# Functional Inequalities Associated  with Cauchy Additive Functional Equation in  Non-Archimedean Spaces

Authors: A. Ebadian; N. Ghobadipour; Th. M. Rassias; M. Eshaghi Gordji
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

We investigate the generalized Hyers-Ulam stability of the functional inequalities∥f((x+y+z)/4)+f((3x−y−4z)/4)+f((4x+3z)/4)∥≤∥2f(x)∥ and ∥f((y−x)/3)+f((x−3z)/3)+f((3x+3z−y)/3)∥≤∥f(x)∥ in non-Archimedean normed spaces in the spirit of the Th. M. Rassias stability approach.


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## Body

## 1. Introduction

Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms.Let(G1,·)be a group and let(G2,*)be a metric group with the metricd(·,·). 					Givenϵ>0, does there exist aδ>0, such that if a mappingh:G1→G2satisfies the inequalityd(h(x·y),h(x)*h(y))<δfor allx,y∈G1, then there exists a homomorphismH:G1→G2withd(h(x),H(x))<ϵfor allx∈G1?In other words, under what condition does there exist a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers [2] gave the first affirmative answer to the question of Ulam for Banach spaces. Let f:E→E′ be a mapping between Banach spaces such that (1.1)‖f(x+y)-f(x)-f(y)‖≤δ
for all x,y∈E, and for some δ>0. Then there exists a unique additive mapping T:E→E′ such that (1.2)‖f(x)-T(x)‖≤δ
for all x∈E. Moreover, if f(tx) is continuous in t∈ℝ for each fixed x∈E, then T is linear. In 1978, Rassias [3] proved the following theorem.Theorem 1.1.
Letf:E→E′ be a mapping from a normed vector space E into a Banach space E′ subject to the inequality
(1.3)‖f(x+y)-f(x)-f(y)‖≤ϵ(‖x‖p+‖y‖p)
for all x,y∈E, where ϵ and p are constants with ϵ>0 and p<1. Then there exists a unique additive mapping T:E→E′ such that
(1.4)‖f(x)-T(x)‖≤2ϵ2-2p‖x‖p
for all x∈E. If p<0 then inequality (1.3) holds for all x,y≠0, and (1.4) for x≠0. Also, if the function t↦f(tx) from ℝ into E′ is continuous in real t for each fixed x∈E, then T is linear.In 1991, Gajda [4] answered the question for the case p>1, which was raised by Rassias. This new concept is known as Hyers-Ulam-Rassias stability of functional equations. The reader is referred to [5–13] for a number of results in this domain of research.In 1994, a generalization of the Rassias theorem was obtained by Găvruţa as follows [14].Suppose(G,+) is an abelian group, E is a Banach space, and that the so-called admissible control function φ:G×G→ℝ satisfies (1.5)φ̃(x,y):=2-1∑n=0∞2-nφ(2nx,2ny)<∞
for all x,y∈G. If f:G→E is a mapping with (1.6)‖f(x+y)-f(x)-f(y)‖≤φ(x,y)
for all x,y∈G, then there exists a unique mapping T:G→E such that T(x+y)=T(x)+T(y) and ∥f(x)-T(x)∥≤φ̃(x,x) for all x,y∈G.During the last decades, several stability problems of functional equations have been investigated by a number of mathematicians, see [15–17] and references therein for more detailed information.By anon-Archimedean field we mean a field K equipped with a function (valuation) |·| from K into [0,1) such that |r|=0 if and only if r=0, |rs|=|r||s|, and |r+s|≤max{|r|,|s|} for all r,s∈K. Clearly |1|=|-1|=1 and |n|≤1 for all n∈ℕ.LetX be a vector space over a scalar field 𝕂 with a non-Archimedean nontrivial valuation |·|. A function ∥·∥:X→ℝ is a non-Archimedean norm (valuation) if it satisfies the following conditions:(i)
∥x∥=0 if and only if x=0;(ii)
∥rx∥=|r|∥x∥(r∈𝕂,x∈X);(iii)
the strong triangle inequality (ultrametric), namely,(1.7)‖x+y‖≤max⁡{‖x‖,‖y‖}(x,y∈X).Then(X,∥·∥) is called a non-Archimedean space. Due to the fact that (1.8)‖xn-xm‖≤max⁡{‖xj+1-xj‖:m≤j≤n-1}(n>m),
a sequence {xn} is Cauchy if and only if {xn+1-xn} converges to zero in a non-Archimedean space. By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent (see [18–22]).Gilányi [23] and Rätz [24] showed that if f satisfies the functional inequality (1.9)‖2f(x)+2f(y)-f(xy-1)‖≤f(xy),
then f satisfies the Jordan-von Neumann functional equation (1.10)2f(x)+2f(y)=f(xy)+f(xy-1).
