AM  Vol.10 No.10 , October 2019
Approximation of Functions by Quadratic Mapping in (β, p)-Banach Space
ABSTRACT
In this paper, we study the functions with values in (β, p)-Banach spaces which can be approximated by a quadratic mapping with a given error.

1. Introduction

The stability problem of functional equations originated from a question of Ulam [1] in 1940 concerning the stability of group homomorphisms.

Give a group ( G 1 , ) and a metric group ( G 2 , , d ) with the metric d ( , ) . Given ε > 0 , does there exist a δ > 0 such that if f : G 1 G 2 satisfies d ( f ( x y ) , f ( x ) f ( y ) ) < δ for all x , y G 1 , then there is a homomorphism g : G 1 G 2 with d ( f ( x ) , g ( x ) ) < ε for all x G 1 ?

Hyers [2] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’s Theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias has provided a lot of influence in the development of what we call generalized Hyers-Ulam-Rassias stability of functional equations. Beginning around 1980, the stability problems of several functional equations and approximate homomorphisms have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [5] - [18] ).

The functional equation

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

is called the quadratic functional equation. Every solution of the quadratic functional equation is said to be a quadratic mapping. The Hyers-Ulam stability for quadratic functional equation was first proved by Skof [5] for mappings acting between a normed space and a Banach space. P. W. Cholewa [6] showed that Skof’s Theorem is also valid if the normed space is replaced with an abelian group.

Now we recall some basic facts concerning ( β , p ) -Banach spaces. We fixed real numbers β with 0 < β 1 and p with 0 < p 1 . Let K = or . Let X be linear space over K . A quasi-β-norm is a real-valued function on X satisfying the following conditions:

(i) x 0, x X ; x = 0 if and only if x = 0 ;

(ii) λ x = | λ | β x , x X , β K ;

(iii) There is a constant K 1 such that x + y K ( x + y ) , x , y X .

The pair ( X , ) is called a quasi-β-normed space if is a quasi-β-norm on X. The smallest possible K is called the module of concavity of . A quasi-β-Banach space is a complete quasi-β-normed space.

A quasi-β-norm is called a ( β , p ) -norm if x + y p x p + y p for all x X . In this case, a quasi- ( β , p ) -Banach space is called a ( β , p ) -Banach space. For more details and related stability results on ( β , p ) -Banach spaces, we refer to [19] [20] . Recently, L. Gǎvruta and P. Gǎvruta [21] studied the approximation of functions in Banach space. In this paper, we will consider this problem in ( β , p ) -Banach spaces and extend previous result for quadratic functional equations.

2. Main Results

Given 0 < β 1 and 0 < p 1 . Throughout this paper we always assume that X is a linear space, Y is a ( β , p ) -Banach space and f : X Y is a mapping.

Definition 2.1. Let f : X Y be a mapping. We say f is Φ-approximable by a quadratic map if there exists a quadratic mapping Q : X Y such that

f ( x ) Q ( x ) Φ ( x ) (1)

for all x X . In this case, we say that Q is the quadratic Φ-approximation of f.

The following result is our main result in this paper.

Theorem 2.2. Let V 1 = { Φ : X + : lim n 4 n β p Φ p ( 1 2 n x ) = 0, x X } and suppose Φ V 1 . Then f is Φ-approximable by a quadratic map if and only if the following two condition hold:

(i) lim n 4 n β p f ( 1 2 n x + 1 2 n y ) + f ( 1 2 n x 1 2 n y ) 2 f ( 1 2 n x ) 2 f ( 1 2 n y ) p = 0 , x , y X ;

(ii) There exists Ψ V 1 such that

f ( 1 2 n x ) 1 4 n f ( x ) p Ψ p ( 1 2 n x ) + 1 4 n β p Φ p ( x ) , x X .

In this case, the quadratic Φ-approximation of f is unique and is given by

Q ( x ) = lim n 4 n f ( 1 2 n x )

for all x X .

Proof. We first assume that f is Φ-approximable by a quadratic map. Then for x , y X , we have

f ( x + y ) Q ( x + y ) Φ ( x + y )

and

f ( x y ) Q ( x y ) Φ ( x y ) .

