��

DOI: 10.2478/s12175-013-0197-z

Math. Slovaca 64 (2014), No. 1, 209–228

ON THE STABILITY

OF SOME FUNCTIONAL EQUATIONS

IN MENGER ϕ-NORMED SPACES

Dorel Mihet* — Reza Saadati**

(Communicated by Anatolij Dvurecenskij )

ABSTRACT. Recently, the authors [MIHET, D.—SAADATI, R.—VAEZ-POUR, S. M.: The stability of an additive functional equation in Menger prob-abilistic ϕ-normed spaces, Math. Slovaca 61 (2011), 817–826] considered the

stability of an additive functional in Menger ϕ-normed spaces. In this paper,we establish some stability results concerning the cubic, quadratic and quarticfunctional equations in complete Menger ϕ-normed spaces via fixed point theory.

c©2014Mathematical Institute

Slovak Academy of Sciences

1. Introduction

The study of stability problems for functional equations originates from aquestion of Ulam [40] concerning the stability of group homomorphisms, affir-matively answered by Hyers [8] and further generalized by Aoki [1] for additivemappings and by Rassias [34] for linear mappings. For some information onHyers-Ulam-Rassias stability for different functional equations the interestedreader is referred to [7,9,10,13,17,37].

In the recent papers [2], [23]–[30] the stability in the settings of probabilisticand fuzzy normed spaces is studied. Radu [33] pointed out that a fixed point al-ternative method can be successfully used to solve the Ulam problem. The fixedpoint method in the study of the probabilistic stability of functional equationsfirstly appears in [25] (see also [28]). In this paper we employ this method to

2010 Mathemat i c s Sub j e c t C l a s s i f i c a t i on: Primary 54E40; Secondary 39B82, 46S50,46S40.Keywords: probabilistic normed space, Hyers-Ulam stability, cubic functional equation, qua-dratic functional equation, quartic functional equation.

DOREL MIHET — REZA SAADATI

investigate the stability of the cubic and quartic functional equations in Mengerprobabilistic ϕ-normed spaces introduced by Golet in [11].

A function F : R → [0, 1] is called a distribution function if it is non-decreasingand left continuous with sup

t∈R

F (t) = 1 and inft∈R

F (t) = 0. The class of all distribu-

tion functions F with F (0) = 0 is denoted by D+. ε0 is the specific distributionfunction defined through

ε0(t) =

{0, t ≤ 0

1, t > 0.

���������� 1.1� A mapping T : [0, 1] × [0, 1] → [0, 1] is a triangular norm(briefly, a t-norm) if T satisfies the following conditions:

(i) T is commutative and associative;

(ii) T (a, 1) = a for all a ∈ [0, 1];

(iii) T (a, b) ≤ T (c, d) whenever a ≤ c and b ≤ d (a, b, c, d ∈ [0, 1]).

Three typical examples of continuous t-norms are TP (a, b) = ab, TL(a, b) =max(a+ b− 1, 0) and TM (a, b) = min(a, b).

Let ϕ be a function defined on the real field R into itself, with the followingproperties:

(a) ϕ(−t) = ϕ(t) for every t ∈ R;

(b) ϕ(1) = 1;

(c) ϕ is strictly increasing and continuous on [0,∞), ϕ(0) = 0 and limα→∞

ϕ(α)= ∞.

Examples of such functions are:

ϕ(α) = |α|; ϕ(α) = |α|p, p ∈ (0,∞); ϕ(α) =2α2n

|α|+ 1, n ∈ N.

���������� 1.2� ([11]) A Menger probabilistic ϕ-normed space is a triple(X, ν, T ), where X is a real vector space, T is a continuous t-norm, and ν isa from X into D+ such that the following conditions hold:

(PN1) νx(t) = ε0(t) for all t > 0 if and only if x = 0;

(PN2) ναx(t) = νx(t

ϕ(α) ) for all x in X, α �= 0 and t > 0;

(PN3) νx+y(t+ s) ≥ T (νx(t), νy(s)) for all x, y ∈ X and t, s ≥ 0.

