R E S E A R C H
Open Access
Approximation on the reciprocal
functional equation in several variables in
matrix non-Archimedean random normed
spaces
Ali Ebadian
1, Somaye Zolfaghari
1, Saeed Ostadbashi
1and Choonkil Park
2**Correspondence:
2Research Institute for Natural
Sciences, Hanyang University, Seoul, 133-791, Republic of Korea Full list of author information is available at the end of the article
Abstract
In this paper, we investigate the generalized Hyers-Ulam stability of a reciprocal type functional equation in several variables in matrix non-Archimedean random normed spaces by direct and fixed point methods.
MSC: 39B52; 47H10; 46S50; 54E70; 39B72; 47H09
Keywords: matrix non-Archimedean random normed spaces; Rassias reciprocal functional equation; general reciprocal functional equations; adjoint and difference functional equations; fixed point alternative; Hyers-Ulam-Rassias stability
1 Introduction
About years ago, Ulam [] raised the well-known stability problem of functional equa-tions. In the next year, Ulam’s problem was partially answered by Hyers [] in Banach spaces. Aoki [] generalized Hyers’ theorem for additive mappings in the year . In the year , a generalized version of the theorem of Hyers for approximately linear mappings was given by Rassias []. During -, Rassias [–] treated the Ulam-Gˇavruta-Rassias stability on linear and non-linear mappings and generalized Hyers’ re-sult. In , a further generalization of the Rassias theorem was obtained by Gˇavruta [], who replaced the boundθ(xp+yp) by a general control functionφ(x,y). The stability
problems of several functional equations have been extensively investigated by a number of mathematicians, posed with creative thinking and critical dissent who have arrived at interesting results (see [–]).
In the year , Ravi and Senthil Kumar [] investigated the generalized Hyers-Ulam stability for the reciprocal functional equation
r(x+y) = r(x)r(y)
r(x) +r(y), (.)
wherer:X→Y is a mapping on the spaces of non-zero real numbers. The reciprocal
functionr(x) =xc is a solution of the functional equation (.).
Later, Raviet al.[] introduced the reciprocal difference functional equation
and the reciprocal adjoint functional equation
r
and investigated the generalized Hyers-Ulam stability for the above two functional equa-tions (.) and (.).
Recently, Raviet al.[] discussed the generalized Hyers-Ulam stability for the general-ized reciprocal functional equation
Very recently, Raviet al.[] obtained the general solution and investigated the
gener-alized Hyers-Ulam stability of a reciprocal type functional equation in several variables of the form
wheremis a positive integer withm≥ in various normed spaces.
Remark . Raviet al.[–] gave some counter-examples for the stability of reciprocal
functional equations in singular cases.
In this paper, we apply a direct method and a fixed point method to investigate the gen-eralized Hyers-Ulam stability of the functional equation (.) in matrix non-Archimedean random normed spaces.
2 Preliminaries
In this section, we recall some definitions and results which will be used later in the article. A triangular norm (shortert-norm) is a binary operation on the unit interval [, ],i.e., a functionT: [, ]×[, ]→[, ] such that for alla,b,c∈[, ] the following four axioms
LetKbe a field. A non-Archimedean absolute value onKis a function| · |fromKinto
() |a| ≥and equality holds if and only ifa= ; () |ab|=|a||b|;
() |a+b| ≤max{|a|,|b|}.
The condition () is called the strict triangle inequality. By (), we have||=|–|= . Thus, by induction, it follows from () that|n| ≤ for each integern. We always assume in ad-dition that| · |is non-trivial,i.e., that there is ana∈Ksuch that|a| = , .
LetXbe a vector space over a scalar fieldKwith a non-Archimedean non-trivial
valua-tion| · |. A function · :X→Ris a non-Archimedean norm (valuation) if it satisfies the following conditions:
(NA) x= if and only ifx= ;
(NA) rx=|r|xfor allr∈Kandx∈X;
(NA) x+y ≤max{x,y}for allx,y∈X(the strong triangle inequality).
Then (X, · ) is called a non-Archimedean space.
Thanks to the inequality
xm–xl ≤max xj+–xj:l≤j≤m–
(m>l)
a sequence {xm} is Cauchy if and only if {xm+ –xm} converges to zero in a
non-Archimedean space. By a complete non-non-Archimedean space we mean one in which every Cauchy sequence is convergent.
In , Hensel [] introduced a normed space, which does not have the Archimedean property.
