Home On certain functional equation in prime rings
Article Open Access

On certain functional equation in prime rings

  • Maja Fošner , Benjamin Marcen EMAIL logo and Joso Vukman
Published/Copyright: April 4, 2022
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 20 Issue 1

Abstract

The purpose of this paper is to prove the following result. Let R be prime ring of characteristic different from two and three, and let F : R → R be an additive mapping satisfying the relation F ( x 3 ) = F ( x 2 ) x − x F ( x ) x + x F ( x 2 ) for all x ∈ R . In this case, F is of the form 4 F ( x ) = D ( x ) + q x + x q for all x ∈ R , where D : R → R is a derivation, and q is some fixed element from the symmetric Martindale ring of quotients of R .

Keywords: prime ring; derivation; Jordan derivation; two-sided centralizer; functional equation
MSC 2010: 16R60; 16W25; 39B05

Introduction

Throughout, R will represent an associative ring with center Z ( R ) . Given an integer n > 1 , a ring R is said to be n -torsion free, if for x ∈ R , n x = 0 implies x = 0 . The commutator x y − y x will be denoted by [ x , y ] . A ring R is prime if for a , b ∈ R , a R b = ( 0 ) implies that either a = 0 or b = 0 and is semiprime in case a R a = ( 0 ) implies a = 0 . We denote by Q m r , Q s , and C the maximal Martindale right ring of quotients, symmetric Martindale ring of quotients, and extended centroid of a semiprime ring R , respectively (see [1], Chapter 2). An additive mapping D : R → R is called a derivation if D ( x y ) = D ( x ) y + x D ( y ) holds for all pairs x , y ∈ R and is called a Jordan derivation in case D ( x 2 ) = D ( x ) x + x D ( x ) is fulfilled for all x ∈ R . A derivation D : R → R is inner in case D is of the form D ( x ) = [ a , x ] for all x ∈ R and some fixed a ∈ R . Every derivation is Jordan derivation. The converse is in general not true. A classical result of Herstein [2] asserts that any Jordan derivation on a prime ring with characteristic different from two is a derivation. A brief proof of Herstein theorem can be found in [3]. In [4], one can find a generalization of Herstein theorem. Cusack [5] generalized Herstein theorem to 2-torsion free semiprime rings (see [6] for an alternative proof). Herstein theorem has been fairly generalized by Beidar et al. [7]. For results related to Herstein theorem, we refer to [8,9, 10,11]. We proceed with the following result proved by Brešar [12] (see [13] for a generalization).

Theorem 1

Let R be a 2-torsion free semiprime ring and let D : R → R be an additive mapping satisfying the relation.

(1) D ( x y x ) = D ( x ) y x + x D ( y ) x + x y D ( x )

for all pairs x , y ∈ R . In this case, D is a derivation.

An additive mapping satisfying the relation (1) on an arbitrary ring is called a Jordan triple derivation. It is easy to prove that any Jordan derivation on a 2-torsion free ring is a Jordan triple derivation, which means that Theorem 1 generalizes Cusack’s generalization of Herstein theorem.

Motivated by Theorem 1, Vukman et al. [14] have proved the following result (see [15] for a generalization).

Theorem 2

Let R be a 2-torsion free semiprime ring and let F : R → R be an additive mapping satisfying the relation

(2) T ( x y x ) = T ( x ) y x − x T ( y ) x + x y T ( x )

for all pairs x , y ∈ R . In this case, F is of the form 2 T ( x ) = q x + x q , where q ∈ Q s ( R ) is some fixed element.

We proceed with the following functional equation:

(3) F ( x y x ) = F ( x y ) x − x F ( y ) x + x F ( y x ) ,

which appears naturally in the proof of Theorem 2 in [16]. One can easily prove that in case we have an additive mapping F : R → R , where R is 2-torsion free semiprime ring, satisfying the relation (3) for all pairs x , y ∈ R , then F is of the form 2 F ( x ) = D ( x ) + a x + x a , where D : R → R is a derivation and a ∈ R some fixed element (see [16] for the details). In [16], one can find the following conjecture.

Conjecture 3

Let R be a 2-torsion free semiprime ring and let F : R → R be an additive mapping satisfying the relation (3) for all pairs x , y ∈ R . In this case, F is of the form 2 F ( x ) = D ( x ) + q x + x q for all x ∈ R , where D : R → R is a derivation and q ∈ Q s ( R ) some fixed element.

By our knowledge, the aforementioned conjecture is still an open question. The substitution y = x in (1), (2), and (3) gives

(4) D ( x 3 ) = D ( x ) x 2 + x D ( x ) x + x 2 D ( x ) ,

(5) F ( x 3 ) = F ( x ) x 2 − x F ( x ) x + x 2 F ( x )

and

(6) F ( x 3 ) = F ( x 2 ) x − x F ( x ) x + x F ( x 2 ) .

The relation (4) has been considered in [7] (actually, much more general situation has been considered). A result related to (5) can be found in [15]. It is our aim in this paper to prove the following result, which is related to the aforementioned conjecture.

Theorem 4

Let R be a prime ring of characteristic different from two and three, and let F : R → R be an additive mapping satisfying the relation

(7) F ( x 3 ) = F ( x 2 ) x − x F ( x ) x + x F ( x 2 )

for all x ∈ R . In this case, F is of the form 4 F ( x ) = D ( x ) + q x + x q , where D : R → R is a derivation, and q ∈ Q s ( R ) is some fixed element.

Main results

As the main tool in this paper, we use the theory of functional identities (Brešar-Beidar-Chebotar theory). The theory of functional identities considers set-theoretic maps on rings that satisfy some identical relations. When treating such relations, one usually concludes that the form of the mappings involved can be described, unless the ring is very special. We refer the reader to [17] for the introductory account on the theory of functional identities, where Brešar presents this theory and its applications to a wider audience and to [18] for the full treatment of this theory.

Let R be an algebra over a commutative ring ϕ and let

(8) p ( x 1 , x 2 , x 3 ) = ∑ π ∈ S 3 x π ( 1 ) x π ( 2 ) x π ( 3 )

be a fixed multilinear polynomial in noncommuting indeterminates x i over ϕ . Here, S 3 stands for the symmetric group of order 3. Let ℒ be a subset of R closed under p , i.e., p ( x ¯ 3 ) ∈ ℒ for all x 1 , x 2 , x 3 ∈ ℒ , where x ¯ 3 = ( x 1 , x 2 , x 3 ) . We shall consider a mapping D : ℒ → R satisfying

(9) F ( p ( x ¯ 3 ) ) = ∑ π ∈ S 3 ( F ( x π ( 1 ) x π ( 2 ) ) x π ( 3 ) − x π ( 1 ) F ( x π ( 2 ) ) x π ( 3 ) + x π ( 1 ) F ( x π ( 2 ) x π ( 3 ) ) )

for all x 1 , x 2 , x 3 ∈ ℒ . Let us mention that the idea of considering the expression [ p ( x ¯ 3 ) , p ( y ¯ 3 ) ] in its proof is taken from [19]. For the proof of Theorem 4, we need Theorem 5, which might be of independent interest.

Theorem 5

Let ℒ be a 6-free Lie subring of R closed under p. If T : ℒ → R is an additive mapping satisfying (9), then there exists q ∈ R such that 4 F ( x ) = D ( x ) + x q + q x for all x ∈ ℒ .

Proof

For any a ∈ R and x ¯ 3 ∈ ℒ 3 , we have

[ p ( x ¯ 3 ) , a ] = p ( [ x 1 , a ] , x 2 , x 3 ) + p ( x 1 , [ x 2 , a ] , x 3 ) + p ( x 1 , x 2 , [ x 3 , a ] ) .

Thus,

(10) F [ p ( x ¯ 3 ) , a ] = F ( p ( [ x 1 , a ] , x 2 , x 3 ) ) + F ( p ( x 1 , [ x 2 , a ] , x 3 ) ) + F ( p ( x 1 , x 2 , [ x 3 , a ] ) ) .

By using (10), it follows that

F [ p ( x ¯ 3 ) , a ] = ∑ π ∈ S 3 F ( [ x π ( 1 ) , a ] x π ( 2 ) ) x π ( 3 ) − ∑ π ∈ S 3 [ x π ( 1 ) , a ] F ( x π ( 2 ) ) x π ( 3 ) + ∑ π ∈ S 3 [ x π ( 1 ) , a ] F ( x π ( 2 ) x π ( 3 ) ) + ∑ π ∈ S 3 F ( x π ( 1 ) [ x π ( 2 ) , a ] ) x π ( 3 ) − ∑ π ∈ S 3 x π ( 1 ) F ( [ x π ( 2 ) , a ] ) x π ( 3 ) + ∑ π ∈ S 3 x π ( 1 ) F ( [ x π ( 2 ) , a ] x π ( 3 ) ) + ∑ π ∈ S 3 F ( x π ( 1 ) x π ( 2 ) ) [ x π ( 3 ) , a ] − ∑ π ∈ S 3 x π ( 1 ) F ( x π ( 2 ) ) [ x π ( 3 ) , a ] + ∑ π ∈ S 3 x π ( 1 ) F ( x π ( 2 ) [ x π ( 3 ) , a ] ) = ∑ π ∈ S 3 F ( [ x π ( 1 ) x π ( 2 ) , a ] ) x π ( 3 ) − ∑ π ∈ S 3 [ x π ( 1 ) , a ] F ( x π ( 2 ) ) x π ( 3 ) + ∑ π ∈ S 3 [ x π ( 1 ) , a ] F ( x π ( 2 ) x π ( 3 ) ) − ∑ π ∈ S 3 x π ( 1 ) F ( [ x π ( 2 ) , a ] ) x π ( 3 ) + ∑ π ∈ S 3 x π ( 1 ) F ( [ x π ( 2 ) x π ( 3 ) , a ] ) + ∑ π ∈ S 3 F ( x π ( 1 ) x π ( 2 ) ) [ x π ( 3 ) , a ] − ∑ π ∈ S 3 x π ( 1 ) F ( x π ( 2 ) ) [ x π ( 3 ) , a ] .

