Kyungpook Mathematical Journal 2018; 58(2): 291-305  
A General Uniqueness Theorem concerning the Stability of AQCQ Type Functional Equations
Yang-Hi Lee and Soon-Mo Jung*
Department of Mathematics Education, Gongju National University of Education, 32553 Gongju, Republic of Korea, e-mail : yanghi2@hanmail.net, Mathematics Section, College of Science and Technology, Hongik University, 30016 Sejong, Republic of Korea, e-mail: smjung@hongik.ac.kr
*Corresponding Author.
Received: August 24, 2017; Accepted: April 12, 2018; Published online: June 23, 2018.
© Kyungpook Mathematical Journal. All rights reserved.

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Abstract

In this paper, we prove a general uniqueness theorem which is useful for proving the uniqueness of the relevant additive mapping, quadratic mapping, cubic mapping, quartic mapping, or the additive-quadratic-cubic-quartic mapping when we investigate the (generalized) Hyers-Ulam stability.

Keywords: uniqueness, stability, Hyers-Ulam stability, generalized Hyers-Ulam stability, AQCQ type functional equation, AQCQ mapping
1. Introduction

From now on, we assume that V and W are vector spaces and a is a fixed real number larger than 1. If a mapping f : VW is given, then we set

f1(x):=fo(ax)-a3fo(x),f2(x):=fe(ax)-a4fe(x),f3(x):=fo(ax)-afo(x),f4(x):=fe(ax)-a2fe(x),Af(x,y):=f(x+y)-f(x)-f(y),Q2f(x,y):=f(x+y)+f(x-y)-2f(x)-2f(y),Cf(x,y):=f(2x+y)-3f(x+y)+3f(y)-f(-x+y)-6f(x),Q4f(x,y):=f(2x+y)-4f(x+y)+6f(y)-4f(-x+y)+f(-2x+y)-24f(x)

for x, yV, where fo and fe denote the odd and even part of f, respectively.

If a mapping f : VW satisfies the equation Af(x, y) = 0, Q2f(x, y) = 0, Cf(x, y) = 0, or Q4f(x, y) = 0 for all x, yV, then it is called an additive mapping, a quadratic mapping, a cubic mapping, or a quartic mapping, respectively. We notice that the real-valued mappings f(x) = ax, g(x) = ax2, h(x) = ax3, and k(x) = ax4 are solutions to Af(x, y) = 0, Q2f(x, y) = 0, Cg(x, y) = 0, and Q4h(x, y) = 0, respectively.

We call a mapping f : VW an additive-quadratic-cubic-quartic mapping if it is expressed as the sum of an additive mapping, a quadratic mapping, a cubic mapping, and a quartic mapping, and vice versa. An additive-quadratic-cubic-quartic type functional equation is just the functional equation, each of whose solutions is an additive-quadratic-cubic-quartic mapping. For example, the real-valued mapping f(x) = ax4 + bx3 + cx2 + dx defined on ℝ is a solution to the additive-quadratic-cubic-quartic type functional equation.

Whenever we investigate the stability problems for additive-quadratic-cubic-quartic type functional equations or others, we encounter some uniqueness problems. To our best of knowledge, however, no author has succeeded in proving even a uniqueness theorem for these cases ([1, 2, 7, 12, 14, 15, 17]) except our papers ([8, 9, 10, 11]). The ideas of the present paper are strongly based on the previous papers [8, 9, 10, 11]. But this paper includes more general results than the previous three papers.

In this paper, a general uniqueness theorem will be proved which is useful for proving the uniqueness of the relevant additive mapping, quadratic mapping, cubic mapping, quartic mapping, or the additive-quadratic-cubic-quartic mapping when we investigate the (generalized) Hyers-Ulam stability. In Section 3, we make use of our uniqueness theorem to improve the stability theorems presented in [5, 13], where the uniqueness of exact solutions have not been proved.