Gilányi [23] and Fechner [25] proved the generalized Hyers-Ulam stability of the functional inequality (1.3).Cho and Kim [26] proved the generalized Hyers-Ulam stability of the following functional inequalities: (1.11)‖f(x-y2-z)+f(y)+2f(z)‖≤‖f(x+y2+z)‖+φ(x,y,z),‖f(x)+f(y)+2f(z)‖≤‖2f(x+y2+z)‖+φ(x,y,z),
which are associated with Jordan-von Neumann-type Cauchy-Jensen additive functional equations.Now, we consider the following functional inequality:(1.12)‖f(x+y+z4)+f(3x-y-4z4)+f(4x+3z4)‖≤‖2f(x)‖,(1.13)‖f(y-x3)+f(x-3z3)+f(3x+3z-y3)‖≤‖f(x)‖,
which is associated with Cauchy additive functional equation.The purpose of this paper is to prove that iff satisfies the inequalities (1.12) and (1.13), which satisfies certain conditions, then f is Cauchy additive, and thus we prove the generalized Hyers-Ulam stability of the functional inequalities (1.12) and (1.13) in non-Archimedean normed spaces.


## 2. Stability of Functional Inequality (1.12)

In this section, we prove the generalized Hyers-Ulam stability of the functional inequality (1.12). Throughout this section, we assume that S is an additive semigroup and X is a complete non-Archimedean space.We need the following lemma in the main results.Lemma 2.1.
Letf:S→X be a mapping such that
(2.1)‖f(x+y+z4)+f(3x-y-4z4)+f(4x+3z4)‖≤‖2f(x)‖
for all x,y,z∈S. If |3|>|2|, then the mapping f is Cauchy additive.Proof.
Lettingx=y=z=0 in (2.1), we get |3|∥f(0)∥≤|2|∥f(0)∥. So, f(0)=0. Letting x=z=0 and replacing y by 4y in (2.1), we get ∥f(y)+f(-y)∥≤|2|∥f(0)∥ for all y∈S. So, f(-y)=-f(y) for all y∈S. Setting x=0 in (2.1), we obtain
(2.2)‖f(y+z4)+f(-y-4z4)+f(3z4)‖≤|2|‖f(0)‖.
So,
(2.3)f(y+z4)+f(-y-4z4)+f(3z4)=0
for all y,z∈S. Replacing y by 2z in (2.3), we get
(2.4)2f(3z4)=f(3z2)
for all z∈S. Using (2.4), we obtain f(2z)=2f(z) and f(4z)=4f(z) for all z∈S. Letting x=0, w1=(y+z)/4 and w2=(y-4z)/4, in (2.1) we get
(2.5)f(w1)+f(w2)=f(w1+w2)
for all w1,w2∈S. Hence, f is additive.Theorem 2.2.
Letφ:S×S×S→ℝ+∪{0} be a function such that
(2.6)lim⁡n→∞max⁡[φ(2nx,2ny,2nz)|2|n,φ(-2nx,-2ny,-2nz)|2|n]=0
for all x,y,z∈S and let the limit
(2.7)φ̃(z):=lim⁡n→∞max⁡{max⁡[φ(0,2.2iz,2iz)|2|i,φ(0,-2.2iz,-2iz)|2|i];0≤i<n}
exists for all z∈S. Suppose that f:S→X with f(0)=0 is a mapping satisfying
(2.8)‖f(x+y+z4)+f(3x-y-4z4)+f(4x+3z4)‖≤|2|‖f(x)‖+φ(x,y,z)
for all x,y,z∈S. Then there exists an additive mapping h:S→X such that
(2.9)‖f(z)-f(-z)2-h(z)‖≤1|2|2φ̃(z)
for all z∈S. Moreover, if
(2.10)lim⁡k→∞lim⁡n→∞max⁡{max⁡[φ(0,2.2iz,2iz)|2|i,φ(0,-2.2iz,-2iz)|2|i];k≤i<n+k}=0
then h is the unique additive mapping satisfying (2.9).Proof.
