1. Funding information: This work was supported by the National Natural Science Foundation of China (12161074) and the Foundation of Education Department of Jiangxi (GJJ212305, GJJ202303, GJJ201813, GJJ201343) of China, the Talent Introduction Research Foundation of Suqian University (106-CK00042/028) and sponsored by Qing Lan Project and Suqian Talent Xiongying Plan of Suqian.

  2. Author contributions: Both the authors have contributed equally towards the formation of the paper.

  3. Conflict of interest: The authors declare that they have no conflict of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no database was generated or analyzed during the current study.

References

[1] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. 141 (1995), 443–551. 10.2307/2118559Suche in Google Scholar

[2] W. K. Hayman, Meromorphic Functions, The Clarendon Press, Oxford, 1964. Suche in Google Scholar

[3] I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter Berlin/Newyork, 1993. 10.1515/9783110863147Suche in Google Scholar

[4] P. Montel, Leçons sur les Familles Normales de Fonctions Analytiques et Leurs Applications, Gauthier-Villars, Paris, 1927, pp. 135–136. Suche in Google Scholar

[5] G. Iyer, On certain functional equations, J. Indian Math. Soc. 3 (1939), 312–315. Suche in Google Scholar

[6] F. Gross, On the equation fn(z)+gn(z)=1, Bull. Amer. Math. Soc. 72 (1966), 86–88. 10.1090/S0002-9904-1966-11429-5Suche in Google Scholar

[7] C. C. Yang and P. Li, On the transcendental solutions of a certain type of non-linear differential equations, Arch. Math. 82 (2004), 442–448. 10.1007/s00013-003-4796-8Suche in Google Scholar

[8] Y. M. Chiang and S. J. Feng, On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane, Ramanujan J. 16 (2008), 105–129. 10.1007/s11139-007-9101-1Suche in Google Scholar

[9] R. G. Halburd and R. J. Korhonen, Difference analog of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl. 314 (2006), 477–487. 10.1016/j.jmaa.2005.04.010Suche in Google Scholar

[10] R. G. Halburd and R. J. Korhonen, Finite-order meromorphic solutions and the discrete Painleve equations, Proc. Lond. Math. Soc. 94 (2007), no. 2, 443–474. 10.1112/plms/pdl012Suche in Google Scholar

[11] P. Li and C. C. Yang, On the nonexistence of entire solutions of certain type of nonlinear differential equations, J. Math. Anal. Appl. 320 (2006), 827–835. 10.1016/j.jmaa.2005.07.066Suche in Google Scholar

[12] L. W. Liao, C. C. Yang, and J. J. Zhang, On meromorphic solutions of certain type of non-linear differential equations, Ann. Fenn. Math. 38 (2013), 581–593. 10.5186/aasfm.2013.3840Suche in Google Scholar

[13] G. Haldar, On entire solutions of system of Fermat-type difference and differential-difference equations, J. Anal. 32 (2024)1519–1543, DOI: https://doi.org/10.1007/s41478-023-00702-3. 10.1007/s41478-023-00702-3Suche in Google Scholar

[14] K. Liu, T. B. Cao, and H. Z. Cao, Entire solutions of Fermat-type differential-difference equations, Arch. Math. 99 (2012), 147–155. 10.1007/s00013-012-0408-9Suche in Google Scholar

[15] K. Liu and L. Z. Yang, A note on meromorphic solutions of Fermat-types equations, An. Stiint. Univ. Al. I. Cuza Lasi Mat. (N. S.). 1 (2016), 317–325. Suche in Google Scholar

[16] G. Haldar, Solutions of Fermat-type partial differential difference equations in C2, Mediterr. J. Math. 20 (2023), 50, DOI: https://doi.org/10.1007/s00009-022-02180-6. 10.1007/s00009-022-02180-6Suche in Google Scholar

[17] G. Haldar, On entire solutions of system of Fermat-type difference and partial differential-difference equations in Cn, Rend. Circ. Mat. Palermo 73 (2024), 1467–1490, DOI: https://doi.org/10.1007/s12215-023-00997-y. 10.1007/s12215-023-00997-ySuche in Google Scholar

[18] G. Haldar and A. Banerjee, Characterizations of entire solutions for the system of Fermat-type binomial and trinomial shift equations in Cn, Demonstr. Math. 56 (2023), 20230104, DOI: https://doi.org/10.1515/dema-2023-0104. 10.1515/dema-2023-0104Suche in Google Scholar

[19] G. Haldar and A. Banerjee, On entire solutions of Fermat-type difference and k-th order partial differential difference equations in several complex variables, Afr. Mat. 35 (2024), 45, DOI: https://doi.org/10.1007/s13370-024-01188-3. 10.1007/s13370-024-01188-3Suche in Google Scholar

[20] G. Haldar and M. B. Ahamed, Entire solutions of several quadratic binomial and trinomial partial differential-difference equations in C2, Anal. Math. Phys. 12 (2022), 113, DOI: https://doi.org/10.1007/s13324-022-00722-5. 10.1007/s13324-022-00722-5Suche in Google Scholar

