A fixed point approach to the non-Archimedean random stability of generalized mixed type AQCQ-functional equations

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Adv. Fixed Point Theory, 4 (2014), No. 3, 420-443 ISSN: 1927-6303

A FIXED POINT APPROACH TO THE NON-ARCHIMEDEAN RANDOM STABILITY OF GENERALIZED MIXED TYPE AQCQ-FUNCTIONAL

EQUATIONS

IZ. EL-FASSI∗, S. KABBAJ

Laboratory LAMA, Harmonic Analysis and functional equations Team,

Department of Mathematics, Faculty of Sciences, University of Ibn Tofail , Kenitra, Morocco

Copyright c2014 El-Fassi and Kabbaj. This is an open access article distributed under the Creative Commons Attribution License, which

permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract. Using the fixed point methods, we prove the generalized Hyers-Ulam stability of a general mixed additive-quadratic-cubic-quartic functional equation.

Keywords: non-Archimedean normed spaces; random Banach spaces; fixed point method, quadratic function; generalized Hyers-Ulam stability.

2010 AMS Subject Classification:46S10, 47H10, 39B82.

1. Introduction

In Ulam [1], the following question concerning the stability of homomorphisms is studied: Let G be a group and let H be a metric group with metric d(., .). Given ε >0, there exists

a δ >0 such that if the function f :G →H satisfies d(f(xy),f(x)f(y))< δ for all x,y∈

G, then there exists a homomorphism a :G→ H with d(f(x),a(x))<ε for all x∈ G. In

1941, Hyers [2] gave the first affirmative partial answer to the question of Ulam in Banach ∗_{Corresponding author}

E-mail address: [email protected] Received February 18, 2014

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spaces. In 1950, Aoki [3] generalized the Hyers theorem for additive mappings. In 1978, Rassias [4] provided a generalized version of the Hyers theorem which permits the Cauchy difference to become unbounded. In 1942, Menger [5] introduced the notion of probabilistic metric spaces. Since then, the theory of probabilistic metric spaces has been developed by many authors in many directions; see [6], [7] and the references therein. The idea of Menger was to use the distribution functions instead of non-negative real numbers as values of the metric. The notion of a probabilistic metric space corresponds to situations when we do not know exactly the distance between two points, but we know probabilities of possible values of this distance. A probabilistic generalization of metric spaces appears to be interested in the investigation of physical quantities, physiological thresholds and some other fields. It is also of fundamental importance in probabilistic functional analysis. In 1962, Serstnev [8] introduced the concept of a probabilistic normed space introduced by means of a definition that was closely modeled on the theory of (classical) normed spaces and used to study the problems of best approximation in statistics. On the other hand, Isac and Rassias [9] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. The stability problems of several various functional equations have been extensively investigated by a number of authors using fixed point methods; see [10-12] and the references therein.

The functional equation

f(x+y) + f(x−y) =2f(x) +2f(y) (1.1) is said to be a quadratic functional equation because the quadratic function f(x) =ax2is a solu-tion of the funcsolu-tional equasolu-tion (1.1). A quadratic funcsolu-tional equasolu-tion was used to characterize inner product spaces [13,14]. It is well known that a function f is a solution of (1.1) if and only if there exists a unique symmetric biadditive function Bsuch that f(x) =B(x,x)for all x; see [14]. The biadditive functionBis given by

B(x,y) = 1

4[f(x+y) + f(x−y)]. (1.2) The functional equation

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is called a cubic functional equation, because the cubic function f(x) =cx3is a solution of the equation (1.3). The general solution and the generalized Hyers-Ulam-Rassias stability for the functional equation (1.3) was discussed by Jun and Kim [15]. They proved that a function f between real vector spaces X andY is a solution of (1.3) if and only if there exists a unique functionC:X3→Y such that f(x) =C(x,x,x)for allx∈X andCis symmetric for each fixed one variable and is additive for fixed two variables. The quartic functional equation

f(x+2y) + f(x−2y)−6f(x) =4[f(x+y) + f(x−y)] +24f(y) (1.4)

was introduced by Rassias [16]. Later Leeet al.[17] remodified Rassias equation and obtained a new quartic functional equation of the form

f(2x+y) + f(2x−y) =4[f(x+y) + f(x−y)] +24f(x)−6f(y) (1.5)

and discussed its general solution. In fact Lee et al. [17] proved that a function f between vector spacesX andY is a solution of (1.5) if and only if there exists a unique symmetric multi - additive functionQ:X4→Y such that f(x) =Q(x,x,x,x)for allx∈X. It is easy to show that the function f(x) =kx4is the solution of (1.4) and (1.5).

A function

f(x) =Q(x1,x2,x3,x4) (1.6)

is called symmetric multi additive if Q is additive with respect to each variable x_i,i=1,2,3,4 in (1.6). A function f is defined as f(x) = β(x)₁₂−α(x), where α(x) = f(2x)−16f(x), β(x) =

f(2x)−4f(x). Further, f satisfies f(2x) =4f(x)and f(2x) =16f(x)is said to be a quadratic - quartic function.

Jun and Kim [18] introduced the following generalized quadratic and additive type functional equation

f(

n

∑

i=1

xi) + (n−2) n

∑

i=1

f(xi) =

_∑

1≤i<j≤j

f(xi+xj) (1.7)

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Hyers-Ulam-Rassias stability for the equation (1.7) whenn=4 was also investigated by Chang et al.[20].

The general solution and the generalized Hyers-Ulam stability for the quadratic and additive type functional equation

f(x+ay) +a f(x−y) = f(x−ay) +a f(x+y) (1.8)

for any positive integerawitha6=−1,0,1 was discussed by Jun and Kim [21]. Recently Najati and Moghimi [22] investigated the generalized Hyers-Ulam-Rassias stability for a quadratic and additive type functional equation of the form

f(2x+y) + f(2x−y) =2f(x+y) +2f(x−y) +2f(2x)−4f(x). (1.9) Very recently, the authors [23,24] investigated a mixed type functional equation of cubic and quartic type and obtained its general solution. The stability of generalized mixed type functional equations of the form

f(x+ky) +f(x−ky) =k2[f(x+y) + f(x−y)] +2(1−k2)f(x) (1.10) for fixed integers k with k6=0,±1 in quasi -Banach spaces was investigated by Gordji and Khodaie [25]. The mixed type functional equation (1.10) is additive, quadratic and cubic. Park et al. [26] proved the generalized Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation (briefly, AQCQ-functional equation):

f(x+2y) +f(x−2y) = 4f(x+y) +4f(x−y)−6f(x) +f(2y) +f(−2y)−4f(y)−4f(−y)

(1.11)

in random normed spaces .

