Home Mathematics Asymptotic behavior of Fréchet functional equation and some characterizations of inner product spaces
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Asymptotic behavior of Fréchet functional equation and some characterizations of inner product spaces

  • Choonkil Park , Mohammad Amin Tareeghee , Abbas Najati EMAIL logo , Yavar Khedmati Yengejeh and Siriluk Paokanta EMAIL logo
Published/Copyright: October 31, 2023
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 56 Issue 1

Abstract

This article presents the general solution f : G → V of the following functional equation:

f ( x ) − 4 f ( x + y ) + 6 f ( x + 2 y ) − 4 f ( x + 3 y ) + f ( x + 4 y ) = 0 , x , y ∈ G ,

where ( G , + ) is an abelian group and V is a linear space. We also investigate its Hyers-Ulam stability on some restricted domains. We apply the obtained results to present some asymptotic behaviors of this functional equation in the framework of normed spaces. Finally, we provide some characterizations of inner product spaces associated with the mentioned functional equation.

Keywords: Hyers-Ulam stability; quadratic function; cubic function; Fréchet functional equation; asymptotic behavior
MSC 2010: 39B82; 39B52; 39B62; 46C15

1 Introduction and preliminaries

The functional equation

(1) ∑ j = 0 n n j ( − 1 ) n − j g ( x + j y ) = 0 , x , y ∈ R ,

where g : R → R , is known as the Fréchet functional equation. For n = 1 , the Fréchet functional equation becomes g ( x + y ) = g ( x ) . Then, g is a constant function. In the case of n = 2 , the Fréchet functional equation appears as g ( x + 2 y ) + g ( x ) = 2 g ( x + y ) , which is the Jensen functional equation. In [1, Theorem 7.20], the general solution of (1) was obtained for n = 3 without assuming any regularity condition on g , where g is a function between two linear spaces. A result from Fréchet [2] states that if a continuous function g : R → R satisfies (1), then g is a polynomial of degree < n . Johnson [3] proved that a normed linear space X is an inner product space if and only if, for some integer n ⩾ 3 ,

∑ j = 0 n n j ( − 1 ) n − j ‖ x + j y ‖ 2 = 0 , x , y ∈ X .

It will be interesting to determine the general solutions of (1) without additional assumptions on g . In this article, we obtain the general solution of (1) for n = 4 without assuming any regularity condition on g : G → V , where ( G , + ) is an abelian group and V is a linear space. Indeed, we deal with the following functional equation:

(2) f ( x ) − 4 f ( x + y ) + 6 f ( x + 2 y ) − 4 f ( x + 3 y ) + f ( x + 4 y ) = 0 , x , y ∈ G .

We investigate the Hyers-Ulam stability of (2) on some restricted domains. The obtained results are used to present some asymptotic behaviors of the functional equation (2) in the framework of normed spaces. We also provide some characterizations of inner product spaces related to (2).

In 1940, Ulam [4] proposed the following question regarding the stability of homomorphisms between groups:

Let ( G 1 , ∗ ) be a group and let ( G 2 , ◇ , d ) be a metric group with a metric d . Given ε > 0 , does there exist a δ > 0 such that if a function f : G 1 → G 2 satisfies the inequality d ( f ( x ∗ y ) , f ( x ) ◇ f ( y ) ) < δ for all x , y ∈ G 1 , then there is a homomorphism h : G 1 → G 2 , with d ( f ( x ) , h ( x ) ) < ε for all x ∈ G 1 ?

One year later, Hyers [5] answered Ulam’s question for the case where G 1 and G 2 are assumed to be Banach spaces. It will also be interesting to study the stability problems of functional equations on restricted domains.

Skof [6] was the first author to study the Hyers-Ulam stability for additive functions on a restricted domain and applied the result to the study of an asymptotic behavior of additive functions. Jung [7] and Rassias [8] investigated the Hyers-Ulam stability for additive and quadratic mappings on restricted domains. The bounds and thus the stability results obtained in [7,8] were improved in [9].

Let us recall that a function f : G → V , where ( G , + ) is an abelian group and V is a linear space, is called

  1. additive, if f ( x + y ) = f ( x ) + f ( y ) for all x , y ∈ G ;

  2. quadratic, if f ( x + y ) + f ( x − y ) = 2 f ( x ) + 2 f ( y ) for all x , y ∈ G ;

  3. cubic, if f ( 2 x + y ) + f ( 2 x − y ) = 2 f ( x + y ) + 2 f ( x − y ) + 12 f ( x ) for all x , y ∈ G .

For more information on functional equations and the concept of Hyers-Ulam stability and its applications, we refer the reader to [1,10–21].

2 General solution of (2)

In this section, ( G , + ) is an abelian group and V denotes a linear space.

Lemma 2.1

Let f : G → V be a function satisfying

(3) f ( x − y ) + 2 f ( x + y ) − f ( y ) − 3 f ( x ) = 0 , x , y ∈ G .

Then, f is additive.

Proof

The functional equation (3) gives us f ( 0 ) = 0 by letting x = y = 0 . Putting x = 0 in (3), we infer that f is odd. Replacing y by − y in (3), we obtain

(4) f ( x + y ) + 2 f ( x − y ) + f ( y ) − 3 f ( x ) = 0 , x , y ∈ G .

Multiplying (3) by − 2 and adding the resultant to (4), we obtain f ( x + y ) = f ( x ) + f ( y ) for all x , y ∈ G . This completes the proof.□

Lemma 2.2

Let f : G → V be an odd function satisfying (2) for all x , y ∈ G . Then, the function g : G → V defined by g ( x ) = f ( 2 x ) − 8 f ( x ) is additive and the function h : G → V defined by h ( x ) = f ( 2 x ) − 2 f ( x ) is cubic.