Puttingx=0 and y=2z in (2.8), we get
(2.11)‖2f(3z4)+f(-3z2)‖≤φ(0,2z,z)
for all z∈S. Replacing z by 4z/3 in (2.11), we obtain
(2.12)‖2f(z)+f(-2z)‖≤φ(0,8z3,4z3)
for all z∈S. Replacing z by -z in (2.12), we get
(2.13)‖2f(-z)+f(2z)‖≤φ(0,-8z3,-4z3)
for all z∈S. Let g(z):=(f(z)-f(-z))/2. It follows from (2.12) and (2.13) that
(2.14)‖g(2z)-2g(z)‖≤1|2|max⁡[φ(0,8z3,4z3),φ(0,-8z3,-4z3)]
for all z∈S. Replacing z by 2n-1z in (2.14), we get
(2.15)‖g(2nz)2n-g(2n-1z)2n-1‖≤1|2|n+1max⁡[φ(0,8.2n-1z3,4.2n-1z3),φ(0,-8.2n-1z3,-4.2n-1z3)]
for all z∈S. It follows from (2.6) and (2.15) that the sequence {g(2nz)/2n} is Cauchy. Since X is complete, we conclude that {g(2nz)/2n} is convergent. Set h(z):=limn→∞(g(2nz)/2n) for all z∈S. Using induction one can show that
(2.16)‖g(2nz)2n-g(z)‖≤1|2|2max⁡{max⁡[φ(0,8.2kz/3,4.2kz/3)|2|k,φ(0,-8.2kz/3,-4.2kz/3)|2|k];0≤k<n}
for all z∈S and n∈ℕ. By taking n to approach infinity in (2.16) and using (2.7) one obtains (2.9).
It follows from (2.8) that
(2.17)‖h(x+y+z4)+h(3x-y-4z4)+h(4x+3z4)‖=lim⁡n→∞1|2|n‖g(2n(x+y+z4))+g(2n(3x-y-4z4))+g(2n(4x+3z4))‖=lim⁡n→∞1|2|n+1‖f(2n(x+y+z))4+f(2n(3x-y-4z))4+f(2n(4x+3z))4-f(2n(-x-y-z))4-f(2n(-3x+y+4z))4-f(2n(-4x-3z))4‖≤1|2|n‖f(2nz)-f(-2nz)‖+lim⁡n→∞1|2|max⁡[φ(2nx,2ny,2nz)|2|n,φ(-2nx,-2ny,-2nz)|2|n]=|2|‖h(z)‖
for all x,y,z∈S. So,
(2.18)‖h(x+y+z4)+h(3x-y-4z4)+h(4x+3z4)‖≤|2|‖h(z)‖
for all x,y,z∈S. By Lemma 2.1, the mapping h:S→X is additive.
Now, letT:S→X be another additive mapping satisfying (2.9). Then we have
(2.19)‖h(z)-T(z)‖=1|2|k‖h(2kz)-T(2kz)‖≤1|2|kmax⁡[‖T(2kz)-g(2kz)‖,‖g(2kz)-h(2kz)‖]≤1|2|2lim⁡k→∞lim⁡n→∞max⁡{max⁡[φ(0,2.2iz,2iz)|2|i,φ(0,-2.2iz,-2iz)|2|i];k≤i<i+k}=0
for all z∈S. Therefore h=T. This completes the proof of the uniqueness of h.Corollary 2.3.
Letp>1 and θ be positive real numbers, and let f:S→X be a mapping satisfying
(2.20)‖f(x+y+z4)+f(3x-y-4z4)+f(4x+3z4)‖≤|2|‖f(x)‖+θ(‖x‖p+‖y‖p+‖z‖p)
for all x,y,z∈S. If |2|<1 then there exists a unique additive mapping h:S→X such that
(2.21)‖f(z)-f(-z)2-h(z)‖≤2θ|2|2‖z‖p
for all z∈S.Proof.