Throughout this paper, we assume thatX be a vector space over a non-Archimedean field_K, (Y,µ,T)is a non-Archimedean random Banach space over_K. Based on fixed point methods,

we prove the generalized stability of the following equation:

f(x+ay) + f(x−ay) = a2[f(x+y) +f(x−y)] +2(1−a2)f(x)+ a4−a2

12 [f(2y) + f(−2y)−4f(y)

−f(−y)]

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for fixed integers a with a6=0,±1 in random normed spaces. In the sequel, we shall adopt the usual terminologies, notions, and conventions of the theory of non-Archimedean random normed spaces (non-ARN-spaces) as in [27, 28, 7]. In this paper, the space of all probability distribution functions is denoted by∆+. Elements of∆+ are functionsF:_R∪ {−∞,∞} →[0,1],

such thatFis left continuous and nondecreasing onRandF(0) =0,F(+∞) =1. It is clear that

the subsetD+:={F∈∆+:l−F(+∞) =1},wherel−f(x) =lim_t_→_x−f(t)is a subset of∆+. The

space∆+ is partially ordered by the usual point-wise ordering of functions,i.e., F≤Gif and

only ifF(t)≤G(t)for allt∈_R. The maximal element for∆+ in this order is the distribution

functionε0given by

ε0(t) =







1, ift>0. 0, ift≤0.

2. Preliminaries

In this section, we give the definition and theorems that are important in this paper.

Theorem 2.1. [29]Let(X,d) be a complete generalized metric space and let J:X →X be a strict contractive mapping with a Lipschitz constant 0<L<1. If there exists a nonnegative integer k such that d(Jk+1x,Jkx)<∞for some x∈X , then the followings are true:

(1) the sequence{Jnx}converge to a fixed point x∗for J,

(2) x∗is the unique fixed point of J in X∗={y∈X,d(Jkx,y)≺∞},

(3) if y∈X∗, then d(y,x∗)≤ 1

1−Ld(Jy,y).

Definition 2.2. [7] A mapping T :[0,1]2 →[0,1] is a continuous triangular norm (briefly, a continuous t-norm) ifT satisfies the following conditions:

(1) T is commutative and associative; (2) T is continuous;

(3) T(a,1) =afor alla∈[0,1];

(4) T(a,b)≤T(c,d)whenevera≤candb≤dfor alla,b,c,d∈[0,1].

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{x_n} is a given sequence of numbers in[0,1], T_in₌₁x_i is defined recurrently byT_i1₌₁x_i=x₁and T_in₌₁x_i=T(T_in₌₁−1x_i,x_n) =T(x₁, ...,x_n)forn≥1. T∞

i=nxiis defined asTi∞=1xn+i. It is known ([31])

that for the Lukasiewicz t-norm the following holds: lim

n→∞

(TL)∞i=1xn+i=1⇔ ∞

∑

n=1

(1−xn)<∞.

Definition 2.3.By a non-Archimedean field, we mean a field IK equipped with a function(valuation)

|.|:K→[0,∞)such that for allr,s∈K, the following conditions hold:

(1) |r|=0 if and only ifr=0; (2) |rs|=|r||s|;

(3) |r+s| ≤max(|r|,|s|)for allr,s∈_K.

Clearly,|1|=| −1|=1 and|n| ≤1 for alln∈_N. The function|.|is called the trivial valuation if|r|=1,∀r∈_K,r6=0, and|0|=0.

Definition 2.4. Let X be a vector space over a scalar field _Kwith a Archimedean non-trivial valuation|.|. A functionk.k:X →_Ris non-Archimedean norm (valuation) if it satisfies the following conditions:

(1) kxk=0 if and only ifx=0;

(2) krxk=|r|kxkfor allr∈_Kandx∈X; (3) kx+yk ≤max(kxk,kyk)for allx,y∈X.

Then,(X,k.k)is called a non-Archimedean space. Due to the fact that

kx_m−x_nk ≤max{kx_j₊₁−x_jk:m≤ j≤n−1},

in which n> m, the sequence {x_n} is Cauchy if and only if {x_n₊₁−x_n} converges to zero in a non-Archimedean normed space. In a complete non-Archimedean space, every Cauchy sequence is convergent.

Definition 2.5. A non-Archimedean random normed space (briefly, non-Archimedean RN-space) is a triple(X,µ,T), where X is a linear space over a non-Archimedean field K,T is a

continuous t-norm, andµis a mapping fromX intoD+such that, the following conditions hold:

(1) µx(t) =ε0(t)for allt>0 if and only ifx=0;

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(3) µx+y(max(t,s))≥T(µx(t),µy(s))for allx,y∈X andt,s≥0.

It is easy to see that if (3) holds, then (3’): µx+y(t+s)≥T(µx(t),µy(s))for all x,y∈X and

t,s≥0.

Every non-Archimedean normed linear space(X,k.k)defines a non-Archimedean RN-space (X,µ,TM)whereµx(t) =_t₊t_k_x_k for allt >0 andx∈X.

Definition 2.6. Let(X,µ,T)be a non-Archimedean RN-space.

(1) A sequence{xn}inXis said to be convergent toxinXif for allt>0, limn→∞µxn−x(t) =

1;

(2) A sequence{xn}inX is said to be Cauchy sequence in X if for eachε >0 andt >0,

there exist a positive integern₀such that for alln≥n₀andp>0, we haveµxn+p−xn(t)>

1−ε;

(3) A non-Archimedean RN-space(X,µ,T)is said to be complete (i.e.,(X,µ,T)is called a

non-Archimedean random Banach space) if every Cauchy sequence in X is convergent to a point inX.

Theorem 2.7. [7]If(X,µ,T)is a non-Archimedean RN-space and{xn}is a sequence such that

xn→x, then limn→∞µxn(t) =µx(t)almost everywhere.

Theorem 2.8. [31]Let f :X →Y be a function satisfying (1.12) for all x,y∈X . If f is even then f is quadratic - quartic.

