# Approximately Quintic and Sextic Mappings Formr-Divisible Groups into Ŝerstnev Probabilistic Banach Spaces: Fixed Point Method

Authors: M. Eshaghi Gordji; Y. J. Cho; M. B. Ghaemi; H. Majani
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

Using the fixed point method, we investigate the stability of the systems of quadratic-cubic and additive-quadratic-cubic functional equations with constant coefficients formr-divisible groups into Ŝerstnev probabilistic Banach spaces.


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## Body

## 1. Introduction and Preliminaries

Thestability problem of functional equations started with the following question concerning stability of group homomorphisms proposed by Ulam [1] during a talk before a Mathematical Colloquium at the University of Wisconsin, Madison, in 1940.Let(G1,·) be a group and (G2,*) a metric group with the metric d(·,·). Given ϵ>0, does there exist a δ>0 such that if a mapping h:G1→G2 satisfies the inequality d(h(x·y),h(x)*h(y))<δ for all x,y∈G1, then there exists a homomorphism H:G1→G2 with d(h(x),H(x))<ϵ for all x∈G1?In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces as follows.IfE and E′ are Banach spaces and f:E→E′ is a mapping for which there is ɛ>0 such that ∥f(x+y)-f(x)-f(y)∥≤ɛ for all x,y∈E, then there is a unique additive mapping L:E→E′ such that ∥f(x)-L(x)∥≤ɛ for all x∈E.Hyers’ Theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference, respectively.The paper of Rassias [5] has provided a lot of influence in the development of what we now call the generalized Hyers-Ulam stability or as Hyers-Ulam-Rassias stability of functional equations. In 1994, a generalization of the Rassias theorem was obtained by Găvruţa [6] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. For more details about the results concerning such problems, the reader is referred to [4, 5, 7–21, 21–30].The functional equation(1.1)f(x+y)+f(x-y)=2f(x)+2f(y)
is related to a symmetric biadditive function [31, 32]. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.1) is called a quadratic function. The Hyers-Ulam stability problem for the quadratic functional equation was solved by Skof [33]. In [8], Czerwik proved the Hyers-Ulam-Rassias stability of (1.1). Eshaghi Gordji and Khodaei [34] obtained the general solution and the generalized Hyers-Ulam-Rassias stability of the following quadratic functional equation: for all a,b∈ℤ∖{0} with a≠±1,±b,(1.2)f(ax+by)+f(ax-by)=2a2f(x)+2b2f(y).Jun and Kim [35] introduced the following cubic functional equation:(1.3)f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+12f(x),
and they established the general solution and the generalized Hyers-Ulam-Rassias stability for the functional equation (1.3). Jun et al. [36] investigated the solution and the Hyers-Ulam stability for the cubic functional equation(1.4)f(ax+by)+f(ax-by)=ab2(f(x+y)+f(x-y))+2a(a2-b2)f(x),
where a,b∈ℤ∖{0} with a≠±1,±b. For other cubic functional equations, see [37].Lee et al. [38] considered the following functional equation:(1.5)f(2x+y)+f(2x-y)=4f(x+y)+4f(x-y)+24f(x)-6f(y).In fact, they proved that a functionf between two real vector spaces X and Y is a solution of (1.5) if and only if there exists a unique symmetric biquadratic function B2:X×X→Y such that f(x)=B2(x,x) for all x∈X. The bi-quadratic function B2 is given by (1.6)B2(x,y)=112(f(x+y)+f(x-y)-2f(x)-2f(y)).Obviously, the functionf(x)=cx4 satisfies the functional equation (1.5), which is called the quartic functional equation. For other quartic functional equations, see [39].Ebadian et al. [40] considered the generalized Hyers-Ulam stability of the following systems of the additive-quartic functional equations:(1.7)f(x1+x2,y)=f(x1,y)+f(x2,y),f(x,2y1+y2)+f(x,2y1-y2)=4f(x,y1+y2)+4f(x,y1-y2)+24f(x,y1)-6f(x,y2),