It follows that

f ( x + y ) + f ( x y ) 2 f ( x ) 2 f ( y ) p f ( x + y ) Q ( x + y ) p + f ( x y ) Q ( x y ) p + 2 f ( x ) 2 Q ( x ) p + 2 f ( y ) 2 Q ( y ) p Φ p ( x + y ) + Φ p ( x y ) + 2 β p Φ p ( x ) + 2 β p Φ p (y)

for all x , y X . Hence

4 n β p f ( 1 2 n x + 1 2 n y ) + f ( 1 2 n x 1 2 n y ) 2 f ( 1 2 n x ) 2 f ( 1 2 n y ) p 4 n β p Φ p ( 1 2 n x + 1 2 n y ) + 4 n β p Φ p ( 1 2 n x 1 2 n y ) + 4 n β p 2 β p Φ p ( 1 2 n x ) + 4 n β p 2 β p Φ p ( 1 2 n y )

for all x , y X . By letting n , we obtain condition (i) since Φ V 1 . Since Q is quadratic, we have

f ( 1 2 n x ) 1 4 n f ( x ) p f ( 1 2 n x ) Q ( 1 2 n x ) p + 1 4 n Q ( x ) 1 4 n f ( x ) p Φ p ( 1 2 n x ) + 1 4 n β p Φ p (x)

for all x X . We take Φ = Ψ V 1 in the first position, then for all x X , we have

f ( 1 2 n x ) 1 4 n f ( x ) p Ψ p ( 1 2 n x ) + 1 4 n β p Φ p (x)

and the condition (ii) holds.

Conversely we suppose that (i) and (ii) hold. It follows from condition (ii) that for all x X , we have

4 n f ( 1 2 n x ) f ( x ) p 4 n β p Ψ p ( 1 2 n x ) + Φ p ( x ) . (2)

Then { 4 n f ( 1 2 n x ) } is a Cauchy sequence. Indeed, by using 1 2 m x replace x, we get

4 n f ( 1 2 n + m x ) f ( 1 2 m x ) p 4 n β p Ψ p ( 1 2 n + m x ) + Φ p ( 1 2 m x ) ,

and by multipling 4 m β p , for all x X , we have

4 n + m f ( 1 2 n + m x ) 4 m f ( 1 2 m x ) p 4 ( n + m ) β p Ψ p ( 1 2 n + m x ) + 4 m Φ p ( 1 2 m x ) .

Hence, for all x X ,

4 n + m f ( 1 2 n + m x ) 4 m f ( 1 2 m x ) p 0

as m , n . Since Y is a ( β , p ) -Banach space, the limit Q ( x ) : = lim n 4 n f ( 1 2 n x ) exists. Let n in relation (2), we get condition (1).

Now we show that Q satisfies the required conditions. From the hypothesis, for all x , y X ,

lim n 4 n β p f ( 1 2 n x + 1 2 n y ) + f ( 1 2 n x 1 2 n y ) 2 f ( 1 2 n x ) 2 f ( 1 2 n y ) p = 0.

Hence for all x , y X ,

Q ( x + y ) + Q ( x y ) 2 Q ( x ) 2 Q ( y ) = 0.

Therefore

Q ( x + y ) + Q ( x y ) = 2 Q ( x ) + 2 Q (y)

and Q is a quadratic map. Now we show the uniqueness of Q. We suppose that Q satisfies

f ( x ) Q ( x ) Φ (x)

for all x X and there exists a Q satisfying

f ( x ) Q ( x ) Φ ( x ) .

Since Q and Q are quadratic mappings, we have

f ( 1 2 n x ) Q ( 1 2 n x ) = f ( 1 2 n x ) 1 4 n Q ( x ) Φ ( 1 2 n x )

for all x X . Hence for all x , y X ,

Q ( x ) Q ( x ) p Q ( x ) 4 n f ( 1 2 n x ) p + 4 n f ( 1 2 n x ) Q ( x ) p 2 4 n β p Φ p ( 1 2 n x ) .

Since Φ V 1 , for all x X , we have

Q ( x ) Q ( x ) p 2 lim n 4 n β p Φ p ( 1 2 n x ) = 0.

Hence for all x X , Q ( x ) = Q ( x ) . This completes the proof. ,

Corollary 2.3. Let φ : X × X [ 0, ) be a mapping satisfying

Φ 1 p ( x , y ) = n = 0 4 n β p φ p ( 1 2 n + 1 x , 1 2 n + 1 y ) <

and

lim n 4 n β p Φ p ( 1 2 n x ) = 0

for all x , y X where Φ ( x ) = Φ 1 ( x , x ) . Suppose f : X Y a function with f ( 0 ) = 0 and satisfying

f ( x + y ) + f ( x y ) 2 f ( x ) 2 f ( y ) p φ p ( x , y ) (3)

for all x , y X . Then there exists a unique quadratic function Q : X Y such that

f ( x ) Q ( x ) Φ ( x ) , x X

which is defined

Q ( x ) = lim n 4 n f ( 1 2 n x )

for all x X .

Proof. Replace x and y by 1 2 x in (3), we have

f ( x ) 4 f ( x 2 ) p φ p ( x 2 , x 2 ) .

