R E S E A R C H
Open Access
Approximation on the reciprocal-cubic
and reciprocal-quartic functional equations
in non-Archimedean fields
Sang Og Kim
1, Beri Venkatachalapathy Senthil Kumar
2and Abasalt Bodaghi
3**Correspondence:
[email protected] 3Young Researchers and Elite Club,
Islamshahr Branch, Islamic Azad University, Islamshahr, Iran Full list of author information is available at the end of the article
Abstract
The aim of this paper is to study the generalized Hyers-Ulam stability of a form of reciprocal-cubic and reciprocal-quartic functional equations in non-Archimedean fields. Some related examples for the singular cases of these new functional equations on an Archimedean field are indicated.
MSC: 39B82; 39B72
Keywords: reciprocal functional equation; reciprocal-cubic functional equation; reciprocal-quartic functional equation; generalized Hyers-Ulam stability;
non-Archimedean field
1 Introduction
The study of the stability of functional equations was instigated by the famous question of Ulam [] during a Mathematical Colloquium at the University of Wiskonsin in the year . In the successive year, Hyers [] provided a partial answer to the question of Ulam. Later, Hyers’s result was extended and generalized for a Cauchy functional equation by Bourgin [], Th.M. Rassias [], Gruber [], Aoki [], J.M. Rassias [] and Găvruta [] in var-ious adaptations. After that several stability articles, many textbooks and research mono-graphs have investigated the result for various functional equations, also for mappings with more general domains and ranges; for instance, see [–] and [].
In , Ravi and Senthil Kumar [] obtained Ulam-Găvruta-Rassias stability for the Rassias reciprocal functional equation
r(x+y) = r(x)r(y)
r(x) +r(y), (.)
wherer:X−→Ris a mapping with X as the space of non-zero real numbers. The re-ciprocal functionr(x) =xc is a solution of the functional equation (.). The functional equation (.) holds good for the ‘reciprocal formula’ of any electric circuit with two resis-tors connected in parallel []. Ravi et al. [] obtained the solution of a new generalized reciprocal-type functional equation in two variables of the form
r(x+y) = kr(x+ (k– )y)r((k– )x+y)
r(x+ (k– )y) +r((k– )x+y), (.)
where k> is a positive integer, and investigated its generalized Hyers-Ulam stability in non-Archimedean fields. Then Senthil Kumar et al. [] found a general solution of a reciprocal-type functional equation
f(x+y) = f(
kx+ky
k )f( kx+ky
k ) f(kx+ky
k ) +f( kx+ky
k )
(.)
and investigated its generalized Hyers-Ulam-Rassias stability in non-Archimedean fields, wherek> ,kandkare positive integers withk=k+kandk=k. The other results
pertaining to the stability of different reciprocal-type functional equations can be found in [–] and [].
For the first time, Kim and Bodaghi [] introduced and studied the Ulam-Găvruta-Rassias stability for the quadratic reciprocal functional equation
f(x+y) +f(x–y) = f(x)f(y)[f(y) +f(x)]
(f(y) –f(x)) . (.)
Then the functional equation (.) was generalized in [] as
f(a+ )x+ay+f(a+ )x–ay=f(x)f(y)[(a+ )
f(y) +af(x)]
((a+ )f(y) –af(x)) , (.)
wherea∈Zwitha= , –. In [], the authors established the generalized Hyers-Ulam-Rassias stability for the functional equation (.) in non-Archimedean fields. Since then Ravi et al. [] investigated the generalized Hyers-Ulam-Rassias stability of a reciprocal-quadratic functional equation of the form
r(x+ y) +r(x+y) =r(x)r(y)[r(x) + r(y) +
r(x)r(y)]
[r(x) + r(y) + r(x)r(y)] (.)
in intuitionistic fuzzy normed spaces; for another form of a reciprocal-quadratic func-tional equation, see [].
In this paper, we introduce the reciprocal-cubic functional equation
c(x+y) +c(x+ y) =c(x)c(y)[c(x) +c(y) + c(x)
c(y)(c(x)+c(y))]
[c(x) + c(y)+ c(x)c(y)]
(.)
and the reciprocal-quartic functional equation
q(x+y) +q(x–y) =q(x)q(y)[q(x) + q(y) +
q(x)q(y)]
[q(y) –q(x)] . (.)
It can be verified that the reciprocal-cubic functionc(x) =
x and the reciprocal-quartic
functionq(x) = x are solutions of the functional equations (.) and (.), respectively.
2 Preliminaries
In this section, we recall the basic concepts of a non-Archimedean field.
