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A system of additive functional equations in complex Banach algebras

  • Siriluk Paokanta , Mehdi Dehghanian EMAIL logo , Choonkil Park EMAIL logo and Yamin Sayyari
Published/Copyright: February 13, 2023
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 56 Issue 1

Abstract

In this article, we solve the system of additive functional equations:

2 f ( x + y ) − g ( x ) = g ( y ) , g ( x + y ) − 2 f ( y − x ) = 4 f ( x )

and prove the Hyers-Ulam stability of the system of additive functional equations in complex Banach spaces. Furthermore, we prove the Hyers-Ulam stability of f -hom-ders in Banach algebras.

Keywords: additive mapping; f-hom-der; fixed point method; Hyers-Ulam stability; system of additive functional equations
MSC 2010: 47B47; 17B40; 39B72; 47H10

1 Introduction

Let ℬ be a complex Banach algebra and f : ℬ → ℬ be a C -linear mapping. Mirzavaziri and Moslehian [1] introduced the concept of f -derivation g : ℬ → ℬ as follows:

(1.1) g ( x y ) = f ( x ) g ( y ) + g ( x ) f ( y )

for all x , y ∈ ℬ .

Park et al. [2] introduced the concept of hom-derivation on ℬ , i.e., g : ℬ → ℬ is a homomorphism and f satisfies (1.1) for all x , y ∈ ℬ . Dehghanian et al. [3] introduced the concept of hom-der g : ℬ → ℬ as follows:

g ( x ) g ( y ) = x g ( y ) + g ( x ) y

for all x , y ∈ ℬ . Kheawborisuk et al. [4] defined and studied hom-ders in fuzzy Banach algebras.

Definition 1.1

Let ℬ be a complex Banach algebra and f : ℬ → ℬ be a homomorphism. A C -linear mapping g : ℬ → ℬ is called an f -hom-der if it satisfies

g ( x ) g ( y ) = f ( x ) g ( y ) + g ( x ) f ( y )

for all x , y ∈ ℬ .

Example 1.2

Let M n be the collection of all n × n complex matrices and g : M n → M n defined by g ( X ) = 2 X and f : M n → M n defined by f ( X ) = X . Then f is a homomorphism, and g is an f -hom-der.

We say that an equation is stable if any function satisfying the equation approximately is near to an exact solution of the equation.

The stability analysis of functional equations emanated from a question of Ulam [5], which was raised in 1940 about the stability of group homomorphisms and then was extended by Hyers [6]. Recently, results on the so-called Hyers-Ulam stability have comfortable the stability conditions. Many mathematicians developed the Hyers results in various directions [7,8,9, 10,11,12, 13,14].

The method provided by Hyers [6], which produces the additive function, will be called a direct method. This method is the most significant and strong tool to concerning the stability of different functional equations. That is, the exact solution of the functional equation is explicitly constructed as a limit of a sequence, starting from the given approximate solution [10,15]. The other significant method is fixed-point theorem, that is, the exact solutions of functional equations and differential equations are explicitly created as fixed points of some certain mappings [16,17,18, 19,20,21, 22,23,24, 25,26,27, 28,29,30, 31,32,33, 34,35].

We remember a fixed-point alternative theorem.

Theorem 1.3

[36] If ( ℬ , d ) is a complete generalized metric space and J : ℬ → ℬ is a strictly contractive mapping, that is,

d ( J u , J v ) ≤ L d ( u , v )

for all u , v ∈ ℬ and a Lipschitz constant L < 1 . Then for each given element u ∈ ℬ , either

d ( J n u , J n + 1 u ) = + ∞ , ∀ n ≥ 0 ,

or

d ( J n u , J n + 1 u ) < + ∞ , ∀ n ≥ n 0 ,

for some positive integer n 0 . Furthermore, if the second alternative holds, then:

  1. the sequence ( J n u ) is convergent to a fixed point v ∗ of J ;

  2. v ∗ is the unique fixed point of J in the set V ≔ { v ∈ ℬ , d ( J n 0 u , v ) < + ∞ } ;

  3. d ( v , v ∗ ) ≤ 1 1 − L d ( v , J v ) for all u , v ∈ V .