Proof

First, we prove g is additive. Since f is odd, replacing x by 3 x − y and y by y − x in (2), we have

(5) f ( 3 x − y ) − 4 f ( 2 x ) + 6 f ( x + y ) − 4 f ( 2 y ) − f ( x − 3 y ) = 0 , x , y ∈ G .

Also, replacing x by − 4 x and y by x + y in (2), we have

(6) − f ( 4 x ) + 4 f ( 3 x − y ) − 6 f ( 2 x − 2 y ) + 4 f ( x − 3 y ) + f ( 4 y ) = 0 , x , y ∈ G .

Multiplying (5) by − 4 and adding the obtained equation to (6), we have

(7) 8 f ( x − 3 y ) − 6 f ( 2 x − 2 y ) − 24 f ( x + y ) − f ( 4 x ) + 16 f ( 2 x ) + f ( 4 y ) + 16 f ( 2 y ) = 0 ,

for all x , y ∈ G . For x = − y , equation (2) obtains us

f ( 3 y ) = 4 f ( 2 y ) − 5 f ( y ) , y ∈ G .

So, by considering x = 0 in (2), we obtain

(8) f ( 4 y ) = 10 f ( 2 y ) − 16 f ( y ) , y ∈ G .

By (8), equation (7) is equivalent to

(9) 8 f ( x − 3 y ) − 6 f ( 2 x − 2 y ) − 24 f ( x + y ) + 6 f ( 2 x ) + 16 f ( x ) + 26 f ( 2 y ) − 16 f ( y ) = 0 ,

for all x , y ∈ G . On the other hand, replacing x by x − 3 y in (2), we have

(10) f ( x − 3 y ) = 4 f ( x − 2 y ) − 6 f ( x − y ) + 4 f ( x ) − f ( x + y ) , x , y ∈ G .

By (9) and (10), we obtain

(11) 32 f ( x − 2 y ) − 48 f ( x − y ) − 32 f ( x + y ) − 6 f ( 2 x − 2 y ) + 6 f ( 2 x ) + 48 f ( x ) + 26 f ( 2 y ) − 16 f ( y ) = 0 , x , y ∈ G .

Also, replacing x by x − 2 y in (2), we have

(12) f ( x − 2 y ) = 4 f ( x − y ) − 6 f ( x ) + 4 f ( x + y ) − f ( x + 2 y ) , x , y ∈ G .

By (11) and (12), we obtain

(13) 80 f ( x − y ) − 6 f ( 2 x − 2 y ) + 96 f ( x + y ) − 32 f ( x + 2 y ) + 6 f ( 2 x ) − 144 f ( x ) + 26 f ( 2 y ) − 16 f ( y ) = 0 , x , y ∈ G .

Replacing x by 2 x and y by 2 y in (13) and applying (8), we have

(14) 8 f ( 2 x + 4 y ) = 5 f ( 2 x − 2 y ) + 24 f ( x − y ) + 24 f ( 2 x + 2 y ) − 21 f ( 2 x ) − 24 f ( x ) + 61 f ( 2 y ) − 104 f ( y ) , x , y ∈ G .

Since f is odd, replacing x by x − y and y by x + y in (9) obtains us

(15) 8 f ( 2 x + 4 y ) = 6 f ( 2 x − 2 y ) + 26 f ( 2 x + 2 y ) + 16 f ( x − y ) − 16 f ( x + y ) − 24 f ( 2 x ) + 6 f ( 4 y ) , x , y ∈ G .

By (14), (15), and (8), it is concluded

[ f ( 2 x − 2 y ) − 8 f ( x − y ) ] + 2 [ f ( 2 x + 2 y ) − 8 f ( x + y ) ] = [ f ( 2 y ) − 8 f ( y ) ] + 3 [ f ( 2 x ) − 8 f ( x ) ] , x , y ∈ G .

This means

g ( x − y ) + 2 g ( x + y ) − g ( y ) − 3 g ( x ) = 0 , x , y ∈ G .

So, by Lemma 2.1, the function g is additive.

Now, we show h is a cubic function. By interchanging x and y in (11), we have

(16) − 32 f ( 2 x − y ) = − 48 f ( x − y ) + 32 f ( x + y ) − 6 f ( 2 x − 2 y ) − 6 f ( 2 y ) − 48 f ( y ) − 26 f ( 2 x ) + 16 f ( x ) , x , y ∈ G .

Replacing x by 2 x and y by 2 y in (16) and multiplying the obtained equation by − 1 2 , we obtain

(17) 16 f ( 4 x − 2 y ) = 24 f ( 2 x − 2 y ) − 16 f ( 2 x + 2 y ) + 3 f ( 4 x − 4 y ) + 3 f ( 4 y ) + 24 f ( 2 y ) + 13 f ( 4 x ) − 8 f ( 2 x ) , x , y ∈ G .

Replacing y by − y in (16), we have

(18) − 32 f ( 2 x + y ) = − 48 f ( x + y ) + 32 f ( x − y ) − 6 f ( 2 x + 2 y ) + 6 f ( 2 y ) + 48 f ( y ) − 26 f ( 2 x ) + 16 f ( x ) , x , y ∈ G .

Also replacing y by − y in (17), we obtain

(19) 16 f ( 4 x + 2 y ) = 24 f ( 2 x + 2 y ) − 16 f ( 2 x − 2 y ) + 3 f ( 4 x + 4 y ) − 3 f ( 4 y ) − 24 f ( 2 y ) + 13 f ( 4 x ) − 8 f ( 2 x ) , x , y ∈ G .