Definition . By a non-Archimedean field, we mean a fieldKequipped with a function
(valuation)| · |fromKinto [,∞) such that|p|= if and only ifp= ,|pq|=|p||q|and |p+q| ≤max{|p|,|q|}for allp,q∈K.
Clearly,||=|–|= and|n| ≤ for alln∈N. We always assume, in addition, that| · |is non-trivial, i.e., there existsa∈Ksuch that|a| = , . Due to the fact that
|pn–pm| ≤max
|pj+–pj|:m≤j≤n–
(n>m),
a sequence{pn}is Cauchy if and only if{pn+–pn}converges to zero in a non-Archimedean
field. By a complete non-Archimedean field, we mean that every Cauchy sequence is con-vergent in the field.
An example of a non-Archimedean valuation is the mapping| · |taking everything but into and||= . This valuation is called trivial. Another example of a non-Archimedean valuation on a fieldAis the mapping
|k|=
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ifk= ,
k ifk> ,
–
k ifk<
for anyk∈A.
Letpbe a prime number. For any non-zero rational numberx=pr mn in whichmandn
are co-prime to the prime numberp, consider thep-adic absolute value|x|p=p–ronQ.
It is easy to check that| · |pis a non-Archimedean norm onQ. The completion ofQwith
respect to| · |p, which is denoted byQp, is said to be thep-adic number field. Note that if p> , then|n|
p= for all integersn.
Throughout this paper, we consider thatXandYare a non-Archimedean field and a
complete non-Archimedean field, respectively. From now on, for a non-Archimedean field
X, we putX∗=X\ {}. For the purpose of simplification, let us define the difference op-eratorsc,q:X∗×X∗−→Yby
c(x,y) =c(x+y) +c(x+ y) –
c(x)c(y)[c(x) +c(y) + c(x)c(y)(c(x)+c(y))]
[c(x) + c(y) + c(x)c(y)]
and
q(x,y) =q(x+y) +q(x–y) –
q(x)q(y)[q(x) + q(y) + q(x)q(y)] [q(y) –q(x)]
for allx,y∈X∗.
Definition . A mappingc:X∗−→Yis called a reciprocal-cubic mapping ifcsatisfies
equation (.). Also, a mappingq:X∗−→Yis called a reciprocal-quartic mapping ifq
3 Hyers-Ulam stability for equations (1.7) and (1.8)
In this section, we investigate the generalized Hyers-Ulam stability of equations (.) and (.) in non-Archimedean fields. We also establish the results pertaining to Hyers-Ulam stability, Hyers-Ulam-Rassias stability and Ulam-Găvruta-Rassias stability controlled by product-sum of powers of norms.
Theorem . Let l∈ {, –}be fixed,and let F:X∗×X∗−→[,∞)be a mapping such
Cauchy. SinceYis complete, this sequence converges to a mappingC:X∗−→Ydefined by
On the other hand, for eachx∈X∗and non-negative integersn, we have
Applying (.) and lettingn→ ∞in inequality (.), we find that inequality (.) holds.
Thus, the mappingCsatisfies (.) and hence it is a reciprocal-cubic mapping. In order to prove the uniqueness ofC, let us consider another reciprocal-cubic mappingC :X∗−→Y satisfying (.). Then
From now on, we assume that||< . The following corollaries are immediate conse-quences of Theorem . concerning the stability of (.).
Corollary . Let> be a constant.If c:X∗−→Ysatisfies|c(x,y)| ≤for all x,y∈
X∗,then there exists a unique reciprocal-cubic mapping C:X∗−→Ysatisfying(.)and |c(x) –C(x)| ≤for all x∈X∗.
Proof DefiningF(x,y) =and applying Theorem . for the casel= –, we get the desired
result.
Corollary . Let≥and r= –be fixed constants.If c:X∗−→Ysatisfies|c(x,y)| ≤
(|x|r+|y|r)for all x,y∈X∗,then there exists a unique reciprocal-cubic mapping C:X∗−→
Ysatisfying(.)and
ists a unique reciprocal-cubic mapping C:X∗−→Ysatisfying(.)and c(x) –C(x)≤
⎧ ⎨ ⎩
||r|x|r, r> –,
||||x|r, r< –
for all x∈X∗.
Proof The required result is obtained by choosingF(x,y) =|x|p|y|q for allx,y∈X∗ in
Theorem ..
Corollary . Let≥and r= –be real numbers and c:X∗−→Ybe a mapping
satis-fying the functional inequality
c(x,y)≤
|x|r|y|r +|x|r+|y|r
for all x,y∈X∗.Then there exists a unique reciprocal-cubic mapping C:X∗−→Y satisfy-ing(.)and
c(x) –C(x)≤
⎧ ⎨ ⎩
||r|x|r, r> –,
|||x|r, r< –
for all x∈X∗.
