On a class of linear functional equations without range condition

Gselmann, Eszter; Kiss, Gergely; Vincze, Csaba

doi:10.1007/s00010-019-00672-7

Cited by 7 publications

(6 citation statements)

References 17 publications

(40 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In what follows we shall study (3). We start with the following result (see also [9] in which a function with values in an arbitrary field of characteristic different from two is considered).…”

Section: Solutions Of (3)mentioning

confidence: 99%

See 1 more Smart Citation

On a general bilinear functional equation

Bahyrycz

Sikorska

2021

Aequat. Math.

View full text Add to dashboard Cite

Let X, Y be linear spaces over a field $${\mathbb {K}}$$ K . Assume that $$f :X^2\rightarrow Y$$ f : X 2 → Y satisfies the general linear equation with respect to the first and with respect to the second variables, that is, for all $$x,x_i,y,y_i \in X$$ x , x i , y , y i ∈ X and with $$a_i,\,b_i \in {\mathbb {K}}{\setminus } \{0\}$$ a i , b i ∈ K \ { 0 } , $$A_i,\,B_i \in {\mathbb {K}}$$ A i , B i ∈ K ($$i \in \{1,2\}$$ i ∈ { 1 , 2 } ). It is easy to see that such a function satisfies the functional equation for all $$x_i,y_i \in X$$ x i , y i ∈ X ($$i \in \{1,2\}$$ i ∈ { 1 , 2 } ), where $$C_1:=A_1B_1$$ C 1 : = A 1 B 1 , $$C_2:=A_1B_2$$ C 2 : = A 1 B 2 , $$C_3:=A_2B_1$$ C 3 : = A 2 B 1 , $$C_4:=A_2B_2$$ C 4 : = A 2 B 2 . We describe the form of solutions and study relations between $$(*)$$ ( ∗ ) and $$(**)$$ ( ∗ ∗ ) .

show abstract

Section: Solutions Of (3)mentioning

confidence: 99%

“…Theorem 8, and so Theorem 9, give a sufficient condition for the existence of a non-constant solution of (3). Some approach for obtaining the necessary conditions in the case Y is a field is given in [9,Section 6]. Therefore it is worth finishing this section with formulating a problem.…”

Section: Example 2 Consider the Following Equationmentioning

confidence: 99%

On a general bilinear functional equation

Bahyrycz

Sikorska

2021

Aequat. Math.

View full text Add to dashboard Cite

show abstract

“…Therefore, the functions ϕ(x) := f (x, 0) − f (0, 0) for x ∈ X, and ψ(y) := f (0, y) − f (0, 0) for y ∈ X, satisfy the conditions (12) and…”

Section: Lemmamentioning

confidence: 99%

“…For the convenience of the reader we recall here a result describing the solutions of (1) (see [6], and also [12], where Y is an arbitrary field of characteristic different from two). Theorem 1.…”

Section: Introductionmentioning

confidence: 99%

On Stability of a General Bilinear Functional Equation

Bahyrycz

Sikorska

2021

Results Math

View full text Add to dashboard Cite

We prove the Hyers–Ulam stability of the functional equation $$\begin{aligned}&f(a_1x_1+a_2x_2,b_1y_1+b_2y_2)=C_{1}f(x_1,y_1)\nonumber \\ \nonumber \\&\quad +C_{2}f(x_1,y_2)+C_{3}f(x_2,y_1)+C_{4}f(x_2,y_2) \end{aligned}$$ f ( a 1 x 1 + a 2 x 2 , b 1 y 1 + b 2 y 2 ) = C 1 f ( x 1 , y 1 ) + C 2 f ( x 1 , y 2 ) + C 3 f ( x 2 , y 1 ) + C 4 f ( x 2 , y 2 ) in the class of functions from a real or complex linear space into a Banach space over the same field. We also study, using the fixed point method, the generalized stability of $$(*)$$ ( ∗ ) in the same class of functions. Our results generalize some known outcomes.

show abstract

“…Early studies of polynomial equations were made by Maurice Fréchet ( [14,15]), while the class above was already considered by W. Harold Wilson ([32]) in the first decades of the previous century. For recent investigations, we refer to the papers [20,[23][24][25]31] and the references therein. Solutions of (1), in the general case when X and Y are certain Abelian groups and the products p i x and q i y in the arguments of the unknown functions are replaced by homomorphisms of X, were determined by László Székelyhidi in [29] (cf., also, [30]).…”

Section: Introductionmentioning

confidence: 99%