Definingφ:S×S×S→X by φ(x,y,z):=θ(∥x∥p+∥y∥p+∥z∥p) we have
(2.22)lim⁡n→∞max⁡[φ(2nx,2ny,2nz)|2|n,φ(-2nx,-2ny,-2nz)|2|n]=lim⁡n→∞θ|2|np|2|n(‖x‖p+‖y‖p+‖z‖p)=0,φ̃(z):=lim⁡n→∞max⁡{max⁡[φ(0,2.2iz,2iz)|2|i,φ(0,-2.2iz,-2iz)|2|i];0≤i<n}=lim⁡n→∞max⁡{|2|(i+1)p+|2|ip|2|iθ‖z‖p;0≤i<n}≤2θ‖z‖p,lim⁡k→∞lim⁡n→∞max⁡{max⁡[φ(0,2.2iz,2iz)|2|i,φ(0,-2.2iz,-2iz)|2|i];k≤i<n+k}≤lim⁡k→∞|2|kpθ‖z‖p=0
for all z∈S.
Applying Theorem2.2, we conclude the required result.Corollary 2.4.
Letψ:ℝ+∪{0}→ℝ+∪{0} be a function satisfying
(2.23)ψ(|2|r)≤ψ(|2|)ψ(r)(r≥0),ψ(|2|)<|2|.
Let θ>0, let S be a normed space and let f:S→X fulfill the inequality
(2.24)‖f(x+y+z4)+f(3x-y-4z4)+f(4x+3z4)‖≤|2|‖f(x)‖+θ[ψ(‖x‖)+ψ(‖y‖)+ψ(‖z‖)]
for all x,y,z∈S. Then there exists a unique additive mapping h:S→X such that
(2.25)‖f(z)-f(-z)2-h(z)‖≤2θ|2|2ψ(‖z‖)
for all z∈S.Proof.
Definingφ:S×S×S→X by φ(x,y,z):=θ[ψ(∥x∥)+ψ(∥y∥)+ψ(∥z∥)] we have
(2.26)lim⁡n→∞max⁡[φ(2nx,2ny,2nz)|2|n,φ(-2nx,-2ny,-2nz)|2|n]≤θlim⁡n→∞max⁡[(ψ(|2|)|2|)n(φ(x,y,z),φ(-x,-y,-z))]=0,φ̃(z):=lim⁡n→∞max⁡{max⁡[φ(0,2.2iz,2iz)|2|i,φ(0,-2.2iz,-2iz)|2|i];0≤i<n}=2θψ(‖z‖),lim⁡k→∞lim⁡n→∞max⁡{max⁡[φ(0,2.2iz,2iz)|2|i,φ(0,-2.2iz,-2iz)|2|i];k≤i<n+k}≤lim⁡k→∞(ψ(|2|)|2|)kψ(‖z‖)=0
for all z∈S.
Applying Theorem2.2, we conclude the required result.Theorem 2.5.
Letφ:S×S×S→ℝ+∪{0} be a function such that
(2.27)lim⁡n→∞max⁡{|2|nφ(2-nx,2-ny,2-n),|2|nφ(-2-nx,-2-ny,-2-nz)}=0
for all x,y,z∈S and let the limit
(2.28)φ̃(z):=lim⁡n→∞max⁡{max⁡[|2|nφ(0,2.2-iz,2-iz),|2|nφ(0,-2.2-iz,-2-iz)];0≤i<n}
exist for all z∈S. Suppose that f:S→X with f(0)=0 is a mapping satisfying
(2.29)‖f(x+y+z4)+f(3x-y-4z4)+f(4x+3z4)‖≤|2|‖f(x)‖+φ(x,y,z)
for all x,y,z∈S. Then there exists an additive mapping h:S→X such that
(2.30)‖f(z)-f(-z)2-h(z)‖≤1|2|φ̃(z)
for all z∈S. Moreover, if
(2.31)lim⁡k→∞lim⁡n→∞max⁡{max⁡[|2|iφ(0,2.2-iz,2-iz),|2|iφ(0,-2.2-iz,-2-iz)];k≤i<n+k}=0
then h is the unique additive mapping satisfying (2.30).Proof.
