Thiết lập tính siêu ổn định của phương trình hàm tuyến tính suy rộng nhiều biến trong không gian tựa chuẩn

l a , 1
 xL [0,1] is quasi-normed space with  2 p . 
 1. ||x || 0 if and only if x 0. p
34 
 Dong Thap University Journal of Science, Vol. 9, No. 5, 2020, 33-41 
 The following corollary is used to study the for all xU . 
hyperstability of generalized linear equations in 2. For every xU , if there exists a 
several variables in quasi-Banach spaces. positive real M such that 
 Corollary 1.4 (Dung and Hang, 2018, * (x ) () M ( n )( x )  (1.8) 
Corollary 2.2). Suppose that n 0
 1. U is a nonempty set, (Y ,||.||, ) is a then the fixed point of satisfying (1.7) 
quasi-Banach space, and :YYUU is a is unique. 
given function, Y U is the set of all mappings The following result is well-known and is 
from U to Y. usually called Aoki-Rolewicz theorem. 
 Theorem 1.5 (Maligranda, 2008, Theorem 
 2. There exist f1,,: fk U U and 
 1). Let (X ,||.||, ) be a quasi-normed space, 
LLU,,: such that for all , Y U 
 1 k p log 2 and |||.|||: X defined by 
and xU , 2 
 1
 nnp
 || (xx ) ( ) || p
 |||x ||| inf   || xi || : x x i : x i X , n 1
 ii 11
 k 
  Li( x ) ||()() f i ( x ) f i ( x ) ||. (1.2) for all xX . Then |||.||| is p -norm on X 
 i 1
 and 
 3. There exist  :U and :UY 1
 ||xxx || ||| ||| || ||, for all xX . 
such that for all xU , 2
 || ()x ()|| x  (). x (1.3) 2. Main results 
 In this section, we establish and prove 
 4. For every xU and  log 2, 
 2 some results on the hyperstability of the 
 generalized linear equations in several 
 * n 
 ():()()xx   (1.4) variables (1.1) in quasi-normed spaces. 
 n 0
 Theorem 2.1. Suppose that 
where  : UU defined by 
 1. , denote the fields of real or 
 k
 complex numbers and (X ,||.||XX , ) is a 
  ():()()x Lii x() f x (1.5) 
 i 1 quasi-normed space over field , (Y ,||.||YY , ) 
 is a quasi-Banach space over field and 
for all  :U and xU . 
 f: X Y is a given mapping. 
 Then we have, 2. n 2 and m are positive integers, 
 1. For every xU , the limit C 0, aij and Li are given 
 parameters for im 1, , , jn 1, , . 
 limn  (xx ) ( ) (1.6) 
 n 
 3. There exist im0 {1, , } and 
exists and the so difined function  :UY is j j {1,  , n } such that a 0, a 0. 
 12 ij01 ij02
a fixed point of satisfying 
 For all ii 0 ,  0, there is jn {1, , } 
  *
 ||() x  ()|| x 4  () x (1.7) satisfying aa  . 
 ij i0 j
 35 
Natural Sciences issue 
 4. There exists p 0 such that For a given large t , (kjj t b ) 0 and 
 m n n x 0, we set xj () k j t b j x , jn 1, , , and 
 p
 Li f  a ij x j C || x j || X (2.1) n
 i 1 j 1 j 1 write s()() t a k t b , im 1, , . Then 
 Y i ij j j
 j 1
 n n n
for all x1, , xn X \{0}. 
 st1( )  aktb 1j ( j j )  akt 1 j j  ab 1 j j 1 
 Then we have j 1 j 1 j 1
 and the inequality (2.1) takes the form 
 mn 
 L f a x 0 (2.2) mn
 i ij j pp
 ij 11 Li f() s i() t x C| k j t b j | || x || X .(2.3) 
 ij 11Y
for all x, , x X \{0}. 