In particular,

(11) F [ p ( x ¯ 3 ) , p ( y ¯ 3 ) ] = ∑ π ∈ S 3 F ( [ x π ( 1 ) x π ( 2 ) , p ( y ¯ 3 ) ] ) x π ( 3 ) − ∑ π ∈ S 3 [ x π ( 1 ) , p ( y ¯ 3 ) ] F ( x π ( 2 ) ) x π ( 3 ) + ∑ π ∈ S 3 [ x π ( 1 ) , p ( y ¯ 3 ) ] F ( x π ( 2 ) x π ( 3 ) ) − ∑ π ∈ S 3 x π ( 1 ) F ( [ x π ( 2 ) , p ( y ¯ 3 ) ] ) x π ( 3 ) + ∑ π ∈ S 3 x π ( 1 ) F ( [ x π ( 2 ) x π ( 3 ) , p ( y ¯ 3 ) ] ) + ∑ π ∈ S 3 F ( x π ( 1 ) x π ( 2 ) ) [ x π ( 3 ) , p ( y ¯ 3 ) ] − ∑ π ∈ S 3 x π ( 1 ) F ( x π ( 2 ) ) [ x π ( 3 ) , p ( y ¯ 3 ) ]

for all x ¯ 3 , y ¯ 3 ∈ ℒ 3 . For i = 1 , 2 , we have

(12) F [ x π ( i ) x π ( i + 1 ) , p ( y ¯ 3 ) ] = − F [ p ( y ¯ 3 ) , x π ( i ) x π ( i + 1 ) ] = ∑ σ ∈ S 3 F ( [ x π ( i ) x π ( i + 1 ) , y σ ( 1 ) y σ ( 2 ) ] ) y σ ( 3 ) − ∑ σ ∈ S 3 [ x π ( i ) x π ( i + 1 ) , y σ ( 1 ) ] F ( y σ ( 2 ) ) y σ ( 3 ) + ∑ σ ∈ S 3 [ x π ( i ) x π ( i + 1 ) , y σ ( 1 ) ] F ( y σ ( 2 ) y σ ( 3 ) ) − ∑ σ ∈ S 3 y σ ( 1 ) F ( [ x π ( i ) x π ( i + 1 ) , y σ ( 2 ) ] ) y σ ( 3 )

(12) + ∑ σ ∈ S 3 y σ ( 1 ) F ( [ x π ( i ) x π ( i + 1 ) , y σ ( 2 ) y σ ( 3 ) ] ) + ∑ σ ∈ S 3 F ( y σ ( 1 ) y σ ( 2 ) ) [ x π ( i ) x π ( i + 1 ) , y σ ( 3 ) ] − ∑ σ ∈ S 3 y σ ( 1 ) F ( y σ ( 2 ) ) [ x π ( i ) x π ( i + 1 ) , y σ ( 3 ) ]

and

F [ x π ( 2 ) , p ( y ¯ 3 ) ] = − F [ p ( y ¯ 3 ) , x π ( 2 ) ] = ∑ σ ∈ S 3 F ( [ x π ( 2 ) , y σ ( 1 ) y σ ( 2 ) ] ) y σ ( 3 ) − ∑ σ ∈ S 3 [ x π ( 2 ) , y σ ( 1 ) ] F ( y σ ( 2 ) ) y σ ( 3 ) + ∑ σ ∈ S 3 [ x π ( 2 ) , y σ ( 1 ) ] F ( y σ ( 2 ) y σ ( 3 ) ) − ∑ σ ∈ S 3 y σ ( 1 ) F ( [ x π ( 2 ) , y σ ( 2 ) ] ) y σ ( 3 ) + ∑ σ ∈ S 3 y σ ( 1 ) F ( [ x π ( 2 ) , y σ ( 2 ) y σ ( 3 ) ] ) + ∑ σ ∈ S 3 F ( y σ ( 1 ) y σ ( 2 ) ) [ x π ( 2 ) , y σ ( 3 ) ] − ∑ σ ∈ S 3 y σ ( 1 ) F ( y σ ( 2 ) ) [ x π ( 2 ) , y σ ( 3 ) ]

for all y ¯ 3 ∈ ℒ 3 . Therefore, (11) can be written as follows:

F [ p ( x ¯ 3 ) , p ( y ¯ 3 ) ] = ∑ π ∈ S 3 ∑ σ ∈ S 3 F ( [ x π ( 1 ) x π ( 2 ) , y σ ( 1 ) y σ ( 2 ) ] ) y σ ( 3 ) x π ( 3 ) − ∑ π ∈ S 3 ∑ σ ∈ S 3 [ x π ( 1 ) x π ( 2 ) , y σ ( 1 ) ] F ( y σ ( 2 ) ) y σ ( 3 ) x π ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 [ x π ( 1 ) x π ( 2 ) , y σ ( 1 ) ] F ( y σ ( 2 ) y σ ( 3 ) ) x π ( 3 ) − ∑ π ∈ S 3 ∑ σ ∈ S 3 y σ ( 1 ) F ( [ x π ( 1 ) x π ( 2 ) , y σ ( 2 ) ] ) y σ ( 3 ) x π ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 y σ ( 1 ) F ( [ x π ( 1 ) x π ( 2 ) , y σ ( 2 ) y σ ( 3 ) ] ) x π ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 F ( y σ ( 1 ) y σ ( 2 ) ) [ x π ( 1 ) x π ( 2 ) , y σ ( 3 ) ] x π ( 3 ) − ∑ π ∈ S 3 ∑ σ ∈ S 3 y σ ( 1 ) F ( y σ ( 2 ) ) [ x π ( 1 ) x π ( 2 ) , y σ ( 3 ) ] x π ( 3 ) − ∑ π ∈ S 3 ∑ σ ∈ S 3 [ x π ( 1 ) , y σ ( 1 ) y σ ( 2 ) y σ ( 3 ) ] F ( x π ( 2 ) ) x π ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 [ x π ( 1 ) , y σ ( 1 ) y σ ( 2 ) y σ ( 3 ) ] F ( x π ( 2 ) x π ( 3 ) ) − ∑ π ∈ S 3 ∑ σ ∈ S 3 x π ( 1 ) F ( [ x π ( 2 ) , y σ ( 1 ) y σ ( 2 ) ] ) y σ ( 3 ) x π ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 x π ( 1 ) [ x π ( 2 ) , y σ ( 1 ) ] F ( y σ ( 2 ) ) y σ ( 3 ) x π ( 3 ) − ∑ π ∈ S 3 ∑ σ ∈ S 3 x π ( 1 ) [ x π ( 2 ) , y σ ( 1 ) ] F ( y σ ( 2 ) y σ ( 3 ) ) x π ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 x π ( 1 ) y σ ( 1 ) F ( [ x π ( 2 ) , y σ ( 2 ) ] ) y σ ( 3 ) x π ( 3 ) − ∑ π ∈ S 3 ∑ σ ∈ S 3 x π ( 1 ) y σ ( 1 ) F ( [ x π ( 2 ) , y σ ( 2 ) y σ ( 3 ) ] ) x π ( 3 ) − ∑ π ∈ S 3 ∑ σ ∈ S 3 x π ( 1 ) F ( y σ ( 1 ) y σ ( 2 ) ) [ x π ( 2 ) , y σ ( 3 ) ] x π ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 x π ( 1 ) y σ ( 1 ) F ( y σ ( 2 ) ) [ x π ( 2 ) , y σ ( 3 ) ] x π ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 x π ( 1 ) F ( [ x π ( 2 ) x π ( 3 ) , y σ ( 1 ) y σ ( 2 ) ] ) y σ ( 3 ) − ∑ π ∈ S 3 ∑ σ ∈ S 3 x π ( 1 ) [ x π ( 2 ) x π ( 3 ) , y σ ( 1 ) ] F ( y σ ( 2 ) ) y σ ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 x π ( 1 ) [ x π ( 2 ) x π ( 3 ) , y σ ( 1 ) ] F ( y σ ( 2 ) y σ ( 3 ) ) − ∑ π ∈ S 3 ∑ σ ∈ S 3 x π ( 1 ) y σ ( 1 ) F ( [ x π ( 2 ) x π ( 3 ) , y σ ( 2 ) ] ) y σ ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 x π ( 1 ) y σ ( 1 ) F ( [ x π ( 2 ) x π ( 3 ) , y σ ( 2 ) y σ ( 3 ) ] ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 x π ( 1 ) F ( y σ ( 1 ) y σ ( 2 ) ) [ x π ( 2 ) x π ( 3 ) , y σ ( 3 ) ] − ∑ π ∈ S 3 ∑ σ ∈ S 3 x π ( 1 ) y σ ( 1 ) F ( y σ ( 2 ) ) [ x π ( 2 ) x π ( 3 ) , y σ ( 3 ) ] + ∑ π ∈ S 3 ∑ σ ∈ S 3 F ( x π ( 1 ) x π ( 2 ) ) [ x π ( 3 ) , y σ ( 1 ) y σ ( 2 ) y σ ( 3 ) ] − ∑ π ∈ S 3 ∑ σ ∈ S 3 x π ( 1 ) F ( x π ( 2 ) ) [ x π ( 3 ) , y σ ( 1 ) y σ ( 2 ) y σ ( 3 ) ]

for all x ¯ 3 , y ¯ 3 ∈ ℒ 3 . On the other hand, by using [ p ( x ¯ 3 ) , p ( y ¯ 3 ) ] = − [ p ( y ¯ 3 ) , p ( x ¯ 3 ) ] , we obtain from aforementioned identity

F [ p ( x ¯ 3 ) , p ( y ¯ 3 ) ] = ∑ π ∈ S 3 ∑ σ ∈ S 3 F ( [ x π ( 1 ) x π ( 2 ) , y σ ( 1 ) y σ ( 2 ) ] ) x π ( 3 ) y σ ( 3 ) − ∑ π ∈ S 3 ∑ σ ∈ S 3 [ x π ( 1 ) , y σ ( 1 ) y σ ( 2 ) ] F ( x π ( 2 ) ) x π ( 3 ) y σ ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 [ x π ( 1 ) , y σ ( 1 ) y σ ( 2 ) ] F ( x π ( 2 ) x π ( 3 ) ) y σ ( 3 ) − ∑ π ∈ S 3 ∑ σ ∈ S 3 x π ( 1 ) F ( [ x π ( 2 ) , y σ ( 1 ) y σ ( 2 ) ] ) x π ( 3 ) y σ ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 x π ( 1 ) F ( [ x π ( 2 ) x π ( 3 ) , y σ ( 1 ) y σ ( 2 ) ] ) y σ ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 F ( x π ( 1 ) x π ( 2 ) ) [ x π ( 3 ) , y σ ( 1 ) y σ ( 2 ) ] y σ ( 3 )