In the sequel, we adopt the usual terminology, notations, and conventions of the theory
of random normed spaces as in [–]. Throughout this paper,+is the space of
dis-tribution functions, that is, the space of all mappingsF:R∪ {–∞,∞} →[, ] such that
Fis left-continuous and non-decreasing onR,F() = , andF(+∞) = .D+is a subset of +consisting of all functionsF∈+for whichl–F(+∞) = , wherel–f(x) denotes the left limit of the functionf at the pointx, that is,l–f(x) =lim
t→x–f(t) . The space+is partially
ordered by the usual point-wise ordering of functions,i.e.,F≤Gif and only ifF(t)≤G(t) for alltinR.
Definition .[] A non-Archimedean random normed space (briefly, NA-RN-space)
is a triple (X,μ,T), whereXis a vector space,Tis a continuoust-norm, andμis a mapping
fromXintoD+such that the following conditions hold:
(RN) μx(t) =ε(t)for allt> if and only ifx= ; (RN) μαx(t) =μx(|αt|)for allx∈X,α= ; (RN) μx+y(max{t,s})≥T(μx(t),μy(s)).
It is easy to see that if (RN) holds, then we have
(RN) μx+y(t+s)≥T(μx(t),μy(s)).
Definition . Let (X,μ,T) be an NA-RN-space.
() A sequence{xn}inXis called a Cauchy sequence if, for every > andλ> , there exists a positive integerNsuch thatμxn–xn+( ) > –λwhenevern≥N.
() An NA-RN-space(X,μ,T)is said to be complete if and only if every Cauchy sequence inXis convergent to a point inX.
We will also use the following notations. The set of allm×n-matrices inXwill be
de-noted byMm,n(X). Whenm=n, the matrixMm,n(X) will be written asMn(X). The symbols ej∈M,n(C) will denote the row vector whosejth component is and the other
compo-nents are . Similarly,Eij∈Mn(C) will denote then×nmatrix whose (i,j)-component is
and the other components are . Then×nmatrix whose (i,j)-component isxand the
other components are will be denoted byEij⊗x∈Mn(X).
Let (X,·) be a normed space. Note that (X,{·n}) is a matrix normed space if and only if (Mn(X), · n) is a normed space for each positive integernandAxBk≤ ABxn
holds forA∈Mk,n,x= [xij]∈Mn(X) andB∈Mn,k, and that (X,{ · n}) is a matrix Banach
space if and only ifXis a Banach space and (X,{ · n}) is a matrix normed space. LetE,Fbe vector spaces. For a given mappingh:E→Fand a given positive integern, definehn:Mn(E)→Mn(F) by
hn
[xij]
=h(xij)
for all [xij]∈Mn(E).
We introduce the concept of matrix non-Archimedean random normed space.
Definition . Let (X,μ,T) be a non-Archimedean random normed space. Then:
() (X,{μ(n)},T)is called a matrix non-Archimedean random normed space if for each positive integern,(Mn(X),{μ(n)},T)is a non-Archimedean random normed space and
μ(AxBk) (t)≥μ(xn)(A·t B)for allt> ,A∈Mk,n(R),x= [xij]∈Mn(X)andB∈Mn,k(R)with A · B = .
() (X,{μ(n)},T)is called a matrix non-Archimedean random Banach space if (X,μ,T) is a non-Archimedean random Banach space and (X,{μ(n)},T) is a matrix non-Archimedean random normed space.
Definition . LetEbe a set. A functiond:E×E→[,∞] is called a generalized metric
onEifdsatisfies the following conditions:
() d(x,y) = if and only ifx=y; () d(x,y) =d(y,x)for allx,y∈E;
() d(x,z)≤d(x,y) +d(y,z)for allx,y,z∈E.
We note that the only one difference of the generalized metric from the usual metric is that the range of the former is permitted to include infinity.
The following theorem is very useful for proving our main results; it is due to Diaz and Margolis [].
Theorem .[] Let(,d)be a complete generalized metric space and letJ :→be
x∈,either
dJnx,Jn+x=∞
for all non-negative integers n or there exists a positive integer nsuch that:
() d(Jnx,Jn+x) <∞for alln≥n;
We note that a mappingr:X→Y satisfies the functional equation (.) if and only if
there exists a reciprocal mappingr:X→Ysatisfying the reciprocal functional equation
(.) [].