In this article, we consider the following system of additive functional equations

(1.2) 2 f ( x + y ) − g ( x ) = g ( y ) , g ( x + y ) − 2 f ( y − x ) = 4 f ( x )

for all x , y ∈ ℬ . The aim of the present article is to solve the system of additive functional equations and prove the Hyers-Ulam stability of f -hom-ders in complex Banach algebras by using the fixed point method.

Throughout this article, assume that ℬ is a complex Banach algebra.

2 Stability of system of additive functional equations

We solve and investigate the system of additive functional equations (1.2) in complex Banach algebras.

Lemma 2.1

Let f , g : ℬ → ℬ be mappings satisfying (1.2) for all x , y ∈ ℬ . Then the mappings f , g : ℬ → ℬ are additive.

Proof

By substituting x = y = 0 in (1.2), we obtain

f ( 0 ) = g ( 0 ) = 0 .

By substituting y = 0 in (1.2), we have

(2.1) 2 f ( x ) = g ( x ) , f ( − x ) = − f ( x )

for all x ∈ ℬ . So

g ( x + y ) = 2 f ( x + y ) = g ( x ) + g ( y )

for all x , y ∈ ℬ . Hence, the mapping g : ℬ → ℬ is additive, and thus by (2.1), the mapping f : ℬ → ℬ is additive.□

By using the fixed-point technique, we prove the Hyers-Ulam stability of the system of additive functional equations (1.2) in complex Banach algebras.

Theorem 2.2

Suppose that Δ : ℬ 2 → [ 0 , ∞ ) is a function such that there exists an L < 1 with

(2.2) Δ x 2 , y 2 ≤ L 2 Δ ( x , y )

for all x , y ∈ ℬ . Let f , g : ℬ → ℬ be mappings satisfying

(2.3) ‖ 2 f ( x + y ) − g ( x ) − g ( y ) ‖ ≤ Δ ( x , y ) , ‖ g ( x + y ) − 2 f ( y − x ) − 4 f ( x ) ‖ ≤ Δ ( x , y )

for all x , y ∈ ℬ . Then there exist unique additive mappings F , G : ℬ → ℬ such that

(2.4) ‖ F ( x ) − f ( x ) ‖ ≤ L 2 + L 4 ( 1 − L ) Δ ( x , x ) ,

(2.5) ‖ G ( x ) − g ( x ) ‖ ≤ L 2 + L 2 ( 1 − L ) Δ ( x , x )

for all x ∈ ℬ .

Proof

By substituting x = y = 0 in (2.3), we obtain

‖ 2 f ( 0 ) − 2 g ( 0 ) ‖ ≤ Δ ( 0 , 0 ) = 0 , ‖ g ( 0 ) − 6 f ( 0 ) ‖ ≤ Δ ( 0 , 0 ) = 0

and so f ( 0 ) = g ( 0 ) = 0 .

By substituting y = x in (2.3), we obtain

‖ 2 f ( 2 x ) − 2 g ( x ) ‖ ≤ Δ ( x , x ) , ‖ g ( 2 x ) − 4 f ( x ) ‖ ≤ Δ ( x , x )

and so

(2.6) g ( x ) − 4 g x 4 ≤ 2 Δ x 4 , x 4 + Δ x 2 , x 2 ≤ L 2 + L 2 Δ ( x , x ) , f ( x ) − 4 f x 4 ≤ Δ x 4 , x 4 + 1 2 Δ x 2 , x 2 ≤ L 2 + L 4 Δ ( x , x )

for all x ∈ ℬ .

Let Γ = { γ : ℬ → ℬ : γ ( 0 ) = 0 } . We define a generalized metric on Γ as follows: d : Γ × Γ ⟶ [ 0 , ∞ ] by

d ( δ , γ ) = inf { μ ∈ R + : ‖ δ ( x ) − γ ( x ) ‖ ≤ μ Δ ( x , x ) , ∀ x ∈ ℬ } ,

and we consider inf ∅ = + ∞ . Then d is a complete generalized metric on Γ (see [37]).

Now, we define the mapping J : ( Γ , d ) → ( Γ , d ) such that

J δ ( x ) ≔ 4 δ x 4

for all x ∈ ℬ .