It follows from (2.14) that
(2.32)‖2g(z2)-g(z)‖≤1|2|max⁡{φ(0,4z3,2z3),φ(0,-4z3,-2z3)}
for all z∈S. Hence,
(2.33)‖2ng(2-nz)-2(n+1)g(2-(n+1)z)‖≤|2|nmax⁡{φ(0,4.2-nz3,2.2-nz3),φ(0,-4.2-nz3,-2.2-nz3)}
for all z∈S. It follows from (2.27) and (2.33) that the sequence {2ng(2-nz)} is a Cauchy sequence for all z∈S. Since X is complete, the sequence {2ng(2-nz)} converges. So, one can define the mapping h:S→X by h(z):=limn→∞{2ng(2-nz)} for all z∈S.
The rest of the proof is similar to the proof of Theorem2.2.Remark 2.6.
We can obtain similar results to Corollary2.3 for p<1 and Corollary 2.4.


## 3. Stability of Functional Inequality (1.13)

We prove the generalized Hyers-Ulam stability of the functional inequality (1.13). Throughout this section, we assume that S is an additive semigroup and X is a complete non-Archimedean space.We need the following lemma in the main results.Lemma 3.1.
Letf:S→X be a mapping such that
(3.1)‖f(y-x3)+f(x-3z3)+f(3x+3z-y3)‖≤‖f(x)‖
for all x,y,z∈S. If f(0)=0, then the mapping f is Cauchy additive.Proof.
Lettingx=y=0 in (3.1), we get
(3.2)‖f(-z)+f(z)‖≤‖f(0)‖=0
for all z∈S. Hence, f(-z)=-f(z) for all z∈S. Letting x=0 and y=6z in (3.1), we get
(3.3)‖f(2z)-2f(z)‖≤‖f(0)‖=0
for all z∈S. Hence,
(3.4)f(2z)=2f(z)
for all z∈S. Letting x=0 and y=9z in (3.1), we get
(3.5)‖f(3z)-f(z)-2f(z)‖≤‖f(0)‖=0
for all z∈S. Hence,
(3.6)f(3z)=3f(z)
for all z∈S. Letting x=0 in (3.1), we get
(3.7)‖f(y3)+f(-z)+f(z-y3)‖≤‖f(0)‖=0
for all x,y,z∈S. So,
(3.8)f(y3)+f(-z)+f(z-y3)=0
for all x,y,z∈S. Let t1=z-y/3 and t2=y/3 in (3.8). Then
(3.9)f(t2)-f(t1+t2)+f(t1)=0
for all t1,t2∈S. So, f is additive.Theorem 3.2.
Letφ:S×S×S→[0,∞) be a function such that
(3.10)lim⁡n→∞φ(2nx,2ny,2nz)|2|n=0
for all x,y,z∈S and let the limit
(3.11)φ̃(z):=lim⁡n→∞max⁡{max⁡[φ(0,6.2iz,2iz)|2|i,|2|φ(0,3.2iz,2iz)|2|i];0≤i<n}
exist for all z∈S. Suppose that f:S→X with f(0)=0 is a mapping satisfying
(3.12)‖f(y-x3)+f(x-3z3)+f(3x+3z-y3)‖≤‖f(x)‖+φ(x,y,z)
for all x,y,z∈S. Then there exists an additive mapping T:S→X such that
(3.13)‖f(z)-T(z)‖≤1|2|φ̃(z)
for all z∈S. Moreover, if
(3.14)lim⁡k→∞lim⁡n→∞max⁡{max⁡[φ(0,6.2iz,2iz)|2|i,|2|φ(0,3.2iz,2iz)|2|i];k≤i<n+k}=0
then T is the unique additive mapping satisfying (3.13).Proof.