2. Main Result

We assume that V is a real vector space and Y is a real normed space. The following somewhat surprising theorem states that if for any given mapping f, there exists a mapping F (near f) with some properties (which are possessed by quadratic, cubic, quartic, or possessed by additive-quadratic-cubic-quartic mappings), then F is uniquely determined.

Theorem 2.1

Assume that a > 1 is a fixed real number and Φ : V {0} → [0,∞) is a function satisfying one of the following properties

limn1anΦ(anx)=0,limnanΦ(xan)=limn1a2nΦ(anx)=0,limna2nΦ(xan)=limn1a3nΦ(anx)=0,limna3nΦ(xan)=limn1a4nΦ(anx)=0,limna4nΦ(xan)=0

for all xV {0}. Suppose f : VY is an arbitrary mapping. If a mapping F : VY satisfies the inequality

f(x)-F(x)Φ(x)

for all xV {0} and if F satisfies each of the following equalities

F1(ax)=aF1(x),F2(ax)=a2F2(x),F3(ax)=a3F3(x),F4(ax)=a4F4(x)

for all xV, then F is given as

F(x)={limn[1a4-a2(f4(anx)a4n-f2(anx)a2n)+1a3-a(f3(anx)a3n-f1(anx)an)]inthecaseof(2.1),limn[1a4-a2(f4(anx)a4n-f2(anx)a2n)+1a3-a(f3(anx)a3n-anf1(xan))]inthecaseof(2.2),limn[1a4-a2(f4(anx)a4n-a2nf2(xan))+1a3-a(f3(anx)a3n-anf1(xan))]inthecaseof(2.3),limn[1a4-a2(f4(anx)a4n-a2nf2(xan))+1a3-a(a3nf3(xan)-anf1(xan))]inthecaseof(2.4),limn[1a4-a2(a4nf4(xan)-a2nf2(xan))+1a3-a(a3nf3(xan)-anf1(xan))]inthecaseof(2.5)

for all xV {0}. In other words, F is the unique mapping satisfying (2.6) and (2.7).

Proof

Suppose a mapping F satisfies (2.6) and (2.7) for a given f : VY .

(i) We consider F1(x) = Fo(ax) – a3Fo(x). When Φ : V {0} → [0,∞) has the property (2.1), we make use of (2.7) to get

F1(x)-1anf1(anx)=1anF1(anx)-f1(anx)1anFo(an+1x)-fo(an+1x)+a3anfo(anx)-Fo(anx)12an(Φ(an+1x)+Φ(-an+1x)+a3Φ(anx)+a3Φ(-anx))0,         as         n,

for all xV {0}. Thus, F1(x)=limn1anfo(anx) for all xV {0} provided Φ has the property (2.1).

When Φ : V {0} → [0,∞) has the property (2.2), (2.3), (2.4), or (2.5), then we use (2.7) to show that

F1(x)-anf1(xan)=anF1(xan)-f1(xan)anFo(axan)-fo(axan)+an+3fo(xan)-Fo(xan)an2(Φ(axan)+Φ(-axan)+a3Φ(xan)+a3Φ(-xan))0,         as         n,

for all xV {0}. Hence, Fo(x)=limnanf1(xan) for all xV {0} provided Φ has the property (2.2), (2.3), (2.4), or (2.5).

(ii) We consider the mapping F2(x) = Fe(ax)–a4Fe(x). When Φ : V {0} → [0,∞) has the property (2.1) or (2.2), then we apply (2.7) to verify

F2(x)-1a2nf2(anx)=1a2nF2(anx)-f2(anx)1a2n(Fe-fe)(an+1x)+1a2na4(Fe-fe)(anx)12a2n(Φ(an+1x)+Φ(-an+1x)+a4Φ(anx)+a4Φ(-anx))0,         as         n,

for all xV {0}. Then, F2(x)=limn1a2nf2(anx) is true for all xV {0} provided Φ has the property (2.1) or (2.2).

