Quadratic (s1,s2)-functional inequality in fuzzy normed space

(1)

J. Math. Comput. Sci. 10 (2020), No. 2, 384-393 https://doi.org/10.28919/jmcs/4174

ISSN: 1927-5307

QUADRATIC (s1,s2)-FUNCTIONAL INEQUALITY IN FUZZY NORMED SPACE

SHALINI TOMAR1,∗, NAWNEET HOODA2

1_{Kanya Mahavidyalaya, Kharkhoda, Sonepat, Haryana, India}

2_{Department of Mathematics, DCRUST, Sonepat, Haryana, India}

Copyright c2020 the author(s). This is an open access article distributed under the Creative Commons Attribution License, which permits

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract. In this paper, we introduce and prove the Generalized Hyers-Ulam stability of Quadratic (s1,s2)

-functional inequality in Fuzzy Normed space using the fixed point method.

Keywords: Generalized Hyers-Ulam(HU) stability; quadratic (s1,s2)-functional inequality; quadratic (s1,s2)

-functional equation; fuzzy normed space.

2010 AMS Subject Classification:46S40, 39B52, 47H10, 39B62, 26E50, 47S40.

1.

INTRODUCTION

Nearly two decades ago, Gl ´anyi [8] proved that any h satisfies the Jordan-von Neumann

functional equation

2h(x) +2h(y) =h(xy) +h(xy−1)

if h satisfies the functional inequality

||2h(x) +2h(y)−h(xy−1)|| ≤ ||h(xy)||.

(1)

∗_{Corresponding author}

E-mail address: s [email protected]

Received June 6, 2019

(2)

Gl ´anyi [9] and Fechner [5] proved the HU stability of the functional inequality (1). Park,

Cho and Han [18] investigated and proved the HU Stability of the Cauchy additive functional

inequality

||h(x) +h(y) +h(z)|| ≤ ||h(x+y+z)||.

(2)

and the Cauchy-Jensen additive functional inequality

||h(x) +h(y) +2h(z)|| ≤ ||2h

x+y

2 +z

||.

(3)

The HU Stability is consequence of study of Ulam’s [1] problem regarding stability of group

homomorphism. A number of mathematicians namely Hyers [10], Aoki [2], Th.M.Rassias

[19],G ˘avruta [7] studied HU Stability under various adaptations. Park [16],[17] introduced

additive ρ-functional inequalities and proved their HU stability in Banach spaces and

non-Archimedean Banach spaces. In this paper,we introduce and prove HU stability of quadratic

(s1,s2)-functional inequality

F(F1(x,y),t)≤min{F(s1F2(x,y),t),F(s2F3(x,y),t)}

(4)

where

F₁(x,y) = f(kx+y)−f(x+ky)−(k2−1)[f(x)−f(y)] F₂(x,y) = (k+1)2f((₍kx+y)

k+1))−f(x+ky)−(k

2₋₁_)[_f_(x)₋_f_(y)]

F₃(x,y) = (k+1)2f((₍kx_k₊+₁y₎))−f(x+ky)−(k+1)2(k2−1)[f(₍_k₊x₁₎)−f(₍_k₊y₁₎)]

in Fuzzy Normed space, wherekis a non zero positive integer;s₁ ands₂are fixed non-zero real numbers with_s11 +_s21<2 .

2.

PRELIMINARIES

The concept of fuzzy norm on a linear space was given by Katsaras [11] in 1984. Since

then until now, the fuzzy norm has been defined in different ways by various mathematicians

[3],[20],[6],[12].

2.1.

Definition ([3],[15]). Let X be a real vector space. A functionF:X×R→[0,1]is called

afuzzy normon X if for alla,b∈X and allr,m,n∈R,

(3)

FN2: a=0 iff F(a,n) = 1 for alln>0;

FN3: F(ra, n) =F(a,_|n_r_|)ifr6=0;

FN4: F(a+b,m+n )≥min{F(a,m),F(b,n)};

FN5: lim_n_→_∞F(a,n) =1,where F(a,.) is a non-decreasing function ofR.

