functional inequality in Banach spaces

Inequalities

Volume 13, Number 1 (2019), 95–104 doi:10.7153/jmi-2019-13-07

THE STABILITY OF AN ADDITIVE

(ρ1,ρ2)–FUNCTIONAL INEQUALITY IN BANACH SPACES

CHOONKILPARK

(Communicated by J. K. Kim)

Abstract. In this paper, we introduce and solve the following additive (ρ1,ρ2)-functional in- equality

f(x+y)−f(x)−f(y)ρ1(f(x+y) +f(x−y)−2f(x)) (1) +

ρ2

2f

x+y 2

−f(x)−f(y) , whereρ1andρ2arefixed nonzero complex numbers with√

2|ρ1|+|ρ2|<1 .

Using thefixed point method and the direct method, we prove the Hyers-Ulam stability of the additive (ρ1,ρ2)-functional inequality (1) in complex Banach spaces.

1. Introduction and preliminaries

The stability problem of functional equations originated from a question of Ulam [26] concerning the stability of group homomorphisms. Hyers [10] gave afirst affir- mative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [2] for additive mappings and by Rassias [24] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias the- orem was obtained by G˘avruta [9] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. The stability of quadratic functional equation was proved by Skof [25] for mappings f :E₁→E₂, where E₁is a normed space andE₂is a Banach space. Cholewa [6] noticed that the theorem of Skof is still true if the relevant domainE₁is replaced by an Abelian group.

Park [21, 22] defined additive ρ-functional inequalities and proved the Hyers- Ulam stability of the additive ρ-functional inequalities in Banach spaces and non- Archimedean Banach spaces. The stability problems of various functional equations have been extensively investigated by a number of authors (see [1,8,12,13,14,15,16, 20]).

We recall a fundamental result infixed point theory.

Mathematics subject classification(2010): Primary 39B62, 47H10, 39B52.

Keywords and phrases: Hyers-Ulam stability, additive (ρ1,ρ2)-functional inequality, fixed point method, direct method, Banach space.

c , Zagreb

Paper JMI-13-07 95

THEOREM1.1 [3,7]Let (X,d) be a complete generalized metric space and let J: X→X be a strictly contractive mapping with Lipschitz constantα<1. Then for each given element x∈X , either

d(Jⁿx,Jⁿ⁺¹x) =∞

for all nonnegative integers n or there exists a positive integer n₀such that (1) d(Jⁿx,Jⁿ⁺¹x)<∞, ∀nn₀;

(2)the sequence{Jⁿx} converges to afixed point y^∗of J ;

(3) y^∗ is the uniquefixed point of J in the set Y ={y∈X|d(Jⁿ⁰x,y)<∞}; (4) d(y,y^∗)_1−α¹ d(y,Jy)for all y∈Y .

In 1996, Isac and Rassias [11] were thefirst to provide applications of stability the- ory of functional equations for the proof of newfixed point theorems with applications.

By usingfixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [4,5,18,19,23]).

In Section 2, we solve the additive(ρ1,ρ2)-functional inequality (1) and prove the Hyers-Ulam stability of the additive(ρ1,ρ2)-functional inequality (1) in Banach spaces by using thefixed point method. In Section 3, we prove the Hyers-Ulam stability of the additive(ρ1,ρ2)-functional inequality (1) in Banach spaces by using the direct method.

Throughout this paper, let X be a real or complex normed space with norm · andY a complex Banach space with norm · . Assume that ρ1 and ρ2 are fixed nonzero complex numbers with√

2|ρ1|+|ρ2|<1.

2. Additive(ρ1,ρ2)-functional inequality(1): afixed point method In this section, we solve and investigate the additive(ρ1,ρ2)-functional inequality (1) in complex Banach spaces.

LEMMA2.1 If a mapping f:X→Y satisfies f(0) =0 and

f(x+y)−f(x)−f(y)ρ1(f(x+y) +f(x−y)−2f(x)) (2) +

ρ2

2f

x+y 2

−f(x)−f(y) for all x,y∈X , then f is additive.

Proof. Assume that f satisfies the inequality (2).