Adding (16), (17), (18), and (19) yields

(20) 16 f ( 4 x + 2 y ) + 16 f ( 4 x − 2 y ) − 32 f ( 2 x + y ) − 32 f ( 2 x − y ) = 3 f ( 4 x + 4 y ) + 3 f ( 4 x − 4 y ) + 2 f ( 2 x + 2 y ) + 2 f ( 2 x − 2 y ) − 16 f ( x + y ) − 16 f ( x − y ) + 26 f ( 4 x ) − 68 f ( 2 x ) + 32 f ( x ) ,

for all x , y ∈ G . So, by (20) and (8), we obtain

[ f ( 4 x + 2 y ) − 2 f ( 2 x + y ) ] + [ f ( 4 x − 2 y ) − 2 f ( 2 x − y ) ] = [ 2 f ( 2 x + 2 y ) − 4 f ( x + y ) ] + [ 2 f ( 2 x − 2 y ) − 4 f ( x − y ) ] + [ 12 f ( 2 x ) − 24 f ( x ) ] , x , y ∈ G .

This means

h ( 2 x + y ) + h ( 2 x − y ) = 2 [ h ( x + y ) + h ( x − y ) ] + 12 h ( x ) , x , y ∈ G .

Therefore, h is a cubic function.□

Lemma 2.3

Let f : G → V be an even function satisfying (2) for all x , y ∈ G . If f ( 0 ) = 0 , then f is a quadratic function.

Proof

Since f is even and f ( 0 ) = 0 , replacing x by − 2 y in equation (2), we obtain

(21) f ( 2 y ) = 4 f ( y ) , y ∈ G .

Replacing x by x − 2 y in (2), we have

(22) f ( x − 2 y ) − 4 f ( x − y ) + 6 f ( x ) − 4 f ( x + y ) + f ( x + 2 y ) = 0 , x , y ∈ G .

Since f is even, replacing x by 2 y and y by x in (22) and applying (21), we obtain

(23) f ( x − y ) + 6 f ( y ) + f ( x + y ) = f ( x − 2 y ) + f ( x + 2 y ) , x , y ∈ G .

Adding (22) and (23), we infer that f ( x + y ) + f ( x − y ) = 2 f ( x ) + 2 f ( y ) , i.e., f is quadratic.□

Lemma 2.4

Each additive, quadratic, and cubic function satisfies equation (2).

Proof

For the case that a function is additive, the proof is obvious. Let Q : G → V be quadratic. We have

(24) 2 Q ( x ) + 2 Q ( y ) = Q ( x + y ) + Q ( x − y ) , x , y ∈ G .

and also

(25) Q ( x + 2 y ) + Q ( x − 2 y ) = 2 Q ( x ) + 8 Q ( y ) , x , y ∈ G .

Multiplying (24) by 4 and adding the resultant to (25), we have

(26) Q ( x − 2 y ) + 6 Q ( x ) + Q ( x + 2 y ) = 4 Q ( x − y ) + 4 Q ( x + y ) , x , y ∈ G .

Replacing x by x + 2 y in (26), we obtain Q that satisfies (2). Now, let C : G → V be cubic. Then, we have

(27) 2 C ( x + y ) + 2 C ( x − y ) + 12 C ( x ) = C ( 2 x + y ) + C ( 2 x − y ) , x , y ∈ G .

It is obvious that C ( 2 x ) = 8 C ( x ) for all x ∈ G . Replacing y by 2 y in (27), we obtain

(28) C ( 2 x + 2 y ) + C ( 2 x − 2 y ) = 2 C ( x + 2 y ) + 2 C ( x − 2 y ) + 12 C ( x ) , x , y ∈ G .

Using C ( 2 x ) = 8 C ( x ) in (28), we obtain

(29) C ( x − 2 y ) − 4 C ( x − y ) + 6 C ( x ) − 4 C ( x + y ) + C ( x + 2 y ) = 0 , x , y ∈ G .

Replacing x by x + 2 y in (29), the function C satisfies (2).□

Theorem 2.5

A function f : G → V satisfies (2) if and only if f has the form f = A + Q + C + f ( 0 ) , where A , Q , C : G → V are additive, quadratic, and cubic, respectively.

Proof

Let f o and f e be the odd and even parts of f , i.e.,

f o ( x ) = f ( x ) − f ( − x ) 2 , f e ( x ) = f ( x ) + f ( − x ) 2 , x ∈ G .

First, we assume that f satisfies (2). Then, f o and f e fulfill (2). Consider the functions g , h : G → V defined by:

g ( x ) = f o ( 2 x ) − 8 f o ( x ) , h ( x ) = f o ( 2 x ) − 2 f o ( x ) , x ∈ G .

Then, by Lemma 2.2, g is additive and h is cubic. Moreover, we have

f o ( x ) = 1 6 [ h ( x ) − g ( x ) ] , x ∈ G .

Also, f e − f ( 0 ) is quadratic by Lemma 2.3. Set

A = − 1 6 g , Q = f e − f ( 0 ) , and C = 1 6 h .

Hence, f = f o + f e = A + Q + C + f ( 0 ) .

Conversely, let f = A + Q + C + f ( 0 ) . Hence, by Lemma 2.4, f satisfies (2).□

3 Stability of (2) on some restricted domains

In this section, X and W denote linear normed spaces and Y is a Banach space. For convenience, we let

D f ( x , y ) ≔ f ( x ) − 4 f ( x + y ) + 6 f ( x + 2 y ) − 4 f ( x + 3 y ) + f ( x + 4 y ) ,

where f is a function between two linear spaces.