FN6: F(a,.) is continuous onR, for a6=0

The pair (X,F) is called afuzzy normed vector space.

2.2.

Definition ([3],[15]).

1. Let (X,F) be a fuzzy normed vector space. A sequence{a_n}in X is said to beconvergent

if ∃ an a∈X such that lim

n→∞F(an−

a,r) =1 for all r >0,where a is the limit of the

sequence{a_n}, denoted byF− lim

n→∞

a_n=a.

2. Let (X,F) be a fuzzy normed vector space. A sequence{an}in X is said to becauchy

if for eachε >0 and each r>0 there exists ann0∈Nsuch that for alln≥n0and all m>0,we haveF(an+m−an,r)>1−ε.

3. The fuzzy norm is said to becompleteif every cauchy sequence is convergent and the

fuzzy normed vector space is called afuzzy Banach space.

4. A mapping f :X →Y where X and Y are fuzzy normed vector spaces is continuous at

a point a₀∈X if for each sequence{an} converging toa0∈X, the sequence {f(an)}

converges to f(a0).If f :X→Y is continuous at each a∈X, then f :X →Y is said to

becontinuouson X.

2.3.

Definition [13]. Let X be a set. A function d:X×X →[0,∞)is called a generalized

metricon X if d satisfies the following conditions:

(1) d(x,y) =0 if and only ifx=yfor allx,y∈X; (2) d(x,y) =d(y,x)for allx,y∈X;

(3) d(x,z)≤d(x,y) +d(y,z)for allx,y,z∈X.

2.4.

Theorem[4]. Let(X,d)be a complete generalized metric space andJ:X →X a strictly

contractive mapping with Lipschitz constantL<1. Then, for allx∈X, eitherd(Jnx,Jn+1x) =∞

(4)

(2) the sequence{Jn_x}_{converges to a fixed point}_y?_{of J;}

(3) y?nis the unique fixed point of J in the setY ={y∈X :d(Jn0x,y)<∞};

(4) d(y,y?)≤(1/(1−L))d(y,Jy)for ally∈Y.

Throughout the paper, suppose thats1ands2are fixed nonzero real numbers with

1

s1+

1

s2

<2

andkis a non zero positive integer. Also X and Y be real fuzzy normed space and fuzzy banach

space respectively with norm F(.,t).

3.

QUADRATIC(s1,s2)-FUNCTIONAL INEQUALITY

3.1.

Lemma. Let f :X→Y be a mapping with f(0)=0 and satisfies (4) for allx,y∈X and all

t>0.Then f is Quadratic.

Proof: Suppose that functionf satisfies (4). By letting x=y in (4), we get

1≤min{F(s1((k+1)2f(x)− f((k+1)x)),t),(s2((k+1)2f(x)−f((k+1)x)),t)}

≤F((s₁+s₂)((k+1)2f(x)−f((k+1)x)),2t) =F

(k+1)2f(x)−f((k+1)x), 2t

(s1+s2)

Therefore,

(k+1)2f(x) = f((k+1)x)

(5)

Now from (4) and (5) we get

F(F1(x,y),t)≤min{F(s1F1(x,y),t),F(s2F1(x,y),t)}

=min{F(F1(x,y), t |s1|

),F(F1(x,y), t |s2|

)}

≤F

F₁(x,y),( 1 |s1|

+ 1 |s2|

)t

2

i.e.

F(F1(x,y),t)≥F(F1(x,y), t

ζ)

whereζ =

n

1 2

1

s1 +

1

s2 o

. Putting _| t

ζ|n−1 instead of t, we get

F

F₁, t

|ζ|n−1

≥F

F₁, t

(5)

Thus,for all n∈Z+ we have, F(F1,t))≥F

F1,_|_ζt_|n

. Since ζ <1, therefore by taking limit

n→∞and using (FN5) , we getF(F1(x,y),t) =1 for all x,y ∈X , and henceF1(x,y) =0.So, f :X →Y is Quadratic.