[21] H. Y. Xu and G. Haldar, Solutions of complex nonlinear functional equations including second-order partial differential and difference equations in C2, Electron. J. Differential Equations 43 (2023), 1–18. 10.58997/ejde.2023.43Suche in Google Scholar

[22] H.Y. Xu and G. Haldar, Entire solutions to Fermat-type difference and partial differential-difference equations in Cn, Electron. J. Differential Equations 2024 (2024), no. 26, 1–21. 10.58997/ejde.2024.26Suche in Google Scholar

[23] H. Y. Xu, H. Li, and X. Ding, Entire and meromorphic solutions for systems of the differential difference equations, Demonstr. Math. 55 (2022), 676–694, DOI: https://doi.org/10.1515/dema-2022-0161. 10.1515/dema-2022-0161Suche in Google Scholar

[24] H. Y. Xu and Y. Y. Jiang, Results on entire and meromorphic solutions for several systems of quadratic trinomial functional equations with two complex variables, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 116 (2022), 8, DOI: https://doi.org/10.1007/s13398-021-01154-9. 10.1007/s13398-021-01154-9Suche in Google Scholar

[25] E. G. Saleeby, On entire and meromorphic solutions of λuk+∑i=1nuzim=1, Complex Var. Theory Appl. 49 (2004), 101–107. 10.1080/02781070310001658056Suche in Google Scholar

[26] E. G. Saleebly, Entire and meromorphic solutions of Fermat-type partial differential equations, Analysis (Munich) 19 (1999), 369–376. 10.1524/anly.1999.19.4.369Suche in Google Scholar

[27] D. Khavinson, A note on entire solutions of the Eiconal equation, Amer. Math. Monthly 102 (1995), 159–161. 10.1080/00029890.1995.11990551Suche in Google Scholar

[28] B. Q. Li, Entire solutions of (uz1)m+(uz2)n=eg, Nagoya Math. J. 178 (2005), 151–162. 10.1017/S0027763000009156Suche in Google Scholar

[29] B. Q. Li, Entire solutions of Eiconal type equations, Arch. Math. 89 (2007), 350–357. 10.1007/s00013-007-2118-2Suche in Google Scholar

[30] L. Xu and T. B. Cao, Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math. 15 (2018), 1–14. 10.1007/s00009-018-1274-xSuche in Google Scholar

[31] H. Y. Xu, K. Y. Zhang, and X. M. Zheng, Entire and meromorphic solutions for several Fermat-type partial differential difference equations in C2, Rocky Mountain J. Math. 52 (2022), no. 6, 2169–2187. 10.1216/rmj.2022.52.2169Suche in Google Scholar

[32] E. G. Saleeby, On complex analytic solutions of certain trinomial functional and partial differential equations, Aequationes Math. 85 (2013), 553–562. 10.1007/s00010-012-0154-xSuche in Google Scholar

[33] H. Li and H. Y. Xu, Solutions for several quadratic trinomial difference equations and partial differential difference equations in C2, Axioms 10 (2021), 126. DOI: https://doi.org/10.3390/axioms10020126. 10.3390/axioms10020126Suche in Google Scholar

[34] T. B. Cao, The growth, oscillation and fixed points of solutions of complex linear differential equations in the unit disc, J. Math. Anal. Appl. 352 (2009), no. 2, 739–748. 10.1016/j.jmaa.2008.11.033Suche in Google Scholar

[35] T. B. Cao and R. J. Korhonen, A new version of the second main theorem for meromorphic mappings intersecting hyperplanes in several complex variables, J. Math. Anal. Appl. 444 (2016), no. 2, 1114–1132. 10.1016/j.jmaa.2016.06.050Suche in Google Scholar

[36] P. C. Hu, P. Li, and C. C Yang, Unicity of Meromorphic Mappings, Advances in Complex Analysis and its Applications, vol. 1. Kluwer Academic Publishers, Dordrecht, Boston, London, 2003. Suche in Google Scholar

[37] P. Lelong, Fonctionnelles Analytiques et Fonctions Entières (n variables), Presses de l’Université de Montréal, Montréal, 1968. Suche in Google Scholar

[38] W. Stoll, Holomorphic Functions of Finite Order in Several Complex Variables, American Mathematical Society, Providence, 1974. Suche in Google Scholar

[39] L. I. Ronkin, Introduction to the Theory of Entire Functions of Several Variables, Moscow: Nauka 1971 (Russian), American Mathematical Society, Providence, 1974. 10.1090/mmono/044Suche in Google Scholar

[40] G. Pólya, On an integral function of an integral function, J. Lond. Math. Soc. 1 (1926), 12–15. 10.1112/jlms/s1-1.1.12Suche in Google Scholar
Received: 2023-12-16
Revised: 2024-06-08
Accepted: 2024-08-06
Published Online: 2024-11-21

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
https://doi.org/10.1515/dema-2024-0052
Schlagwörter für diesen Artikel
functions of several complex variables; Fermat-type equations; entire solutions; Nevanlinna theory
Creative Commons
BY 4.0

Artikel in diesem Heft

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 20.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/dema-2024-0052/html
Button zum nach oben scrollen