Theorem 3.1

Take ε ⩾ 0 and d > 0 . Consider an odd function f : X → Y satisfying one of the following conditions:

(30) ‖ D f ( x , y ) ‖ ⩽ ε , x , y ∈ X : ‖ x + y ‖ ⩾ d ;

(31) ‖ D f ( x , y ) ‖ ⩽ ε , x , y ∈ X : min { ‖ x ‖ , ‖ y ‖ } ⩾ d ;

(32) ‖ D f ( x , y ) ‖ ⩽ ε , x , y ∈ X : ‖ x ‖ ⩾ d ;

(33) ‖ D f ( x , y ) ‖ ⩽ ε , x , y ∈ X : ‖ y ‖ ⩾ d .

Then, there is a unique additive function A : X → Y such that we have

(34) ‖ f ( 2 x ) − 8 f ( x ) − A ( x ) ‖ ⩽ 59 6 ε , x ∈ X .

Proof

We assume that f is satisfying (30) (we have a similar argument if f satisfies (31), (32), or (33)) . Since f is odd, replacing x by − 3 y in (30), we have

(35) ‖ f ( 3 y ) − 4 f ( 2 y ) + 5 f ( y ) ‖ ⩽ ε , ‖ y ‖ ⩾ d .

Also, by putting x = − 4 y in (30), we obtain

(36) ‖ f ( 4 y ) − 4 f ( 3 y ) + 6 f ( 2 y ) − 4 f ( y ) ‖ ⩽ ε , ‖ y ‖ ⩾ d .

Multiplying (35) by 4 and adding the obtained inequality to (36), we conclude

(37) ‖ f ( 4 y ) − 10 f ( 2 y ) + 16 f ( y ) ‖ ⩽ 5 ε , ‖ y ‖ ⩾ d .

So, by considering g ( y ) = f ( 2 y ) − 8 f ( y ) for all y ∈ X , (37) is equivalent to

(38) ‖ g ( 2 y ) − 2 g ( y ) ‖ ⩽ 5 ε , ‖ y ‖ ⩾ d .

Replacing y by 2 n y , n ∈ N ∪ { 0 } , in (38), and multiplying the reached inequality by 2 − ( n + 1 ) , we have

(39) 1 2 n + 1 g ( 2 n + 1 y ) − 1 2 n g ( 2 n y ) ⩽ 5 ε 2 n + 1 , ‖ y ‖ ⩾ d .

By (39), for integer numbers n ≥ m ≥ 0 , we derive

(40) 1 2 n + 1 g ( 2 n + 1 y ) − 1 2 m g ( 2 m y ) ⩽ ∑ i = m n 5 ε 2 i + 1 , ‖ y ‖ ⩾ d .

From (40), we can see { 1 2 n g ( 2 n y ) } n ∈ N is a Cauchy sequence for all y ∈ Y . So, by the completeness of Y , it is convergent. Now, the function A : X → Y with A ( x ) = lim n → ∞ 1 2 n g ( 2 n x ) is well defined. By (30), we can obtain

(41) ‖ g ( x ) − 4 g ( x + y ) + 6 g ( x + 2 y ) − 4 g ( x + 3 y ) + g ( x + 4 y ) ‖ ⩽ 9 ε , ‖ x + y ‖ ⩾ d .

Replacing x and y by 2 n x and 2 n y in (41) and multiplying the obtained inequality by 2 − n , and then letting n → ∞ , we conclude that A satisfies (2) for all x , y with x + y ≠ 0 . It follows from (35) that

(42) ‖ g ( 3 y ) − 4 g ( 2 y ) + 5 g ( y ) ‖ ⩽ 9 ε , ‖ y ‖ ⩾ d .

Because A ( 0 ) = 0 , (42) yields A ( 3 y ) − 4 A ( 2 y ) + 5 A ( y ) = 0 for all y ∈ X . Since A is an odd function, the last equation implies

A ( x ) − 4 A ( 0 ) + 6 A ( − x ) − 4 A ( − 2 x ) + A ( − 3 x ) = 0 , x ∈ X .

This means that A satisfies (2) for x + y = 0 . Therefore, A satisfies (2) for all x , y ∈ X . So, the mapping x ↦ A ( 2 x ) − 8 A ( x ) is additive by Lemma 2.2. On the other hand, by the definition of A , we obtain A ( 2 x ) = 2 A ( x ) . Hence, A is additive. Letting m = 0 and n → ∞ in (40), we have

‖ g ( y ) − A ( y ) ‖ ⩽ 5 ε , ‖ y ‖ ⩾ d .

Therefore,

(43) ‖ A ( y − 2 x ) − g ( y − 2 x ) ‖ ⩽ 5 ε , ‖ y − 2 x ‖ ⩾ d ; ‖ 4 g ( 2 y − x ) − 4 A ( 2 y − x ) ‖ ⩽ 20 ε , ‖ 2 y − x ‖ ⩾ d ; ‖ 4 g ( x + 4 y ) − 4 A ( x + 4 y ) ‖ ⩽ 20 ε , ‖ x + 4 y ‖ ⩾ d ; ‖ A ( 2 x + 5 y ) − g ( 2 x + 5 y ) ‖ ⩽ 5 ε , ‖ 2 x + 5 y ‖ ⩾ d .

Replacing x by y − 2 x and y by x + y in (41), we obtain

(44) ‖ g ( y − 2 x ) − 4 g ( 2 y − x ) + 6 g ( 3 y ) − 4 g ( x + 4 y ) + g ( 2 x + 5 y ) ‖ ⩽ 9 ε , ‖ 2 y − x ‖ ⩾ d .

Let y ∈ X and choose x ∈ X such that ‖ x ‖ ⩾ d + 4 ‖ y ‖ . Then,

min { ‖ y − 2 x ‖ , ‖ 2 y − x ‖ , ‖ x + 4 y ‖ , ‖ 2 x + 5 y ‖ } ⩾ d .