When Φ : V {0} → [0,∞) satisfies (2.3), (2.4) or (2.5), we get

F2(x)-a2nf2(xan)=a2nF2(xan)-f2(xan)a2n(Fe-fe)(xan-1)+a2na4(Fe-fe)(xan)a2n2(Φ(xan-1)+Φ(-xan-1)+a4Φ(xan)+a4Φ(-xan))0,         as         n,

for all xV {0}. Thus, F2(x)=limna2nf2(xan) for all xV {0} provided Φ has the property (2.3), (2.4) or (2.5). (iii) We now consider F3(x) = Fo(ax) – aFo(x). When Φ : V {0} → [0,∞) has the property (2.1), (2.2) or (2.3), then we make use of (2.7) to see

F3(x)-1a3nf3(anx)=1a3nF3(anx)-f3(anx)1a3n(Fo-fo)(an+1x)+1a3na(Fo-fo)(anx)12a3n(Φ(an+1x)+Φ(-an+1x)+aΦ(anx)+aΦ(-anx))0,         as         n,

for all xV {0}. Hence, F3(x)=limn1a3nf3(anx) holds for all xV {0} provided Φ has the property (2.1), (2.2), or (2.3).

When Φ : V {0} → [0,∞) has the property (2.4) or (2.5), we then obtain

F3(x)-a3nf3(xan)=a3nF3(xan)-f3(xan)a3n(Fo-fo)(xan-1)+a2na(Fo-fo)(xan)a3n2(Φ(xan-1)+Φ(-xan-1)+aΦ(xan)+aΦ(-xan))0,         as         n,

for all xV {0}. Therefore, we have F3(x)=limna3nf3(xan) for all xV {0} provided Φ satisfies (2.4) or (2.5).

(iv) Finally, we consider F4(x) = Fe(ax) – a2Fe(x). If Φ : V {0} → [0,∞) has the property (2.1), (2.2), (2.3) or (2.4), then it follows from (2.7) that

F4(x)-1a4nf4(anx)=1a4nF4(anx)-f4(anx)1a4n(Fe-fe)(an+1x)+1a4na2(fe-Fe)(anx)12a4n(Φ(an+1x)+Φ(-an+1x)+a2Φ(anx)+a2Φ(-anx))0,         as         n,

for all xV {0} provided Φ has the property (2.1), (2.2), (2.3), or (2.4). That is, F4(x)=limn1a4nf4(anx) holds for all xV {0}.

Now, we deal with the case when Φ : V {0} → [0,∞) satisfies (2.5). It holds that

F4(x)-a4nf4(xan)=a4nF4(xan)-f4(xan)a4n(Fe-fe)(xan-1)+a4na2(fe-Fe)(xan)a4n2(Φ(xan-1)+Φ(-xan-1)+a2Φ(xan)+a2Φ(-xan))0,         as         n,

for all xV {0}. Thus, it holds that F4(x)=limna4nf4(xan) for all xV {0} provided Φ satisfies (2.5).

Consequently, since F(x)=F4(x)-F2(x)a4-a2+F3(x)-F1(x)a3-a, F is expresses as one of equalities in (2.8) and F is uniquely determined in each case.

3. Applications

Theorem 2.1 seems to be impractical for applications in general cases. Thus, it is necessary to introduce some corollaries which are easily applicable to the uniqueness problems for the generalized Hyers-Ulam stability. For the exact definition of the generalized Hyers-Ulam stability, we refer the reader to [3, 6].

Corollary 3.1

Assume that a > 1 is a fixed real number and φ : V {0} → [0,∞) satisfies either

Φ(x):=i=01aiφ(aix)<         forallxV{0}

or

Φ(x):=i=0a4iφ(xai)<         forallxV{0}.

Suppose f : VY is an arbitrary mapping. If a mapping F : VY satisfies (2.6) for all xV {0} and (2.7) for all xV, then F is uniquely determined.

Proof

When φ satisfies (3.1), it is obvious that

limn1anΦ(anx)=limni=01an+iφ(an+ix)=limni=n1aiφ(aix)=0,

i.e., Φ has the property (2.1) for all xV {0}. For the case of (3.2), it is clear that

limna4nΦ(xan)=limni=0a4n+4iφ(xan+i)=limni=na4iφ(xai)=0,

i.e., Φ has the property (2.5) for all xV {0}. Hence, our assertion is true in view of Theorem 2.1.