Actually, let δ , γ ∈ ( Γ , d ) be given such that d ( δ , γ ) = μ . Then

‖ δ ( x ) − γ ( x ) ‖ ≤ μ Δ ( x , x )

for all x ∈ ℬ . Hence,

‖ J δ ( x ) − J γ ( x ) ‖ = 4 δ x 4 − 4 γ x 4 ≤ 4 μ Δ x 4 , x 4 ≤ L 2 μ Δ ( x , x )

for all x ∈ ℬ . It follows that d ( J δ ( x ) , J γ ( x ) ) ≤ L 2 μ . So

d ( J δ ( x ) , J γ ( x ) ) ≤ L 2 d ( δ , γ )

for all x ∈ ℬ and all δ , γ ∈ Γ .

It follows from (2.6) that d ( f , J f ) ≤ L 2 + L 4 and d ( g , J g ) ≤ L 2 + L 2 .

By using the fixed-point alternative we deduce the existence of a unique fixed point of J and a unique fixed point of J , that is, the existence of mappings F , G : ℬ → ℬ , respectively, such that

F ( x ) = 4 F x 4 , G ( x ) = 4 G x 4

with the following property: there exist μ , η ∈ ( 0 , ∞ ) satisfying

‖ f ( x ) − F ( x ) ‖ ≤ μ Δ ( x , x ) , ‖ g ( x ) − G ( x ) ‖ ≤ η Δ ( x , x )

for all x ∈ ℬ .

Since lim n → ∞ d ( J n f , F ) = 0 and lim n → ∞ d ( J n g , G ) = 0 ,

lim n → ∞ 4 n f x 4 n = F ( x ) , lim n → ∞ 4 n g x 4 n = G ( x )

for all x ∈ ℬ .

Next, d ( f , F ) ≤ 1 1 − L d ( f , J f ) and d ( g , G ) ≤ 1 1 − L d ( g , J g ) , which imply

‖ f ( x ) − F ( x ) ‖ ≤ L 2 + L 4 ( 1 − L ) Δ ( x , x ) , ‖ g ( x ) − G ( x ) ‖ ≤ L 2 + L 2 ( 1 − L ) Δ ( x , x )

for all x ∈ ℬ .

By using (2.2) and (2.3), we conclude that

‖ 2 F ( x + y ) − G ( x ) − G ( y ) ‖ = lim n → ∞ 4 n 2 f x + y 4 n + g x 4 n − g y 4 n ≤ lim n → ∞ 4 n Δ x 4 n , y 4 n ≤ lim n → ∞ L 2 n Δ ( x , y ) = 0

and

‖ G ( x + y ) − 2 F ( y − x ) − 4 F ( x ) ‖ = lim n → ∞ 4 n g x + y 4 n − 2 f y − x 4 n − 4 f x 4 n ≤ lim n → ∞ 4 n Δ x 4 n , y 4 n ≤ lim n → ∞ L 2 n Δ ( x , y ) = 0

for all x , y ∈ ℬ , since L < 1 . Hence,

2 F ( x + y ) − G ( x ) = G ( y ) , G ( x + y ) − 2 F ( y − x ) = 4 F ( x )

for all x , y ∈ ℬ . Therefore, by Lemma 2.1, the mappings F , G : ℬ → ℬ are additive.□

Corollary 2.3

Let f , g : ℬ → ℬ be mappings satisfying

‖ 2 f ( x + y ) − g ( x ) − g ( y ) ‖ ≤ ‖ x y ‖ , ‖ g ( x + y ) − 2 f ( y − x ) − 4 f ( x ) ‖ ≤ ‖ x y ‖

for all x , y ∈ ℬ . Then there exist unique additive mappings F , G : ℬ → ℬ such that

‖ F ( x ) − f ( x ) ‖ ≤ 3 8 ‖ x ‖ 2 , ‖ G ( x ) − g ( x ) ‖ ≤ 3 4 ‖ x ‖ 2

for all x ∈ ℬ .

Proof

The proof follows from Theorem 2.2 by taking L = 1 2 and Δ ( x , y ) = ‖ x y ‖ for all x , y ∈ ℬ .□

Corollary 2.4

Let p and θ be nonnegative real numbers with p > 1 and f , g : ℬ → ℬ be mappings satisfying

‖ 2 f ( x + y ) − g ( x ) − g ( y ) ‖ ≤ θ ( ‖ x ‖ p + ‖ y ‖ p ) , ‖ g ( x + y ) − 2 f ( y − x ) − 4 f ( x ) ‖ ≤ θ ( ‖ x ‖ p + ‖ y ‖ p )

for all x , y ∈ ℬ . Then there exist unique additive mappings F , G : ℬ → ℬ such that

‖ F ( x ) − f ( x ) ‖ ≤ 2 p + 2 2 p ( 2 p − 2 ) θ ‖ x ‖ p , ‖ G ( x ) − g ( x ) ‖ ≤ 2 ( 2 p + 2 ) 2 p ( 2 p − 2 ) θ ‖ x ‖ p

for all x ∈ ℬ .