and the quadratic-cubic functional equations:(1.8)f(x,2y1+y2)+f(x,2y1-y2)=2f(x,y1+y2)+2f(x,y1-y2)+12f(x,y1),f(x,y1+y2)+f(x,y1-y2)=2f(x,y1)+2f(x,y2).For more details about the results concerning mixed type functional equations, the readers are referred to [41–44].Recently, Ghaemi et al. [45] investigated the stability of the following systems of quadratic-cubic functional equations:(1.9)f(ax1+bx2,y)+f(ax1-bx2,y)=2a2f(x1,y)+2b2f(x2,y),f(x,ay1+by2)+f(x,ay1-by2)=ab2(f(x,y1+y2)+f(x,y1-y2))+2a(a2-b2)f(x,y1),
and additive-quadratic-cubic functional equations:(1.10)f(ax1+bx2,y,z)+f(ax1-bx2,y,z)=2af(x1,y,z),f(x,ay1+by2,z)+f(x,ay1-by2,z)=2a2f(x,y1,z)+2b2f(x,y2,z),f(x,y,az1+bz2)+f(x,y,az1-bz2)=ab2(f(x,y,z1+z2)+f(x,y,z1-z2))+2a(a2-b2)f(x,y,z1)
in PN-spaces (see Definition 1.6), where a,b∈ℤ∖{0} with a≠±1,±b. The function f:ℝ×ℝ→ℝ given by f(x,y)=cx2y3 is a solution of the system (1.9). In particular, letting y=x, we get a quintic function g:ℝ→ℝ in one variable given by g(x):=f(x,x)=cx5. Also, it is easy to see that the function f:ℝ×ℝ×ℝ→ℝ defined by f(x,y,z)=cxy2z3 is a solution of the system (1.10). In particular, letting y=z=x, we get a sextic function h:ℝ→ℝ in one variable given by h(x):=f(x,x,x)=cx6.The proof of the following propositions are evident.Proposition 1.1.
Let X and Y be real linear spaces. If a functionf:X×X→Y satisfies the system  (1.9), then  f(λx,μy)=λ2μ3f(x,y) for all  x,y∈X and rational numbers  λ,μ.Proposition 1.2.
Let X and Y be real linear spaces. If a functionf:X×X×X→Y satisfies the system  (1.10), then  f(λx,μy,ηz)=λμ2η3f(x,y,z) for all  x,y,z∈X and rational numbers  λ,μ,η.For our main results, we introduce Banach’s fixed point theorem and related results. For the proof of Theorem1.3, refer to [46] and also Chapter 5 in [29] and, for more fixed point theory and other nonlinear methods, refer to [28, 47]. Especially, in 2003, Radu [27] proposed a new method for obtaining the existence of exact solutions and error estimations, based on the fixed point alternative (see also [48–53]).Let(X,d) be a generalized metric space. We say that an operator T:X→X satisfies a Lipschitz condition with Lipschitz constant L if there exists a constant L≥0 such that d(Tx,Ty)≤Ld(x,y) for all x,y∈X. If the Lipschitz constant L is less than 1, then the operator T is called a strictly contractive operator.Note that the distinction between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity. We recall the following theorem by Margolis and Diaz.Theorem 1.3 (see [27, 46]).
Suppose that(Ω,d) is a complete generalized metric space and  T:Ω→Ω is a strictly contractive mapping with Lipschitz constant  L. Then, for any  x∈Ω, either
(1.11)d(Tmx,Tm+1x)=∞
for all m≥0 or there exists a natural number m0 such that(1)
d(Tmx,Tm+1x)<∞ for all m≥m0;(2)
the sequence{Tmx} is convergent to a fixed point y* of T;(3)
y* is the unique fixed point of T in Λ={y∈Ω:d(Tm0x,y)<∞};(4)
d(y,y*)≤(1/1-L)d(y,Ty) for all y∈Λ.The PN-spaces were first defined by Šerstnev in 1963 (see [54]). Their definition was generalized by Alsina et al. in [55]. In this paper, we follow the definition of probabilistic space briefly as given in [56] (also, see [57]).Definition 1.4.