3 Generalized Hyers-Ulam stability of (1.5): direct method
Theorem . Letϕ:Xm→D+be a function such that there existsα∈Rwith <|α|<||
all t> .If a function f :X→Y satisfies the functional inequality
μ(Dmfnn) ([xij],...,[xmij])(t)≥
n
i,j=
for all x= [xij], . . . ,xm= [xmij]∈Mn(X)and all t> ,then there exists a unique reciprocal mapping r:X→Y which satisfies(.)and the inequality
μ(fnn)([xij])–rn([xij])(t)≥T
for allx∈X, allt> andı≥. Sincelimı→∞Tκ∞=ı(ϕ x and all t> .If a function f :X→Y satisfies the functional inequality
μ(Dmfnn) ([xij],...,[xmij])(t)≥
n
i,j=
for all x= [xij], . . . ,xm= [xmij]∈Mn(X)and all t> ,then there exists a unique reciprocal mapping r:X→Y which satisfies(.)and the inequality
μ(fnn)([xij])–rn([xij])(t)≥T
by Theorem ., we arrive at
4 Generalized Hyers-Ulam stability of equation (1.5): fixed point method
Theorem . Suppose that the mapping f :X→Y satisfies the inequality
μ(Dmfnn) ([xij],...,[xmij])(t)≥
n
i,j=
φxij,...,xmij(t) (.)
for all x= [xij], . . . ,xm= [xmij]∈Mn(X)and all t> ,whereφ:Xm→D+is a given func-tion.If there exists L< such that
φx
,...,xm (t)≥φx,...,xm
L||t
(.)
for all x, . . . ,xm∈X and all t> ,then there exists a unique reciprocal mapping r:X→Y which satisfies(.)and the inequality
μ(fnn)([xij])–rn([xij])(t)≥T
φxij,...,xij
–L nL|||m– |t
:i,j= , . . . ,n
(.)
for all x= [xij]∈Mn(X)and all t> .
Proof Letn= . Then (.) is equivalent to
μDmf(x,...,xm)(t)≥φx,...,xm(t) (.)
for allx, . . . ,xm∈Xand allt> .
Define a setSby
S={h:X→Y|his a function}
and introduce the generalized metricdonSas follows:
d(g,h) =inf λ∈R+:μg(x)–h(x)(λt)≥φx,...,x(t),∀x∈X,∀t>
, (.)
where, as usual,inf∅= +∞. It is easy to show that (S,d) is complete (see [], Lemma .).
Define a mappingσ:S→Sby
σh(x) = h
x
(x∈X) (.)
for allh∈S. We claim thatσ is strictly contractive onS. For any giveng,h∈S, let gh∈
[,∞] be an arbitrary constant withd(g,h)≤ gh. Hence
d(g,h)≤ gh ⇒ μg(x)–h(x)( ght)≥φx,...,x(t), ∀x∈X,∀t>
⇒ μ
g(x)–h(x)( ght)≥φ x ,...,x
||t, ∀x∈X,∀t>
⇒ μ
g(x)–h(x)( ght)≥φx,...,x
Lt
, ∀x∈X,∀t>
⇒ μ
Therefore, we see that
d(σg,σh)≤Ld(g,h), for allg,h∈S
that is,σis strictly contractive mapping ofS, with the Lipschitz constantL< . Now, replacingxiby x fori= , . . . ,min (.), we get
μf(x)–
f(x)(t)≥φ x ,...,x
|m– |t
≥φx,...,x
L|||m– |t
for allx∈Xand allt> . Hence (.) implies thatd(f,σf)≤L|||m– |<∞. Hence by ap-plying the fixed point alternative Theorem ., there exists a functionr:X→Ysatisfying the following:
() ris a fixed point ofσ, that is,
r(x) = r
x
(.)
for allx∈X. The mappingris the unique fixed point ofσ in the set
M= g∈S|d(f,g) <∞.
This implies that ris the unique mapping satisfying (.) such that there exists ∈ (,∞)satisfying
μf(x)–r(x)( t)≥φx,...,x(t), ∀x∈X,∀t> .
() d(σnf,r)→asn→ ∞. Thus we have
lim
n→∞μnf(xn)–r(x)(t) = (.)
for allx∈Xand allt> .
() d(r,f)≤–Ld(f,σf)which implies
d(r,f)≤L|||m– |
–L . (.)