Proof

The proof follows from Theorem 2.2 by taking L = 2 2 p and Δ ( x , y ) = θ ( ‖ x ‖ p + ‖ y ‖ p ) for all x , y ∈ ℬ .□

3 Stability of F -hom-ders in Banach algebras

In this section, by using the fixed-point technique, we prove the Hyers-Ulam stability of F -hom-ders in complex Banach algebras.

Lemma 3.1

[38] Let ℬ be a complex Banach algebra and ℱ : ℬ → ℬ be an additive mapping such that ℱ ( α x ) = α ℱ ( x ) for all α ∈ T 1 ≔ { ζ ∈ C : ∣ ζ ∣ = 1 } and all x ∈ ℬ . Then ℱ is C -linear.

Lemma 3.2

Let f , g : ℬ → ℬ be mappings satisfying

(3.1) 2 f ( λ ( x + y ) ) − g ( λ x ) = λ g ( y ) , g ( λ ( x + y ) ) − 2 f ( λ ( y − x ) ) = 4 λ f ( x )

for all x , y ∈ ℬ and all λ ∈ T 1 . Then the mappings f , g : ℬ → ℬ are C -linear.

Proof

If we substitute λ = 1 in (3.1), then f and g are additive by Lemma 2.1.

By substituting y = 0 in (3.1), we have

(3.2) 2 f ( λ x ) = λ g ( x ) , g ( λ x ) = 2 λ f ( x )

for all x ∈ ℬ and all λ ∈ T 1 , since the mappings f and g are additive. By substituting λ = 1 in (3.2), we obtain

2 f ( x ) = g ( x ) , g ( x ) = 2 f ( x ) ,

and so

2 f ( λ x ) = 2 λ f ( x ) , g ( λ x ) = λ g ( x )

for all x ∈ ℬ and all λ ∈ T 1 . So by Lemma 3.2, the mappings f and g are C -linear.□

Theorem 3.3

Suppose that Δ : ℬ 2 → [ 0 , ∞ ) is a function such that there exists an L < 1 with

(3.3) Δ ( x , y ) ≤ L 4 Δ ( 2 x , 2 y ) ≤ L 2 Δ ( 2 x , 2 y )

for all x , y ∈ ℬ . Let f , g : ℬ → ℬ be mappings satisfying

(3.4) ‖ 2 f ( λ ( x + y ) ) − g ( λ x ) − λ g ( y ) ‖ ≤ Δ ( x , y ) , ‖ g ( λ ( x + y ) ) − 2 f ( λ ( y − x ) ) − 4 λ f ( x ) ‖ ≤ Δ ( x , y ) ,

(3.5) ‖ f ( x y ) − f ( x ) f ( y ) ‖ ≤ Δ ( x , y ) ,

(3.6) ‖ g ( x ) g ( y ) − f ( x ) g ( y ) − g ( x ) f ( y ) ‖ ≤ Δ ( x , y )

for all x , y ∈ ℬ , and all λ ∈ T 1 . Then there exist unique mappings F , G : ℬ → ℬ satisfying (2.4) and (2.5) and F is a homomorphism and G is an F-hom-der.

Proof

Let λ = 1 in (3.4). Since

Δ ( x , y ) ≤ L 4 Δ ( 2 x , 2 y ) ≤ L 2 Δ ( 2 x , 2 y )

for all x , y ∈ ℬ , by Theorem 2.2, there exist unique mappings F , G : ℬ → ℬ satisfying (2.4) and (2.5), which are given by

lim n → ∞ 4 n f x 4 n = F ( x ) , lim n → ∞ 4 n g x 4 n = G ( x )

for all x ∈ ℬ .