 1 n From (2.3), we gain 
Proof. Without any loss of generality, we may 
 mn
assume that i 1 and ()a is the row pp
 0 11jn Lfx1 ( )  Lfstxi() i ( ) C | ktb j j | || x || X .
satisfying Condition (3). For im 1, , , let ij 21Y
 i Dividing the two sides of the above inequality 
 n
 n
denote the hyperplane atij j 0 in . For by | L1 |, we obtain 
 j 1
 m
kn 1, , , let ck, be the coordinate plane 
 Li
 1.f ( x ) f() si ( t ) x 
t 0 in . Then is the hyperplane i 2 L
k 1 1 Y
 n
 n
at1 jj 0. By the hypothesis on ()a11jn , it C pp
j 1 |kj t b j | || x || X , 
 || L1 j 1
follows that 1 i (im  2, , ) and 
 L C
 1, ck. So, we get set L : 1, L : i and C :. Then 
 1 i L || L
 nm 1 1
 1,\\ c k i . we can use (2.1) as the form 
 ki 12
 mn
Choose an element pp (2.4) 
 Lfstxi() i() fx( ) C | ktb j j | || x || X .
 nm ij 21Y
 (,,).kk1 n(\)\ 1 c , k i 
 ki 12 Since kk1, ,n 0 we have limkjj t b , 
 t 
Obviously, (,,)kk1  n satisfies n
 p
 for all jn 1, , . Define t: C | k j t b j | , 
 n 
 j 1
 ak 0
  1 jj so that 
 j 1
 lim t 0. (2.5) 
 kj 0, j 1,  , n t 
 n We can suppose that t is sufficiently large so 
 a k 0, i 2,  , m .
  ij j that 0 1. 
 j 1 t
 XX\{0} \{0}
 Define mapping TYYt : by 
 Keep the hypothesis on ()a11jn in mind, 
 n m
there exists bb,, such that ab 1. 
 1 n  1 jj Tt()() x  L i() s i t x 
 j 1 i 2
36 
 Dong Thap University Journal of Science, Vol. 9, No. 5, 2020, 33-41 
 X \{0} n
for all xX \{0} and  Y . We set m
 n m 2 p p (2.8) 
  t t(x )  Y | L i |.| s i ( t ) | t || x || X .
 p i 2
  t(xx ) t || || X (2.6) 
 Indeed, if n 0 , then (2.8) holds by (2.6). 
for all xX \{0}. The inequality (2.4) can be Suppose that (2.8) holds for nk , that is, 
 k
written as m
  k(x )  m 2 | L |.| s ( t ) | p || x || p . 
 ||T f () x f ()|| x (). x t t Y i i t X
 t Y t i 2
This proves that (1.3) is satisfied. We have 
 XX\{0} \{0}
 k 1
 Define mapping  t : by 
 tt ()x 
 m
  (x ) m 2 | L |  s ( t ) x (2.7) k
 t Y i() i t()  t t ()x 
 i 2
 k
 X \{0} mm
 m 22 m p p
for all xX \{0} and  . This proves 
 Y|L i |  Y | L i |.| s i ( t ) | t || s i ( t ) x || X
 ii 22 
that (2.7) has the form as (1.5), where Li is 
 mmk
replaced by  m 2 |L |. Furthermore, for all m 22 p m p p
 Yi Y|L i |.| s i ( t ) | .  Y | L i |.| s i ( t ) | t || x || X
 ii 22 
, Y X \{0} , xX \{0} and Remark 1.2 (3), 
 m k 1
we have m 2 p p
  Y|L i |.| s i ( t ) | t || x || X . 
 i 2
 ||Tt ( x ) T t ( x ) || Y 
 So, (2.8) holds for all n . 