Now, it follows from (43) and (44)

(45) ‖ 6 g ( 3 y ) + A ( y − 2 x ) − 4 A ( 2 y − x ) − 4 A ( x + 4 y ) + A ( 2 x + 5 y ) ‖ ⩽ 59 ε .

Since A is additive, (45) yields

‖ g ( 3 y ) − A ( 3 y ) ‖ ⩽ 59 6 ε .

Hence, we obtain (34). The uniqueness of A is a simple consequence of (34).□

Theorem 3.2

Assume that an odd function f : X → Y satisfies one of the conditions (30)–(33) for some ε ⩾ 0 and d > 0 . Then, there is a unique cubic function C : X → Y such that

(46) ‖ f ( 2 x ) − 2 f ( x ) − C ( x ) ‖ ⩽ 71 42 ε , x ∈ X .

Proof

Let f satisfy (30). As we shown in the proof of Theorem 3.1, f satisfies (37). Then,

(47) ‖ h ( 2 y ) − 8 h ( y ) ‖ ⩽ 5 ε , ‖ y ‖ ⩾ d ,

where h ( y ) = f ( 2 y ) − 2 f ( y ) . Replacing y by 2 n y , n ⩾ 0 , in (47) and multiplying the resultant inequality by 8 − ( n + 1 ) , we have

1 8 n + 1 h ( 2 n + 1 y ) − 1 8 n h ( 2 n y ) ⩽ 5 ε 8 n + 1 , ‖ y ‖ ⩾ d .

So, we derive

(48) 1 8 n + 1 h ( 2 n + 1 y ) − 1 8 m h ( 2 m y ) ⩽ ∑ i = m n 5 ε 8 i + 1 , ‖ y ‖ ⩾ d , n ⩾ m ⩾ 0 .

From (48), we can infer that { 1 8 n h ( 2 n y ) } n is a Cauchy sequence for all y ∈ Y , and then, it is convergent by completeness of Y . Now, the function C : X → Y given by C ( x ) = lim n → ∞ 1 8 n h ( 2 n x ) is well defined. By (30), we can obtain

(49) ‖ h ( x ) − 4 h ( x + y ) + 6 h ( x + 2 y ) − 4 h ( x + 3 y ) + h ( x + 4 y ) ‖ ⩽ 3 ε , ‖ x + y ‖ ⩾ d .

Replacing x and y by 2 n x and 2 n y in (49) and multiplying the obtained inequality by 8 − n , and then taking n → ∞ , we conclude that C satisfies (2) for x , y with x + y ≠ 0 . It follows from (35) that

(50) ‖ h ( 3 y ) − 4 h ( 2 y ) + 5 h ( y ) ‖ ⩽ 3 ε , ‖ y ‖ ⩾ d .

Because C ( 0 ) = 0 , (50) yields C ( 3 y ) − 4 C ( 2 y ) + 5 C ( y ) = 0 for all y ∈ X . Since C is an odd function, the last equation implies

C ( x ) − 4 C ( 0 ) + 6 C ( − x ) − 4 C ( − 2 x ) + C ( − 3 x ) = 0 , x ∈ X .

This means that C satisfies (2) for x + y = 0 . Therefore, C satisfies (2) for all x , y ∈ X . Since C is odd, we obtain the map x → C ( 2 x ) − 2 C ( x ) is a cubic function by Lemma 2.2. On the other hand, we have C ( 2 x ) = 8 C ( x ) by the definition of C . So C is cubic. Letting m = 0 and n → ∞ in (48), we obtain

‖ h ( y ) − C ( y ) ‖ ⩽ 5 7 ε , ‖ y ‖ ⩾ d .

Therefore,

(51) ‖ C ( y − 2 x ) − h ( y − 2 x ) ‖ ⩽ 5 7 ε , ‖ y − 2 x ‖ ⩾ d ; ‖ 4 h ( 2 y − x ) − 4 C ( 2 y − x ) ‖ ⩽ 20 7 ε , ‖ 2 y − x ‖ ⩾ d ; ‖ 4 h ( x + 4 y ) − 4 C ( x + 4 y ) ‖ ⩽ 20 7 ε , ‖ x + 4 y ‖ ⩾ d ; ‖ C ( 2 x + 5 y ) − h ( 2 x + 5 y ) ‖ ⩽ 5 7 ε , ‖ 2 x + 5 y ‖ ⩾ d .

Replacing x by y − 2 x and y by x + y in (49), we obtain

(52) ‖ h ( y − 2 x ) − 4 h ( 2 y − x ) + 6 h ( 3 y ) − 4 h ( x + 4 y ) + h ( 2 x + 5 y ) ‖ ⩽ 3 ε , ‖ 2 y − x ‖ ⩾ d .

Let y ∈ X and choose x ∈ X such that ‖ x ‖ ⩾ d + 4 ‖ y ‖ . Then,

min { ‖ y − 2 x ‖ , ‖ 2 y − x ‖ , ‖ x + 4 y ‖ , ‖ 2 x + 5 y ‖ } ⩾ d .

Now, it follows from (51) and (52)

(53) ‖ 6 h ( 3 y ) + C ( y − 2 x ) − 4 C ( 2 y − x ) − 4 C ( x + 4 y ) + C ( 2 x + 5 y ) ‖ ⩽ 71 7 ε .

Since C satisfies (2), it follows from (53) that

‖ h ( 3 y ) − C ( 3 y ) ‖ ⩽ 71 42 ε .

Hence, we obtain (46). The uniqueness of A is a simple consequence of (46).□

Theorem 3.3

Take ε ⩾ 0 , d > 0 and consider an even function f : X → Y satisfying one of the conditions (30)–(33). Then, there is a unique quadratic function Q : X → Y , such that

(54) ‖ f ( x ) − Q ( x ) − f ( 0 ) ‖ ⩽ 4 9 ε , x ∈ X .