Corollary 3.2

Assume that a > 1 is a fixed real number and the functions φ, ψ : V {0} → [0,∞) satisfy each of the following conditions

i=0aiψ(xai)<,i=01a2iφ(aix)<,Φ˜(x):=i=0aiφ(xai)<,Ψ˜(x):=i=01a2iψ(aix)<

for all xV {0}. Suppose f : VY is an arbitrary mapping. If a mapping F : VY satisfies the inequality

f(x)-F(x)Φ˜(x)+Ψ˜(x)

for all xV {0} and if F satisfies each condition of (2.7) for all xV, then F is uniquely determined.

Proof

We set Φ(x) := Φ̃(x) + Ψ̃(x) and then use (3.3) to obtain

1a4nΦ(a2nx)=i=01a4n-iφ(a2n-ix)+i=01a4n+2iψ(a2n+ix)

for all xV {0}. We make change of the summation indices in the preceding equality with j = i – 2n and k = 2n + i to get

1a4nΦ(a2nx)=1a2nj=-2najφ(xaj)+k=2n1a2kψ(akx)=1a2ni=12n1aiφ(aix)+1a2ni=0aiφ(xai)+i=2n1a2iψ(aix)=1ani=1n-1aian1a2iφ(aix)+i=n2naia2n1a2iφ(aix)+1a2nΦ˜(x)+i=2n1a2iψ(aix)1ani=11a2iφ(aix)+i=n1a2iφ(aix)+1a2nΦ˜(x)+i=2n1a2iψ(aix)

for any xV {0}. Hence, we obtain

limn1a4nΦ(a2nx)=0

for all xV {0}.

On the other hand, we make use of the above equality to prove

limn1a4n+2Φ(a2n+1x)=1a2limn1a4nΦ(a2nax)=0

for all xV {0}. Using the two equalities above, we get

limn1a2nΦ(anx)=0

for all xV {0}.

Similarly, it holds that

a2nΦ(xa2n)=i=0a2n+iφ(xa2n+i)+i=01a2i-2nψ(ai-2nx)

for all xV {0}. If we make change of the summation indices in the last equality with j = i + 2n and k = i – 2n, then we get

a2nΦ(xa2n)=j=-2najφ(xaj)+1a2nk=-2n1a2kψ(akx)=i=2naiφ(xai)+1a2ni=12na2iψ(xai)+1a2ni=01a2iψ(aix)=i=2naiφ(xai)+1ani=1n-1aianaiψ(xai)+i=n2naia2naiψ(xai)+1a2nΨ˜(x)i=2naiφ(xai)+1ani=1aiψ(xai)+i=naiψ(xai)+1a2nΨ˜(x)

for any xV {0}. Thus, we obtain

limna2nΦ(xa2n)=0,limna2n+1Φ(xa2n+1)=alimna2nΦ(1a2nxa)=0

for any xV {0}. Hence, it holds that

limnanΦ(xan)=0

for xV {0}.

Theorem 2.1 implies that our conclusion for this corollary is true.

Corollary 3.3

Assume that a > 1 is a fixed real number and the functions φ, ψ : V {0} → [0,∞) satisfy each of the following conditions

i=0a2iψ(xai)<,i=01a3iφ(aix)<,Φ˜(x):=i=0a2iφ(xai)<,Ψ˜(x):=i=01a3iψ(aix)<

for all xV {0}. Suppose f : VY is an arbitrary mapping. If a mapping F : VY satisfies the inequality

f(x)-F(x)Φ˜(x)+Ψ˜(x)

for all xV {0} and each of the conditions in (2.7) for all xV, then F is uniquely determined.