Adistance distribution function (d.d.f.) is a nondecreasing function F from ℝ¯+ into [0,1] that satisfies F(0)=0, F(+∞)=1 and F is left-continuous on (0,+∞), where ℝ¯+:=[0,+∞].Forward, the space of distance distribution functions is denoted byΔ+ and the set of all F in Δ+ with limt→+∞-F(t)=1 by D+. The space Δ+ is partially ordered by the usual pointwise ordering of functions, that is, F≤G if and only if F(x)≤G(x) for all x in ℝ¯+. For any a≥0, ɛa+ is the d.d.f. given by (1.12)ɛa+(t)={0,ift≤a,1,ift>a.Definition 1.5.
Atriangle function is a binary operation on Δ+, that is, a function τ:Δ+×Δ+→Δ+ that is associative, commutative, non-decreasing in each place, and has ɛ0 as the identity, that is, for all F,G and H in Δ+,(TF1)
τ(τ(F,G),H)=τ(F,τ(G,H));(TF2)
τ(F,G)=τ(G,F);(TF3)
F≤G⇒τ(F,H)≤τ(G,H);(TF4)
τ(F,ɛ0)=τ(ɛ0,F)=F.Typical continuous triangle function is(1.13)ΠT(F,G)(x)=T(F(x),G(x)),
where T is a continuous t-norm, that is, a continuous binary operation on [0,1] that is commutative, associative, non-decreasing in each variable, and has 1 as the identity. For example, we introduce the following: (1.14)M(x,y)=min⁡(x,y)
for all x,y∈[0,1] is a continuous and maximal t-norm, namely, for any t-norm T, M≥T. Also, note that ΠM is a maximal triangle function, that is, for all triangle function τ, ΠM≥τ.Definition 1.6.
AŠerstnev probabilistic normed space (Šerstnev PN-space) is a triple (X,ν,τ), where X is a real vector space, τ is continuous triangle function, and ν is a mapping (the probabilistic norm) from X into Δ+ such that, for all choice of p,q∈X and a∈ℝ+, the following conditions hold:(N1)
ν(p)=ɛ0, if and only if p=θ(θ is the null vector in X);(N2)
ν(ap)(t)=ν(p)(t/|a|);(N3)
ν(p+q)≥τ(ν(p),ν(q)).Let(X,ν,τ) be a PN-space and {xn} a sequence in X. Then {xn} is said to be convergent if there exists x∈X such that (1.15)lim⁡n→∞ν(xn-x)(t)=1
for all t>0. In this case, the point x is called the limit of {xn}. The sequence {xn} in (X,ν,τ) is called a Cauchy sequence if, for any ɛ>0 and δ>0, there exists a positive integer n0 such that ν(xn-xm)(δ)>1-ɛ for all m,n≥n0. Clearly, every convergent sequence in a PN-space is a Cauchy sequence. If each Cauchy sequence is convergent in a PN-space (X,ν,τ), then (X,ν,τ) is called a probabilistic Banach space (PB-space).For more details about the results concerning stability of the functional equations on PN-spaces, the readers are referred to [58–61].In this paper, by using the fixed point method, we establish the stability of the systems (1.9) and (1.10) form r-divisible groups into Šerstnev PB-space.


## 2. Mail Results

We start our work by the following theorem which investigates the stability problem for the system of the functional equations (1.9) form r-divisible groups into Šerstnev PB-space by using fixed point methods.Theorem 2.1.