 mm
 L s()() t x L s t x By using (2.8) with  log 2, we gain 
 i()() i i i 2Y
 ii 22Y *
 t ()x 
 m 
 ()()n  x
 [Li ( s i()() t x ) L i ( s i t x )]  tt
 i 2 Y n 0
 n
 m
 m  m 2 |L |.| s ( t ) | p || x || p . (2.9) 
 m 2 |L |.||  s ( t ) x  s ( t ) x || .  Y i i t X
 Y i()() i i Y ni 02 
 i 2
 n n n
It proves that (1.2) is satisfied when is sti()()  aktb ijj j  akt ijj  ab ijj and 
 j 1 j 1 j 1
replaced by n
 akij j 0 for all im 2, , , we have 
For all xX \{0} we have j 1
 nn
 tt ()x 
 lim |si ( t ) | lim a ij k j t a ij b j . 
 m tt 
 jj 11
  mp 2 |L |. || s ( t ) x ||
 Y i t i X m
 i 2 mp 2
 So, we gain limY |L i |.| s i ( t ) | 0. We 
 m t 
 m 2 p p i 2
  Y|L i |.| s i ( t ) | t || x || X . 
 i 2 choose a large positive integer t such that 
 m
By induction, we will show that for all mp 2
 Y|L i |.| s i ( t ) | 1. (2.10) 
xX \{0}, n i 2
 Then 
 37 
Natural Sciences issue 
 n
 m mn 
  mp 2 |L |.| s ( t ) | l
  Y i i LTi t f a ij x j
 ni 02 ij 11
 Y
 1 mnl
  (2.11) m 2 p p
 m   |L |.| s ( t ) | C || x || . 
 mp 2 Yii j X
 1 Y |L i |.| s i ( t ) | i 21j
 i 2
 We have 
 mn
 By using (2.9) and (2.11), we have l 1
 LTi t f() a ij x j 
 p ij 11
 Y
 * tX||x ||
 t ()x 
 m  mm n
 l
 mp 2 L L T f() s() t a x 
 1 Y |L i |.| s i ( t ) | i  k t k  ij j
 i 1 k 2 j 1
 i 2 Y
 m m n
for all xX \{0}. This proves that (1.4) l
 Lk  LT i t f()  a ij s k() t x j 
 k 2 i 1 j 1
is satisfied. Y
 According to Corollary 1.4, with a large m m n
  m 2 |L | LTl f a s ( t ) x 
positive integer t , there exists a fixed point Y k  i t()  ij k j
 k 2 i 1 j 1
 Y
 ft : X Y of Tt f t()() x f t x satisfying 
 m ml n
  m 22 m p p
 ||ftY ( x ) f ( x ) || Y|L k | Y  | L i |.| s i ( t ) |  C || s k ( t ) x j || X
 ki 2 2j 1
 *
 4 (x ) l
 t mm 
 m 22|L |.| s ()| t p . m ||.|()| L s t p
 4  ||x || p Y k k Y i i 
 tX (2.12) ki 22 
 m  n
 mp 2 p
 1 Y |L i |.| s i ( t ) | .Cx||jX || 
 i 2 j 1
 mnl 1
for all xX \{0}. Furthermore, by (1.6) m 2 p p
 Y|L i |.| s i ( t ) | C || x j || X . 
we obtain ij 21
 n
 ftt( x ) lim T f ( x ). (2.13) 
 n So, (2.14) holds for all r . 
 By using (2.10), we gain 
By induction, we will show that for all 
 mnr
xX \{0}, r m 2 p p
 limY |L i |.| s i ( t ) | C || x j || X 0. (2.15) 
 r 
 ij 21
 mn 
 r From (2.13), (2.14), (2.15), Remark 1.2 (2) and 
 LTi t f a ij x j
 ij 11 Theorem 1.5, we obtain 
 Y
 mn
 mnr 
 m 2 p p
 Li f t a ij x j 
 Y|L i |.| s i ( t ) | C || x j || X . (2.14) 
  ij 11
 ij 21 Y
 mn
Indeed, if r 0 , then (2.14) holds by (2.1). r 1 
 lim LTi t f a ij x j 
Suppose that (2.14) holds for rl , that is, r 
 ij 11
 Y
38 
 Dong Thap University Journal of Science, Vol. 9, No. 5, 2020, 33-41 
 mn 2. n 2 and m are positive integers, 
 r 1 
 lim LTi t f a ij x j C 0, a and L are given 
 r  ij i
 ij 11 
 Y parameters for im 1, , , jn 1, , . 
 mn
 r 1 
 3. There exist im0 {1, , } and 
 lim LTi t f a ij x j 
 r 
 ij 11 j j {1,  , n } such that a 0, a 0. 