Theorem 2.9. [31]Let f :X →Y be a function satisfying (1.12) for all x,y∈X . If f is odd then f is additive - cubic.

Theorem 2.10. [31]Let f :X →Y be a function satisfying (1.12) for all x,y∈X If and only if there exists functions A:X→Y , B:X2→Y , C:X3→Y and D:X4→Y such that

f(x) =A(x) +B(x,x) +C(x,x,x) +D(x,x,x,x) (2.1)

for all x∈X , where A is additive, B is symmetric bi-additive, C is symmetric for each fixed one

variable and is additive for fixed two variables and D is is symmetric multi - additive.

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In the rest of the paper let f :X →Y and we define

∆f(x,y) = f(x+ay) +f(x−ay)−a2[f(x+y) + f(x−y)]−2(1−a2)f(x)

−a

4₋_a2

12 [f(2y) +f(−2y)−4f(y)−f(−y)], whereain_Z− {−1,0,1}.

Now using fixed point approach to the non-Archimedean RN-space under arbitrary t-norm, we prove the stability of generalized mixed type of AQCQ- functional equations∆f(x,y) =0.

Theorem 3.1. Let _Kbe a non-Archimedean field, X be a vector space over _Kand(Y,µ,TM)

be a non-Archimedean random Banach space over_K. Let ϕ:X2→D+ (ϕ(x,y)is denoted by ϕx,y) be a function such that for someλ ∈R,0<λ <4

ϕ2x,2y(λt)≥ϕx,y(t), (3.1)

for all x,y∈X and t>0. If f :X→Y be an even mapping such that

µ_∆_f₍_x_,_y₎(t)≥ϕx,y(t); (3.2)

and f(0) =0, then there exists a unique quadratic mapping Q:X →Y satisfying (1.12) and

µ_f₍₂_x₎₋₁₆_f₍_x₎₋_Q₍_x₎(t)≥ψx(

4−λ

4 t), (3.3)

whereψx(t) =TM(ϕ0,x(a 2_t

12),ϕx,x( (a2−1)t

12 ),ϕ0,2x(

(a4−a2)t

6 ),ϕax,x(

(a4−a2)t

12 )).Moreover

Q(x) = lim

n→∞

1 4n(f(2

n+1_x)₋₁₆_f₍₂n_x)).

Proof.Using the evenness of f and (3.2), we get

µ

f(x+ay)+f(x−ay)−a2₍_f₍_x₊_y₎₊_f₍_x₋_y₎₎₋₂₍₁₋_a2₎_f₍_x₎₋a4−a2

12 (2f(2y)−8f(y))

(t)≥ϕx,y(t). (3.4)

for allx∈X andt>0. Interchangingxandyin (3.4), we obtain

µ

f(ax+y)+f(ax−y)−a2₍_f₍_x₊_y₎₊_f₍_x₋_y₎₎₋₂₍₁₋_a2₎_f₍_y₎₋a4−a2

12 (2f(2x)−8f(x))

(t)≥ϕy,x(t). (3.5)

for allx,y∈X andt>0. Lettingy=0 in (3.5), we get

µ

2f(ax)−2a2_f₍_x₎₋a4−a2

12 (2f(2x)−8f(x))

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Puttingy=xin (3.5), we obtain

µ

f((a+1)x)+f((a−1)x)−a2_f₍₂_x₎₋₂₍₁₋_a2₎_f₍_x₎₋a4−a2

12 (2f(2x)−8f(x))

(t)≥ϕx,x(t) (3.7)

forx∈X andt >0. Replacingxby 2xin (3.6) we get

µ

2f(2ax)−2a2_f₍₂_x₎₋a4−a2

12 (2f(4x)−8f(2x))

(t)≥ϕ0,2x(t). (3.8)

forx∈X andt >0. Settingybyaxin (3.5), we obtain

µ

f(2ax)−a2₍_f₍₍₁₊_a₎_x₎₊_f₍₍₁₋_a₎_x₎₎₋₂₍₁₋_a2₎_f₍_ax₎₋a4−a2

12 (2f(2x)−8f(x))

(t)≥ϕax,x(t). (3.9)

Hence, (3.6), (3.7), (3.8) and (3.9) imply that

µ₆₄_f₍_x₎₋₂₀_f₍₂_x₎₊_f₍₄_x₎(t)≥ TM(ϕ0,x(

a2t

12),ϕx,x(

(a2−1)t

12 ),ϕ0,2x(

(a4−a2)t

6 ),

ϕax,x(

(a4−a2)t

12 )) =ψx(t)

(3.10)

forx∈X andt >0. Now we putF(x) = f(2x)−16f(x). It follows that

µ_f₍₄_x₎₋₁₆_f₍₂_x₎₋₄₍_f₍₂_x₎₋₁₆_f₍_x₎₎(t)≥ψx(t),

which implies that

µ_F(2x)−4F(x)(t)≥ψx(t). (3.11)

Now, we define the set S byS:={F :X →Y}and introduce a generalized metric on Sas follows

d_ψ(F,G) =inf{ε∈_R+:µF(x)−G(x)(εt)≥ψx(t),∀x∈X,∀t>0}. (3.12)

Then, it is easy to verify that(S,d_ψ)is complete (see [30]). We define an operatorJ:S→Sby JL(x) =L(2₄x),for allx∈X.LetF,G∈Sandε∈_R+be an arbitrary constant withdψ(F,G)≤ε,

that is,

µ_F(x)−G(x)(εt)≥ψx(t) (3.13)

for allx∈X andt>0.Then

µ_JF(x)−JG(x)( λ εt

4 ) =µF(42x)−

G(2x) 4

(λ εt

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for all x ∈X and t >0, that is, d_ψ(JF,JG) ≤ λ ε

4 . We hence conclude that dψ(JF,JG)≤ λ

4dψ(F,G)for any F,G∈S. As 0<λ <4, then operator J is strictly contractive. It follows

from (3.11) that

µ_JF(x)−F(x)( εt

4) =µF(42x)−F(x)

(εt

4) =µF(2x)−4F(x)(εt)≥ψx(t) (3.15) for allx∈X andt>0, that is,d_ψ(JF,F)≤ ε

4<∞.Using Theorem 2.1, we deduce existence of

a fixed point ofJ, that is, the existence of mappingQ:X →Y satisfying the following: (1) Qis a fixed point ofJsuch that limn→∞dψ(JnF,Q) =0.This implies the equality

Q(x) = lim

n→∞

JnF(x) = lim

n→∞

F(2nx)

4n =_nlim_→_∞

1 4n(f(2

n+1_x)₋

16f(2nx))

andQ(2x) =4Q(x)for allx∈X. AlsoQis the unique fixed point ofJon the set

S∗={G∈S:dψ(F,G)<∞}.