Proof

Let f satisfy (30). Since f is an even function, replacing x by − 2 y in (30), we obtain

(55) ‖ f ( 2 y ) − 4 f ( y ) + 3 f ( 0 ) ‖ ⩽ ε 2 , ‖ y ‖ ⩾ d .

Similarly, replacing x by − 3 y in (30), we obtain

(56) ‖ f ( 3 y ) − 4 f ( 2 y ) + 7 f ( y ) − 4 f ( 0 ) ‖ ⩽ ε , ‖ y ‖ ⩾ d .

Replacing y by 2 n y , n ∈ N ∪ { 0 } , in (55) and multiplying the resultant inequality by 4 − ( n + 1 ) , we have

1 4 n + 1 f ( 2 n + 1 y ) − 1 4 n f ( 2 n y ) + 3 4 n + 1 f ( 0 ) ⩽ ε 2 2 n + 3 , ‖ y ‖ ⩾ d .

Then, for integers n ⩾ m ⩾ 0 , we obtain

(57) 1 4 n + 1 f ( 2 n + 1 y ) − 1 4 m f ( 2 m y ) + ∑ i = m n 3 4 i + 1 f ( 0 ) ⩽ ∑ i = m n ε 2 2 i + 3 , ‖ y ‖ ⩾ d .

It follows from (57) that the sequence { 1 4 n f ( 2 n y ) } n ∈ N is Cauchy for all y ∈ Y , and then it is convergent by the completeness of Y . Now, the function Q : X → Y given by Q ( x ) = lim n → ∞ 1 4 n f ( 2 n x ) is well defined. Replacing x and y by 2 n x and 2 n y in (30) and multiplying the resultant inequality by 4 − n , and taking the limit as n tends to ∞ , we infer that Q satisfies (2) for all x , y ∈ X with x + y ≠ 0 . It follows from (56) that Q ( 3 y ) − 4 Q ( 2 y ) + 7 Q ( y ) = 0 for all y ∈ X . Since Q ( 0 ) = 0 and Q is an even function, the last equation implies

Q ( x ) − 4 Q ( 0 ) + 6 Q ( − x ) − 4 Q ( − 2 x ) + Q ( − 3 x ) = 0 , x ∈ X .

This means Q satisfies (2) for x + y = 0 . Therefore, Q satisfies (2) for all x , y ∈ X . So, Q is quadratic by Lemma 2.3. Taking m = 0 and n → ∞ in (57), we obtain

‖ f ( y ) − Q ( y ) − f ( 0 ) ‖ ⩽ ε 6 , ‖ y ‖ ⩾ d .

Therefore,

(58) ‖ Q ( y − 2 x ) − f ( y − 2 x ) + f ( 0 ) ‖ ⩽ ε 6 , ‖ y − 2 x ‖ ⩾ d ; ‖ 4 f ( 2 y − x ) − 4 Q ( 2 y − x ) − 4 f ( 0 ) ‖ ⩽ 2 3 ε , ‖ 2 y − x ‖ ⩾ d ; ‖ 4 f ( x + 4 y ) − 4 Q ( x + 4 y ) − 4 f ( 0 ) ‖ ⩽ 2 3 ε , ‖ x + 4 y ‖ ⩾ d ; ‖ Q ( 2 x + 5 y ) − f ( 2 x + 5 y ) + f ( 0 ) ‖ ⩽ ε 6 , ‖ 2 x + 5 y ‖ ⩾ d .

Replacing x by y − 2 x and y by x + y in (30), we obtain

(59) ‖ f ( y − 2 x ) − 4 f ( 2 y − x ) + 6 f ( 3 y ) − 4 f ( x + 4 y ) + f ( 2 x + 5 y ) ‖ ⩽ ε , ‖ 2 y − x ‖ ⩾ d .

Let y ∈ X and choose x ∈ X such that ‖ x ‖ ⩾ d + 4 ‖ y ‖ . Then,

min { ‖ y − 2 x ‖ , ‖ 2 y − x ‖ , ‖ x + 4 y ‖ , ‖ 2 x + 5 y ‖ } ⩾ d .

Now, it follows from (58) and (59)

(60) ‖ 6 f ( 3 y ) + Q ( y − 2 x ) − 4 Q ( 2 y − x ) − 4 Q ( x + 4 y ) + Q ( 2 x + 5 y ) − 6 f ( 0 ) ‖ ⩽ 8 3 ε .

Since Q satisfies (2), it follows from (60) that

‖ f ( 3 y ) − Q ( 3 y ) − f ( 0 ) ‖ ⩽ 4 9 ε .

Hence, we obtain (54). The uniqueness of Q is a simple consequence of (54).□

Theorem 3.4

Take ε ⩾ 0 , d > 0 and consider a function f : X → Y satisfying one of the conditions (30)–(33). Then, there exist unique additive, quadratic, and cubic functions A , Q , C : X → Y such that

(61) ‖ f ( x ) − A ( x ) − Q ( x ) − C ( x ) − f ( 0 ) ‖ ⩽ 149 63 ε , x ∈ X .

Proof

We may assume that f satisfies (30). Let f e and f o be the even and odd parts of f . It is clear that f e and f o satisfy (30) for all x , y ∈ X with ‖ x + y ‖ ⩾ d . By Theorems 3.1, 3.2, and 3.3, we have additive, quadratic, and cubic functions A ˜ , Q , C ˜ : X → Y such that

‖ A ˜ ( x ) − f o ( 2 x ) + 8 f o ( x ) ‖ ⩽ 59 6 ε , ‖ f o ( 2 x ) − 2 f o ( x ) − C ˜ ( x ) ‖ ⩽ 71 42 ε , ‖ f e ( x ) − Q ( x ) − f ( 0 ) ‖ ⩽ 4 9 ε , x ∈ X .