− ∑ π ∈ S 3 ∑ σ ∈ S 3 x π ( 1 ) F ( x π ( 2 ) ) [ x π ( 3 ) , y σ ( 1 ) y σ ( 2 ) ] y σ ( 3 ) − ∑ π ∈ S 3 ∑ σ ∈ S 3 [ x π ( 1 ) x π ( 2 ) x π ( 3 ) , y σ ( 1 ) ] F ( y σ ( 2 ) ) y σ ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 [ x π ( 1 ) x π ( 2 ) x π ( 3 ) , y σ ( 1 ) ] F ( y σ ( 2 ) y σ ( 3 ) ) − ∑ π ∈ S 3 ∑ σ ∈ S 3 y σ ( 1 ) F ( [ x π ( 1 ) x π ( 2 ) , y σ ( 2 ) ] ) x π ( 3 ) y σ ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 y σ ( 1 ) [ x π ( 1 ) , y σ ( 2 ) ] F ( x π ( 2 ) ) x π ( 3 ) y σ ( 3 ) − ∑ π ∈ S 3 ∑ σ ∈ S 3 y σ ( 1 ) [ x π ( 1 ) , y σ ( 2 ) ] F ( x π ( 2 ) x π ( 3 ) ) y σ ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 y σ ( 1 ) x π ( 1 ) F ( [ x π ( 2 ) , y σ ( 2 ) ] ) x π ( 3 ) y σ ( 3 ) − ∑ π ∈ S 3 ∑ σ ∈ S 3 y σ ( 1 ) x π ( 1 ) F ( [ x π ( 2 ) x π ( 3 ) , y σ ( 2 ) ] ) y σ ( 3 ) − ∑ π ∈ S 3 ∑ σ ∈ S 3 y σ ( 1 ) F ( x π ( 1 ) x π ( 2 ) ) [ x π ( 3 ) , y σ ( 2 ) ] y σ ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 y σ ( 1 ) x π ( 1 ) F ( x π ( 2 ) ) [ x π ( 3 ) , y σ ( 2 ) ] y σ ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 y σ ( 1 ) F ( [ x π ( 1 ) x π ( 2 ) , y σ ( 2 ) y σ ( 3 ) ] ) x π ( 3 ) − ∑ π ∈ S 3 ∑ σ ∈ S 3 y σ ( 1 ) [ x π ( 1 ) , y σ ( 2 ) y σ ( 3 ) ] F ( x π ( 2 ) ) x π ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 y σ ( 1 ) [ x π ( 1 ) , y σ ( 2 ) y σ ( 3 ) ] F ( x π ( 2 ) x π ( 3 ) ) − ∑ π ∈ S 3 ∑ σ ∈ S 3 y σ ( 1 ) x π ( 1 ) F ( [ x π ( 2 ) , y σ ( 2 ) y σ ( 3 ) ] ) x π ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 y σ ( 1 ) x π ( 1 ) F ( [ x π ( 2 ) x π ( 3 ) , y σ ( 2 ) y σ ( 3 ) ] ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 y σ ( 1 ) F ( x π ( 1 ) x π ( 2 ) ) [ x π ( 3 ) , y σ ( 2 ) y σ ( 3 ) ] − ∑ π ∈ S 3 ∑ σ ∈ S 3 y σ ( 1 ) x π ( 1 ) F ( x π ( 2 ) ) [ x π ( 3 ) , y σ ( 2 ) y σ ( 3 ) ] + ∑ π ∈ S 3 ∑ σ ∈ S 3 F ( y σ ( 1 ) y σ ( 2 ) ) [ x π ( 1 ) x π ( 2 ) x π ( 3 ) , y σ ( 3 ) ] − ∑ π ∈ S 3 ∑ σ ∈ S 3 y σ ( 1 ) F ( y σ ( 2 ) ) [ x π ( 1 ) x π ( 2 ) x π ( 3 ) , y σ ( 3 ) ]

for all x ¯ 3 , y ¯ 3 ∈ ℒ 3 . By comparing so obtained identities, we arrive at

(13) 0 = ∑ π ∈ S 3 ∑ σ ∈ S 3 F ( [ x π ( 1 ) x π ( 2 ) , y σ ( 1 ) y σ ( 2 ) ] ) y σ ( 3 ) x π ( 3 ) − F ( x π ( 1 ) x π ( 2 ) ) y σ ( 1 ) y σ ( 2 ) y σ ( 3 ) x π ( 3 ) + F ( y σ ( 1 ) y σ ( 2 ) ) x π ( 1 ) x π ( 2 ) y σ ( 3 ) x π ( 3 ) − y σ ( 1 ) F ( y σ ( 2 ) ) x π ( 1 ) x π ( 2 ) y σ ( 3 ) x π ( 3 ) − x π ( 1 ) F ( y σ ( 1 ) y σ ( 2 ) ) x π ( 2 ) y σ ( 3 ) x π ( 3 ) − y σ ( 1 ) x π ( 1 ) F ( [ y σ ( 2 ) y σ ( 3 ) , x π ( 2 ) ] ) x π ( 3 ) + x π ( 1 ) y σ ( 1 ) F ( y σ ( 2 ) ) x π ( 2 ) y σ ( 3 ) x π ( 3 ) + x π ( 1 ) F ( x π ( 2 ) ) y σ ( 1 ) y σ ( 2 ) y σ ( 3 ) x π ( 3 ) + y σ ( 1 ) F ( x π ( 1 ) x π ( 2 ) ) y σ ( 2 ) y σ ( 3 ) x π ( 3 ) − y σ ( 1 ) x π ( 1 ) F ( x π ( 2 ) ) y σ ( 2 ) y σ ( 3 ) x π ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 F ( [ y σ ( 1 ) y σ ( 2 ) , x π ( 1 ) x π ( 2 ) ] ) x π ( 3 ) y σ ( 3 ) − F ( y σ ( 1 ) y σ ( 2 ) ) x π ( 1 ) x π ( 2 ) x π ( 3 ) y σ ( 3 ) + F ( x π ( 1 ) x π ( 2 ) ) y σ ( 1 ) y σ ( 2 ) x π ( 3 ) y σ ( 3 ) − x π ( 1 ) F ( x π ( 2 ) ) y σ ( 1 ) y σ ( 2 ) x π ( 3 ) y σ ( 3 ) − y σ ( 1 ) F ( x π ( 1 ) x π ( 2 ) ) y σ ( 2 ) x π ( 3 ) y σ ( 3 ) − x π ( 1 ) y σ ( 1 ) F ( [ x π ( 2 ) x π ( 3 ) , y σ ( 2 ) ] ) y σ ( 3 ) + y σ ( 1 ) x π ( 1 ) F ( x π ( 2 ) ) y σ ( 2 ) x π ( 3 ) y σ ( 3 ) + y σ ( 1 ) F ( y σ ( 2 ) ) x π ( 1 ) x π ( 2 ) x π ( 3 ) y σ ( 3 ) + x π ( 1 ) F ( y σ ( 1 ) y σ ( 2 ) ) x π ( 2 ) x π ( 3 ) y σ ( 3 ) − x π ( 1 ) y σ ( 1 ) F ( y σ ( 2 ) ) x π ( 2 ) x π ( 3 ) y σ ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 x π ( 1 ) y σ ( 1 ) F ( [ x π ( 2 ) x π ( 3 ) , y σ ( 2 ) y σ ( 3 ) ] ) + x π ( 1 ) y σ ( 1 ) x π ( 2 ) x π ( 3 ) F ( y σ ( 2 ) ) y σ ( 3 ) − x π ( 1 ) y σ ( 1 ) x π ( 2 ) x π ( 3 ) F ( y σ ( 2 ) y σ ( 3 ) ) + x π ( 1 ) y σ ( 1 ) y σ ( 2 ) y σ ( 3 ) F ( x π ( 2 ) x π ( 3 ) ) − x π ( 1 ) y σ ( 1 ) x π ( 2 ) F ( y σ ( 2 ) ) y σ ( 3 ) x π ( 3 ) + x π ( 1 ) y σ ( 1 ) x π ( 2 ) F ( y σ ( 2 ) y σ ( 3 ) ) x π ( 3 ) − x π ( 1 ) y σ ( 1 ) y σ ( 2 ) y σ ( 3 ) F ( x π ( 2 ) ) x π ( 3 ) − x π ( 1 ) F ( [ x π ( 2 ) , y σ ( 1 ) y σ ( 2 ) ] ) y σ ( 3 ) x π ( 3 ) + x π ( 1 ) y σ ( 1 ) F ( [ x π ( 2 ) , y σ ( 2 ) ] ) y σ ( 3 ) x π ( 3 ) − x π ( 1 ) y σ ( 1 ) F ( [ x π ( 2 ) , y σ ( 2 ) y σ ( 3 ) ] ) x π ( 3 ) − x π ( 1 ) F ( [ y σ ( 1 ) y σ ( 2 ) , x π ( 2 ) ] ) x π ( 3 ) y σ ( 3 ) + x π ( 1 ) y σ ( 1 ) y σ ( 2 ) F ( x π ( 2 ) ) x π ( 3 ) y σ ( 3 ) − x π ( 1 ) y σ ( 1 ) y σ ( 2 ) F ( x π ( 2 ) x π ( 3 ) ) y σ ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 y σ ( 1 ) x π ( 1 ) F ( [ y σ ( 2 ) y σ ( 3 ) , x π ( 2 ) x π ( 3 ) ] ) + y σ ( 1 ) x π ( 1 ) y σ ( 2 ) y σ ( 3 ) F ( x π ( 2 ) ) x π ( 3 ) − y σ ( 1 ) x π ( 1 ) y σ ( 2 ) y σ ( 3 ) F ( x π ( 2 ) x π ( 3 ) ) + y σ ( 1 ) x π ( 1 ) x π ( 2 ) x π ( 3 ) F ( y σ ( 2 ) y σ ( 3 ) ) − y σ ( 1 ) x π ( 1 ) y σ ( 2 ) F ( x π ( 2 ) ) x π ( 3 ) y σ ( 3 ) + y σ ( 1 ) x π ( 1 ) y σ ( 2 ) F ( x π ( 2 ) x π ( 3 ) ) y σ ( 3 ) − y σ ( 1 ) x π ( 1 ) x π ( 2 ) x π ( 3 ) F ( y σ ( 2 ) ) y σ ( 3 )