(2)d_ψ(F,Q)≤ ₁₋1_Ld_ψ(JF,F)implies the inequalityd_ψ(F,Q)≤ 1 1−λ

4

. Then

µ_F₍_x₎₋_Q₍_x₎( 4t

4−λ)≥ψx(t),

which implies that

µ_F₍_x₎₋_Q₍_x₎(t)≥ψx(

4−λ

4 t). (3.16)

It follows from (3.1) and (3.2) that

µ∆Q(x,y)(t)≥ lim

n→∞

TM(ϕx,y(

4nt

λn+1),ϕx,y(

4nt

16λn)) =1.

Hence∆Q(x,y) =0 andµ_f₍₂_x₎₋₁₆_f₍_x₎₋_Q₍_x₎(t)≥ψx(4−₄λt)for allx∈Xandt>0. This completes

the proof.

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µ_∆_f(x,y)(t)≥ϕx,y(t) (3.18)

and f(0) =0, then there exists a unique quartic mapping D:X→Y satisfying (1.12) and

µ_f₍₂_x₎₋₄_f₍_x₎₋_D₍_x₎(t)≥ψx(

16−λ

16 t), (3.19)

12),ϕx,x( (a2−1)t

12 ),ϕ0,2x(

(a4−a2)t

6 ),ϕax,x(

(a4−a2)t

12 )),Moreover

D(x) = lim

n→∞

1 16n(f(2

n+1_x)₋₄_f₍₂n_x)).

Proof.Using Theorem 3.1, we haveµ_f₍₄_x₎₋₄_f₍₂_x₎₋₁₆₍_f₍₂_x₎₋₄_f₍_x₎₎(t)≥ψx(t).It follows that

µ_G(2x)−16G(x)(t)≥ψx(t) (3.20)

whereG(x) = f(2x)−4f(x),for all x∈X andt >0. Now, we define the set S by S:={G: X →Y}and introduce a generalized metric onSas follows

d_ψ(F,G) =inf{ε∈_R+:µ_F₍_x₎₋_G₍_x₎(εt)≥ψx(t),∀x∈X,∀t>0}. (3.21)

Then, it is easy to verify that(S,d_ψ)is complete (see [30]). We define an operatorJ:S→Sby JL(x) =L(2₁₆x),for allx∈X.LetF,G∈Sandε∈_R+be an arbitrary constant withdψ(F,G)≤ε,

that is,

µ_F(x)−G(x)(εt)≥ψx(t) (3.22)

µ_JF₍_x₎₋_JG₍_x₎(λ εt

16 ) =µF(162x)−

G(2x) 16

(λ εt

16 ) =µF(2x)−G(2x)(λ εt)≥ψ2x,2x(λt)≥ψx(t) (3.23) for all x ∈X and t >0, that is, d_ψ(JF,JG) ≤ λ ε

16. We hence conclude that dψ(JF,JG)≤ λ

16dψ(F,G) for any F,G∈S. As 0<λ <16, then operator J is strictly contractive. It

fol-lows from (3.20) that

µJG(x)−G(x)( εt

16) =µG(162x)−G(x)

(εt

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for allx∈X andt>0, that is,d_ψ(JG,G)≤ ε

16 <∞.By Theorem 2.1, we deduce existence of

a fixed point ofJ, that is, the existence of mappingD:X →Y satisfying the following: (1)Dis a fixed point ofJsuch that limn→∞dψ(JnG,D) =0 This implies the equality

D(x) = lim

n→∞

JnG(x) = lim

n→∞

F(2nx)

16n =_nlim_→_∞

1 16n(f(2

n+1_x)₋₄_f₍₂n_x))

andD(2x) =16D(x)for allx∈X. AlsoDis the unique fixed point ofJon the set M:={G∈S:dψ(F,G)<∞}.

(2)d_ψ(G,D)≤ ₁₋1_Ld_ψ(JG,G)implies the inequalityd_ψ(G,D)≤ 1 1−λ

16

. Then

µ_G(x)−D(x)(

16t

16−λ)≥ψx(t),

which implies that

µ_G₍_x₎₋_D₍_x₎(t)≥ψx(

16−λ

16 t), (3.25)

µ_∆D₍_x_,_y₎(t)≥ lim

n→∞

T_M(ϕ₂n+1_x_,₂n+1_y(16nt),ϕ₂n_x_,₂n_y₎(

16nt 4 ))

≥ lim

n→∞

T_M(ϕx,y(

16nt

λn+1),ϕx,y(

16nt

4λn)) =1.

Then∆D(x,y) =0 andµ_f(2x)−4f(x)−D(x)(t)≥ψx(16₁₆−λt)for allx∈Xandt>0. This completes

the proof.

ϕ2x,2y(λt)≥ϕx,y(t), (3.26)

µ_∆_f(x,y)(t)≥ϕx,y(t); (3.27)

and f(0) =0, then there exists a unique quadratic mapping Q:X →Y and a unique quartic mapping D:X →Y satisfying (1.12) and

µ_f₍_x₎₋_Q₍_x₎₋_D₍_x₎(t)≥TM(ψx(

3(16−λ)

(13)

12),ϕx,x( (a2−1)t

12 ),ϕ0,2x(

(a4−a2)t

6 ),ϕax,x(

(a4−a2)t

12 )).