By using (3.3), we conclude that

‖ 2 F ( λ ( x + y ) ) − G ( λ x ) − λ G ( y ) ‖ = lim n → ∞ 4 n 2 f λ ( x + y ) 4 n − g λ x 4 n − λ g y 4 n ≤ lim n → ∞ 4 n Δ x 4 n , y 4 n ≤ lim n → ∞ L 2 n Δ ( x , y ) = 0

and

‖ G ( λ ( x + y ) ) − 2 F ( λ ( y − x ) ) − 4 λ F ( x ) ‖ = lim n → ∞ 4 n g λ ( x + y ) 4 n − 2 f λ ( y − x ) 4 n − 4 λ f x 4 n ≤ lim n → ∞ 4 n Δ x 4 n , y 4 n ≤ lim n → ∞ L 2 n Δ ( x , y ) = 0

for all x , y ∈ ℬ , since L < 1 . Hence,

2 F ( λ ( x + y ) ) − G ( λ x ) = λ G ( y ) , G ( λ ( x + y ) ) − 2 F ( λ ( y − x ) ) = 4 λ F ( x )

for all x , y ∈ ℬ and all λ ∈ T 1 . Therefore, by Lemma 3.2, the mappings F , G : ℬ → ℬ are C -linear.

It follows from (3.5) that

‖ F ( x y ) − F ( x ) F ( y ) ‖ = lim n → ∞ 4 n f x y 4 n − f x 2 n f y 2 n ≤ lim n → ∞ 4 n Δ x 2 n , y 2 n ≤ lim n → ∞ L n Δ ( x , y ) = 0

for all x , y ∈ ℬ . So

F ( x y ) = F ( x ) F ( y )

for all x , y ∈ ℬ . Thus, F is a homomorphism.

It follows from (3.6) that

‖ G ( x ) G ( y ) − F ( x ) G ( y ) − G ( x ) F ( y ) ‖ = lim n → ∞ 4 n g x 2 n g y 2 n − f x 2 n g y 2 n − g x 2 n f y 2 n ≤ lim n → ∞ 4 n Δ x 2 n , y 2 n ≤ lim n → ∞ L n Δ ( x , y ) = 0

for all x , y ∈ ℬ . So

G ( x ) G ( y ) = F ( x ) G ( y ) + G ( x ) F ( y )

for all x , y ∈ ℬ . Thus, the C -linear mapping G is an F -hom-der.□

Corollary 3.4

Let p and q be nonnegative real numbers with p + q > 2 and f , g : ℬ → ℬ be mappings satisfying

‖ 2 f ( λ ( x + y ) ) − g ( λ x ) − λ g ( y ) ‖ ≤ ‖ x ‖ p ‖ y ‖ q , ‖ g ( λ ( x + y ) ) − 2 f ( λ ( y − x ) ) − 4 λ f ( x ) ‖ ≤ ‖ x ‖ p ‖ y ‖ q

and

‖ f ( x y ) − f ( x ) f ( y ) ‖ ≤ ‖ x ‖ p ‖ y ‖ q ,

‖ g ( x y ) − f ( x ) g ( y ) − g ( x ) f ( y ) ‖ ≤ ‖ x ‖ p ‖ y ‖ q

for all x , y ∈ ℬ and all λ ∈ T 1 . Then there exist unique mappings F , G : ℬ → ℬ such that F is a homomorphism and G is an F-hom-der and

‖ F ( x ) − f ( x ) ‖ ≤ 2 p + 1 + 4 2 p + q ( 2 p + q − 4 ) ‖ x ‖ p + q , ‖ G ( x ) − g ( x ) ‖ ≤ 2 ( 2 p + 1 + 4 ) 2 p + 1 ( 2 p + q − 4 ) ‖ x ‖ p + q

for all x ∈ ℬ .