Lets∈{-1,1} be fixed. Let G be an r-divisible group and (Y,ν,ΠT) a Šerstnev PB-space. Let ϕ,ψ:G×G×G→D+ be two functions such that
(2.1)Φ(x,y)(t):=ΠT{ϕ(a(s-1)/2x,0,a(s-1)/2y)(2a(5s-1)/2t),ψ(a(s+1)/2x,a(s-1)/2y,0)(2a(5s+5)/2t)}
for all x,y∈G and, for some 0<k<a10s,
(2.2)Φ(asx,asy)(ka-2st)≥Φ(x,y)(t),lim⁡n→∞ϕ(asnx1,asnx2,asny)(a-5snt)=lim⁡n→∞ψ(asnx,asny1,asny2)(a-5snt)=1
for all x,y,x1,x2,y1,y2∈G and t>0. If f:G×G→Y is a function such that f(0,y)=0 for all y∈G and
(2.3)ν(f(ax1+bx2,y)+f(ax1-bx2,y)-2a2f(x1,y)-2b2f(x2,y))(t)≥ϕ(x1,x2,y),(2.4)ν(f(x,ay1+by2)+f(x,ay1-by2)-ab2f(x,y1+y2)-ab2f(x,y1-y2)-2a(a2-b2)f(x,y1))(t)≥ψ(x,y1,y2)
for all x,y,x1,x2,y1,y2∈G, then there exists a unique quintic function T:G×G→Y satisfying the system (1.9) and
(2.5)ν(f(x,y)-T(x,y))(t)≥Φ(x,y)((1-ka-10s)t)
for all x,y∈G.Proof.
Puttingx1=2x and x2=0 and replacing y by 2y in (2.3), we get
(2.6)ν(f(2ax,2y)-a2f(2x,2y))(t)≥ϕ(2x,0,2y)(2t)
for all x,y∈G. Putting y1=2y and y2=0 and replacing x by 2ax in (2.4), we get
(2.7)ν(f(2ax,2ay)-a3f(2ax,2y))(t)≥ψ(2ax,2y,0)(2t)
for all x,y∈G. Thus, we have
(2.8)ν(f(2ax,2ay)-a5f(2x,2y))(t)≥ΠT{ϕ(2x,0,2y)(2a-3t),ψ(2ax,2y,0)(2t)}
for all x,y∈G. Replacing x, y by x/2, y/2 in (2.8), we have
(2.9)ν(f(ax,ay)-a5f(x,y))(t)≥ΠT{ϕ(x,0,y)(2a-3t),ψ(ax,y,0)(2t)}
for all x,y∈G. It follows from (2.9) that
(2.10)ν(a-5f(ax,ay)-f(x,y))(t)≥ΠT{ϕ(x,0,y)(2a2t),ψ(ax,y,0)(2a5t)},ν(a5f(a-1x,a-1y)-f(x,y))(t)≥ΠT{ϕ(a-1x,0,a-1y)(2a-3t),ψ(ax,a-1y,0)(2t)}
for all x,y∈G. So we have
(2.11)ν(a-5sf(asx,asy)-f(x,y))(t)≥Φ(x,y)(t)
for all x,y∈G. Let S be the set of all mappings h:G×G→Y with h(0,x)=0 for all x∈G, and define a generalized metric on S as follows:
(2.12)d(h,k)=inf⁡{u∈R+:ν(h(x,y)-k(x,y))(ut)≥Φ(x,y)(t),∀x,y∈G,∀t>0},
where, as usual, inf∅=+∞. The proof of the fact that (S,d) is a complete generalized metric space, can be shown in [48, 62].