Letting y=x in (2), we get f(2x)−2f(x)|ρ1| · f(2x)−2f(x) and so f(2x) =2f(x)for allx∈X, since |ρ1|<1. Thus

fx 2

=1

2f(x) (3)

for allx∈X.

It follows from (2) and (3) that

f(x+y)−f(x)−f(y)ρ1(f(x+y) +f(x−y)−2f(x)) +

ρ2

2f

x+y 2

−f(x)−f(y)

=ρ1(f(x+y) +f(x−y)−2f(x)) +ρ2(f(x+y)−f(x)−f(y)) and so

(1− |ρ2|)f(x+y)−f(x)−f(y)|ρ1| · f(x+y) +f(x−y)−2f(x) (4) for allx,y∈X.

Letting z=x+yandw=x−yin (4), we get (1− |ρ2|)

f(z)−f z+w

2

−f z−w

2

|ρ1| ·

f(z) +f(w)−2f z+w

2

and so 1

2(1− |ρ2|)f(z+w) +f(z−w)−2f(z)|ρ1| · f(z+w)−f(z)−f(w) for allz,w∈X.

It follows from (4) and (5) that 1

2(1− |ρ2|)²f(x+y)−f(x)−f(y)|ρ1|²f(x+y)−f(x)−f(y) for allx,y∈X. Since√

2|ρ1|+|ρ2|<1, f(x+y) =f(x)+f(y)for allx,y∈X. Thus f is additive.

Using thefixed point method, we prove the Hyers-Ulam stability of the additive (ρ1,ρ2)-functional inequality (2) in complex Banach spaces.

THEOREM2.2 Let ϕ^:X2→[0,∞)be a function such that there exists an L<1 with ϕ₂^x,y

2

L

2ϕ(x,y) (5)

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

f(x+y)−f(x)−f(y)ρ1(f(x+y) +f(x−y)−2f(x)) (6) +

ρ2

2f

x+y

2

−f(x)−f(y)

+ϕ(x,y) for all x,y∈X . Then there exists a unique additive mapping A:X→Y such that

f(x)−A(x) L

2(1−L)(1− |ρ1|)ϕ(x,x) for all x∈X .

Proof. Lettingy=xin (6), we get

(1− |ρ1|)f(2x)−2f(x)ϕ(x,x) (7) for allx∈X.

Consider the set

S:={h:X→Y, h(0) =0} and introduce the generalized metric onS:

d(g,h) =inf{μ∈R+:g(x)−h(x)μϕ(x,x), ∀x∈X}, where, as usual, infφ= +∞. It is easy to show that(S,d)is complete (see [17]).

Now we consider the linear mapping J:S→S such that Jg(x):=2gx

2 for allx∈X.

Let g,h∈S be given such thatd(g,h) =ε^{. Then} g(x)−h(x)εϕ(x,x) for allx∈X. Hence

Jg(x)−Jh(x)=2gx 2

−2hx

22εϕ^x₂,x 2

2ε^L₂ϕ(x,x) =Lεϕ(x,x)

for allx∈X. So d(g,h) =ε implies that d(Jg,Jh)Lε. This means that d(Jg,Jh)Ld(g,h)

for allg,h∈S.

It follows from (7) that f(x)−2fx

2 1

1− |ρ1|ϕ^x₂,x 2

L

2(1− |ρ1|)ϕ(x,x) for allx∈X. So d(f,J f)_2(1−|ρ^L _1|).

By Theorem1.1, there exists a mappingA:X→Y satisfying the following:

(1)Ais afixed point ofJ, i.e.,

A(x) =2Ax 2

(8)

for allx∈X. The mappingAis a uniquefixed point ofJ in the set M={g∈S:d(f,g)<∞}.

This implies thatAis a unique mapping satisfying (8) such that there exists aμ∈(0,∞) satisfying

f(x)−A(x)μϕ(x,x) for allx∈X;

(2)d(J^lf,A)→0 asl→∞. This implies the equality

l→∞lim2ⁿfx 2ⁿ

=A(x)

for allx∈X;

(3)d(f,A)_1−L¹ d(f,J f), which implies f(x)−A(x) L

2(1−L)(1− |ρ1|)ϕ(x,x) for allx∈X.