From (.), (.), and (.), we have
μ
nDmf(xn,...,xmn)(t)≥φxn,...,xmn
||nt≥φx,...,xm
L n
t
for allx, . . . ,xm∈Xand allt> . Sincelimn→∞φx,...,xm((L)nt) = ,rsatisfies (.).
We note thatej∈M,n(R) means that thejth component is and the others are zero,Eij∈ Mn(R) means that the (i,j)-component is and the others are zero, andEij⊗x∈Mn(X)
means that the (i,j)-component isxand the others are zero. SinceμEpq(n)⊗x(t) =μx(t), we
have
μ([nxij)](t) =μ(n)n
i,j=Eij⊗xij(t)≥T
μ(Eijn)⊗xij(tij) :i,j= , , . . . ,n
=Tμxij(tij) :i,j= , , . . . ,n
,
wheret=ni,j=tij. Soμ[(xnij)](t)≥T(μxij( t
By (.),
μ(fnn)([xij])–rn([xij])(t)≥T
μf(xij)–r(xij)
t n
:i,j= , . . . ,n
≥T
φxij,...,xij
–L nL|||m– |t
:i,j= , . . . ,n
for allx= [xij]∈Mn(X) and allt> . Thusr:X→Yis a unique reciprocal mapping
satis-fying (.).
Theorem . Suppose that the mapping f :X→Y satisfies the inequality
μ(Dmfnn) ([xij],...,[xmij])(t)≥
n
i,j=
φxij,...,xmij(t) (.)
for all x= [xij], . . . ,xm= [xmij]∈Mn(X)and all t> ,whereφ:Xm→D+is a given func-tion.If there exists L< such that
φx,...,xm(t)≥φx,...,xm |
|
L t
(.)
for all x, . . . ,xm∈X and all t> ,then there exists a unique reciprocal mapping r:X→Y which satisfies(.)and the inequality
μ(fnn)([xij])–rn([xij])(t)≥T
φxij,...,xij
–L n|||m– |t
:i,j= , . . . ,n
(.)
for all x= [xij]∈Mn(X)and all t> .
Proof The proof is similar to the proof of Theorem ..
Corollary . Letγ ∈ {–, }be fixed.Let f :X→Y be a mapping and let there exist real
numbers q= –andθ≥with–γ <qγ such that
μ(Dmfnn) ([xij],...,[xmij])(t)≥ t
t+ni,j=θ(xijq+· · ·+xmijq)
for all x= [xij], . . . ,xm= [xmij]∈Mn(X)and all t> .Then there exists a unique reciprocal mapping r:X→Y such that
μ(fnn)([xij])–rn([xij])(t)≥
⎧ ⎨ ⎩
T((||q+–)t+(|θ|mnq+–)|||tm–|xijq :i,j= , , . . . ,n) forγ = ,
T((–||q+)t+(–θmn||q+|)||tm–|xijq :i,j= , , . . . ,n) forγ = –
for all x= [xij]∈Mn(X)and all t> .
Proof If we chooseφx,...,xm(t) = t+θ(xq+t···+xmq), for allt> and allx, . . . ,xm∈X, then
by Theorem ., we arrive at
μ(fnn)([xij])–rn([xij])(t)≥T
(||q+– )t
for allx= [xij]∈Mn(X), allt> ,γ= ,L=||–q–, and using Theorem ., we arrive at
μ(fnn)([xij])–rn([xij])(t)≥T
( –||q+)t
( –||q+)t+θmn|||m– |xijq :i,j= , , . . . ,n
for allx= [xij]∈Mn(X), allt> ,γ= – andL=||q+.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally in drafting this manuscript and giving the main proofs. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Urmia University, P.O. Box 165, Urmia, Iran.2Research Institute for Natural Sciences,
Hanyang University, Seoul, 133-791, Republic of Korea.
Acknowledgements
C Park was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology.