Proof. Using Theorems 3.1 and 3.2, we see that there exists a unique quadratic functionQ1:

X →Y and a unique quartic functionD₁:X →Y such that µ_f₍₂_x₎₋₄_f₍_x₎₋_D₁₍_x₎(t)≥ψx(16₁₆−λt)

andµ_f(2x)−16f(x)−Q1(x)(t)≥ψx( 4−λ

4 t).Then

µ_f₍₂_x₎₋₄_f₍_x₎₋_D₁₍_x₎₋_[_f₍₂_x₎₋₁₆_f₍_x₎₋_Q₁₍_x_)](t)≥TM(µf(2x)−4f(x)−D1(x)(t),µf(2x)−16f(x)−Q1(x)(t)),

which implies thatµ₁₂_f₍_x₎₋_D₁₍_x₎₊_Q₁₍_x₎(t)≥TM(ψx(16₁₆−λt),ψx(4−₄λt)).Hence, we have

µ_f₍_x₎₋1

12D1(x)+121Q1(x)(

1

12t)≥TM(ψx( 16−λ

16 t),ψx( 4−λ

4 t)).

It follows thatµ_f₍_x₎₋_Q₍_x₎₋_D₍_x₎(t)≥TM(ψx(3(16₄−λ)t),ψx(3(4−λ)t)),whereQ(x) =−₁₂1Q1(x)

andD(x) = ₁₂1D1(x).

ϕ2x,2y(λt)≥ϕx,y(t), (3.29)

for all x,y∈X and t>0. If f :X→Y be an odd mapping with f(0) =0and satisfying

µ_∆_f(x,y)(t)≥ϕx,y(t) (3.30)

for all x,y∈X and t>0. Then there exists a unique additive mapping A:X →Y satisfying (1.12) and

µ_f₍₂_x₎₋₈_f₍_x₎₋_A₍_x₎(t)≥βx(

2−λ

2 t), (3.31)

whereβx(t) =TM(φx(₂t),αx(t)),

φx(t) =TM(ϕx,x(

a2t

2 ),ϕ2x,x((a

2₋_1)t),

ϕx,2x((a4−a2)t),

TM(ϕ(1+a)x,x((a4−a2)t),ϕ(1−a)x,x((a4−a2)t))),

and

αx(t) =TM(ϕ2x,2x((a2−1)t),ϕx,2x(

a2t

2 ),ϕx,x((a

4₋_a2_)t),

ϕx,3x((a4−a2)t),

T_M(ϕ₍₁₊₂_a₎_x_,_x((a4−a2)t),ϕ₍₁₋₂_a₎_x_,_x((a4−a2)t))),

Moreover A(x) =limn→∞

1

(14)

Proof.Using the oddness of f and (3.30), we get

µ_f₍_x₊_ay₎₊_f₍_x₋_ay₎₋_a2₍_f₍_x₊_y₎₊_f₍_x₋_y₎₎₋₂₍₁₋_a2₎_f₍_x₎(t)≥ϕx,y(t). (3.32)

Replacingybyxin (3.32), we obtain

µ_f₍₍₁₊_a₎_x₎₊_f₍₍₁₋_a₎_x₎₋_a2_f₍₂_x₎₋₂₍₁₋_a2₎_f₍_x₎(t)≥ϕx,x(t). (3.33)

Replacingxby 2xin (3.33), we obtain

µ_f₍₂₍₁₊_a₎_x₎₊_f₍₂₍₁₋_a₎_x₎₋_a2_f₍₄_x₎₋₂₍₁₋_a2₎_f₍₂_x₎(t)≥ϕ2x,2x(t). (3.34)

Again replacing(x,y)by(2x,x)in (3.32), we get

µ_f₍₍₂₊_a₎_x₎₊_f₍₍₂₋_a₎_x₎₋_a2₍_f₍₃_x₎₊_f₍_x₎₎₋₂₍₁₋_a2₎_f₍₂_x₎(t)≥ϕ2x,x(t) (3.35)

for allx∈X andt>0.Replacingyby 2xin (3.32), we obtain

µ_f₍₍₁₊₂_a₎_x₎₊_f₍₍₁₋₂_a₎_x₎₋_a2₍_f₍₃_x₎₋_f₍_x₎₎₋₂₍₁₋_a2₎_f₍_x₎(t)≥ϕ_x_,₂_x(t) (3.36)

for allx∈X andt>0.Replacingyby 3xin (3.32), we get

µ_f₍₍₁₊₃_a₎_x₎₊_f₍₍₁₋₃_a₎_x₎₋_a2₍_f₍₄_x₎₋_f₍₂_x₎₎₋₂₍₁₋_a2₎_f₍_x₎(t)≥ϕ_x_,₃_x(t) (3.37)

for allx∈X andt>0.Replacing(x,y)by((1+a)x,x)in (3.32), we obtain

µ_f₍₍₁₊₂_a₎_x₎₊_f₍_x₎₋_a2₍_f₍₍₂₊_a₎_x₎₊_f₍_ax₎₎₋₂₍₁₋_a2₎_f₍₍₁₊_a₎_x₎(t)≥ϕ₍₁₊_a₎_x_,_x(t) (3.38)

for allx∈X andt>0.Again replacing(x,y)by((1−a)x,x)in (3.32), we obtain

µ_f₍₍₁₋₂_a₎_x₎₊_f₍_x₎₋_a2₍_f₍₍₂₋_a₎_x₎₋_f₍_ax₎₎₋₂₍₁₋_a2₎_f₍₍₁₋_a₎_x₎(t)≥ϕ₍₁₋_a₎_x_,_x(t) (3.39)

for allx∈X andt>0.By (3.38) and (3.39), we get

µ_f₍₍₁₊₂_a₎_x₎₊_f₍₍₁₋₂_a₎_x₎₊₂_f₍_x₎₋_a2₍_f₍₍₂₊_a₎_x₎₊_f₍₍₂₋_a₎_x₎₎₋₂₍₁₋_a2₎₍_f₍₍₁₊_a₎_x₎₊_f₍₍₁₋_a₎_x₎₎(t)

≥T_M(ϕ₍₁₊_a₎_x_,_x(t),ϕ₍₁₋_a₎_x_,_x(t))

(3.40)

for allx∈X andt>0.Replacing(x,y)by((1−2a)x,x)in (3.32), we obtain

(15)

Again replacing(x,y)by((1+2a)x,x)in (3.32), we get

µ_f₍₍₁₊₃_a₎_x₎₊_f₍₍₁₊_a₎_x₎₋_a2₍_f₍₍₂₊₂_a₎_x₎₊_f₍₂_ax₎₎₋₂₍₁₋_a2₎_f₍₍₁₊₂_a₎_x₎(t)≥ϕ(1+2a)x,x(t) (3.42)

By (3.41) and (3.42), we obtain

µ_f₍₍₁₊₃_a₎_x₎₊_f₍₍₁₋₃_a₎_x₎₊_f₍₍₁₊_a₎_x₎₊_f₍₍₁₋_a₎_x₎₋_a2₍_f₍₍₂₊₂_a₎_x₎₊_f₍₍₂₋₂_a₎_x₎₎

−2(1−a2₎₍_f₍₍₁₊₂_a₎_x₎₊_f₍₍₁₋₂_a₎_x₎₎(t)≥T_M(ϕ₍₁₊_a₎_x_,_x(t),ϕ₍₁₋_a₎_x_,_x(t)).