(13) − y σ ( 1 ) F ( [ y σ ( 2 ) , x π ( 1 ) x π ( 2 ) ] ) x π ( 3 ) y σ ( 3 ) + y σ ( 1 ) x π ( 1 ) F ( [ y σ ( 2 ) , x π ( 2 ) ] ) x π ( 3 ) y σ ( 3 ) − y σ ( 1 ) x π ( 1 ) F ( [ y σ ( 2 ) , x π ( 2 ) x π ( 3 ) ] ) y σ ( 3 ) − y σ ( 1 ) F ( [ x π ( 1 ) x π ( 2 ) , y σ ( 2 ) ] ) y σ ( 3 ) x π ( 3 ) − y σ ( 1 ) x π ( 1 ) x π ( 2 ) F ( y σ ( 2 ) y σ ( 3 ) ) x π ( 3 ) + y σ ( 1 ) x π ( 1 ) x π ( 2 ) F ( y σ ( 2 ) ) y σ ( 3 ) x π ( 3 )

for all x ¯ 3 , y ¯ 3 ∈ ℒ 3 . By using the theory of functional identities, we can conclude that

(14) ∑ π ∈ S 2 ∑ σ ∈ S 2 F ( [ x π ( 1 ) x π ( 2 ) , y σ ( 1 ) y σ ( 2 ) ] ) − F ( x π ( 1 ) x π ( 2 ) ) y σ ( 1 ) y σ ( 2 ) + F ( y σ ( 1 ) y σ ( 2 ) ) x π ( 1 ) x π ( 2 ) − y σ ( 1 ) F ( y σ ( 2 ) ) x π ( 1 ) x π ( 2 ) + x π ( 1 ) F ( x π ( 2 ) ) y σ ( 1 ) y σ ( 2 ) = x 1 p 1 ( x 2 , y 1 , y 2 ) + x 2 p 2 ( x 1 , y 1 , y 2 ) + y 1 p 3 ( x 1 , x 2 , y 2 ) + y 2 p 4 ( x 1 , x 2 , y 1 ) + λ p ( x 1 , x 2 , y 1 , y 2 )

and

(15) ∑ π ∈ S 2 ∑ σ ∈ S 2 F ( [ x π ( 2 ) x π ( 3 ) , y σ ( 2 ) y σ ( 3 ) ] ) − x π ( 2 ) x π ( 3 ) F ( y σ ( 2 ) y σ ( 3 ) ) + y σ ( 2 ) y σ ( 3 ) F ( x π ( 2 ) x π ( 3 ) ) + x π ( 2 ) x π ( 3 ) F ( y σ ( 2 ) ) y σ ( 3 ) − y σ ( 2 ) y σ ( 3 ) F ( x π ( 2 ) ) x π ( 3 ) = q 1 ( x 3 , y 2 , y 3 ) x 2 + q 2 ( x 2 , y 2 , y 3 ) x 3 + q 3 ( x 2 , x 3 , y 3 ) y 2 + q 4 ( x 2 , x 3 , y 2 ) y 3 + λ q ( x 2 , x 3 , y 2 , y 3 ) .

We also have

∑ π ∈ S 2 ∑ σ ∈ S 2 F ( [ y σ ( 1 ) y σ ( 2 ) , x π ( 1 ) x π ( 2 ) ] ) − F ( y σ ( 1 ) y σ ( 2 ) ) x π ( 1 ) x π ( 2 ) + F ( x π ( 1 ) x π ( 2 ) ) y σ ( 1 ) y σ ( 2 ) − x π ( 1 ) F ( x π ( 2 ) ) y σ ( 1 ) y σ ( 2 ) + y σ ( 1 ) F ( y σ ( 2 ) ) x π ( 1 ) x π ( 2 ) = x 1 p 1 ′ ( x 2 , y 1 , y 2 ) + x 2 p 2 ′ ( x 1 , y 1 , y 2 ) + y 1 p 3 ′ ( x 1 , x 2 , y 2 ) + y 2 p 4 ′ ( x 1 , x 2 , y 1 ) + λ p ′ ( x 1 , x 2 , y 1 , y 2 )

and

∑ π ∈ S 2 ∑ σ ∈ S 2 F ( [ y σ ( 2 ) y σ ( 3 ) , x π ( 2 ) x π ( 3 ) ] ) − y σ ( 2 ) y σ ( 3 ) F ( x π ( 2 ) x π ( 3 ) ) + x π ( 2 ) x π ( 3 ) F ( y σ ( 2 ) y σ ( 3 ) ) − x π ( 2 ) x π ( 3 ) F ( y σ ( 2 ) ) y σ ( 3 ) + y σ ( 2 ) y σ ( 3 ) F ( x π ( 2 ) ) x π ( 3 ) = q 1 ′ ( x 3 , y 2 , y 3 ) x 2 + q 2 ′ ( x 2 , y 2 , y 3 ) x 3 + q 3 ′ ( x 2 , x 3 , y 3 ) y 2 + q 4 ′ ( x 2 , x 3 , y 2 ) y 3 + λ q ′ ( x 2 , x 3 , y 2 , y 3 )

for all x ¯ 3 , y ¯ 3 ∈ ℒ 3 and p i , p i ′ , q i , q i ′ : ℒ 3 → R , i = 1 , 2 , 3 , 4 and λ p , λ p ′ , λ q , λ q ′ : ℒ 4 → C ( ℒ ) . By comparing identities (14) and (15), we arrive at

(16) ∑ π ∈ S 2 ∑ σ ∈ S 2 ( − F ( x π ( 1 ) x π ( 2 ) ) y σ ( 1 ) y σ ( 2 ) + F ( y σ ( 1 ) y σ ( 2 ) ) x π ( 1 ) x π ( 2 ) − y σ ( 1 ) F ( y σ ( 2 ) ) x π ( 1 ) x π ( 2 ) + x π ( 1 ) F ( x π ( 2 ) ) y σ ( 1 ) y σ ( 2 ) + x π ( 1 ) x π ( 2 ) F ( y σ ( 1 ) y σ ( 2 ) ) − y σ ( 1 ) y σ ( 2 ) F ( x π ( 1 ) x π ( 2 ) ) + y σ ( 1 ) y σ ( 2 ) F ( x π ( 1 ) ) x π ( 2 ) − x π ( 1 ) x π ( 2 ) F ( y σ ( 1 ) ) y σ ( 2 ) ) = x π ( 1 ) p 1 ( x π ( 2 ) , y σ ( 1 ) , y σ ( 2 ) ) + x π ( 2 ) p 2 ( x π ( 1 ) , y σ ( 1 ) , y σ ( 2 ) ) + y σ ( 1 ) p 3 ( x π ( 1 ) , x π ( 2 ) , y σ ( 2 ) ) + y σ ( 2 ) p 4 ( x π ( 1 ) , x π ( 2 ) , y σ ( 1 ) ) + λ 1 ( x π ( 1 ) , x π ( 2 ) , y σ ( 1 ) , y σ ( 2 ) ) − q 1 ( x π ( 2 ) , y σ ( 1 ) , y σ ( 2 ) ) x π ( 1 ) − q 2 ( x π ( 1 ) , y σ ( 1 ) , y σ ( 2 ) ) x π ( 2 ) − q 3 ( x π ( 1 ) , x π ( 2 ) , y σ ( 2 ) ) y σ ( 1 ) − q 4 ( x π ( 1 ) , x π ( 2 ) , y σ ( 1 ) ) y σ ( 2 ) − μ 1 ( x π ( 1 ) , x π ( 2 ) , y σ ( 1 ) , y σ ( 2 ) )

for all x ¯ 3 , y ¯ 3 ∈ ℒ 3 and p i , p i ′ , q i , q i ′ : ℒ 3 → R , i = 1 , 2 , 3 , 4 and λ p , λ p ′ , λ q , λ q ′ : ℒ 4 → C ( ℒ ) . By using equation (16) and the theory of functional identities, it follows that

∑ σ ∈ S 2 ( F ( y σ ( 1 ) y σ ( 2 ) ) x π ( 1 ) − y σ ( 1 ) F ( y σ ( 2 ) ) x π ( 1 ) ) + q 2 ( x π ( 1 ) , y σ ( 1 ) , y σ ( 2 ) ) = x π ( 1 ) m 1 ( y σ ( 1 ) , y σ ( 2 ) ) + y σ ( 1 ) m 2 ( x π ( 1 ) , y σ ( 2 ) ) + y σ ( 2 ) m 3 ( x π ( 1 ) , y σ ( 1 ) ) + λ m ( x π ( 1 ) , y σ ( 1 ) , y σ ( 2 ) )

for all x ¯ 3 , y ¯ 3 ∈ ℒ 3 , q 2 : ℒ 3 → R , m i : ℒ 2 → R , i = 1 , 2 , 3 and λ m : ℒ 3 → C ( ℒ ) . Now setting x π ( 1 ) , y σ ( 1 ) , y σ ( 2 ) = x in the aforementioned equation, we obtain

(17) 2 F ( x 2 ) x − 2 x F ( x ) x + q 2 ( x , x , x ) = x m 1 ( x , x ) + x m 2 ( x , x ) + x m 3 ( x , x ) + λ m ( x , x , x )

for all x ∈ R .

On the other hand, using equation (16) and the theory of functional identities, we arrive at

∑ σ ∈ S 2 ( x π ( 1 ) F ( y σ ( 1 ) y σ ( 2 ) ) − x π ( 1 ) F ( y σ ( 1 ) ) y σ ( 2 ) ) − p 2 ( x π ( 1 ) , y σ ( 1 ) , y σ ( 2 ) ) = n 1 ′ ( y σ ( 1 ) , y σ ( 2 ) ) x π ( 1 ) + n 2 ′ ( x π ( 1 ) , y σ ( 2 ) ) y σ ( 1 ) + n 3 ′ ( x π ( 1 ) , y σ ( 1 ) ) y σ ( 2 ) + λ n ′ ( x π ( 1 ) , y σ ( 1 ) , y σ ( 2 ) )

x ¯ 3 , y ¯ 3 ∈ ℒ 3 , p 2 : ℒ 3 → R , n i ′ : ℒ 2 → R , i = 1 , 2 , 3 and λ n ′ : ℒ 3 → C ( ℒ ) .