Then,

f ( x ) + 1 6 A ˜ ( x ) − 1 6 C ˜ ( x ) − Q ( x ) − f ( 0 ) ⩽ 149 63 ε , x ∈ X .

So we obtain (61) by letting A ( x ) = − 1 6 A ˜ ( x ) and C ( x ) = 1 6 C ˜ ( x ) for all x ∈ X . To prove the uniqueness of A , Q , and C , let A ′ , Q ′ , C ′ : X → Y be additive, quadratic, and cubic functions, respectively, satisfying (61). Let φ = A − A ′ , θ = Q − Q ′ , and ψ = C − C ′ . We show φ = θ = ψ = 0 . By (61), we have

(62) ‖ φ ( x ) + θ ( x ) + ψ ( x ) ‖ ⩽ ‖ f ( x ) − A ′ ( x ) − Q ′ ( x ) − C ′ ( x ) − f ( 0 ) ‖ + ‖ A ( x ) + Q ( x ) + C ( x ) + f ( 0 ) − f ( x ) ‖ ⩽ 298 63 ε ,

for all x ∈ X . Therefore,

(63) lim n → ∞ 1 8 n ‖ φ ( 2 n x ) + θ ( 2 n x ) + ψ ( 2 n x ) ‖ = 0 , x ∈ X .

Since φ , ψ , and θ are additive, quadratic, and cubic, respectively, we have

φ ( 2 n x ) + θ ( 2 n x ) + ψ ( 2 n x ) = 2 n φ ( x ) + 4 n θ ( x ) + 8 n ψ ( x ) , x ∈ X .

So (63) obtains ψ ( x ) = 0 for all x ∈ X , and (62) yields

(64) ‖ φ ( x ) + θ ( x ) ‖ ⩽ 298 63 ε , x ∈ X .

Therefore,

lim n → ∞ 1 4 n ‖ φ ( 2 n x ) + θ ( 2 n x ) ‖ = 0 , x ∈ X .

This implies that θ ( x ) = 0 for all x ∈ X . Hence, by (64), we infer that the additive function φ is bounded, and this yields φ = 0 .□

Remark 3.5

Since

{ ( x , y ) ∈ X × X : ‖ x + y ‖ ⩾ 2 d } ⊆ { ( x , y ) ∈ X × X : ‖ x ‖ + ‖ y ‖ ⩾ 2 d } ⊆ { ( x , y ) ∈ X × X : max { ‖ x ‖ , ‖ y ‖ } ⩾ d } ,

the aforementioned results remain valid if the condition ‖ x + y ‖ ⩾ d in (30) is replaced by ‖ x ‖ + ‖ y ‖ ⩾ d or max { ‖ x ‖ , ‖ y ‖ } ⩾ d .

Now, we can prove the following corollary concerning an asymptotic property of the functional equation (2).

Corollary 3.6

Let f : X → W be a function. Then, the following statements are equivalent:

  1. lim ‖ x ‖ + ‖ y ‖ → ∞ D f ( x , y ) = 0 ;

  2. lim ‖ x + y ‖ → ∞ D f ( x , y ) = 0 ;

  3. lim min { ‖ x ‖ , ‖ y ‖ } → ∞ D f ( x , y ) = 0 ;

  4. lim max { ‖ x ‖ , ‖ y ‖ } → ∞ D f ( x , y ) = 0 ;

  5. lim ‖ x ‖ → ∞ D f ( x , y ) = 0 ;

  6. lim ‖ y ‖ → ∞ D f ( x , y ) = 0 ;

  7. f = A + Q + C + f ( 0 ) , where A , Q , C : X → W are additive, quadratic, and cubic, respectively.

Corollary 3.7

Let φ : X × X → [ 0 , + ∞ ) . A function f : X → W satisfies

D f ( x , y ) = 0 , x , y ∈ X

if one of the following conditions holds:

  1. lim min { ‖ x ‖ , ‖ y ‖ } → ∞ φ ( x , y ) = + ∞ , limsup min { ‖ x ‖ , ‖ y ‖ } → ∞ φ ( x , y ) ‖ D f ( x , y ) ‖ < ∞ ;

  2. lim max { ‖ x ‖ , ‖ y ‖ } → ∞ φ ( x , y ) = + ∞ , limsup max { ‖ x ‖ , ‖ y ‖ } → ∞ φ ( x , y ) ‖ D f ( x , y ) ‖ < ∞ ;

  3. lim ‖ x ‖ + ‖ y ‖ → ∞ φ ( x , y ) = + ∞ , limsup ‖ x ‖ + ‖ y ‖ → ∞ φ ( x , y ) ‖ D f ( x , y ) ‖ < ∞ ;

  4. lim ‖ x + y ‖ → ∞ φ ( x , y ) = + ∞ , limsup ‖ x + y ‖ → ∞ φ ( x , y ) ‖ D f ( x , y ) ‖ < ∞ ;

  5. lim ‖ x ‖ → ∞ φ ( x , y ) = + ∞ , limsup ‖ x ‖ → ∞ φ ( x , y ) ‖ D f ( x , y ) ‖ < ∞ ;

  6. lim ‖ y ‖ → ∞ φ ( x , y ) = + ∞ , limsup ‖ y ‖ → ∞ φ ( x , y ) ‖ D f ( x , y ) ‖ < ∞ .

Corollary 3.8

Let ε ⩾ 0 , p < 0 , and f : X → Y be a function satisfying

‖ D f ( x , y ) ‖ ⩽ ε ( ‖ x ‖ p + ‖ y ‖ p ) , x , y ∈ X ⧹ { 0 } .