(3.43)

Using (3.33), (3.35), (3.36), and (3.40), we see that

µ_[4_f₍₂_x₎₋₅_f₍_x₎₋_f₍₃_x_)](t)≥TM(ϕx,x(

a2t

2 ),ϕ2x,x((a

2₋_1)t),

ϕx,2x((a4−a2)t),

TM(ϕ(1+a)x,x((a4−a2)t),ϕ(1−a)x,x((a4−a2)t)))

=φx(t).

(3.44)

It follows from (3.34), (3.36), (3.33), (3.37), and (3.43) that

µ[f(4x)−2f(2x)−2f(3x)+6f(x)](t)≥TM(ϕ2x,2x((a2−1)t),ϕx,2x(

a2t

2 ),ϕx,x((a

4₋_a2_)t),

ϕx,3x((a4−a2)t),TM(ϕ(1+2a)x,x((a4−a2)t),

ϕ(1−2a)x,x((a4−a2)t)))

=αx(t)

(3.45)

for allx∈X andt>0.From (3.44) and (3.45) we have

µ_[_f₍₄_x₎₋₁₀_f₍₂_x₎₊₁₆_f₍_x_)](t) =µ₂_f₍₃_x₎₋₈_f₍₂_x₎₊₁₀_f₍_x₎₊_f₍₄_x₎₋₂_f₍₃_x₎₋₂_f₍₂_x₎₊₆_f₍_x₎(t)

≥T_M(µ₂_f₍₃_x₎₋₈_f₍₂_x₎₊₁₀_f₍_x₎(t),µ_f₍₄_x₎₋₂_f₍₃_x₎₋₂_f₍₂_x₎₊₆_f₍_x₎(t))

≥T_M(φx(

t

2),αx(t)) =βx(t), which implies that

µ_[_f₍₄_x₎₋₈_f₍₂_x₎₋₂₍_f₍₂_x₎₋₈_f₍_x_))](t)≥βx(t). (3.46)

PuttingH(x) = f(2x)−8f(x), we see that

(16)

for allx∈Xandt>0.New we define the setPbyP:={H:X→Y}and introduce a generalized metric onPas follows

d_β(F,G) =inf{ε∈_R+:µF(x)−G(x)(εt)≥βx(t),∀x∈X,∀t>0}. (3.48)

Then, it is easy to verify that(P,d_β)is complete (see [30]). We define an operatorJ:P→Pby JL(x) =L(2₂x),for allx∈X.LetF,H∈Pandε∈R+be an arbitrary constant withdβ(F,H)≤ε,

that is,

µ_F₍_x₎₋_H₍_x₎(εt)≥βx(t) (3.49)

µ_JF(x)−JH(x)( λ εt

2 ) =µF(22x)−

H(2x) 2

(λ εt

2 ) =µF(2x)−H(2x)(λ εt)≥β2x,2x(λt)≥βx(t) (3.50) for all x ∈ X and t >0, that is, d_β(JF,JH) ≤ λ ε

2 . We hence conclude that dβ(JF,JH)≤ λ

2dβ(F,H)for any F,H ∈P. As 0<λ <2, then operator J is strictly contractive. It follows

from (3.47) that

µ_JH₍_x₎₋_H₍_x₎(εt

2) =µH(22x)−H(x)

(εt

2) =µH(2x)−2H(x)(εt)≥βx(t) (3.51) for allx∈X andt>0, that is,d_β(JH,H)≤ ε

2 <∞.Using Theorem 2.1, we deduce existence

of a fixed point ofJ, that is, the existence of mappingA:X→Y satisfying the following: (1)Ais a fixed point ofJsuch that limn→∞dβ(J

n_H,_{A) =}_{0 This implies the equality}

A(x) = lim

n→∞

JnH(x) = lim

n→∞

H(2nx)

2n =_nlim_→_∞

1 2n(f(2

n+1_x)₋

8f(2nx))

andA(2x) =2A(x)for allx∈X. AlsoAis the unique fixed point ofJon the set

P∗:={H∈S:d_β(F,H)<∞}.

(2) d_β(H,A)≤ ₁₋1_Ld_β(JH,H)implies the inequalityd_β(H,A)≤ 1 1−λ

2

. Then

µ_H₍_x₎₋_A₍_x₎( 2t

2−λ)≥βx(t),

which implies that

µ_H₍_x₎₋_A₍_x₎(t)≥βx(

2−λ

(17)

µ_∆A(x,y)(t)≥ lim

n→∞

TM(ϕx,y(

2nt

λn+1),ϕx,y(

2nt

8λn)) =1.

Then∆A(x,y) =0 andµ_f(2x)−8f(x)−A(x)(t)≥βx(2−₂λt)for allx∈X andt>0. This completes

the proof.

ϕ2x,2y(λt)≥ϕx,y(t), (3.53)

µ_∆_f₍_x_,_y₎(t)≥ϕx,y(t) (3.54)

for all x,y∈X and t>0. Then there exists a unique cubic mapping C:X →Y satisfying (1.12) and

µ_f(2x)−2f(x)−C(x)(t)≥βx(

8−λ

8 t), (3.55)

whereβx(t) =TM(φx(₂t),αx(t)),

φx(t) =TM(ϕx,x(

a2t

2 ),ϕ2x,x((a

2₋_1)t),

ϕx,2x((a4−a2)t),

T_M(ϕ₍₁₊_a₎_x_,_x((a4−a2)t),ϕ₍₁₋_a₎_x_,_x((a4−a2)t))),

and

a2t

2 ),ϕx,x((a

4₋_a2_)t),

ϕx,3x((a4−a2)t),

TM(ϕ(1+2a)x,x((a4−a2)t),ϕ(1−2a)x,x((a4−a2)t))),

Moreover C(x) =lim_n_→_∞1

8n(f(2n+1x)−2f(2nx)).