Now setting x π ( 1 ) , y σ ( 1 ) , y σ ( 2 ) = x , we obtain

(18) 2 x F ( x 2 ) − 2 x F ( x ) x − p 2 ( x , x , x ) = n 1 ′ ( x , x ) x + n 2 ′ ( x , x ) x + n 3 ′ ( x , x ) x + λ n ′ ( x , x , x )

for all x ∈ R . Equation (13) can now be rewritten as follows:

0 = ∑ π ∈ S 3 ∑ σ ∈ S 3 ( x π ( 1 ) p 1 ( x π ( 2 ) , y σ ( 1 ) , y σ ( 2 ) ) y σ ( 3 ) x π ( 3 ) + x π ( 2 ) p 2 ( x π ( 1 ) , y σ ( 1 ) , y σ ( 2 ) ) y σ ( 3 ) x π ( 3 ) + y σ ( 1 ) p 3 ( x π ( 1 ) , x π ( 2 ) , y σ ( 2 ) ) y σ ( 3 ) x π ( 3 ) + y σ ( 2 ) p 4 ( x π ( 1 ) , x π ( 2 ) , y σ ( 1 ) ) y σ ( 3 ) x π ( 3 ) + λ p ( x π ( 1 ) , x π ( 2 ) , y σ ( 1 ) , y σ ( 2 ) ) y σ ( 3 ) x π ( 3 ) + x π ( 1 ) y σ ( 1 ) F ( y σ ( 2 ) ) x π ( 2 ) y σ ( 3 ) x π ( 3 ) − x π ( 1 ) F ( y σ ( 1 ) y σ ( 2 ) ) x π ( 2 ) y σ ( 3 ) x π ( 3 ) − y σ ( 1 ) x π ( 1 ) F ( [ y σ ( 2 ) y σ ( 3 ) , x π ( 2 ) ] ) x π ( 3 ) + y σ ( 1 ) F ( x π ( 1 ) x π ( 2 ) ) y σ ( 2 ) y σ ( 3 ) x π ( 3 ) − y σ ( 1 ) x π ( 1 ) F ( x π ( 2 ) ) y σ ( 2 ) y σ ( 3 ) x π ( 3 ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 ( x π ( 1 ) p 1 ′ ( x π ( 2 ) , y σ ( 1 ) , y σ ( 2 ) ) x π ( 3 ) y σ ( 3 ) + x π ( 2 ) p 2 ′ ( x π ( 1 ) , y σ ( 1 ) , y σ ( 2 ) ) x π ( 3 ) y σ ( 3 ) + y σ ( 1 ) p 3 ′ ( x π ( 1 ) , x π ( 2 ) , y σ ( 2 ) ) x π ( 3 ) y σ ( 3 ) + y σ ( 2 ) p 4 ′ ( x π ( 1 ) , x π ( 2 ) , y σ ( 1 ) ) x π ( 3 ) y σ ( 3 ) ) + λ p ′ ( x π ( 1 ) , x π ( 2 ) , y σ ( 1 ) , y σ ( 2 ) ) x π ( 3 ) y σ ( 3 ) + y σ ( 1 ) x π ( 1 ) F ( x π ( 2 ) ) y σ ( 2 ) x π ( 3 ) y σ ( 3 ) − y σ ( 1 ) F ( x π ( 1 ) x π ( 2 ) ) y σ ( 2 ) x π ( 3 ) y σ ( 3 ) − x π ( 1 ) y σ ( 1 ) F ( [ x π ( 2 ) x π ( 3 ) , y σ ( 2 ) ] ) y σ ( 3 ) + x π ( 1 ) F ( y σ ( 1 ) y σ ( 2 ) ) x π ( 2 ) x π ( 3 ) y σ ( 3 ) − x π ( 1 ) y σ ( 1 ) F ( y σ ( 2 ) ) x π ( 2 ) x π ( 3 ) y σ ( 3 ) ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 ( x π ( 1 ) y σ ( 1 ) q 1 ( x π ( 3 ) , y σ ( 2 ) , y σ ( 3 ) ) x π ( 2 ) + x π ( 1 ) y σ ( 1 ) q 2 ( x π ( 2 ) , y σ ( 2 ) , y σ ( 3 ) ) x π ( 3 ) + x π ( 1 ) y σ ( 1 ) q 3 ( x π ( 2 ) , x π ( 3 ) , y σ ( 3 ) ) y σ ( 2 ) + x π ( 1 ) y σ ( 1 ) q 4 ( x π ( 2 ) , x π ( 3 ) , y σ ( 2 ) ) y σ ( 3 ) + x π ( 1 ) y σ ( 1 ) λ q ( x π ( 2 ) , x π ( 3 ) , y σ ( 2 ) , y σ ( 3 ) ) − x π ( 1 ) y σ ( 1 ) x π ( 2 ) F ( y σ ( 2 ) ) y σ ( 3 ) x π ( 3 ) + x π ( 1 ) y σ ( 1 ) x π ( 2 ) F ( y σ ( 2 ) y σ ( 3 ) ) x π ( 3 ) − x π ( 1 ) F ( [ x π ( 2 ) , y σ ( 1 ) y σ ( 2 ) ] ) y σ ( 3 ) x π ( 3 ) + x π ( 1 ) y σ ( 1 ) F ( [ x π ( 2 ) , y σ ( 2 ) ] ) y σ ( 3 ) x π ( 3 ) − x π ( 1 ) y σ ( 1 ) F ( [ x π ( 2 ) , y σ ( 2 ) y σ ( 3 ) ] ) x π ( 3 ) − x π ( 1 ) F ( [ y σ ( 1 ) y σ ( 2 ) , x π ( 2 ) ] ) x π ( 3 ) y σ ( 3 ) + x π ( 1 ) y σ ( 1 ) y σ ( 2 ) F ( x π ( 2 ) ) x π ( 3 ) y σ ( 3 ) − x π ( 1 ) y σ ( 1 ) y σ ( 2 ) F ( x π ( 2 ) x π ( 3 ) ) y σ ( 3 ) ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 ( y σ ( 1 ) x π ( 1 ) q 1 ′ ( x π ( 3 ) , y σ ( 2 ) , y σ ( 3 ) ) x π ( 2 ) + y σ ( 1 ) x π ( 1 ) q 2 ′ ( x π ( 2 ) , y σ ( 2 ) , y σ ( 3 ) ) x π ( 3 ) + y σ ( 1 ) x π ( 1 ) q 3 ′ ( x π ( 2 ) , x π ( 3 ) , y σ ( 3 ) ) y σ ( 2 ) + y σ ( 1 ) x π ( 1 ) q 4 ′ ( x π ( 2 ) , x π ( 3 ) , y σ ( 2 ) ) y σ ( 3 ) + y σ ( 1 ) x π ( 1 ) λ q ′ ( x π ( 2 ) , x π ( 3 ) , y σ ( 2 ) , y σ ( 3 ) ) − y σ ( 1 ) x π ( 1 ) y σ ( 2 ) F ( x π ( 2 ) ) x π ( 3 ) y σ ( 3 ) + y σ ( 1 ) x π ( 1 ) y σ ( 2 ) F ( x π ( 2 ) x π ( 3 ) ) y σ ( 3 ) − y σ ( 1 ) F ( [ y σ ( 2 ) , x π ( 1 ) x π ( 2 ) ] ) x π ( 3 ) y σ ( 3 ) + y σ ( 1 ) x π ( 1 ) F ( [ y σ ( 2 ) , x π ( 2 ) ] ) x π ( 3 ) y σ ( 3 ) − y σ ( 1 ) x π ( 1 ) F ( [ y σ ( 2 ) , x π ( 2 ) x π ( 3 ) ] ) y σ ( 3 ) − y σ ( 1 ) F ( [ x π ( 1 ) x π ( 2 ) , y σ ( 2 ) ] ) y σ ( 3 ) x π ( 3 ) − y σ ( 1 ) x π ( 1 ) x π ( 2 ) F ( y σ ( 2 ) y σ ( 3 ) ) x π ( 3 ) + y σ ( 1 ) x π ( 1 ) x π ( 2 ) F ( y σ ( 2 ) ) y σ ( 3 ) x π ( 3 ) )

for all x ¯ 3 , y ¯ 3 ∈ ℒ 3 , p i , p i ′ , q i , q i ′ : ℒ 3 → R , i = 1 , 2 , 3 , 4 and λ p , λ p ′ , λ q , λ q ′ : ℒ 4 → C ( ℒ ) . By using the theory of functional identities and exposing everything from the same side (left), the aforementioned equation can now be rewritten as follows:

(19) 0 = ∑ π ∈ S 3 ∑ σ ∈ S 3 x π ( 1 ) ( p 1 ( x π ( 2 ) , y σ ( 1 ) , y σ ( 2 ) ) y σ ( 3 ) x π ( 3 ) − F ( y σ ( 1 ) y σ ( 2 ) ) x π ( 2 ) y σ ( 3 ) x π ( 3 ) + y σ ( 1 ) F ( y σ ( 2 ) ) x π ( 2 ) y σ ( 3 ) x π ( 3 ) + p 1 ′ ( x π ( 2 ) , y σ ( 1 ) , y σ ( 2 ) ) x π ( 3 ) y σ ( 3 ) − y σ ( 1 ) F ( [ x π ( 2 ) x π ( 3 ) , y σ ( 2 ) ] ) y σ ( 3 ) + F ( y σ ( 1 ) y σ ( 2 ) ) x π ( 2 ) x π ( 3 ) y σ ( 3 ) − y σ ( 1 ) F ( y σ ( 2 ) ) x π ( 2 ) x π ( 3 ) y σ ( 3 ) + y σ ( 1 ) q 1 ( x π ( 3 ) , y σ ( 2 ) , y σ ( 3 ) ) x π ( 2 )