Proof ConsideringF(x,y) =(|x|r|y|r + (|x|r+|y|r)) in Theorem ., one can find the
re-sult.
We have the following result which is analogous to Theorem . for the functional equa-tion (.). We include the proof for the sake of completeness.
Theorem . Let l∈ {, –}be fixed,and let G:X∗×X∗−→[,∞)be a mapping such
that
lim n→∞
lnG
x
ln+l+
, y
ln+l+
= (.)
for all x,y∈X∗.Suppose that q:X∗−→Yis a mapping satisfying the inequality
q(x,y)≤G(x,y) (.)
for all x,y∈X∗.Then there exists a unique reciprocal-quartic mapping Q:X∗−→Ysuch that
q(x) –Q(x)≤sup
jl+
l–
F
x
jl+l+ , x
jl+l+
:j∈N∪ {}
(.)
Proof Replacing (x,y) by (x,x) in (.), we get
Here, we bring some corollaries regarding the stability of functional equation (.) which are a direct consequence of Theorem ..
Proof ChoosingG(x,y) =δ|x|α|y|αin Theorem ., one can derive the desired result.
Corollary . Letδ≥andα= –be real numbers and q:X∗−→Ybe a mapping
satisfying the functional inequality
Dq(x,y)≤δ
|x|α|y|α +|x|α+|y|α
for all x,y∈X∗.Then there exists a unique reciprocal-quartic mapping Q:X∗−→Y satis-fying(.)and
q(x) –Q(x)≤
⎧ ⎨ ⎩
δ
||α|x|α, α> –, δ|||x|α, α< –
for all x∈X∗.
Proof The proof follows immediately by takingG(x,y) =δ(|x|α|y|α + (|x|α+|y|α)) in
The-orem ..
4 Related examples
In this section, applying the idea of the well-known counter-example provided by Gajda [], we show that Corollary . forr= – and Corollary . forα= – do not hold inR with usual| · |. Note that (R,| · |) is an Archimedean field.
Consider the function
ϕ(x) =
⎧ ⎨ ⎩
δ
x forx∈(,∞),
δ, otherwise, (.)
whereϕ:R∗−→R. Letf :R∗−→Rbe defined by
f(x) = ∞
n=
–nϕ–nx (.)
for allx∈R∗.
Theorem . If the function f :R∗−→Rdefined in(.)satisfies the functional inequality
f(x,y)≤
δ
|x|–+|y|– (.)
for all x,y∈X,then there do not exist a reciprocal-cubic mapping c:R∗−→Rand a con-stantμ> such that
f(x) –c(x)≤μ|x|– (.)
Proof First, we are going to show thatf satisfies (.). By computation, we have
Therefore, we see thatf is bounded by δ
onR. If|x|
–+|y|–≥, then the left-hand
side of (.) is less than δ. Now, suppose that <|x|–+|y|–< . Hence, there exists a positive integerksuch that
k > . The last inequalities imply that x
for allx,y∈R∗. Therefore, inequality (.) holds. We claim that the reciprocal-cubic func-tional equation (.) is not stable forr= – in Corollary .. Assume that there exists a reciprocal-cubic mappingc:R∗−→Rsatisfying (.). Therefore,
f(x)≤(μ+ )|x|–. (.)
which contradicts (.). This completes the proof.
Also, letg:R∗−→Rbe defined by
Theorem . If the function g:R∗−→Rdefined in(.)satisfies the functional inequality
g(x,y)≤ constantβ> such that
g(x) –q(x)≤β|x|– (.)
for all x∈R∗.
Proof Let us first prove thatgsatisfies (.).
g(x)=
integermsuch that
for allx,y∈R∗. This shows that inequality (.) holds. Here, we prove that the reciprocal-quartic functional equation (.) is not stable forα= – in Corollary .. Assume that there exists a reciprocal-quartic mappingq:R∗−→Rsatisfying (.). Hence
g(x)≤(β+ )|x|–. (.)
On the other hand, we can choose a positive integerkwithkλ>β+ . Ifx∈(, k–), then
–nx∈(,∞) for alln= , , , . . . ,k– , and so
g(x)= ∞
n=
φ(–nx)
n ≥
k–
n= nλ
x
n = kλ
x > (β+ )x –,
which contradicts (.). Therefore, the reciprocal-quartic functional equation (.) is not
stable in the caseα= – in Corollary . for (R,| · |).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The study presented here was carried out in collaboration between all authors. BVSK suggested writing the current article. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Hallym University, Chuncheon, 24252, Republic of Korea.2C. Abdul Hakeem College of
Engineering and Technology, Melvisharam, Tamil Nadu 632509, India.3Young Researchers and Elite Club, Islamshahr Branch, Islamic Azad University, Islamshahr, Iran.