Proof.Using Theorem 3.4, we have

µ_[_f₍₄_x₎₋₂_f₍₂_x₎₋₈₍_f₍₂_x₎₋₂_f₍_x_))](t)≥βx(t). (3.56)

PuttingK(x) = f(2x)−2f(x), we find that

(18)

for allx∈Xandt>0.New we define the setNbyN:={K:X→Y}and introduce a generalized metric onNas follows

d_β(F,G) =in f{ε∈_R+:µF(x)−G(x)(εt)≥βx(t),∀x∈X,∀t>0}. (3.58)

Then, it is easy to verify that(N,d_β)is complete (see [30]). We define an operatorJ:N→Nby JL(x) =L(2₈x),for allx∈X.LetF,K∈Nandε∈R+be an arbitrary constant withdβ(F,K)≤ε,

that is,

µ_F₍_x₎₋_K₍_x₎(εt)≥βx(t) (3.59)

µ_JF(x)−JK(x)( λ εt

8 ) =µF(82x)−

K(2x) 8

(λ εt

8 ) =µF(2x)−K(2x)(λ εt)≥β2x,2x(λt)≥βx(t) (3.60) for all x∈ X and t >0, that is, d_β(JF,JK) ≤ λ ε

8 . We hence conclude that dβ(JF,JK)≤ λ

8dβ(F,K) for any F,K∈N. As 0<λ <8, then operatorJ is strictly contractive. It follows

from (3.57) that

µ_JK₍_x₎₋_K₍_x₎(εt

8) =µK(82x)−K(x)

(εt

8) =µK(2x)−8K(x)(εt)≥βx(t) (3.61) for allx∈Xandt>0, that is,d_β(JK,K)≤ ε

8<∞.Using Theorem 2.1, we deduce existence of

a fixed point ofJ, that is, the existence of mappingC:X→Y satisfying the following: (1)Cis a fixed point ofJsuch that limn→∞dβ(J

n_K,_{C) =}_{0. This implies the equality}

C(x) = lim

n→∞

JnK(x) = lim

n→∞

K(2nx)

2n =_nlim_→_∞

1 8n(f(2

n+1_x)₋₂_f₍₂n_x))

andC(2x) =8C(x)for allx∈X. AlsoCis the unique fixed point ofJon the set

N∗:={K∈S:d_β(F,K)<∞}.

(2) d_β(K,C)≤₁₋1_Ld_β(JK,K)implies the inequalityd_β(K,C)≤ 1 1−λ

8

Then

µ_K₍_x₎₋_C₍_x₎( 8t

8−λ)≥βx(t),

which implies that

µ_K₍_x₎₋_C₍_x₎(t)≥βx(

8−λ

(19)

µ_∆C₍_x_,_y₎(t)≥ lim

n→∞

T_M(ϕx,y(

8nt

λn+1),ϕx,y(

8nt

2λn)) =1.

Then∆C(x,y) =0 andµ_f₍₂_x₎₋₂_f₍_x₎₋_C₍_x₎(t)≥βx(8−₈λt)for allx∈X andt>0. This completes

the proof.

ϕ2x,2y(λt)≥ϕx,y(t), (3.63)

µ_∆_f₍_x_,_y₎(t)≥ϕx,y(t); (3.64)

Then there exists a unique additive mapping A:X →Y and a unique cubic mapping C:X→Y satisfying (1.12) and

µ_f₍_x₎₋_A₍_x₎₋_C₍_x₎(t)≥TM(βx(

3(8−λ)

4 t),βx(3(2−λ)t)), (3.65) whereβx(t) =TM(φx(₂t),αx(t)),

φx(t) =TM(ϕx,x(

a2t

2 ),ϕ2x,x((a

2₋_1)t),

ϕx,2x((a4−a2)t),

T_M(ϕ₍₁₊_a₎_x_,_x((a4−a2)t),ϕ₍₁₋_a₎_x_,_x((a4−a2)t))),

and

a2t

2 ),ϕx,x((a

4₋_a2

)t),ϕx,3x((a4−a2)t),

T_M(ϕ₍₁₊₂_a₎_x_,_x((a4−a2)t),ϕ₍₁₋₂_a₎_x_,_x((a4−a2)t))).

Proof. Using Theorems 3.4 and 3.5, there exists a unique additive functionA1:X →Y and a

unique cubic functionC₁:X →Y such thatµ_f₍₂_x₎₋₈_f₍_x₎₋_A₁₍_x₎(t)≥βx(2−₂λt)and

µ_f₍₂_x₎₋₂_f₍_x₎₋_C₁₍_x₎(t)≥βx(

8−λ

(20)

for allx∈X andt>0. Then

µ_f₍₂_x₎₋₂_f₍_x₎₋_C₁₍_x₎₋_[_f₍₂_x₎₋₈_f₍_x₎₋_A₁₍_x_)](t)≥TM(µf(2x)−2f(x)−C1(x)(t),µf(2x)−8f(x)−A1(x)(t)),

which implies thatµ₆_f₍_x₎₋_C₁₍_x₎₊_A₁₍_x₎(t)≥TM(βx(8−₈λt),βx(2−₂λt)).Hence

µ_f₍_x₎₋1

6C1(x)+16A1(x)(

1

6t)≥TM(βx( 8−λ

8 t),βx( 2−λ

2 t)).

It follows thatµ_f₍_x₎₋_A₍_x₎₋_C₍_x₎(t)≥TM(βx(3(8₄−λ)t),βx(3(2−λ)t))for allx∈X andt>0, where

A(x) =−1₆A₁(x)andC(x) =1₆C₁(x).