Received: 17 May 2015 Accepted: 30 September 2015
References
1. Ulam, SM: A Collection of the Mathematical Problems. Interscience, New York (1960)
2. Hyers, DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci.27, 222-224 (1941) 3. Aoki, T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn.2, 64-66 (1950) 4. Rassias, TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc.72, 297-300 (1978) 5. Rassias, JM: On approximation of approximately linear mappings by linear mappings. J. Funct. Anal.46, 126-130
(1982)
6. Rassias, JM: On approximation of approximately linear mappings by linear mappings. Bull. Sci. Math.108, 445-446 (1984)
7. Rassias, JM: Solution of a problem of Ulam. J. Approx. Theory57, 268-273 (1989)
8. Gˇavruta, P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl.184, 431-436 (1994)
9. Cho, YJ, Eshaghi Gordji, M, Zolfaghari, S: Solutions and stability of generalized mixed typeQCfunctional equations in random normed spaces. J. Inequal. Appl.2010, Article ID 403101 (2010). doi:10.1155/2010/403101
10. Cholewa, PW: Remarks on the stability of functional equations. Aequ. Math.27, 76-86 (1984)
11. Ebadian, A, Zolfaghari, S: Stability of a mixed additive and cubic functional equation in several variables in non-Archimedean spaces. Ann. Univ. Ferrara58, 291-306 (2012)
12. Eshaghi Gordji, M, Ebadian, A, Zolfaghari, S: Stability of a functional equation deriving from cubic and quartic functions. Abstr. Appl. Anal.2008, Article ID 801904 (2008)
13. Eshaghi Gordji, M, Kaboli-Gharetapeh, S, Park, C, Zolfaghari, S: Stability of an additive-cubic-quartic functional equation. Adv. Differ. Equ.2009, Article ID 395693 (2009)
14. Eshaghi Gordji, M, Kaboli-Gharetapeh, S, Rassias, JM, Zolfaghari, S: Solution and stability of a mixed type additive, quadratic and cubic functional equation. Adv. Differ. Equ.2009, Article ID 826130 (2009). doi:10.1155/2009/826130 15. Eshaghi Gordji, M, Moslehian, MS, Kaboli-Gharetapeh, S, Zolfaghari, S: Stability of a mixed type additive, quadratic,
cubic and quartic functional equation. In: Khan, A, et al. (eds.) Nonlinear Analysis and Variational Problems, pp. 65-80. Springer, Berlin (2009)
16. Eshaghi Gordji, M, Zolfaghari, S, Rassias, JM, Bavand-Savadkouhi, M: Solution and stability of a mixed type cubic and quartic functional equation in quasi-Banach spaces. Abstr. Appl. Anal.2009, Article ID 417473 (2009).
doi:10.1155/2009/417473
17. Zolfaghari, S: Stability of generalizedQCA-functional equation inp-Banach spaces. Int. J. Nonlinear Anal. Appl.1, 84-99 (2010)
18. Zolfaghari, S, Ebadian, A, Ostadbashi, S, De La Sen, M, Eshaghi Gordji, M: A fixed point approach to the Hyers-Ulam stability of anAQfunctional equation in probabilistic modular spaces. Int. J. Nonlinear Anal. Appl.4(2), 1-14 (2013) 19. Ravi, K, Senthil Kumar, BV: Ulam-Gˇavruta-Rassias stability of Rassias reciprocal functional equation. Glob. J. Appl. Math.
Math. Sci.3, 57-79 (2010)
20. Ravi, K, Rassias, JM, Senthil Kumar, BV: Ulam stability of reciprocal difference and adjoint functional equations. Aust. J. Math. Anal. Appl.8(1), 13 (2011)
21. Ravi, K, Rassias, JM, Senthil Kumar, BV: Ulam stability of generalized reciprocal functional equation in several variables. Int. J. Appl. Math. Stat.19, 1-19 (2010)
22. Ravi, K, Thandapani, E, Senthil Kumar, BV: Solution and stability of a reciprocal type functional equation in several variables. J. Nonlinear Sci. Appl.7, 18-27 (2014)
23. Ravi, K, Rassias, JM, Senthil Kumar, BV: Generalized Hyers-Ulam stability and examples of non-stability of reciprocal type functional equation in several variables. Indian J. Sci.2(3), 13-22 (2013)
25. Hensel, K: Über eine neue Begrundung der Theorie der algebraischen Zahlen. Jahresber. Dtsch. Math.-Ver.6, 83-88 (1897)
26. Chang, SS, Cho, YJ, Kang, SM: Nonlinear Operator Theory in Probabilistic Metric Spaces. Nova Science Publishers, New York (2001)
27. Mihet, D, Radu, D: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl.343(1), 567-572 (2008)
28. Šerstnev, AN: On the notion of a random normed space. Dokl. Akad. Nauk SSSR149(2), 280-283 (1963) (in Russian) 29. Schweizer, B, Sklar, A: Probabilistic Metric Spaces. North-Holland, New York (1983)