3.2.

Theorem. LetΨ:X2→[0,∞)be a function such that

Ψ(x,y)≤ L

(k+1)2Ψ((k+1)x,(k+1)y)

for someL<1 and for all x,y∈X. Let f :X→Y be a mapping with f(0)=0 and satisfying

min{F(F1(x,y),t),_t₊_Ψt₍_x_,_y₎} ≤

min{F(s1F2(x,y),t),F(s2F3(x,y),t)}

(6)

where

F₁(x,y) = f(kx+y)−f(x+ky)−(k2−1)[f(x)−f(y)]

F₂(x,y) = (k+1)2f((₍kx_k₊+₁y₎))−f(x+ky)−(k2−1)[f(x)−f(y)]

F3(x,y) = (k+1)2f((₍kx_k₊+₁y₎))− f(x+ky)−(k+1)2(k2−1)[f(₍_k₊x₁₎)−f(₍_k₊y₁₎)]for allx,y∈X

and allt >0. Then Q(x) =F−lim

n→∞(k+1)

2n_f x

(k+1)n

exists for all x ∈ X and defines a

Quadratic mappingQ:X →Y such that

F(f(x)−Q(x),t)≥ (2−2L)(k+1)t (2−2L)(k+1)t+ηΨ(x,x)

(7)

for allx∈X,t>0, whereη = n

1 |s1|+

1 |s2|

o

.

Proof: Letx=yin (6), we get

t

t+Ψ(x,x) ≤min{F(s1((k+1)

2_f_(x)₋ _f_((k₊₁_)x))_,_t)_,_F(s

2((k+1)2f(x)−f((k+1)x)),t)} ≤minnF(k+1)2f(x)−f((k+1)x), t

|s1|

,F(k+1)2f(x)− f((k+1)x), t

|s2|

o

≤F

(k+1)2f(x)−f((k+1)x),

1

|s1| + 1

|s2|

t

2

i.e.

F

f(x)−(k+1)2f

x

(k+1)

, ηt

2(k+1)

≥ t t+Ψ(x,x)

(8)

Now let us consider the set

(6)

and a generalized metric on S,such that

d(g,h) =in fε∈R+:F(g(x)−h(x),εt)≥ t

t+Ψ(x,x), f or all x∈X, f or all t>0

,

wherein f(Ψ) = +∞. Next, using lemma 2.1([14]) we can say that (S,d) is Complete.Now,let

us consider a linear mappingA:S→Ssuch that

Ag(x) = (k+1)2g x (k+1)

for allx∈X. Letg,h∈Swithd(g,h) =γ. Then

F(g(x)−h(x),γt)≥ t

t+Ψ(x,x)

for allx∈X,t>0. Therefore,

F(Ag(x)−Ah(x),Lγt) =F

(k+1)2g

x

(k+1)

−(k+1)2h

x

(k+1)

,Lγt

=F

g

x

(k+1)

−h

x

(k+1)

, Lγt

(k+1)2

≥

Lt

(k+1)2 Lt

(k+1)2+Ψ(₍_k₊x₁₎,₍_k₊x₁₎)

≥

Lt

(k+1)2 Lt

(k+1)2+₍_k₊L₁₎2Ψ(x,x)

= t

t+Ψ(x,x)

for allx∈X,t>0. Henced(Ag,Ah) =Lγ, i.e.d(Ag,Ah) =Ld(g,h)for allg,h∈S. Also using

(8), we can say that

d(f,A f)≤ η

2(k+1).

Now, by Theorem (2.4), there exists a mappingQ:X →Y such that:

1. Qis a fixed point of A, i.e.,

Q(x) = (k+1)2Q

x

(k+1)

(9)

for allx∈X. Since the mappingQis a unique fixed point of A in the set

T = (g∈S:d(f,g)<∞),

thusQis a unique mapping satisfying (9) such that there exists aε∈(0,∞)satisfying

F(f(x)−Q(x),εt)≥ t

t+Ψ(x,x)

(7)

2. d(Anf,Q)→0 asn→∞.This implies

Q(x) =F−lim

n→∞

(k+1)2nf

x

(k+1)n

f or all x∈X.