Now, we consider the mappingJ:S→S defined by
(2.13)Jh(x,y):=a-5sh(asx,asy)
for all h∈S and x,y∈G. Let f,g∈S such that d(f,g)<ɛ. Then it follows that
(2.14)ν(Jg(x,y)-Jf(x,y))(kua-10st)=ν(a-5sg(asx,asy)-a-5sf(asx,asy))(kua-10st)=ν(g(asx,asy)-f(asx,asy))(kua-2st)≥Φ(asx,asy)(ka-2st)≥Φ(x,y)(t),
that is, if d(f,g)<ɛ, then we have d(Jf,Jg)<ka-10sɛ. This means that
(2.15)d(Jf,Jg)≤ka-10sd(f,g)
for all f,g∈S; that is, J is a strictly contractive self-mapping on S with the Lipschitz constant ka-10s. It follows from (2.11) that
(2.16)ν(Jf(x,y)-f(x,y))(t)≥Φ(x,y)(t)
for all x,y∈G and t>0, which implies that d(Jf,f)≤1. From Theorem 1.3, it follows that there exists a unique mapping T:G×G→Y such that T is a fixed point of J, that is, T(asx,asy)=a5sT(x,y) for all x,y∈G. Also, we have d(Jmg,T)→0 as m→∞, which implies the equality
(2.17)lim⁡m→∞a-5smf(asmx,asmy)=T(x)
for all x,y∈G. It follows from (2.3) that
(2.18)ν(T(ax1+bx2,y)+T(ax1-bx2,y)-2a2T(x1,y)-2b2T(x2,y))(t)=lim⁡n→∞ν(a-5snf(asn(ax1+bx2),asny)+a-5snf(asn(ax1-bx2),asny)-2a-5sna2f(asnx1,asny)-2a-5snb2f(asnx2,asny))(t)≥lim⁡n→∞ϕ(asnx1,asnx2,asny)(a-5snt)=1
for all x1,x2,y∈G. Also, it follows from (2.4) that
(2.19)ν(T(x,ay1+by2)+T(x,ay1-by2)-ab2(T(x,y1+y2)-T(x,y1-y2))-2a(a2-b2)T(x,y1))(t)=lim⁡n→∞ν(a-5snf(asnx,asn(ay1+by2))+a-5snf(asnx,asn(ay1-by2))-a-5snab2f(asnx,asn(y1+y2))-a-5snab2f(asnx,asn(y1-y2))-2a-5sna(a2-b2)f(asnx,asny1))(t)≥lim⁡n→∞a-5snψ(asnx,asny1,asny2)(t)=1
for all x,y1,y2∈G. This means that T satisfies (1.9); that is, T is quintic.
According to the fixed point alternative, sinceT is the unique fixed point of J in the set Ω={g∈S:d(f,g)<∞}, T is the unique mapping such that
(2.20)ν(f(x,y)-T(x,y))(ut)≥Φ(x,y)(t)
for all x,y∈G and t>0. Using the fixed point alternative, we obtain
(2.21)d(f,T)≤11-Ld(f,Jf)≤11-ka-10s,
which implies the inequality
(2.22)ν(f(x,y)-T(x,y))(t1-ka-10s)≥Φ(x,y)(t)
for all x,y∈G and t>0. Therefore, we have
(2.23)ν(f(x,y)-T(x,y))(t)≥Φ(x,y)((1-ka-10s)t)
for all x,y∈XG and t>0. This completes the proof.Now, we investigate the stability problem for the system of the functional equations (1.10) form r-divisible groups into Šerstnev PB-space by using the fixed point theorem.Theorem 2.2.
Lets∈{-1,1} be fixed. Let G be an r-divisible group and (Y,ν,ΠT) a Šerstnev PB-space. Let Φ,Ψ,Υ:G×G×G×G→D+ be functions such that
(2.24)Θ(x,y,z)(t):=ΠT{Υ(a(s+1)/2x,a(s+1)/2y,a(s+1)/2z,0)(2a3s+3t),ΠT{Ψ(a(s+1)/2x,a(s+1)/2y,0,a(s-1)/2z)(2a3s+6t),Φ(a(s-1)/2x,0,a(s-1)/2y,a(s-1)/2z)(2a3s+8t)}}
for all x,y,z∈G and, for some 0<k<a6s,
(2.25)Φ(asx,asy,asz)(kt)≥Φ(x,y,z)(t);lim⁡n→∞Φ(asnx1,asnx2,asny,asnz)(a-6snt)=lim⁡n→∞Ψ(asnx,asny1,asny2,asnz)(a-6snt)=lim⁡n→∞Υ(asnx,asny,asnz1,asnz2)(a-6snt)=1
for all x,y,x1,x2,y1,y2,z1,z2∈G. If f:G×G×G→Y is a function such that f(x,0,z)=0 for all x,z∈G and
(2.26)ν(f(ax1+bx2,y,z)+f(ax1-bx2,y,z)-2af(x1,y,z))(t)≥Φ(x1,x2,y,z)(t),(2.27)ν(f(x,ay1+by2,z)+f(x,ay1-by2,z)-2a2f(x,y1,z)-2b2f(x,y2,z))(t)≥Ψ(x,y1,y2,z)(t),(2.28)ν(-ab2(f(x,y,z1+z2)+f(x,y,z1-z2))-2a(a2-b2)f(x,y,z1)f(x,y,az1+bz2)+f(x,y,az1-bz2)-ab2(f(x,y,z1+z2)+f(x,y,z1-z2))-2a(a2-b2)f(x,y,z1))(t)≥Υ(x,y,z1,z2)(t)
for all x,y,x1,x2,y1,y2,z1,z2∈G, then there exists a unique quintic function T:G×G×G→Y satisfying (1.10) and
(2.29)ν(f(x,y,z)-T(x,y,z))(t)≥Θ(x,y,z)((1-ka-6s)t)
for all x,y,z∈G.Proof.