Proof

The proof follows from Theorem 3.3 by taking Δ ( x , y ) = ‖ x ‖ p ‖ y ‖ q for all x , y ∈ ℬ . By choosing L = 2 2 − p − q , we obtain the desired result.□

Corollary 3.5

Let p and θ be nonnegative real numbers with p > 2 and f , g : ℬ → ℬ be mappings satisfying

‖ 2 f ( λ ( x + y ) ) − g ( λ x ) − λ g ( y ) ‖ ≤ θ ( ‖ x ‖ p + ‖ y ‖ p ) , ‖ g ( λ ( x + y ) ) − 2 f ( λ ( y − x ) ) − 4 λ f ( x ) ‖ ≤ θ ( ‖ x ‖ p + ‖ y ‖ p )

and

‖ f ( x y ) − f ( x ) f ( y ) ‖ ≤ θ ( ‖ x ‖ p + ‖ y ‖ p ) ,

‖ g ( x y ) − f ( x ) g ( y ) − g ( x ) f ( y ) ‖ ≤ θ ( ‖ x ‖ p + ‖ y ‖ p )

for all x , y ∈ ℬ and all λ ∈ T 1 . Then there exist unique mappings F , G : ℬ → ℬ such that F is a homomorphism and G is an F-hom-der and

‖ F ( x ) − f ( x ) ‖ ≤ 2 ( 2 p + 4 ) 2 p ( 2 p − 4 ) θ ‖ x ‖ p , ‖ G ( x ) − g ( x ) ‖ ≤ 4 ( 2 p + 4 ) 2 p ( 2 p − 4 ) θ ‖ x ‖ p

for all x ∈ ℬ .

Proof

The proof follows from Theorem 3.3 by taking Δ ( x , y ) = θ ( ‖ x ‖ p + ‖ y ‖ p ) for all x , y ∈ ℬ . By choosing L = 2 2 − p , we obtain the desired result.□

4 Conclusion

In this article, we introduced a new system of additive functional equations in complex Banach spaces and the concept of f -hom-der in complex Banach algebras and proved the Hyers-Ulam stability of the new system of additive functional equations in complex Banach spaces and the Hyers-Ulam stability of f -hom-ders in complex Banach algebras.

Acknowledgements

We would like to express our sincere gratitude to the anonymous referee for his/her helpful comments that will help to improve the quality of the manuscript.

  1. Funding information: The authors declare that there is no funding available for this article.

  2. Author contributions: The authors equally participated in conception of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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

  4. Human and animal rights: We would like to mention that this article does not contain any studies with animals and does not involve any studies over human being.

  5. Data availability statement: Not applicable.

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Received: 2022-07-14
Revised: 2022-08-19
Accepted: 2022-08-31
Published Online: 2023-02-13