Proof

We set Φ(x) := Φ̃(x) + Ψ̃(x) and then use (3.5) to show

1a6nΦ(a2nx)=i=01a6n-2iφ(a2n-ix)+i=01a6n+3iψ(a2n-ix)

for all xV {0}. We change the summation indices in the preceding equality with j = i – 2n and k = 2n + i to get

1a6nΦ(a2nx)=1a2nj=-2na2jφ(xaj)+k=2n1a3kψ(akx)=1a2ni=12n1a2iφ(aix)+1a2ni=0a2iφ(xai)+i=2n1a3iψ(aix)=1ani=1n-11a2i+nφ(aix)+i=n2n1a2n+2iφ(aix)+1a2nΦ˜(x)+i=2n1a3iψ(aix)1ani=11a3iφ(aix)+i=n1a3iφ(aix)+1a2nΦ˜(x)+i=2n1a3iψ(aix)

for any xV {0}. Hence, we get

limn1a6nΦ(a2nx)=0

for all xV {0}.

On the other hand, it follows from the above equality that

limn1a6n+3Φ(a2n+1x)=1a3limn1a6nΦ(a2nax)=0

for each xV {0}. By two equalities above, it holds that

limn1a3nΦ(anx)=0

for all xV {0}.

Similarly, we have

a4nΦ(xa2n)=i=0a4n+2iφ(xa2n+i)+i=01a3i-4nψ(ai-2nx)

for all xV {0}. If we change the summation indices in the last equality with j = i + 2n and k = i – 2n, then we get

a4nΦ(xa2n)=j=2na2jφ(xaj)+1a2nk=-2n1a3kψ(akx)=i=2na2iφ(xai)+1a2ni=12na3iψ(xai)+1a2ni=01a3iψ(aix)=i=2na2iφ(xai)+1ani=1n-1aiana2iψ(xai)+i=n2naia2na2iψ(xai)+1a2nΨ˜(x)i=2na2iφ(xai)+1ani=1a2iψ(xai)+i=na2iψ(xai)+1a2nΨ˜(x)

for any xV {0}. Thus, we obtain

limna4nΦ(xa2n)=0,limna4n+2Φ(xa2n+1)=a2limna4nΦ(1a2nxa)=0

for every xV {0}. Hence, we have

limna2nΦ(xan)=0

for each xV {0}. Theorem 2.1 implies that our assertion is true.

Corollary 3.4

Assume that a > 1 is a fixed real number and the functions φ, ψ : V {0} → [0,∞) satisfy each of the following conditions

i=0a3iψ(xai)<,i=01a4iφ(aix)<,Φ˜(x):=i=0a3iφ(xai)<,Ψ˜(x):=i=01a4iψ(aix)<

for all xV {0}. Suppose f : VY is an arbitrary mapping. If a mapping F : VY satisfies the inequality

f(x)-F(x)Φ˜(x)+Ψ˜(x)

for all xV {0} as well as the conditions in (2.7) for all xV, then F is uniquely determined.

Proof

Let us set Φ(x) := Φ̃(x) + Ψ̃(x) and use (3.7) to get

1a4nΦ(anx)=i=01a4n-3iφ(an-ix)+i=01a4n+4iψ(an+ix)

for all xV {0}. We make change of the summation indices in the preceding equality with j = in and k = n + i to get

1a4nΦ(anx)=1a4nj=-na3jφ(xaj)+k=n1a4kψ(akx)=1a4ni=1n1a3iφ(aix)+1a4ni=0a3iφ(xai)+i=n1a4iψ(aix)=1a3ni=1naian1a4iφ(aix)+1a4nΦ˜(x)+i=n1a4iψ(aix)1a3ni=11a4iφ(aix)+1a4nΦ˜(x)+i=n1a4iψ(aix)

for any xV {0}. Hence, we get

limn1a4nΦ(anx)=0

for all xV {0}.