(19) + y σ ( 1 ) q 2 ( x π ( 2 ) , y σ ( 2 ) , y σ ( 3 ) ) x π ( 3 ) + y σ ( 1 ) q 3 ( x π ( 2 ) , x π ( 3 ) , y σ ( 3 ) ) y σ ( 2 ) + y σ ( 1 ) q 4 ( x π ( 2 ) , x π ( 3 ) , y σ ( 2 ) ) y σ ( 3 ) + y σ ( 1 ) λ q ( x π ( 2 ) , x π ( 3 ) , y σ ( 2 ) , y σ ( 3 ) ) − y σ ( 1 ) x π ( 2 ) F ( y σ ( 2 ) ) y σ ( 3 ) x π ( 3 ) + y σ ( 1 ) x π ( 2 ) F ( y σ ( 2 ) y σ ( 3 ) ) x π ( 3 ) − F ( [ x π ( 2 ) , y σ ( 1 ) y σ ( 2 ) ] ) y σ ( 3 ) x π ( 3 ) + y σ ( 1 ) F ( [ x π ( 2 ) , y σ ( 2 ) ] ) y σ ( 3 ) x π ( 3 ) − y σ ( 1 ) F ( [ x π ( 2 ) , y σ ( 2 ) y σ ( 3 ) ] ) x π ( 3 ) − F ( [ y σ ( 1 ) y σ ( 2 ) , x π ( 2 ) ] ) x π ( 3 ) y σ ( 3 ) + y σ ( 1 ) y σ ( 2 ) F ( x π ( 2 ) ) x π ( 3 ) y σ ( 3 ) − y σ ( 1 ) y σ ( 2 ) F ( x π ( 2 ) x π ( 3 ) ) y σ ( 3 ) ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 x π ( 2 ) ( p 2 ( x π ( 1 ) , y σ ( 1 ) , y σ ( 2 ) ) y σ ( 3 ) x π ( 3 ) + p 2 ′ ( x π ( 1 ) , y σ ( 1 ) , y σ ( 2 ) ) x π ( 3 ) y σ ( 3 ) ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 x π ( 3 ) ( y σ ( 3 ) λ p ′ ( x π ( 1 ) , x π ( 2 ) , y σ ( 1 ) , y σ ( 2 ) ) ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 y σ ( 1 ) ( p 3 ( x π ( 1 ) , x π ( 2 ) , y σ ( 2 ) ) y σ ( 3 ) x π ( 3 ) − x π ( 1 ) F ( [ y σ ( 2 ) y σ ( 3 ) , x π ( 2 ) ] ) x π ( 3 ) + F ( x π ( 1 ) x π ( 2 ) ) y σ ( 2 ) y σ ( 3 ) x π ( 3 ) − x π ( 1 ) F ( x π ( 2 ) ) y σ ( 2 ) y σ ( 3 ) x π ( 3 ) + p 3 ′ ( x π ( 1 ) , x π ( 2 ) , y σ ( 2 ) ) x π ( 3 ) y σ ( 3 ) − F ( x π ( 1 ) x π ( 2 ) ) y σ ( 2 ) x π ( 3 ) y σ ( 3 ) + x π ( 1 ) F ( x π ( 2 ) ) y σ ( 2 ) x π ( 3 ) y σ ( 3 ) + x π ( 1 ) q 1 ′ ( x π ( 3 ) , y σ ( 2 ) , y σ ( 3 ) ) x π ( 2 ) + x π ( 1 ) q 2 ′ ( x π ( 2 ) , y σ ( 2 ) , y σ ( 3 ) ) x π ( 3 ) + x π ( 1 ) q 3 ′ ( x π ( 2 ) , x π ( 3 ) , y σ ( 3 ) ) y σ ( 2 ) + x π ( 1 ) q 4 ′ ( x π ( 2 ) , x π ( 3 ) , y σ ( 2 ) ) y σ ( 3 ) + x π ( 1 ) λ q ′ ( x π ( 2 ) , x π ( 3 ) , y σ ( 2 ) , y σ ( 3 ) ) − x π ( 1 ) y σ ( 2 ) F ( x π ( 2 ) ) x π ( 3 ) y σ ( 3 ) + x π ( 1 ) y σ ( 2 ) F ( x π ( 2 ) x π ( 3 ) ) y σ ( 3 ) − F ( [ y σ ( 2 ) , x π ( 1 ) x π ( 2 ) ] ) x π ( 3 ) y σ ( 3 ) + x π ( 1 ) F ( [ y σ ( 2 ) , x π ( 2 ) ] ) x π ( 3 ) y σ ( 3 ) − x π ( 1 ) F ( [ y σ ( 2 ) , x π ( 2 ) x π ( 3 ) ] ) y σ ( 3 ) − F ( [ x π ( 1 ) x π ( 2 ) , y σ ( 2 ) ] ) y σ ( 3 ) x π ( 3 ) − x π ( 1 ) x π ( 2 ) F ( y σ ( 2 ) y σ ( 3 ) ) x π ( 3 ) + x π ( 1 ) x π ( 2 ) F ( y σ ( 2 ) ) y σ ( 3 ) x π ( 3 ) ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 y σ ( 2 ) ( p 4 ( x π ( 1 ) , x π ( 2 ) , y σ ( 1 ) ) y σ ( 3 ) x π ( 3 ) + p 4 ′ ( x π ( 1 ) , x π ( 2 ) , y σ ( 1 ) ) x π ( 3 ) y σ ( 3 ) ) + ∑ π ∈ S 3 ∑ σ ∈ S 3 y σ ( 3 ) ( x π ( 3 ) λ p ( x π ( 1 ) , x π ( 2 ) , y σ ( 1 ) , y σ ( 2 ) ) )

for all x ¯ 3 , y ¯ 3 ∈ ℒ 3 , p i , p i ′ , q i , q i ′ : ℒ 3 → R , i = 1 , 2 , 3 , 4 and λ p , λ p ′ , λ q , λ q ′ : ℒ 4 → C ( ℒ ) . Now by using the theory of functional identities and exposing x 2 from the left side, we obtain

0 = ∑ π ∈ S 2 ∑ σ ∈ S 3 p 2 ( x π ( 1 ) , y σ ( 1 ) , y σ ( 2 ) ) y σ ( 3 ) x π ( 3 ) + p 2 ′ ( x π ( 1 ) , y σ ( 1 ) , y σ ( 2 ) ) x π ( 3 ) y σ ( 3 )

for all x ¯ 3 , y ¯ 3 ∈ ℒ 3 and p 2 , p 2 ′ : ℒ 3 → R . Again by using the theory of functional identities and exposing everything from the right side, we obtain

0 = ∑ σ ∈ S 2 p 2 ( x π ( 1 ) , y σ ( 1 ) , y σ ( 2 ) )

and

0 = ∑ σ ∈ S 2 p 2 ′ ( x π ( 1 ) , y σ ( 1 ) , y σ ( 2 ) ) .

Therefore,

0 = 2 p 2 ( x , x , x ) = 2 p 2 ′ ( x , x , x )

for all x ∈ ℒ and p 2 , p 2 ′ : ℒ 3 → R . Equation (18) can now be rewritten as follows:

(20) 2 x F ( x 2 ) − 2 x F ( x ) x = n 1 ′ ( x , x ) x + n 2 ′ ( x , x ) x + n 3 ′ ( x , x ) x + λ n ′ ( x , x , x )

for all x ∈ ℒ , n i ′ : ℒ 2 → R , i = 1 , 2 , 3 and λ n ′ : ℒ 3 → C ( ℒ ) . Now using the theory of functional identities and exposing x 1 in (19) from the left side,

0 = ∑ π ∈ S 2 ∑ σ ∈ S 3 ( p 1 ( x π ( 2 ) , y σ ( 1 ) , y σ ( 2 ) ) y σ ( 3 ) x π ( 3 ) − F ( y σ ( 1 ) y σ ( 2 ) ) x π ( 2 ) y σ ( 3 ) x π ( 3 ) + y σ ( 1 ) F ( y σ ( 2 ) ) x π ( 2 ) y σ ( 3 ) x π ( 3 ) + p 1 ′ ( x π ( 2 ) , y σ ( 1 ) , y σ ( 2 ) ) x π ( 3 ) y σ ( 3 ) − y σ ( 1 ) F ( [ x π ( 2 ) x π ( 3 ) , y σ ( 2 ) ] ) y σ ( 3 ) + F ( y σ ( 1 ) y σ ( 2 ) ) x π ( 2 ) x π ( 3 ) y σ ( 3 ) − y σ ( 1 ) F ( y σ ( 2 ) ) x π ( 2 ) x π ( 3 ) y σ ( 3 ) + y σ ( 1 ) q 1 ( x π ( 3 ) , y σ ( 2 ) , y σ ( 3 ) ) x π ( 2 ) + y σ ( 1 ) q 2 ( x π ( 2 ) , y σ ( 2 ) , y σ ( 3 ) ) x π ( 3 ) + y σ ( 1 ) q 3 ( x π ( 2 ) , x π ( 3 ) , y σ ( 3 ) ) y σ ( 2 ) + y σ ( 1 ) q 4 ( x π ( 2 ) , x π ( 3 ) , y σ ( 2 ) ) y σ ( 3 ) + y σ ( 1 ) λ q ( x π ( 2 ) , x π ( 3 ) , y σ ( 2 ) , y σ ( 3 ) ) + y σ ( 1 ) x π ( 2 ) F ( y σ ( 2 ) y σ ( 3 ) ) x π ( 3 ) − y σ ( 1 ) x π ( 2 ) F ( y σ ( 2 ) ) y σ ( 3 ) x π ( 3 ) − F ( [ x π ( 2 ) , y σ ( 1 ) y σ ( 2 ) ] ) y σ ( 3 ) x π ( 3 )

+ y σ ( 1 ) F ( [ x π ( 2 ) , y σ ( 2 ) ] ) y σ ( 3 ) x π ( 3 ) − y σ ( 1 ) F ( [ x π ( 2 ) , y σ ( 2 ) y σ ( 3 ) ] ) x π ( 3 ) − F ( [ y σ ( 1 ) y σ ( 2 ) , x π ( 2 ) ] ) x π ( 3 ) y σ ( 3 ) + y σ ( 1 ) y σ ( 2 ) F ( x π ( 2 ) ) x π ( 3 ) y σ ( 3 ) − y σ ( 1 ) y σ ( 2 ) F ( x π ( 2 ) x π ( 3 ) ) y σ ( 3 ) )

for all x ¯ 3 , y ¯ 3 ∈ ℒ 3 and p i , p i ′ , q i , q i ′ : ℒ 3 → R , i = 1 , 2 , 3 , 4 and λ p , λ p ′ , λ q , λ q ′ : ℒ 4 → C ( ℒ ) . The aforementioned equation can now be rewritten as (everything exposing from the right side) follows:

0 = ∑ π ∈ S 2 ∑ σ ∈ S 3 ( y σ ( 1 ) q 1 ( x π ( 3 ) , y σ ( 2 ) , y σ ( 3 ) ) ) x π ( 2 ) + ∑ π ∈ S 2 ∑ σ ∈ S 3 ( y σ ( 1 ) q 2 ( x π ( 2 ) , y σ ( 2 ) , y σ ( 3 ) ) + p 1 ( x π ( 2 ) , y σ ( 1 ) , y σ ( 2 ) ) y σ ( 3 ) − F ( y σ ( 1 ) y σ ( 2 ) ) x π ( 2 ) y σ ( 3 ) + y σ ( 1 ) F ( y σ ( 2 ) ) x π ( 2 ) y σ ( 3 ) − F ( [ x π ( 2 ) , y σ ( 1 ) y σ ( 2 ) ] ) y σ ( 3 ) − y σ ( 1 ) x π ( 2 ) F ( y σ ( 2 ) ) y σ ( 3 ) + y σ ( 1 ) x π ( 2 ) F ( y σ ( 2 ) y σ ( 3 ) ) + y σ ( 1 ) F ( [ x π ( 2 ) , y σ ( 2 ) ] ) y σ ( 3 ) − y σ ( 1 ) F ( [ x π ( 2 ) , y σ ( 2 ) y σ ( 3 ) ] ) ) x π ( 3 ) + ∑ π ∈ S 2 ∑ σ ∈ S 3 ( λ q ( x π ( 2 ) , x π ( 3 ) , y σ ( 2 ) , y σ ( 3 ) ) ) y σ ( 1 ) + ∑ π ∈ S 2 ∑ σ ∈ S 3 ( y σ ( 1 ) q 3 ( x π ( 2 ) , x π ( 3 ) , y σ ( 3 ) ) ) y σ ( 2 ) + ∑ π ∈ S 2 ∑ σ ∈ S 3 ( p 1 ′ ( x π ( 2 ) , y σ ( 1 ) , y σ ( 2 ) ) x π ( 3 ) − F ( [ y σ ( 1 ) y σ ( 2 ) , x π ( 2 ) ] ) x π ( 3 ) − y σ ( 1 ) F ( [ x π ( 2 ) x π ( 3 ) , y σ ( 2 ) ] ) + F ( y σ ( 1 ) y σ ( 2 ) ) x π ( 2 ) x π ( 3 ) − y σ ( 1 ) F ( y σ ( 2 ) ) x π ( 2 ) x π ( 3 ) + y σ ( 1 ) q 4 ( x π ( 2 ) , x π ( 3 ) , y σ ( 2 ) ) + y σ ( 1 ) y σ ( 2 ) F ( x π ( 2 ) ) x π ( 3 ) − y σ ( 1 ) y σ ( 2 ) F ( x π ( 2 ) x π ( 3 ) ) ) y σ ( 3 )

for all x ¯ 3 , y ¯ 3 ∈ ℒ 3 and p i , p i ′ , q i , q i ′ : ℒ 3 → R , i = 1 , 2 , 3 , 4 and λ p , λ p ′ , λ q , λ q ′ : ℒ 4 → C ( ℒ ) . Now by using the theory of functional identities, we obtain from the aforementioned equation:

q 1 ( x , x , x ) = q 3 ( x , x , x ) = 0

for all x ∈ ℒ and q 1 , q 3 : ℒ 3 → R . Equation (17) can now be rewritten as follows:

(21) 2 F ( x 2 ) x − 2 x F ( x ) x = x m 1 ( x , x ) + x m 2 ( x , x ) + x m 3 ( x , x ) + λ m ( x , x , x )

for all x ∈ ℒ , m i : ℒ 2 → R , i = 1 , 2 , 3 and λ m : ℒ 3 → C ( ℒ ) . Since ℒ is 6-free, after a finite number of steps using equations (20) and (21), we arrive at

2 ∑ ( F ( x y ) − x F ( y ) ) = x f ( y ) + y g ( x ) + λ ( x , y )

2 ∑ ( F ( x ) y − F ( x y ) ) = h ( y ) x + k ( x ) y + μ ( x , y )

for all x , y ∈ R , f , g , h , k : ℒ → R and λ , μ : ℒ 2 → C ( ℒ ) . Therefore, we obtain

(22) 2 F ( x y ) + 2 F ( y x ) − 2 x F ( y ) − 2 y F ( x ) = x f ( y ) + y g ( x ) + λ ( x , y )

and

(23) 2 F ( x ) y + 2 F ( y ) x − 2 F ( x y ) − 2 F ( y x ) = h ( y ) x + k ( x ) y + μ ( x , y ) .

For all, x , y ∈ R , f , g , h , k : ℒ → R and λ , μ : ℒ 2 → C ( ℒ ) . Replacing the roles of denotations x and y in (22) and comparing so obtained identities leads to 0 = x f ( y ) + y g ( x ) − y f ( x ) − x g ( y ) + λ ( x , y ) − λ ( y , x ) , which yields f ( x ) = g ( x ) and λ ( x , y ) = λ ( y , x ) for all x , y ∈ ℒ , f , g : ℒ → R and λ : ℒ 2 → C ( ℒ ) . Putting x for y in (22) leads to

(24) 4 F ( x 2 ) = 4 x F ( x ) + 2 x f ( x ) + λ ( x , x ) .

Using the same arguments, it follows from (23) that h ( x ) = k ( x ) and μ ( x , y ) = μ ( y , x ) for all x , y ∈ ℒ , h , k : ℒ → R and μ : ℒ 2 → C ( ℒ ) . Therefore,

4 F ( x 2 ) = 4 F ( x ) x − 2 k ( x ) x − μ ( x , x ) .

Comparing the aforementioned relations gives

0 = x ( 4 F ( x ) + 2 f ( x ) ) + ( − 4 ( x ) + 2 k ( x ) ) x + λ ( x , x ) + μ ( x , x ) .

Hence, there exists r ∈ R and λ : ℒ → C ( ℒ ) such that

4 F ( x ) + 2 f ( x ) = r x + λ ( x ) .

Considering 2 f ( x ) = − 4 F ( x ) + r x + λ ( x ) in (24) gives

(25) 4 F ( x 2 ) = x r x + x λ ( x ) + λ ( x , x ) .

Replacing y for x and x for x 2 in (22) gives

4 F ( x 3 ) = 2 x 2 F ( x ) + 2 x F ( x 2 ) + x 2 f ( x ) + x f ( x 2 ) + λ ( x 2 , x ) .

Using (7) in the aforementioned relation leads to

4 F ( x 2 ) x − 4 x F ( x ) x + 4 x F ( x 2 ) = 2 x 2 F ( x ) + 2 x F ( x 2 ) + x 2 f ( x ) + x f ( x 2 ) + λ ( x 2 , x ) .

Using (24) in the aforementioned relation leads to

4 x f ( x ) x + 3 x λ ( x , x ) = 2 x f ( x 2 ) + 2 λ ( x 2 , x ) .

Considering 2 f ( x ) = − 4 F ( x ) + r x + λ ( x ) and using (25) in the aforementioned relation gives

0 = − 8 x F ( x ) x + x r x 2 + x 2 r x + 3 x 2 λ ( x ) + 4 x λ ( x , x ) − x λ ( x 2 ) − 2 λ ( x 2 , x ) .

The complete linearization of this relation and using the theory of functional identities leads to

0 = − 8 F ( x ) x + r x 2 + x r x + 3 x λ ( x ) + 4 λ ( x , x ) − λ ( x 2 )

and

0 = − 8 F ( x ) + r x + x r + 3 λ ( x ) .

Therefore,

(26) 8 F ( x ) = r x + x r + 3 λ ( x ) .

By substituting the aforementioned equation in (7), we obtain

0 = − 3 λ ( x 3 ) − 6 x λ ( x 2 ) − 3 x 2 λ ( x ) .

Since ℒ is a 6-free subset of R , the aforementioned identity implies λ ( x 3 ) = 0 , λ ( x 2 ) = 0 , λ ( x ) = 0 for all x ∈ R . From equation (26), we obtain

(27) 8 F ( x 2 ) = r x 2 + x 2 r .

Right (left) multiplication of the relation (26) by x gives, respectively,

(28) 8 F ( x ) x = r x 2 + x r x

and

(29) 8 x F ( x ) = x r x + x 2 r .

The relations (27)–(29) imply that the additive mapping F satisfies the relation

8 F ( x 2 ) = 8 F ( x ) x + 8 x F ( x ) − 2 x r x .

The aforementioned equation can now be rewritten as follows:

4 F ( x 2 ) = 4 F ( x ) x + 4 x F ( x ) − x r x

and

(30) 4 F ( x 2 ) = 4 F ( x ) x + 4 x F ( x ) − 2 x q x ,

where r = 2 q . Let us now introduce the mapping D : R → R by

(31) D ( x ) = 4 F ( x ) − q x − x q .

Obviously, the mapping D is additive. It is our aim to prove that D is a Jordan derivation. Putting x 2 for x in the aforementioned relation, we obtain

D ( x 2 ) = 4 F ( x 2 ) − q x 2 − x 2 q ,

which gives, after considering the relation (30), the relation

(32) D ( x 2 ) = 4 F ( x ) x + 4 x F ( x ) − 2 x q x − q x 2 − x 2 q .

Right (left) multiplication of the relation (31) by x gives, respectively,

(33) D ( x ) x = 4 F ( x ) x − q x 2 − x q x

and

(34) x D ( x ) = 4 x F ( x ) − x q x − x 2 q .

The relations (32)–(34) imply that the additive mapping D satisfies the relation

D ( x 2 ) = D ( x ) x + x D ( x )

for all x ∈ R . In other words, D is a Jordan derivation on R . According to Herstein theorem, one can conclude that D is a derivation, which completes the proof of the theorem.□

We are now in the position to prove Theorem 4.

Proof of Theorem 4

The complete linearization of (7) gives us (9). First, suppose that R is not a PI ring (satisfying the standard polynomial identity of degree less than 6). According to Theorem 5, then there exists q ∈ R such that 4 F ( x ) = D ( x ) + x q + q x for all x ∈ ℒ .

Assume now that R is a PI ring. It is well known that in this case R has a nonzero center (see [20]). Let c be a nonzero central element. Picking any x ∈ R and setting x 1 = x 2 = c x and x 3 = x in (9), we obtain

3 F ( c 2 x 3 ) = F ( c 2 x 2 ) x + 2 c F ( c x 2 ) x − c 2 x F ( x ) x − 2 c x F ( c x ) x + x F ( c 2 x 2 ) + 2 c x F ( c x 2 ) .

Next, setting x 1 = x 2 = c and x 3 = x 3 in (9), we obtain

3 F ( c 2 x 3 ) = F ( c 2 ) x 3 + 4 c F ( c x 3 ) − c x 3 F ( c ) − c F ( c ) x 3 − c 2 F ( x 2 ) x + c 2 x F ( x ) x − c 2 x F ( x 2 ) + x 3 F ( c 2 )

for all x ∈ R . Comparing both identities, we obtain

0 = F ( c 2 ) x 3 + 4 c F ( c x 3 ) − c x 3 F ( c ) − c F ( c ) x 3 − c 2 F ( x 2 ) x + c 2 x F ( x ) x − c 2 x F ( x 2 ) + x 3 F ( c 2 ) .

Next, setting x 1 = x 2 = x and x 3 = c 2 in (9), we obtain

(35) 3 F ( c 2 x 2 ) = 2 c 2 F ( x 2 ) + 2 F ( c 2 x ) x − c 2 F ( x ) x − x F ( c 2 ) x − c 2 x F ( x ) + 2 x F ( c 2 x ) .

Next, setting x 1 = x 2 = x and x 3 = c in (9), we obtain

(36) 3 F ( c x 2 ) = 2 c F ( x 2 ) + 2 F ( c x ) x − c F ( x ) x − x F ( c ) x − c x F ( x ) + 2 x F ( c x ) .