Puttingx1=2x and x2=0 and replacing y,z by 2y,2z in (2.26), we get
(2.30)ν(f(2ax,2y,2z)-af(2x,2y,2z))(12t)≥Φ(2x,0,2y,2z)(t)
for all x,y,z∈G. Putting y1=2y and y2=0 and replacing x,z by 2ax,2z in (2.27), we get
(2.31)ν(f(2ax,2ay,2z)-a2f(2ax,2y,2z))(12t)≥Ψ(2ax,2y,0,2z)(t)
for all x,y,z∈G. Putting z1=2z and z2=0 and replacing x,y by 2ax,2ay in (2.28), we get
(2.32)ν(f(2ax,2ay,2az)-a3f(2ax,2ay,2z))(12t)≥Υ(2ax,2ay,2z,0)(t),
for all x,y,z∈G. Thus,
(2.33)ν(f(2ax,2ay,2az)-a6f(2x,2y,2z))(t)≥ΠT{Υ(2ax,2ay,2z,0)(2t),ΠT{Ψ(2ax,2y,0,2z)(2a3t),Φ(2x,0,2y,2z)(2a5t)}}
for all x,y,z∈G. Replacing x, y, and z by x/2, y/2, and z/2 in (2.33), we have
(2.34)ν(f(ax,ay,az)-a6f(x,y,z))(t)≥ΠT{Υ(ax,ay,z,0)(2t),ΠT{Ψ(ax,y,0,z)(2a3t),Φ(x,0,y,z)(2a5t)}}
for all x,y,z∈G. It follows from (2.34) that
(2.35)ν(a-6f(ax,ay,az)-f(x,y,z))(t)≥ΠT{Υ(ax,ay,z,0)(2a6t),ΠT{Ψ(ax,y,0,z)(2a9t),Φ(x,0,y,z)(2a11t)}},ν(a6f(a-1x,a-1y,a-1z)-f(x,y,z))(t)≥ΠT{Υ(x,y,a-1z,0)(2at),ΠT{Ψ(x,a-1y,0,a-1z)(2a3t)Φ(a-1x,0,a-1y,a-1z),(2a5t)}}
for all x,y,z∈G. Thus, we have
(2.36)ν(a-6sf(asx,asy,asz)-f(x,y,z))(t)≥Θ(x,y,z)(t)
for all x,y,z∈G. Let S be the set of all mappings h:X×X×X→Y with h(x,0,z)=0 for all x,z∈G, and define a generalized metric on S as follows:
(2.37)d(h,k)=inf⁡{u∈R+:ν(h(x,y,z)-k(x,y,z))(ut)≥Θ(x,y,z)(t),∀x,y,z∈G,t>0},
where, as usual, inf∅=+∞. The proof of the fact that (S,d) is a complete generalized metric space can be shown in [48, 62].