It follows from (5) and (6) that A(x+y)−A(x)−A(y)

=lim

n→∞2ⁿ f

x+y 2ⁿ

−fx 2ⁿ

−fy

2ⁿ

lim

n→∞2ⁿ|ρ1| f

x+y 2ⁿ

+f

x−y 2ⁿ

−2fx 2ⁿ

+lim

n→∞2ⁿ|ρ2| 2f

x+y

2ⁿ⁺¹

−f x 2ⁿ

−fy

2ⁿ

+lim

n→∞2ⁿϕ₂^x_n, y 2ⁿ

=ρ1(A(x+y) +A(x−y)−2A(x))+ ρ2

2A

x+y 2

−A(x)−A(y) for allx,y∈X. So

A(x+y)−A(x)−A(y)ρ1(A(x+y) +A(x−y)−2A(x)) +

ρ2

2A

x+y

2

−A(x)−A(y) for allx,y∈X. By Lemma2.1, the mappingA:X→Y is additive.

COROLLARY2.3 Let r>1andθ be nonnegative real numbers, and let f:X→Y be a mapping satisfying f(0) =0 and

f(x+y)−f(x)−f(y)ρ1(f(x+y) +f(x−y)−2f(x)) (9) +

ρ2

2f

x+y

2

−f(x)−f(y)

+θ(x^r+y^r)

for all x,y∈X . Then there exists a unique additive mapping A:X→Y such that f(x)−A(x) 2θ

(2^r−2)(1− |ρ1|)x^r for all x∈X .

Proof. The proof follows from Theorem2.2by taking ϕ(x,y) =θ(x^r+y^r) for allx,y∈X. ChoosingL=2^1−r, we obtain the desired result.

THEOREM2.4 Let ϕ^:X2→[0,∞)be a function such that there exists an L<1 with ϕ(x,y)2Lϕ₂^x,y

2

for all x,y∈X . Let f :X→Y be a mapping satisfying f(0) =0 and(6). Then there exists a unique additive mapping A:X→Y such that

f(x)−A(x) 1

2(1−L)(1− |ρ1|)ϕ(x,x) for all x∈X .

Proof. Let(S,d)be the generalized metric space defined in the proof of Theorem 2.2.

Now we consider the linear mapping J:S→S such that Jg(x):=1

2g(2x) for allx∈X.

It follows from (7) that f(x)−1

2f(2x)

1

2(1− |ρ1|)ϕ(x,x) for allx∈X.

The rest of the proof is similar to the proof of Theorem2.2.

COROLLARY2.5 Let r<1 and θ be positive real numbers, and let f :X→Y be a mapping satisfying f(0) =0 and(9). Then there exists a unique additive mapping A:X→Y such that

f(x)−A(x) 2θ

(2−2^r)(1− |ρ1|)x^r for all x∈X .

Proof. The proof follows from Theorem2.4by taking ϕ(x,y) =θ(x^r+y^r) for allx,y∈X. ChoosingL=2^r−1, we obtain the desired result.

REMARK2.6. Ifρ is a real number such that √

2|ρ1|+|ρ2|<1 andY is a real Banach space, then all the assertions in this section remain valid.

3. Additive (ρ1,ρ2)-functional inequality(1): a direct method

In this section, we prove the Hyers-Ulam stability of the additive(ρ1,ρ2)-functional inequality (2) in complex Banach spaces by using the direct method.

THEOREM3.1 Let ϕ^:X2→[0,∞)be a function such that Ψ(x,y):=

∑

^∞

j=1

2^jϕ₂^x_j, y 2^j

<∞ (10)

for all x,y∈X . Let f :X→Y be a mapping satisfying f(0) =0 and(6). Then there exists a unique additive mapping A:X→Y such that

f(x)−A(x) 1

2(1− |ρ1|)Ψ(x,x) (11) for all x∈X .