 Y 12 ij01 ij02
 mnr
 m 2 p p For all ii 0 ,  0, there is jn {1, , } 
 lim Y |L i |.| s i ( t ) | C || x j || X 
 r  satisfying aa  . 
 ij 21 ij i0 j
 0. 
 4. There exists pp,, such that 
It means that 1 n
 mn pp1 n 0 and 
 Li f t a ij x j 0. (2.16) 
  mn n
 ij 11 p j
 Li f a ij x j C|| x j || X 
 ij 11 j 1
So, ft satisfies (2.2) with a large positive Y
integer t . Letting t in (2.12) and using 
 for all x, , x X \{0}. 
(2.5), we gain 1 n
  Then we have 
 lim ||ftY ( x ) f ( x ) || 
 t mn 
 p L f a x 0 
 4 tX ||x || i ij j
 lim  ij 11 
 t m
 1  mp 2 |L |.| s ( t ) |
 Y i i for all x, , x X \{0}. 
 i 2 1 n
 0. n
 p j
 Proof. Set t:| C k j t b j | . Then 
It follows that lim ||ftY ( x ) f ( x ) || 0. So j 1
 t 
 n
 limft ( x ) f ( x ). (2.17) 
 t lim t 0 since p j 0. The proof of 
 t 
 j 1
Letting t in (2.16) and using (2.17), we Theorem 2.2 is now the same as the that of 
 mn Theorem 2.1. 
gain Li f a ij x j 0 for all 
 ij 11 We apply the established result to prove 
 some results of Zhang (2015). 
x1, , xn X \{0}. So, f satisfies (2.2). 
 Corollary 2.3 (Zhang, 2015, Theorem 
 We continue to present an extension of 
 1.6). Suppose that 
(Zhang, 2015, Theorem 1.7) from normed 
spaces to quasi-normed spaces. 1. denote the fields of real or 
 Theorem 2.2. Suppose that complex numbers and (X ,||.||X ) is a normed 
 space over field , (Y ,||.|| ) is a Banach 
 1. , denote the fields of real or Y
complex numbers and (X ,||.|| , ) is a space over field and f: X Y is a 
 XX given mapping. 
quasi-normed space over field , (Y ,||.||YY , ) 
is a quasi-Banach space over field and 2. and are positive integers, 
f: X Y is a given mapping. and are given 
 parameters for 
 39 
Natural Sciences issue 
 3. There exist im0 {1, , } and gain AAi  1. Furthermore, f satisfies (2.1) 
j j {1,  , n } such that a 0, a 0. 
 12 ij01 ij02 for all x, y X \{0}, L1 1, LA2 and 
For all ii 0 ,  0, there is jn {1, , } 
 LB3 . So all assumptions of Theorem 2.1 
satisfying aa  . 
 ij i0 j are satisfied. Then Proposition 2.4 follows from 
 4. There exists p 0 such that Theorem 2.1. 
 m n n
 p Proposition 2.5. Let denote the fields of 
 Li f  a ij x j C || x j || X 
 i 1 j 1 j 1 real or complex numbers, is a 
 Y
 quasi-normed space over field 
for all x1, , xn X \{0}. is a quasi-Banach space ab, \{0}, 
 Then and let satisfy 
 mn n
 n i i
 ( 1)C f ( ix y ) n ! f ( x ) 
 Li f a ij x j 0  n
 ij 11 i 1 Y
 pp
 for all x1, , xn X \{0}. c(|| x ||XX || y || ) 
Proof. The normed spaces are the quasi- for all Then satisfies the equation 
normed spaces when  1. So, all n
 ( 1)n iC i f ( ix y ) n ! f ( x ) 0 
assumptions of Theorem 2.1 are satisfied. Then  n
 i 1
Corollary 2.3 follows from Theorem 2.1. 
 for all x, y X \{0}. 