© 2023 the author(s), published by De Gruyter

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

Articles in the same Issue

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  2. A novel class of bipolar soft separation axioms concerning crisp points
  3. Duality for convolution on subclasses of analytic functions and weighted integral operators
  4. Existence of a solution to an infinite system of weighted fractional integral equations of a function with respect to another function via a measure of noncompactness
  5. On the existence of nonnegative radial solutions for Dirichlet exterior problems on the Heisenberg group
  6. Hyers-Ulam stability of isometries on bounded domains-II
  7. Asymptotic study of Leray solution of 3D-Navier-Stokes equations with exponential damping
  8. Semi-Hyers-Ulam-Rassias stability for an integro-differential equation of order 𝓃
  9. Jordan triple (α,β)-higher ∗-derivations on semiprime rings
  10. The asymptotic behaviors of solutions for higher-order (m1, m2)-coupled Kirchhoff models with nonlinear strong damping
  11. Approximation of the image of the Lp ball under Hilbert-Schmidt integral operator
  12. Best proximity points in ℱ-metric spaces with applications
  13. Approximation spaces inspired by subset rough neighborhoods with applications
  14. A numerical Haar wavelet-finite difference hybrid method and its convergence for nonlinear hyperbolic partial differential equation
  15. A novel conservative numerical approximation scheme for the Rosenau-Kawahara equation
  16. Fekete-Szegö functional for a class of non-Bazilevic functions related to quasi-subordination
  17. On local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) -preinvexity involving local fractional integral operators with Mittag-Leffler kernel
  18. On some geometric results for generalized k-Bessel functions
  19. Convergence analysis of M-iteration for 𝒢-nonexpansive mappings with directed graphs applicable in image deblurring and signal recovering problems
  20. Some results of homogeneous expansions for a class of biholomorphic mappings defined on a Reinhardt domain in ℂn
  21. Graded weakly 1-absorbing primary ideals
  22. The existence and uniqueness of solutions to a functional equation arising in psychological learning theory
  23. Some aspects of the n-ary orthogonal and b(αn,βn)-best approximations of b(αn,βn)-hypermetric spaces over Banach algebras
  24. Numerical solution of a malignant invasion model using some finite difference methods
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https://doi.org/10.1515/dema-2022-0165
Keywords for this article
additive mapping; f-hom-der; fixed point method; Hyers-Ulam stability; system of additive functional equations
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BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. A novel class of bipolar soft separation axioms concerning crisp points
  3. Duality for convolution on subclasses of analytic functions and weighted integral operators
  4. Existence of a solution to an infinite system of weighted fractional integral equations of a function with respect to another function via a measure of noncompactness
  5. On the existence of nonnegative radial solutions for Dirichlet exterior problems on the Heisenberg group
  6. Hyers-Ulam stability of isometries on bounded domains-II
  7. Asymptotic study of Leray solution of 3D-Navier-Stokes equations with exponential damping
  8. Semi-Hyers-Ulam-Rassias stability for an integro-differential equation of order 𝓃
  9. Jordan triple (α,β)-higher ∗-derivations on semiprime rings
  10. The asymptotic behaviors of solutions for higher-order (m1, m2)-coupled Kirchhoff models with nonlinear strong damping
  11. Approximation of the image of the Lp ball under Hilbert-Schmidt integral operator
  12. Best proximity points in ℱ-metric spaces with applications
  13. Approximation spaces inspired by subset rough neighborhoods with applications
  14. A numerical Haar wavelet-finite difference hybrid method and its convergence for nonlinear hyperbolic partial differential equation
  15. A novel conservative numerical approximation scheme for the Rosenau-Kawahara equation
  16. Fekete-Szegö functional for a class of non-Bazilevic functions related to quasi-subordination
  17. On local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) -preinvexity involving local fractional integral operators with Mittag-Leffler kernel
  18. On some geometric results for generalized k-Bessel functions
  19. Convergence analysis of M-iteration for 𝒢-nonexpansive mappings with directed graphs applicable in image deblurring and signal recovering problems
  20. Some results of homogeneous expansions for a class of biholomorphic mappings defined on a Reinhardt domain in ℂn
  21. Graded weakly 1-absorbing primary ideals
  22. The existence and uniqueness of solutions to a functional equation arising in psychological learning theory
  23. Some aspects of the n-ary orthogonal and b(αn,βn)-best approximations of b(αn,βn)-hypermetric spaces over Banach algebras
  24. Numerical solution of a malignant invasion model using some finite difference methods
  25. Increasing property and logarithmic convexity of functions involving Dirichlet lambda function
  26. Feature fusion-based text information mining method for natural scenes
  27. Global optimum solutions for a system of (k, ψ)-Hilfer fractional differential equations: Best proximity point approach
  28. The study of solutions for several systems of PDDEs with two complex variables
  29. Regularity criteria via horizontal component of velocity for the Boussinesq equations in anisotropic Lorentz spaces
  30. Generalized Stević-Sharma operators from the minimal Möbius invariant space into Bloch-type spaces
  31. On initial value problem for elliptic equation on the plane under Caputo derivative
  32. A dimension expanded preconditioning technique for block two-by-two linear equations
  33. Asymptotic behavior of Fréchet functional equation and some characterizations of inner product spaces
  34. Small perturbations of critical nonlocal equations with variable exponents
  35. Dynamical property of hyperspace on uniform space