Now, we consider the mappingJ:S→S defined by
(2.38)Jh(x,y,z):=a-6sh(asx,asy,asz)
for all h∈S and x,y,z∈G. Let f,g∈S be such that d(f,g)<ɛ. Then we have
(2.39)ν(Jg(x,y,z)-Jf(x,y,z))(kua-6st)=ν(a-6sg(asx,asy,asz)-a-6sf(asx,asy,asz))(kua-6st)=ν(g(asx,asy,asz)-f(asx,asy,asz))(kut)≥Θ(asx,asy,asz)(kt)≥Θ(x,y,z)(t),
that is, if d(f,g)<ɛ, then we have d(Jf,Jg)<ka-6sɛ. This means that
(2.40)d(Jf,Jg)≤ka-6sd(f,g)
for all f,g∈S; that is, J is a strictly contractive self-mapping on S with the Lipschitz constant ka-6s. It follows from (2.36) that
(2.41)ν(Jf(x,y,z)-f(x,y,z))(t)≥Θ(x,y,z)(t)
for all x,y,z∈G and all t>0, which implies that d(Jf,f)≤1. From Theorem 1.3, it follows that there exists a unique mapping T:G×G×G→Y such that T is a fixed point of J, that is, T(asx,asy,asz)=a6sT(x,y,z) for all x,y,z∈G. Also, d(Jmg,T)→0 as m→∞, which implies the equality
(2.42)lim⁡m→∞a-6smf(asmx,asmy,asmz)=T(x)
for all x∈X. It follows from (2.26), (2.27), and (2.28) that
(2.43)ν(T(ax1+bx2,y,z)+T(ax1-bx2,y,z)-2aT(x1,y,z))(t)=lim⁡n→∞ν(a-6snf(asn(ax1+bx2),asny,asnz)+a-6snf(asn(ax1-bx2),asny,asnz)-2aa-6snf(asnx1,asny,asnz))(t)≥lim⁡n→∞Φ(asnx1,asnx2,asny,asnz)(a-6snt)=1,ν(T(x,ay1+by2,z)+T(x,ay1-by2,z)-2a2T(x,y1,z)-2b2T(x,y2,z))(t)=lim⁡n→∞ν(a-6snf(asnx,asn(ay1+by2),asnz)+f(asnx,asnay1-asnby2,asnz)-2a2a-6snf(asnx,asny1,asnz)+2b2a-6snf(asnx,asny2,asnz))(t)≥lim⁡n→∞Ψ(asnx,asny1,asny2,asnz)(a-6snt)=1,ν(T(x,y,az1+bz2)+T(x,y,az1-bz2)-ab2(T(x,y,z1+z2)-T(x,y,z1-z2))-2a(a2-b2)T(x,y,z1))(t)=lim⁡n→∞ν(a-6snf(asnx,asny,asn(az1+bz2))+a-6snf(asnx,asny,asn(az1-bz2))-ab2a-6sn(f(asnx,asny,asn(z1+z2))+f(asnx,asny,asn(z1-z2)))-2aa-6sn(a2-b2)f(asnx,asny,asnz1))(t)≥lim⁡n→∞Υ(asnx,asny,asnz1,asnz2)(a-6snt)=1
for all x,y,x1,x2,y1,y2,z1,z2∈G. This means that T satisfies (1.10); that is, T is sextic.
According to the fixed point alternative, sinceT is the unique fixed point of J in the set Ω={g∈S:d(f,g)<∞}, T is the unique mapping such that
(2.44)ν(f(x,y,z)-T(x,y,z))(ut)≥Θ(x,y,z)(t)
for all x,y,z∈G and t>0. Using the fixed point alternative, we obtain
(2.45)d(f,T)≤11-Ld(f,Jf)≤11-ka-6s,
which implies the inequality
(2.46)ν(f(x,y,z)-T(x,y,z))(t1-ka-6s)≥Θ(x,y,z)(t)
for all x,y,z∈G and t>0. So
(2.47)ν(f(x,y,z)-T(x,y,z))(t)≥Θ(x,y,z)((1-ka-6s)t)
for all x,y,z∈G and t>0. This completes the proof.



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*Source: 572062-2011-11-30.xml*