Then, D f ( x , y ) = 0 for all x , y ∈ X .

Corollary 3.9

Let ε ⩾ 0 , min { p , q } < 0 and f : X → Y be a function satisfying

‖ D f ( x , y ) ‖ ⩽ ε ‖ x ‖ p ‖ y ‖ q , x , y ∈ X ⧹ { 0 } .

Then, D f ( x , y ) = 0 for all x , y ∈ X .

4 Some characterizations of inner product spaces

Jordan and von Neumann [22] established that in order to a normed linear space X to be an inner product space, it is necessary and sufficient that the following condition be satisfied:

(65) ‖ x + y ‖ 2 + ‖ x − y ‖ 2 = 2 ‖ x ‖ 2 + 2 ‖ y ‖ 2 , x , y ∈ X .

Some characterizations of inner product spaces could be found in [3,23–26].

Theorem 4.1

Let X ≠ { 0 } be a normed linear space such that

(66) ‖ x ‖ p + 6 ‖ x + 2 y ‖ q + ‖ x + 4 y ‖ r = 4 ‖ x + y ‖ α + 4 ‖ x + 3 y ‖ β , x , y ∈ X ,

for some real numbers p , q , r , α , β ∈ ( 0 , + ∞ ) . Then, X is an inner product space.

Proof

Letting x = − 2 y in (66), we obtain

(67) 2 p ‖ y ‖ p + 2 r ‖ y ‖ r = 4 ‖ y ‖ α + 4 ‖ y ‖ β , y ∈ X .

Choosing ‖ y ‖ = 1 , 2 , 1 2 , 1 4 in (67), we obtain

(68) 2 p + 2 r = 8 ;

(69) 4 p + 4 r = 4 ( 2 α + 2 β ) ;

(70) 2 1 − α + 2 1 − β = 1 ;

(71) 2 − p + 2 − r = 4 1 − α + 4 1 − β .

Let

t = 2 p , s = 2 r , 2 − α = z , 2 − β = w .

Then by (68), (69) and (70), we obtain

(72) 32 z w − t s z w = 1 .

On the other hand, (68), (70), and (71) yield

(73) t s − 8 t s z w = 8 .

By (73) and (72), one obtains t s = 256 z w . Then, (72) yields 16 z w = 1 , and so t s = 16 . Since t + s = 8 and 2 ( z + w ) = 1 , we conclude that t = s = 4 and z = w = 1 4 . Therefore, p = r = α = β = 2 .

Now, letting y = 0 and choosing ‖ x ‖ = 2 in (66), one obtains q = 2 . Define f : X → R by f ( x ) = ‖ x ‖ 2 . Then, (66) means that f satisfies (2) for all x , y ∈ X , and we conclude f is quadratic by Lemma 2.3. Thus, ‖ . ‖ fulfills (65). Hence, X is an inner product space.□

Corollary 4.2

A normed linear space X ≠ { 0 } is an inner product space if and only if

‖ x ‖ 2 + 6 ‖ x + 2 y ‖ 2 + ‖ x + 4 y ‖ 2 = 4 ‖ x + y ‖ 2 + 4 ‖ x + 3 y ‖ 2 , x , y ∈ X .

Theorem 4.3

Let X be a normed linear space and ϕ : [ 0 , + ∞ ) → R be continuous with ϕ ( 0 ) = 0 , ϕ ( 1 ) ≠ 0 and satisfy

(74) ϕ ( ‖ x ‖ ) + 6 ϕ ( ‖ x + 2 y ‖ ) + ϕ ( ‖ x + 4 y ‖ ) = 4 ϕ ( ‖ x + y ‖ ) + 4 ϕ ( ‖ x + 3 y ‖ ) , x , y ∈ X .

Then, X is an inner product space.

Proof

Define f : X → R by f ( x ) ≔ ϕ ( ‖ x ‖ ) . It is clear that f is even and fulfills (2). By Lemma 2.3, f is quadratic. Then,

ϕ ( r ‖ x ‖ ) = ϕ ( ‖ r x ‖ ) = f ( r x ) = r 2 f ( x ) = r 2 ϕ ( ‖ x ‖ ) ,

for all nonnegative rational r and all x ∈ X . Choosing x with ‖ x ‖ = 1 , we obtain ϕ ( r ) = r 2 ϕ ( 1 ) for all nonnegative rational r . Since ϕ is continuous, we infer that ϕ ( t ) = t 2 ϕ ( 1 ) for all t ⩾ 0 . So, (74) becomes

‖ x ‖ 2 + 6 ‖ x + 2 y ‖ 2 + ‖ x + 4 y ‖ 2 = 4 ‖ x + y ‖ 2 + 4 ‖ x + 3 y ‖ 2 , x , y ∈ X .

Hence, X is an inner product space by Corollary 4.2.□

5 Conclusion

We presented the general solutions of the Fréchet functional equation:

∑ j = 0 n n j ( − 1 ) n − j g ( x + j y ) = 0 ,

for functions g : G → V in the case of n = 4 , where ( G , + ) is an abelian group and V is a linear space. We also investigated its Hyers-Ulam stability on some restricted domains. The obtained results have been used to present some asymptotic behaviors of this functional equation in the framework of normed spaces. Finally, we provided some characterizations of inner product spaces associated with the mentioned functional equation.

Acknowledgment

The authors would like to thank the reviewers for their valuable suggestions and comments.

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

  2. Data availability statement: Not applicable.

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Received: 2022-11-14
Revised: 2023-05-07
Accepted: 2023-07-05
Published Online: 2023-10-31

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

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

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https://doi.org/10.1515/dema-2023-0265
Keywords for this article
Hyers-Ulam stability; quadratic function; cubic function; Fréchet functional equation; asymptotic behavior
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  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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