3. d(f,Q)≤ 1

1−Ld(f,A f), which impliesd(f,Q)≤

η

2(k+1)−2(k+1)L.And thus inequality (7)

is proved.Now by

minnF(k+1)2nF₁ x (k+1)n,

y (k+1)n

,(k+1)2nt, t

t+Ψ(₍_k₊x₁₎n, y

(k+1)n)

o

≤min

n

F

(k+1)2ns1F2

x

(k+1)n, y

(k+1)n

,(k+1)2nt

,

F(k+1)2ns₂F₃ x

(k+1)n, y

(k+1)n

,(k+1)2nto

for all x, y∈X, allt>0 and alln∈N.Now,by (6)

min

n

F

(k+1)2nF₁

x

(k+1)n, y

(k+1)n

,t

,₍ t/(k+1)2n

t/(k+1)2n₎₊₍_Ln_/₍_k₊₁₎2n₎_Ψ₍_x_,_y₎

o

≤minnF(k+1)2ns₁F₂ x (k+1)n,

y (k+1)n

,t,

(10)

F

(k+1)2ns2F3

x

(k+1)n,

y (k+1)n

,t

o

Since lim

n→∞

t/(k+1)2n

(t/(k+1)2n_{) + (L}n_/_(k₊₁₎2n₎_Ψ_(x_,_y) =1 for all x,y∈X, allt>0,therefore

by lemma (3.1) the mappingC:X →Y is Quadratic.

3.3.

Corollary. Letς ≥0 andpbe a real number withp>2.Let X be a normed vector space

with norm ||.||and (Y,N) be a fuzzy normed vector space. Let f :X →Y be a mapping with

f(0) =0 and

min

n

F(F1(x,y),t),

t

t+ς(||x||p+||y||p) o

≤min{F(s1F2(x,y),t),F(s2F3(x,y),t)

o

(11)

where F₁(x,y),F₂(x,y) and F₃(x,y) are as defined earlier for all x,y∈X and all t >0. Then

Q(x) =F−lim

n→∞(k+1)

2n_f x

(k+1)n

exists for all x∈X and a Quadratic mappingC:X →Y

such that

F(f(x)−Q(x),t)≥ ((k+1)

p₋_(k₊₁₎2_)(k₊₁_)t

((k+1)p₋_(k₊₁₎2_)(k₊₁_)t₊_{η ς}_||(k₊₁_)x||p

(12)

for allx∈X,t>0, whereη =_|_s1 1|+

1 |s2|.

Proof: The proof follows from above Theorem by taking Ψ(x,y) =ς(||x||p+||y||p) for all

(8)

3.4.

Theorem. LetΨ:X2→[0,∞)be a function such that

Ψ(x,y)≤(k+1)2LΨ( x

(k+1), y (k+1))

for some L<1 and for all x,y ∈ X. Let f :X →Y be a mapping with f(0)=0 and satisfying

(6). ThenQ(x) =F−lim

n→∞

1

(k+1)2nf((k+1)

n_x)_{exists for all x} _∈_{X and defines a Quadratic}

mappingC:X→Y such that

F(f(x)−Q(x),t)≥ (2−2L)(k+1) 2_t (2−2L)(k+1)2_t₊_η_Ψ_(x_,_x)

(13)

for allx∈X,t>0, whereη =_|_s11_|+_|_s21_|. Proof: It follows from (8) that, F

f(x)−₍ 1

k+1)2f((k+1)x), ηt

2(k+1)2

≥_t₊ t

Ψ(x,x)

for allx∈X and allt>0.Now consider linear mappingA:S→Ssuch that

Ag(x) = 1

(k+1)2f((k+1)x)

for allx∈X,where (S,d) is the generalized metric space as defined in previous theorem. Then

d(f,A f)≤ η

2(k+1)2.Hence

d(f,C)≤ η

2(k+1)2₋₂_(k₊₁₎2_L

which proves inequality (13).Rest of the proof can be generated from (3.2).