ϕ2x,2y(λt)≥ϕx,y(t), (3.66)

for all x,y∈X and t>0. If f :X→Y be a mapping with f(0) =0and satisfying

µ_∆_f₍_x_,_y₎(t)≥ϕx,y(t). (3.67)

Then there exists a unique additive mapping A:X→Y , a unique quadratic mapping Q:X→Y , a unique cubic mapping C:X→Y and a unique quartic mapping D:X→Y such that

µ_f(x)−A(x)−Q(x)−C(x)−D(x)(t)≥TM{TM(ψex(

3(16−λ)

4 t),ψex(3(4−λ)t)),

TM(βe_x(

3(8−λ)

4 t),βex(3(2−λ)t))},

(3.68)

where

e

ψx(t) =TM(ϕ0e,x(

a2t

12),ϕex,x(

(a2−1)t

12 ),ϕ0e,2x(

(a4−a2)t

6 ),ϕeax,x(

(a4−a2)t 12 )),

e

βx(t) =TM(φex(

t

2),αex(t)), e

φx(t) =TM(ϕex,x(

a2t

2 ),ϕe2x,x((a

2₋_1)t),

e

ϕx,2x((a4−a2)t),

TM(ϕe(1+a)x,x((a

4₋_a2_)t_),

e

ϕ(1−a)x,x((a4−a2)t))),

e

αx(t) =TM(ϕ2ex,2x((a

2₋_1)t_),

e

ϕx,2x(

a2t

2 ),ϕex,x((a

4₋_a2_)t),

e

ϕx,3x((a4−a2)t),

T_M(ϕe(1+2a)x,x((a

4₋_a2

)t),ϕe(1−2a)x,x((a

4₋_a2

(21)

andϕ_ex,y(t) =TM(ϕx,y(2t),ϕ−x,−y(2t))for all x∈X and t>0.

Proof. Let fe(x) = 1₂(f(x) + f(−x)) for all x∈ X. Then fe(0) = 0, fe(−x) = fe(x) and 2∆fe(x,y) =∆f(x,y) +∆f(−x,−y).Hence

µ₂_∆_fe₍_x_,_y₎(t) =µ_∆_f₍_x_,_y₎₊_∆_f₍₋_x_,−_y₎(t)≥TM(µ∆f(x,y)(t),µ∆f(−x,−y)(t)),

which implies that

µ_∆_fe₍_x_,_y₎(t)≥T_M(µ_∆_f(x,y)(2t),µ∆f(−x,−y)(2t))≥TM(ϕx,y(2t),ϕ−x,−y(2t)) =:ϕex,y(t).

Using Theorem 3.3 there exist a unique quadratic mapping Q:X →Y and a unique quartic mappingD:X →Y such that

µ_fe₍_x₎₋_Q₍_x₎₋_D₍_x₎(t)≥T_M(ψ_ex(

3(16−λ)

4 t),ψex(3(4−λ)t)) (3.69)

whereψex(t) =TM(ϕe0,x( a2t

12),ϕex,x(

(a2−1)t

12 ),ϕe0,2x(

(a4−a2)t

6 ),ϕeax,x(

(a4−a2)t

12 )).for allx∈X andt>

0. Again fo(x) = 1₂(f(x)− f(−x)) for all x∈X. Then fo(0) = 0, fo(−x) = −fo(x) and 2∆fo(x,y) =∆f(x,y)−∆f(−x,−y).Hence

µ₂_∆_fo₍_x_,_y₎(t) =µ_∆_f(x,y)−∆f(−x,−y)(t)≥TM(µ∆f(x,y)(t),µ∆f(−x,−y)(t)).

Using Theorem 3.6, we see that there exist a unique additive mappingA:X →Y and a unique cubic mappingC:X →Y such that

µ_fo₍_x₎₋_A₍_x₎₋_C₍_x₎(t)≥T_M(βe_x(

3(8−λ)

4 t),βex(3(2−λ)t)), (3.70) whereβe_x(t) =T_M(φe_x(₂t),α_e_x(t)),

e

φx(t) =TM(ϕex,x(

a2t

2 ),ϕ2e x,x((a

2₋_1)t),

e

ϕx,2x((a4−a2)t),

T_M(ϕe(1+a)x,x((a

4₋_a2_)t_),

e

ϕ₍₁₋_a₎_x_,_x((a4−a2)t))),

and

e

αx(t) =TM(ϕe2x,2x((a

2₋

1)t),ϕex,2x(

a2t

2 ),ϕex,x((a

4₋_a2

)t),ϕex,3x((a

4₋_a2

)t),

T_M(ϕe(1+2a)x,x((a

4₋_a2_)t),

e

ϕ₍₁₋₂_a₎_x_,_x((a4−a2)t)))

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In the following, we give a corollary which was obtained in [32].

Corollary 3.8. Let_Kbe a non-Archimedean field, X be a vector space over_Kand(Y,µ,TM)

be a non-Archimedean random Banach space over_K. Let ϕ:X2→D+ (ϕ(x,y)is denoted by ϕx,y) be a function such that for some λ ∈R, 0<λ <2ϕ2x,2y(λt)≥ϕx,y(t), for all x,y∈X

and t>0. If f :X→Y be a mapping with f(0) =0and satisfying

µ_f(x+2y)+f(x−2y)−4(f(x+y)+f(x−y))+6f(x)−f(2y)−f(−2y)+4f(y)+4f(−y)(t)≥ϕx,y(t).

Then there exists a unique additive mapping A:X→Y , a unique quadratic mapping Q:X→Y , a unique cubic mapping C:X→Y and a unique quartic mapping D:X→Y such that

µ_f₍_x₎₋_A₍_x₎₋_Q₍_x₎₋_C₍_x₎₋_D₍_x₎(t)≥TM{TM(ψex(

3(16−λ)

4 t),ψex(3(4−λ)t)),

TM(_βe_x(

3(8−λ)

4 t),βex(3(2−λ)t))}, whereψ_ex(t) =TM(ϕe0,x(

t

3),ϕex,x( t

4),ϕe0,2x(2t),ϕe2x,x(t)),βex(t) =TM(φex( t

2),αex(t)),

e

φx(t) =TM(ϕex,x(2t),ϕe2x,x(3t),ϕex,2x(12t),TM(ϕe3x,x(12t),ϕe(−x,x(12t))),

e

αx(t) =TM(ϕe2x,2x(3t),ϕex,2x(2t),ϕex,x(12t),ϕex,3x(12t),TM(ϕe5x,x(12t),ϕe−3x,x(12t))),

andϕ_ex,y(t) =TM(ϕx,y(2t),ϕ−x,−y(2t)).

Conflict of Interests

The authors declare that there is no conflict of interests. Acknowledgement

The authors are thankful to the anonymous referees for valuable suggestions.

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