Similarly, we obtain

a3nΦ(xan)=i=0a3n+3iφ(xan+i)+i=01a4i-3nψ(ai-nx)

for all xV {0}. If we change the summation indices in the last equality with j = i + n and k = in, then we get

a3nΦ(xan)=j=na3jφ(xaj)+1a3nk=-n1a4kψ(akx)=i=na3iφ(xai)+1a3ni=1na4iψ(xai)+1a3ni=01a4iψ(aix)=i=na3iφ(xai)+1a2ni=1naiana3iψ(xai)+1a3nΨ˜(x)i=na3iφ(xai)+1a2ni=1a3iψ(xai)+1a3nΨ˜(x)

for any xV {0}. Thus, we get

limna3nΦ(xan)=0

for each xV {0}. Theorem 2.1 implies that our conclusion is true.

The following corollary states that if, for any given mapping f, there exists an additive, a quadratic, a cubic, a quartic, or an additive-quadratic-quartic mapping F near f, then F is uniquely determined.

Corollary 3.5

Assume that a > 1 is a fixed rational number and a function φ : V {0} → [0,∞) satisfies the condition (3.1) or (3.2). Suppose f : VY is an arbitrary mapping. If an additive, a quadratic, a cubic, a quartic, or an additive-quadratic-cubic-quartic mapping F : VY satisfies the inequality (2.6), then F is uniquely determined.

The following corollaries are immediate consequences of Corollaries 3.2, 3.3, and 3.4, respectively, because each of additive, quadratic, cubic, quartic, and additive-quadratic-cubic-quartic mapping satisfies the conditions in (2.7) for any given rational number a > 1.

Corollary 3.6

Assume that a > 1 is a fixed rational number and φ, ψ : V {0} → [0,∞) satisfy each of the conditions in (3.3). Suppose f : VY is an arbitrary mapping. If an additive, a quadratic, a cubic, a quartic, or an additive-quadratic-cubic-quartic mapping F : VY satisfies (3.4), then F is uniquely determined.

Corollary 3.7

Assume that a > 1 is a fixed rational number and φ, ψ : V {0} → [0,∞) satisfy each of the conditions in (3.5). Suppose f : VY is an arbitrary mapping. If an additive, a quadratic, a cubic, a quartic, or an additive-quadratic-cubic-quartic mapping F : VY satisfies (3.6), then F is uniquely determined.

Corollary 3.8

Assume that a > 1 is a fixed rational number and φ, ψ : V {0} → [0,∞) satisfy each of the conditions in (3.7). Suppose f : VY is an arbitrary mapping. If an additive, a quadratic, a cubic, a quartic, or an additive-quadratic-cubic-quartic mapping F : VY satisfies (3.8), then F is uniquely determined.

If we set Φ(x) := θ||x||p for some constants pR{1, 2, 3, 4} and θ > 0, then

Φhas the property {(2.1)for p<1,(2.2)for 1<p<2,(2.3)for 2<p<3,(2.4)for 3<p<4,(2.5)for p>4.

Hence, by Theorem 2.1, we have the following corollary concerning the Hyers-Ulam-Rassias stability. (For the exact definition of the Hyers-Ulam-Rassias stability, we refer to [4, 16, 18].)

Corollary 3.9

Let p ∉ {1, 2, 3, 4} and θ > 0 be real constants, let X and Y be real normed spaces, and let f : XY be an arbitrary mapping. If a mapping F : XY satisfies the inequality

f(x)-F(x)θxp

for all xX{0} as well as (2.7) for all xX, then F is uniquely determined.

Since each of additive, quadratic, cubic, quartic, or additive-quadratic-cubic-quartic mappings satisfies the conditions in (2.7), using Corollary 3.9, we can easily prove the following corollary.

Corollary 3.10

Let p ∉ {1, 2, 3, 4} and θ > 0 be real constants, let X and Y be real normed spaces, and let f : XY be an arbitrary mapping. If an additive, a quadratic, a cubic, a quartic, or an additive-quadratic-cubic-quartic mapping F : XY satisfies the inequality

f(x)-F(x)θxp

for all xX{0}, then F is uniquely determined.

Acknowledgements

Soon-Mo Jung was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2016R1D1A1B03931061).

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