3.5.

Corollary. Letς ≥0 and pbe a real number with 0<p<2.Let X be a normed vector

space with norm||.||and (Y,N) be a fuzzy normed vector space. Let f :X →Y be a mapping

with f(0) =0 and satisfying (11). ThenQ(x) =F− lim

n→∞

1

(k+1)2nf((k+1)

n_x)_{exists for all x}

∈X and a Quadratic mappingC:X→Y such that

F(f(x)−Q(x),t)≥ ((k+1)

2₋_(k₊₁₎p_)t

((k+1)2₋_(k₊₁₎p_)t₊_{η ς}_||x||p

(14)

for allx∈X,t>0, whereη =_|_s11_|+_|_s21_|.

Proof: The proof follows from above Theorem by taking Ψ(x,y) =ς(||x||p+||y||p) for all

x,y∈X andL=|k+1|p−2_{and we get desired result.}

CONFLICT OF INTERESTS

(9)

REFERENCES

[1] Ulam S.M., Problems in Modern Mathematics, Science Editions, JohnWiley and Sons, New York, USA, 1964.

[2] Aoki T., On the stability of the linear transformation in Banach spaces.J. Math. Soc. Jpn., 2 (1950), 64-66.

[3] Bag T. and Samanta S. K., Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11(2003), 687

-705.

[4] Diaz J.B., Margolis B., A fixed point theorem of the alternative for contraction on a generalized complete

metric space, Bull. Amer. Math. Soc., 74 (1968), 305-309.

[5] Fechner W., Stability of a functional inequalities associated with the Jordan-von Neumann functional equation,

Aequationes Math., 71 (2006), 149-161.

[6] Felbin C., Finite dimensional fuzzy normed linear spaces, Fuzzy Sets Syst. 48 (1992), 239-248.

[7] Gavruta P., A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math.

Anal. Appl., 184(3) (1994), 431-436.

[8] Gl ´anyi A., Eine zur Parallelogrammgleichung ˜aquivalente Ungleichung, Aequationes Math., 62 (2001),

303-309.

[9] Gl ´anyi A., On a problem by K. Nikodem, Math. Inequal. Appl., 5 (2002), 707-710.

[10] Hyers D.H., On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27(4) (1941),

222-224.

[11] Katsaras A.K., Fuzzy topological vector spaces II, Fuzzy Sets Syst. 12 (1984), 143-154.

[12] Kim H.,Lee J.and Son E., Approximate functional inequalities by additive mappings, J. Math. Inequal. 6

(2012), 461-471.

[13] Luxemburg W.A.J., On the convergence of successive approximation in the theory of ordinary differential

equation, Proc. K. Ned. Aked. Wet., Ser. A., Indag. Math., 20 (1958), 540-546.

[14] Mihet D. and Radu V., On the stability of the additive Cauchy functional equation in random normed spaces,

J. Math. Anal. Appl. 343 (2008), 567-572.

[15] Mirmostafaee A. K.,Mirzavaziri M.and Moslehian M. S., Fuzzy stability of the Jensen functional equation,

Fuzzy Sets Syst. 159 (2008), 730- 738.

[16] Park C., Additiveρ−functional inequalities and equations, J. Math. Inequal. 9 (2015), 17-26.

[17] Park C., Additiveρ−functional inequalities in non-Archimedean normed spaces, J. Math. Inequal. 9 (2015),

397 - 407.

[18] Park C.,Cho Y. and Han M., Stability of functional inequalities associated with Jordan-von Neumann type

additive functional equations, J. Inequal. Appl. 2007 (2007), Art. ID 41820.

[19] Rassias Th. M., On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72(2)

(10)

[20] Xiao J.Z. and Zhu X. H., Fuzzy normed spaces of operators and its completeness, Fuzzy Sets Syst. 133