Proof. Lettingy=xin (6), we get

(1− |ρ1|)f(2x)−2f(x)ϕ(x,x) (12) for allx∈X. So

f(x)−2fx

2 1

1− |ρ1|ϕ^x₂,x 2

for allx∈X. Hence 2^lfx 2^l

−2^mf x

2^mm−1

∑

j=l

2^jfx 2^j

−2^j+1f x 2^{j+1 m−1}

∑

j=l

2^j

2(1− |ρ1|)ϕ_2j+1^x , x 2^j+1

(13)

for all nonnegative integersmandl withm>l and allx∈X. It follows from (13) that the sequence{2^kf(₂^xk)} is Cauchy for all x∈X. Since Y is complete, the sequence {2^kf(₂^xk)} converges. So one can define the mappingA:X→Y by

A(x):=lim

k→∞2^kfx 2^k

for allx∈X. Moreover, lettingl=0 and passing the limitm→∞in (13), we get (11).

It follows from (6) and (10) that A(x+y)−A(x)−A(y)

=lim

n→∞2ⁿ f

x+y 2ⁿ

−fx 2ⁿ

−fy

2ⁿ

lim

n→∞2ⁿ|ρ1| f

x+y

2ⁿ

+f

x−y

2ⁿ

−2fx 2ⁿ

+lim

n→∞2ⁿ|ρ2| 2f

x+y

2ⁿ⁺¹

−f x 2ⁿ

−fy

2ⁿ

+lim

n→∞2ⁿϕ₂^x_n, y 2ⁿ

=ρ1(A(x+y) +A(x−y)−2A(x))+ ρ2

2A

x+y 2

−A(x)−A(y) for allx,y∈X. So

A(x+y)−A(x)−A(y)ρ1(A(x+y) +A(x−y)−2A(x)) +

ρ2

2A

x+y 2

−A(x)−A(y) for allx,y∈X. By Lemma2.1, the mappingA:X→Y is additive.

Now, let T:X→Y be another additive mapping satisfying (11). Then we have A(x)−T(x) =2^qAx

2^q

−2^qTx 2^q 2^qAx

2^q

−2^qfx

2^q+2^qT x 2^q

−2^qfx 2^q

2^q

1− |ρ1|Ψ x 2^q, x

2^q ,

which tends to zero asq→∞for allx∈X. So we can conclude thatA(x) =T(x)for allx∈X. This proves the uniqueness of A.

COROLLARY3.2 Let r>1andθ be nonnegative real numbers, and let f:X→Y be a mapping satisfying f(0) =0 and(9). Then there exists a unique additive mapping A:X→Y such that

f(x)−A(x) 2θ

(2^r−2)(1− |ρ1|)x^r for all x∈X .

THEOREM3.3 Let ϕ ^:X2→[0,∞) be a function and let f :X →Y be a mapping satisfying f(0) =0,(6)and

Ψ(x,y):=

∑

^∞

j=0

1

2^jϕ(2^jx,2^jy)<∞

for all x,y∈X . Then there exists a unique additive mapping A:X→Y such that f(x)−A(x) 1

2(1− |ρ1|)Ψ(x,x) (14) for all x∈X .

Proof. It follows from (12) that f(x)−1

2f(2x)

1

2(1− |ρ1|)ϕ(x,x) for allx∈X. Hence

1

2^l f(2^lx)− 1

2^mf(2^mx) ^m−1

∑

j=l

1 2^jf

2^jx

− 1 2^j+1f

2^j+1x ^m−1

∑

j=l

1

2^j+1(1− |ρ1|)ϕ(2^jx,2^jx) (15) for all nonnegative integersmandl withm>l and allx∈X. It follows from (15) that the sequence{₂¹nf(2ⁿx)} is Cauchy for all x∈X. SinceY is complete, the sequence {₂¹nf(2ⁿx)} converges. So one can define the mapping A:X→Y by

A(x):=lim

n→∞

1 2ⁿf(2ⁿx)

for allx∈X. Moreover, lettingl=0 and passing the limitm→∞in (15), we get (14).

The rest of the proof is similar to the proof of Theorem3.1.

COROLLARY3.4 Let r<1 and θ be positive real numbers, and let f :X→Y be a mapping satisfying f(0) =0 and(9). Then there exists a unique additive mapping A:X→Y such that

f(x)−A(x) 2θ

(2−2^r)(1− |ρ1|)x^r for all x∈X .