 We continue to apply established results 
to some special cases. The next is an extension Proof. We set A1 : (1;1), Aii : ( ,1) for all 
of the result of Zhang (2015) from the 
 in {2, , } and An 1 : (1,0). For all 
normed spaces to the quasi-normed spaces. 
 i {2,n 1} and  , we gain AAi  1. 
Proposition 2.4. Let , denote the fields of Furthermore, f satisfies (2.1) for all 
 n 0
real or complex numbers, (X ,||.||XX , ) is a 
 x, y X \{0} and LLL12,,, n are ( 1) Cn , 
quasi-normed space over field , (Y ,||.||YY , ) n 11
 (  1)Cn , ,1, respectively, and Lnn 1 !. 
is a quasi-Banach space , ab, \{0}, So all assumptions of Theorem 2.1 are 
AB,, c 0, p 0 and let f: X Y satisfy satisfied. Then Proposition 2.5 follows from 
 Theorem 2.1. 
 ||f ( ax by ) Af ( x ) Bf ( y ) ||Y 
 Proposition 2.6. Let denote the fields of 
 c(|| x ||pp || y || ) 
 XX real or complex numbers, is a 
for all x, y X \{0}. Then f satisfies the equation quasi-normed space over field 
 f( ax by ) Af ( x ) Bf ( y ) 0 is a quasi-Banach space 
 and let satisfy 
for all x, y X \{0}. 
 ||f ( x y ) f ( y z ) f ( x z ) 
Proof. We set A1 : ( a , b ), A2 : (1,0) and 
 f( x ) f ( y ) f ( z ) f ( x y z ) ||Y
A3 : (0,1). For all i {2,3} and  , we p p p
 c(|| x ||XXX || y || || z || ) 
40 
 Dong Thap University Journal of Science, Vol. 9, No. 5, 2020, 33-41 
for all x, y , z X \{0}. Then f satisfies N. V. Dung and V. T. L. Hang. (2018). The 
the equation generalized hyperstability of general 
 f()()() x y f y z f x z linear equations in quasi-Banach spaces. 
 J. Math. Anal. Appl., 462(1), 131-147. 
 f()()()() x f y f z f x y z 
 D. H. Hyers. (1941). On the stability of the 
 for all x, y , z X \{0}. linear functional equation. Proc. Natl. 
 Acad. Sci. USA. (27), 222-224. 
Proof. We set A : (1,1,0), A : (0,1,1), 
 1 2 N. Kalton. (2003). Quasi-Banach spaces. In: 
A3 : (1,0,1), A4 : (1,0,0), A5 : (0,1,0), W.B. Johnson, and J. Lindenstrauss 
 (Eds.). Handbook of the Geometry of 
A6 : (0,0,1) and A7 : (1,1,1). For all 
 Banach Spaces. Elsevier, Amsterdam. (2), 
i {2,3,4,5,6,7} and  , we gain AA  . 
 i 1 1099-1130. 
Furthermore, f satisfies (2.1) for all 
 N. J. Kalton, N. T. Peck, and J. W. Roberts. 
x, y , z X \{0} and LLL 1, 
 1 2 3 (1984). An F-space sampler. London 
LLLL4 5 6 7 1. So all assumptions of mathematical society lecture note series, 
Theorem 2.1 are satisfied. Then Proposition Cambridge University Press. (89), 15-32. 
2.6 follows from Theorem 2.1. L. Maligranda. (2008). Tosio Aoki (1910-
 Acknowledgments: This article is 1989). In International symposium on 
supported by Dong Thap University with Banach and Function Spaces II. 
Topics Student scientific research code Yokohama Publishers, Yokohama. 1-23. 
SPD2019.02.13./. G. Maksa and Z. Pales. (2001). Hyperstability 
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 41