  36. Some notes on graded weakly 1-absorbing primary ideals
  37. On the problem of detecting source points acting on a fluid
  38. Integral transforms involving a generalized k-Bessel function
  39. Ruled real hypersurfaces in the complex hyperbolic quadric
  40. On the monotonic properties and oscillatory behavior of solutions of neutral differential equations
  41. Approximate multi-variable bi-Jensen-type mappings
  42. Mixed-type SP-iteration for asymptotically nonexpansive mappings in hyperbolic spaces
  43. On the equation fn + (f″)m ≡ 1
  44. Results on the modified degenerate Laplace-type integral associated with applications involving fractional kinetic equations
  45. Characterizations of entire solutions for the system of Fermat-type binomial and trinomial shift equations in ℂn#
  46. Commentary
  47. On I. Meghea and C. S. Stamin review article “Remarks on some variants of minimal point theorem and Ekeland variational principle with applications,” Demonstratio Mathematica 2022; 55: 354–379
  48. Special Issue on Fixed Point Theory and Applications to Various Differential/Integral Equations - Part II
  49. On Cauchy problem for pseudo-parabolic equation with Caputo-Fabrizio operator
  50. Fixed-point results for convex orbital operators
  51. Asymptotic stability of equilibria for difference equations via fixed points of enriched Prešić operators
  52. Asymptotic behavior of resolvents of equilibrium problems on complete geodesic spaces
  53. A system of additive functional equations in complex Banach algebras
  54. New inertial forward–backward algorithm for convex minimization with applications
  55. Uniqueness of solutions for a ψ-Hilfer fractional integral boundary value problem with the p-Laplacian operator
  56. Analysis of Cauchy problem with fractal-fractional differential operators
  57. Common best proximity points for a pair of mappings with certain dominating property
  58. Investigation of hybrid fractional q-integro-difference equations supplemented with nonlocal q-integral boundary conditions
  59. The structure of fuzzy fractals generated by an orbital fuzzy iterated function system
  60. On the structure of self-affine Jordan arcs in ℝ2
  61. Solvability for a system of Hadamard-type hybrid fractional differential inclusions
  62. Three solutions for discrete anisotropic Kirchhoff-type problems
  63. On split generalized equilibrium problem with multiple output sets and common fixed points problem
  64. Special Issue on Computational and Numerical Methods for Special Functions - Part II
  65. Sandwich-type results regarding Riemann-Liouville fractional integral of q-hypergeometric function
  66. Certain aspects of Nörlund ℐ-statistical convergence of sequences in neutrosophic normed spaces
  67. On completeness of weak eigenfunctions for multi-interval Sturm-Liouville equations with boundary-interface conditions
  68. Some identities on generalized harmonic numbers and generalized harmonic functions
  69. Study of degenerate derangement polynomials by λ-umbral calculus
  70. Normal ordering associated with λ-Stirling numbers in λ-shift algebra
  71. Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operators
  72. Compositions of positive integers with 2s and 3s
  73. Kinematic-geometry of a line trajectory and the invariants of the axodes
  74. Hahn Laplace transform and its applications
  75. Discrete complementary exponential and sine integral functions
  76. Special Issue on Recent Methods in Approximation Theory - Part II
  77. On the order of approximation by modified summation-integral-type operators based on two parameters
  78. Bernstein-type operators on elliptic domain and their interpolation properties
  79. A class of strongly convergent subgradient extragradient methods for solving quasimonotone variational inequalities
  80. Special Issue on Recent Advances in Fractional Calculus and Nonlinear Fractional Evaluation Equations - Part II
  81. Application of fractional quantum calculus on coupled hybrid differential systems within the sequential Caputo fractional q-derivatives
  82. On some conformable boundary value problems in the setting of a new generalized conformable fractional derivative
  83. A certain class of fractional difference equations with damping: Oscillatory properties
  84. Weighted Hermite-Hadamard inequalities for r-times differentiable preinvex functions for k-fractional integrals
  85. Special Issue on Recent Advances for Computational and Mathematical Methods in Scientific Problems - Part II
  86. The behavior of hidden bifurcation in 2D scroll via saturated function series controlled by a coefficient harmonic linearization method
  87. Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves
  88. Petri net analysis of a queueing inventory system with orbital search by the server
  89. Asymptotic stability of an epidemiological fractional reaction-diffusion model
  90. On the stability of a strongly stabilizing control for degenerate systems in Hilbert spaces
  91. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part I
  92. New conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in terms of generalized conformable fractional operators via majorization
  93. Pell-Lucas polynomials for numerical treatment of the nonlinear fractional-order Duffing equation
  94. Impacts of Brownian motion and fractional derivative on the solutions of the stochastic fractional Davey-Stewartson equations
  95. Some results on fractional Hahn difference boundary value problems
  96. Properties of a subclass of analytic functions defined by Riemann-Liouville fractional integral applied to convolution product of multiplier transformation and Ruscheweyh derivative
  97. Special Issue on Development of Fuzzy Sets and Their Extensions - Part I
  98. The cross-border e-commerce platform selection based on the probabilistic dual hesitant fuzzy generalized dice similarity measures
  99. Comparison of fuzzy and crisp decision matrices: An evaluation on PROBID and sPROBID multi-criteria decision-making methods
  100. Rejection and symmetric difference of bipolar picture fuzzy graph
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