R E F E R E N C E S

[1] M. ADAM,On the stability of some quadratic functional equation, J. Nonlinear Sci. Appl.4(2011), 50–59.

[2] T. AOKI,On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan2(1950), 64–66.

[3] L. C˘ADARIU, V. RADU,Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math.4, no. 1, Art. ID 4 (2003).

[4] L. C˘ADARIU, V. RADU,On the stability of the Cauchy functional equation: afixed point approach, Grazer Math. Ber.346 (2004), 43–52.

[5] L. CADARIU˘ , V. RADU,Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory Appl.2008, Art. ID 749392 (2008).

[6] P. W. CHOLEWA,Remarks on the stability of functional equations, Aequationes Math.27 (1984), 76–86.

[7] J. DIAZ, B. MARGOLIS,Afixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Am. Math. Soc.74 (1968), 305–309.

[8] G. Z. ESKANDANI, P. GAVRUTAˇ ,Hyers-Ulam-Rassias stability of pexiderized Cauchy functional equation in 2-Banach spaces, J. Nonlinear Sci. Appl.5(2012), 459–465.

[9] P. GˇAVRUTA,A generalization of the Hyers-Ulam-Rassias stability of approximately additive map- pings, J. Math. Anal. Appl.184 (1994), 431–436.

[10] D. H. HYERS,On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A.27 (1941), 222–224.

[11] G. ISAC, TH. M. RASSIAS,Stability of ψ-additive mappings: Applications to nonlinear analysis, Internat. J. Math. Math. Sci.19 (1996), 219–228.

[12] B. KHOSRAVI, M. B. MOGHIMI, A. NAJATI,Generalized Hyers-Ulam stability of a functional equa- tion of Hosszu type, Nonlinear Funct. Anal. Appl.23 (2018), 157–166.

[13] G. KIM, H. SHIN,Approximately quadratic mappings in non-Archimedean fuzzy normed spaces, Nonlinear Funct. Anal. Appl.23 (2018), 369–380.

[14] Y. LEE, S. JUNG,A general theorem on the fuzzy stability of a class of functional equations including quadratic-additive functional equations, Nonlinear Funct. Anal. Appl.23 (2018), 353–368.

[15] Y. MANAR, E. ELQORACHI, TH. M. RASSIAS,Hyers-Ulam stability of the Jensen functional equa- tion in quasi-Banach spaces, Nonlinear Funct. Anal. Appl.15 (2010), 581–603.

[16] Y. MANAR, E. ELQORACHI, TH. M. RASSIAS,On the Hyers-Ulam stability of the quadratic and Jensen functional equations on a restricted domain, Nonlinear Funct. Anal. Appl.15 (2010), 647–

655.

[17] D. MIHET¸, V. RADU,On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl.343 (2008), 567–572.

[18] C. PARK,Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory Appl.2007, Art. ID 50175 (2007).

[19] C. PARK,Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: afixed point approach, Fixed Point Theory Appl.2008, Art. ID 493751 (2008).

[20] C. PARK,Orthogonal stability of a cubic-quartic functional equation, J. Nonlinear Sci. Appl.5(2012), 28–36.

[21] C. PARK,Additiveρ-functional inequalities and equations, J. Math. Inequal.9(2015), 17–26.

[22] C. PARK,Additiveρ-functional inequalities in non-Archimedean normed spaces, J. Math. Inequal.9 (2015), 397–407.

[23] V. RADU,Thefixed point alternative and the stability of functional equations, Fixed Point Theory4 (2003), 91–96.

[24] TH. M. RASSIAS,On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc.72 (1978), 297–300.

[25] F. SKOF,Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano53 (1983), 113–129.

[26] S. M. ULAM,A Collection of the Mathematical Problems, Interscience Publ. New York, 1960.

(Received August 31, 2016) Choonkil Park

Research Institute for Natural Sciences Hanyang University Seoul 04763, Republic of Korea e-mail:baak@@hanyang.ac.kr

Journal of Mathematical Inequalities www.ele-math.com

jmi@ele-math.com