Lettingx=0 and y=6z in (3.12), we get
(3.15)‖f(2z)+2f(-z)‖≤φ(0,6z,z)
for all z∈S. Putting x=0 and y=3z in (3.12), we get
(3.16)‖2f(z)+2f(-z)‖≤|2|φ(0,3z,z)≤φ(0,3z,z)
for all z∈S. It follows from (3.15) and (3.16) that
(3.17)‖f(2z)-2f(z)‖≤max⁡{φ(0,6z,z),φ(0,3z,z)}
for all z∈S. Replacing z by 2n-1z in (3.17), we get
(3.18)‖f(2nz)2n-f(2n-1z)2n-1‖≤1|2|nmax⁡[φ(0,6.2n-1z,2n-1z),φ(0,3.2n-1z,2n-1z)]
for all z∈S. It follows from (3.10) and (3.18) that the sequence {f(2nz)/2n} is Cauchy. Since X is complete, we conclude that {f(2nz)/2n} is convergent. Set T(z):=limn→∞(f(2nz)/2n) for all z∈S. Using induction one can show that
(3.19)‖f(2nz)2n-f(z)‖≤1|2|max⁡{max⁡[φ(0,6.2kz,2kz)|2|k,|2|φ(0,3.2kz,2kz)|2|k];0≤k<n}
for all z∈S and n∈ℕ. By taking n to approach infinity in (3.19) and using (3.11) one obtains (3.13).
Replacingx,  y, and z by 2nx,2ny, and 2nz, respectively, in (3.12) we get
(3.20)‖f(2n(y-x)3.2n)+f(2n(x-3z)3.2n)+f(2n(3x+3z-y)3.2n)‖≤‖f(2nx)2n‖+1|2|nφ(2nx,2ny,2nz)
for all x,y,z∈S. Taking the limit as n→infty and using (3.10) we get
(3.21)‖T(y-x3)+T(x-3z3)+T(3x+3z-y3)‖≤‖T(x)‖
for all x,y,z∈S. By Lemma 3.1, the mapping T:S→X is additive.
IfT′ is another additive mapping satisfying (3.13), then
(3.22)‖T(z)-T′(z)‖=1|2|k‖T(2kz)-T′(2kz)‖≤1|2|kmax⁡[‖T(2kz)-f(2kz)‖,‖f(2kz)-T′(2kz)‖]≤1|2|lim⁡k→∞lim⁡n→∞max⁡{max⁡[φ(0,6.2iz,2iz)|2|i,|2|φ(0,3.2iz,2iz)|2|i];k≤i<i+k}=0
for all z∈S. Therefore T′=T. This proves the uniqueness of T. Hence, the mapping T:S→X is a unique additive mapping satisfying (3.13).Corollary 3.3.
Letp>1 and θ be positive real numbers, and let f:S→X be a mapping satisfying
(3.23)‖f(y-x3)+f(x-3z3)+f(3x+3z-y3)‖≤‖f(x)‖+θ(‖x‖p+‖y‖p+‖z‖p)
for all x,y,z∈S. If |2|<1 then there exists a unique additive mapping T:S→X such that
(3.24)‖f(z)-T(z)‖≤2θ|2|‖z‖p
for all z∈S.Proof.
Lettingφ(x,y,z):=θ(∥x∥p+∥y∥p+∥z∥p) in Theorem 3.2, we obtain the result.Corollary 3.4.
Letψ:[0,∞)→[0,∞) be a function satisfying
(3.25)ψ(|2|r)≤ψ(|2|)ψ(r)(r≥0),ψ(|2|)<|2|.
Let θ>0, let S be a normed space, and let f:S→X fulfill the inequality
(3.26)‖f(y-x3)+f(x-3z3)+f(3x+3z-y3)‖≤‖f(x)‖+θ[ψ(‖x‖)+ψ(‖y‖)+ψ(‖z‖)]
for all x,y,z∈S. Then there exists a unique additive mapping T:S→X such that
(3.27)‖f(z)-T(z)‖≤2θ|2|ψ(‖z‖)
for all z∈S.Proof.
If we defineφ(x,y,z):=θ[ψ(∥x∥)+ψ(∥y∥)+ψ(∥z∥)] in Theorem 3.2, then we get the result.Remark 3.5.
We can formulate a similar statement to Theorem3.2 in which we can define the sequence T(z):=limn→∞2nf(z/2n) under suitable conditions on the function φ and ψ and then obtain similar results to Corollary 3.3 for p<1 and Corollary 3.4.



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*Source: 929824-2011-06-08.xml*