In case x = c , we arrive at F ( c 3 ) = 2 c F ( c 2 ) − c 2 F ( c ) . Next, setting x 1 = x 2 = c and x 3 = x in (9), we obtain

(37) 3 F ( c 2 x ) = F ( c 2 ) x + 4 c F ( c x ) − c x F ( c ) − c 2 F ( x ) − c F ( c ) x + x F ( c 2 ) .

Setting x 1 = x 2 = c and x 3 = x 2 in (9), we obtain

3 F ( c 2 x 2 ) = F ( c 2 ) x 2 + 4 c F ( c x 2 ) − c x 2 F ( c ) − c 2 F ( x 2 ) − c F ( c ) x 2 + x 2 F ( c 2 ) .

From the aforementioned equation and (36), we obtain

9 F ( c 2 x 2 ) = 3 F ( c 2 ) x 2 + 8 c 2 F ( x 2 ) + 8 c F ( c x ) x − 4 c 2 F ( x ) x − 4 c x F ( c ) x − 4 c 2 x F ( x ) + 8 c x F ( c x ) − 3 c x 2 F ( c ) − 3 c 2 F ( x 2 ) − 3 c F ( c ) x 2 + 3 x 2 F ( c 2 ) .

Comparing aforementioned equation with (35) and using (37), we obtain

(38) 0 = c 2 F ( x 2 ) − c 2 F ( x ) x − c 2 x F ( x ) − F ( c 2 ) x 2 − x 2 F ( c 2 ) + c F ( c ) x 2 + x F ( c 2 ) x + c x 2 F ( c ) .

Complete linearization of the the aforementioned equation and setting x 1 = c and x 2 = x , we obtain

0 = 2 c 2 F ( c x ) + c 2 F ( c ) x − 2 c 3 F ( x ) + c 2 x F ( c ) − c x F ( c 2 ) − c F ( c 2 ) x .

Setting c = c 2 in (38) we obtain

0 = c 4 F ( x 2 ) − c 4 F ( x ) x − c 4 x F ( x ) − F ( c 4 ) x 2 − x 2 F ( c 4 ) + c 2 F ( c 2 ) x 2 + x F ( c 4 ) x + c 2 x 2 F ( c 2 ) .

On other hand, multiplying equation (38) with c 2 , we obtain

0 = c 4 F ( x 2 ) − c 4 F ( x ) x − c 4 x F ( x ) − c 2 F ( c 2 ) x 2 − c 2 x 2 F ( c 2 ) + c 3 F ( c ) x 2 + c 2 x F ( c 2 ) x + c 3 x 2 F ( c ) .

Comparing the aforementioned two equations and using F ( c 4 ) = 3 c 2 F ( c 2 ) − 2 c 3 F ( c ) , we obtain

(39) 0 = − F ( c 2 ) x 2 − x 2 F ( c 2 ) + c F ( c ) x 2 + c x 2 F ( c ) + 2 x F ( c 2 ) x − 2 c x F ( c ) x .

On the other hand, adding the same two equations together, we obtain

0 = 2 c 2 F ( x 2 ) − 2 c 2 F ( x ) x − 2 c 2 x F ( x ) − 3 F ( c 2 ) x 2 − 3 x 2 F ( c 2 ) + 3 c F ( c ) x 2 + 3 c x 2 F ( c ) + 4 x F ( c 2 ) x − 2 c x F ( c ) x .

The aforementioned equation can be rewritten as follows:

0 = 2 c 2 F ( x 2 ) − 2 c 2 F ( x ) x − 2 c 2 x F ( x ) − 3 F ( c 2 ) x 2 − 3 x 2 F ( c 2 ) + 3 c F ( c ) x 2 + 3 c x 2 F ( c ) + 6 x F ( c 2 ) x − 2 x F ( c 2 ) x − 6 c x F ( c ) x + 4 c x F ( c ) x .

Using the aforementioned equation and (39), we obtain

0 = 2 c 2 F ( x 2 ) − 2 c 2 F ( x ) x − 2 c 2 x F ( x ) + 4 c x F ( c ) x − 2 x F ( c 2 ) x .

The aforementioned equation can now be rewritten as follows:

c 2 F ( x 2 ) = c 2 F ( x ) x + c 2 x F ( x ) + x F ( c 2 ) x − 2 c x F ( c ) x

and

4 c 2 F ( x 2 ) = 4 c 2 F ( x ) x + 4 c 2 x F ( x ) − 2 x ( 2 F ( c 2 ) − 4 c F ( c ) ) x .

Setting 2 F ( c 2 ) − 4 c F ( c ) = q , we arrive at

(40) 4 c 2 F ( x 2 ) = 4 c 2 F ( x ) x + 4 c 2 x F ( x ) − 2 x q x .

If q = 0 than F ( x ) is derivation, on the other hand, if q ≠ 0 , we can conclude as follows. Let us now introduce the mapping D : R → R by

(41) D ( x ) = 4 F ( x ) − q x − x q .

Obviously, the mapping D is additive. It is our aim to prove that D is a Jordan derivation. Putting x 2 for x in the aforementioned relation, we obtain

c 2 D ( x 2 ) = 4 c 2 F ( x 2 ) − c 2 q x 2 − c 2 x 2 q ,

which gives, after considering the relation (40), the relation

(42) c 2 D ( x 2 ) = 4 c 2 F ( x ) x + 4 c 2 x F ( x ) − 2 c 2 x q x − c 2 q x 2 − c 2 x 2 q .

Right (left) multiplication of the relation (41) by x gives, respectively,

(43) D ( x ) x = 4 F ( x ) x − q x 2 − x q x

and

(44) x D ( x ) = 4 x F ( x ) − x q x − x 2 q .

The relations (42)–(44) imply that the additive mapping D satisfies the relation

c 2 D ( x 2 ) = c 2 D ( x ) x + c 2 x D ( x )

for all x ∈ R , whence it follows D ( x 2 ) = D ( x ) x + x D ( x ) for all x ∈ R . In other words, D is a Jordan derivation on R . According to Herstein theorem, one can conclude that D is a derivation, which completes the proof of the theorem.□

Acknowledgements

The author wishes to thank the referees for their useful comments.

  1. Conflict of interest: The authors state no conflict of interest.

References

[1] K. I. Beidar, E. S. Martindale III, and A. V. Mikhalev, Rings with Generalized Identities, Marcel Dekker, Inc., New York, 1996. Search in Google Scholar

[2] I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104–1119. 10.1090/S0002-9939-1957-0095864-2Search in Google Scholar

[3] M. Brešar and J. Vukman, Jordan derivations on prime rings, Bull. Aust. Math. Soc. 37 (1988), 321–322. 10.1017/S0004972700026927Search in Google Scholar

[4] M. Brešar and J. Vukman, Jordan (θ,ϕ)-derivations, Glas. Mat. Ser. III 16 (1991), 13–17. Search in Google Scholar

[5] J. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc. 53 (1975), 321–324. 10.1090/S0002-9939-1975-0399182-5Search in Google Scholar

[6] M. Brešar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 104 (1988), 1003–1006. 10.1090/S0002-9939-1988-0929422-1Search in Google Scholar

[7] K. I. Beidar, M. Brešar, M. A. Chebotar, and W. S. Martindale III, On Herstein’s Lie map Conjectures II, J. Algebra 238 (2001), 239–264. 10.1006/jabr.2000.8628Search in Google Scholar

[8] M. Fošner, B. Marcen, and N. Širovnik, On certain identity related to Herstein theorem on Jordan derivations, Mediterr. J. Math. 13 (2016), 537–556. 10.1007/s00009-014-0512-0Search in Google Scholar

[9] M. Fošner, B. Marcen, and J. Vukman, A result in the spirit of Herstein theorem, Glas. Mat. Ser. III 53 (2018), 73–95. 10.3336/gm.53.1.06Search in Google Scholar

[10] M. Fošner, N. Širovnik, and J. Vukman, A result related to Herstein theorem, Bull. Malays. Math. Sci. Soc. 39 (2016), 885–899. 10.1007/s40840-015-0196-zSearch in Google Scholar

[11] J. Vukman, Some remarks on derivations in semiprime rings and standard operator algebras, Glas. Mat. Ser. III 46 (2011), 43–48. 10.3336/gm.46.1.07Search in Google Scholar

[12] M. Brešar, Jordan mappings of semiprime rings, J. Algebra 127 (1989), 218–228. 10.1016/0021-8693(89)90285-8Search in Google Scholar

[13] C. K. Liu and W. K. Shiue, Generalized Jordan triple (θ,ϕ)-derivations on semiprime rings, Taiwanese J. Math. 11 (2007), 1397–1406. 10.11650/twjm/1500404872Search in Google Scholar

[14] J. Vukman, I. Kosi-Ulbl, and D. Eremita, On certain equations in rings, Bull. Aust. Math. Soc. 71 (2005), 53–60. 10.1017/S0004972700038004Search in Google Scholar

[15] M. Fošner and J. Vukman, On some equations in prime rings, Monatsh. Math. 152 (2007), 135–150. 10.1007/s00605-007-0464-6Search in Google Scholar

[16] N. Peršin and J. Vukman, On certain functional equation arising from (m,n)-Jordan centralizers in prime rings, Glas. Mat. Ser. III 47 (2012), 119–132. 10.3336/gm.47.1.09Search in Google Scholar

[17] M. Brešar, Functional identities: A survey, Contemp. Math. 259 (2000), 93–109. 10.1090/conm/259/04089Search in Google Scholar

[18] M. Brešar, M. A. Chebotar, and W. S. Martindale III, Functional Identities, Birkhauser Verlag, P. O. BOX, CH-4010 Basel, Switzerland, 2007. 10.1007/978-3-7643-7796-0Search in Google Scholar

[19] K. I. Beidar and Y. Fong, On additive isomorphisms of prime rings preserving polynomials, J. Algebra 217 (1999), 650–667. 10.1006/jabr.1998.7833Search in Google Scholar

[20] P. Šemrl, Ring derivations on standard operator algebras, J. Funct. Anal. 112 (1993), 318–324. 10.1006/jfan.1993.1035Search in Google Scholar
Received: 2021-10-07
Accepted: 2022-01-10
Published Online: 2022-04-04

© 2022 Maja Fošner et al., published by De Gruyter

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

Articles in the same Issue

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
https://doi.org/10.1515/math-2022-0002
Keywords for this article
prime ring; derivation; Jordan derivation; two-sided centralizer; functional equation
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 22.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2022-0002/html
Scroll to top button