Dividing by 4 β p , we have

1 4 f ( x ) f ( x 2 ) p 1 4 β p φ p ( x 2 , x 2 ) . (4)

Replacing x by 1 2 x in (4), we get

1 4 f ( x 2 ) f ( x 4 ) p 1 4 β p φ p ( x 4 , x 4 ) . (5)

Then we have

1 4 2 f ( x ) f ( 1 2 2 x ) p = 1 4 2 f ( x ) 1 4 f ( x 2 ) p + 1 4 f ( x 2 ) f ( 1 2 2 x ) p 1 4 2 β p φ p ( x 2 , x 2 ) + 1 4 β p φ p ( x 4 , x 4 ) = 1 4 2 β p [ φ p ( x 2 , x 2 ) + 4 β p φ p ( x 4 , x 4 ) ] 1 4 2 β p Φ p (x)

for all x X . We claim that

1 4 m f ( x ) f ( 1 2 m x ) p 1 4 m β p Φ p ( x ) . (6)

holds for all m 1 and x X . When m = 1 , this is obviously by (4). Suppose (6) holds when m = k , i.e. for all x X ,

1 4 k f ( x ) f ( 1 2 k x ) p 1 4 k β p Φ p ( x ) .

Then for m = k + 1 , we have

1 4 k + 1 f ( x ) f ( 1 2 k + 1 x ) p 1 4 k + 1 f ( x ) 1 4 k f ( x 2 ) p + 1 4 k f ( x 2 ) f ( 1 2 k + 1 x ) p 1 4 ( k + 1 ) β p [ φ p ( x 2 , x 2 ) + 4 β p Φ p ( x 2 ) ] 1 4 ( k + 1 ) β p Φ p (x)

for all x X . By induction, (6) is true for all m 1 and x X . Replacing ( x , y ) by ( 1 2 n x , 1 2 n y ) in (3) and multiplying both side by 4 n β p , we have

4 n β p f ( 1 2 n x + 1 2 n y ) + f ( 1 2 n x 1 2 n y ) 2 f ( 1 2 n x ) 2 f ( 1 2 n y ) p 4 n β p φ p ( 1 2 n x , 1 2 n y ) .

Since

Φ 1 p ( x , y ) = n = 0 4 n β p φ p ( 1 2 n + 1 x , 1 2 n + 1 y ) < ,

we have

lim n 4 n β p φ p ( 1 2 n + 1 x , 1 2 n + 1 y ) = 0

for all x , y X . Hence for all x , y X ,

lim n 4 n β p f ( 1 2 n x + 1 2 n y ) + f ( 1 2 n x 1 2 n y ) 2 f ( 1 2 n x ) 2 f ( 1 2 n y ) p = 0.

It follows from Theorem 2.2 (with Ψ = 0 there) that there exists a unique quadratic function Q such that

f ( x ) Q ( x ) Φ (x)

for all x X . ,

Theorem 2.4. Let V 2 = { Φ : X + : lim n 1 4 n β p Φ p ( 2 n x ) = 0 , x X } . Suppose Φ V 2 . Then f is Φ-approximable by a quadratic map if and only if the following two condition

(i) lim n 1 4 n β p f ( 2 n x + 2 n y ) + f ( 2 n x 2 n y ) 2 f ( 2 n x ) 2 f ( 2 n y ) p = 0 ;

(ii) There exists a Ψ V 2 such that

f ( 2 n x ) 4 n f ( x ) p Ψ p ( 2 n x ) + 4 n β p Φ p (x)

hold for all x , y X . In this case, the quadratic Φ-approximation of f is unique and is given by

Q ( x ) = lim n 1 4 n f ( 2 n x ) , x X .

Proof. The proof is similar to that of Theorem 2.2 and we omit it. ,

Corollary 2.5. Let φ : X × X [ 0, ) be a mapping such that

Φ 1 p ( x , y ) = n = 0 4 ( n + 1 ) β p φ p ( 2 n x , 2 n y ) <

for all x , y X . Let Φ ( x ) = Φ 1 ( x , x ) . Suppose lim n 1 4 n β p Φ p ( 2 n x ) = 0 all x X . Let f : X Y a function with f ( 0 ) = 0 and satisfying

f ( x + y ) + f ( x y ) 2 f ( x ) 2 f ( y ) p φ p ( x , y )

for all x , y X . Then there exists a unique quadratic function Q : X Y such that

f ( x ) Q ( x ) Φ (x)

for all x X .

Proof. The proof is similar to that of Corollary 2.3 and we omit it. ,

Funding

This article is partially supported by NSFC (11871303 and 11671133) and NSF of Shandong Province (ZR2019MA039).

Cite this paper
Chi, X. , Bao, L. and Wang, L. (2019) Approximation of Functions by Quadratic Mapping in (&beta;, p)-Banach Space. Applied Mathematics, 10, 817-825. doi: 10.4236/am.2019.1010058.
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