Acknowledgements
The authors would like to thank the referee for careful reading of the paper and giving some useful suggestions.
Received: 30 January 2017 Accepted: 23 February 2017 References
1. Ulam, SM: Topological spaces. In: Problems in Modern Mathematics, Chapter VI. Wiley-Interscience, New York (1964) 2. Hyers, DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA27, 222-224 (1941)
3. Bourgin, DG: Classes of transformations and bordering transformations. Bull. Am. Math. Soc.57, 223-237 (1951) 4. Rassias, TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc.72, 297-300 (1978) 5. Gruber, PM: Stability of isometries. Trans. Am. Math. Soc.245, 263-277 (1978)
6. Aoki, T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn.2, 64-66 (1950) 7. Rassias, JM: On approximation of approximately linear mappings by linear mappings. J. Funct. Anal.46, 126-130
(1982)
8. G˘avruta, P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mapppings. J. Math. Anal. Appl.184, 431-436 (1994)
9. Aczel, J: Lectures on Functional Equations and Their Applications, vol. 19. Academic Press, New York (1966) 10. Aczel, J: Functional Equations: History, Applications and Theory. Reidel, Dordrecht (1984)
11. Alsina, C: On the stability of a functional equation. In: General Inequalities, vol. 5, pp. 263-271. Birkhäuser, Basel (1987) 12. Bodaghi, A: Approximate mixed type additive and quartic functional equation. Bol. Soc. Parana. Mat.35(1), 43-56
(2017)
13. Bodaghi, A: Intuitionistic fuzzy stability of the generalized forms of cubic and quartic functional equations. J. Intell. Fuzzy Syst.30, 2309-2317 (2016)
14. Bodaghi, A: Stability of a mixed type additive and quartic function equation. Filomat28(8), 1629-1640 (2014) 15. Czerwik, S: Functional Equations and Inequalities in Several Variables. World Scientific, Singapore (2002) 16. Hyers, DH, Isac, G, Rassias, TM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel (1998) 17. Ravi, K, Rassias, JM, Arunkumar, M, Kodandan, R: Stability of a generalized mixed type additive, quadratic, cubic and
quartic functional equation. J. Inequal. Pure Appl. Math.10(4), Article ID 114 (2009)
18. Ravi, K, Senthil Kumar, BV: Ulam-G˘avruta-Rassias stability of Rassias reciprocal functional equation. Glob. J. Appl. Math. Math. Sci.3(1-2), 57-79 (2010)
19. Ravi, K, Senthil Kumar, BV: Stability and geometrical interpretation of reciprocal functional equation. Asian J. Curr. Eng. Maths1(5), 300-304 (2012)
20. Ravi, K, Rassias, JM, Senthil Kumar, BV: Ulam stability of a generalized reciprocal type functional equation in non-Archimedean fields. Arab. J. Math.4, 117-126 (2015)
22. Bodaghi, A, Narasimman, P, Rassias, JM, Ravi, K: Ulam stability of the reciprocal functional equation in non-Archimedean fields. Acta Math. Univ. Comen.85(1), 113-124 (2016)
23. Ravi, K, Rassias, JM, Senthil Kumar, BV: Ulam stability of reciprocal difference and adjoint functional equations. Aust. J. Math. Anal. Appl.8(1), Article ID 13 (2011)
24. Ravi, K, Rassias, JM, Senthil Kumar, BV: Ulam stability of generalized reciprocal functional equation in several variables. Int. J. Appl. Math. Stat.19, Article ID D10 (2010)
25. Ravi, K, Thandapani, E, Senthil Kumar, BV: Stability of reciprocal type functional equations. Panam. Math. J.21(1), 59-70 (2011)
26. Bodaghi, A, Kim, SO: Approximation on the quadratic reciprocal functional equation. J. Funct. Spaces Appl.2014, Article ID 532463 (2014)
27. Bodaghi, A, Ebrahimdoost, Y: On the stability of quadratic reciprocal functional equation in non-Archimedean fields. Asian-Eur. J. Math.9(1), Article ID 1650002 (2016)
28. Ravi, K, Rassias, JM, Kumar, BVS, Bodaghi, A: Intuitionistic fuzzy stability of a reciprocal-quadratic functional equation. Int. J. Appl. Sci. Math.1(1), 9-14 (2014)
29. Bodaghi, A, Rassias, JM, Park, C: Fundamental stabilities of an alternative quadratic reciprocal functional equation in non-Archimedean fields. Proc. Jangjeon Math. Soc.18(3), 313-320 (2015)