We note that if ϕ(t) = |t| one obtains the definition of a random normedspace (RN space) or Menger probabilistic normed space, see [38]. An importantexample is given by the triple (X, ν, TM), where (X, ‖ · ‖) is a normed space and

νx(t) =t

t+ ‖x‖p , p ∈ (0, 1].

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STABILITY OF SOME FUNCTIONAL EQUATIONS IN MENGER ϕ-NORMED SPACES

If (X, ν, T ) is a probabilistic ϕ-normed space under a continuous t-norm Tsuch that T ≥ TL, then

V ={V (ε, λ) : ε > 0, λ ∈ (0, 1)

}, V (ε, λ) =

{x ∈ X : νx(ε) > 1− λ

}is a complete system of neighborhoods of null vector for linear topology τ on Xgenerated by the ϕ-norm ν.

���������� 1.3� Let (X, ν, T ) be a Menger probabilistic ϕ-normed space.

(i) A sequence {xn} in X is said to be convergent to x in X in the topologyτ if for every t > 0 and ε > 0, there exists positive integer N such thatνxn−x(t) > 1− ε whenever n ≥ N .

(ii) A sequence {xn} in X is called Cauchy if for every t > 0 and ε > 0, thereexists positive integer N such that νxn−xm

(t) > 1− ε whenever n,m ≥ N .

(iii) A Menger probabilistic ϕ-normed space (X, ν, T ) is said to be complete ifevery Cauchy sequence in X is convergent to a point in X.

The notion of generalized metric space has been introduced by Luxemburgin [22], by allowing the value +∞ for the distance mapping. The followinglemma (Luxemburg-Jung theorem) will be used in the proof of Theorem 2.1.

�� 1.4� ([19]) Let (X, d) be a complete generalized metric space andT : X → X be a strict contraction with the Lipschitz constant k, such thatd(x0, T (x0)) < +∞ for some x0 ∈ X. Then T has a unique fixed point inthe set Y :=

{y ∈ X : d(x0, y) < ∞}

and the sequence (Tn(x))n∈N convergesto the fixed point x∗ for every x ∈ Y . Moreover, d(x0, T (x0)) ≤ δ impliesd(x∗, x0) ≤ δ

1−k .

Let X be a linear space, (Y, ν, TM) be a complete Menger probabilisticϕ-normed space andG be a mapping fromX×R into [0, 1], such thatG(x, ·)∈D+

for all x. Consider the set E :={g : X → Y : g(0) = 0

}and the mapping dG

defined on E × E by

dG(g, h) = inf{u ∈ R

+ : νg(x)−h(x)(ut) ≥ G(x, t) for all x ∈ X and t > 0}

where, as usual, inf φ = +∞. The following lemma can be proved as in [23]:

�� 1.5� (cf. [23,25]) dG is a complete generalized metric on E.

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DOREL MIHET — REZA SAADATI

2. Probabilistic stability of the equation 1af(bx) = f(x)

Our first result is the following stability theorem in random ϕ-normed spaces:

���� � 2.1� Let X be a real linear space, a, b ∈ R, a �= 0, f be a mappingfrom X into a complete Menger probabilistic ϕ-normed space (Y, ν, TM) withf(0) = 0 and let G : X → D+ be a mapping with the property

∃α ∈ (0, ϕ(a)) ∀x ∈ X ∀t ∈ (0,∞) : G(bx, αt) ≥ G(x, t). (2.1)

If

ν 1a f(bx)−f(x)(t) ≥ G(x, t) (x ∈ X, t > 0),

then there exists a unique mapping g : X → X such that g(bx) = ag(x), for allx ∈ X and

νg(x)−f(x)(t) ≥ G

(x,

ϕ(a)− α

ϕ(a)t

)(x ∈ X, t > 0).

Moreover,

g(x) = limn→∞

f(bnx)

an(x ∈ X).

P r o o f. Consider the set

E :={g : X → Y : g(0) = 0

}and the mapping dG defined on E ×E by

dG(g, h) = inf{u ∈ R+ : νg(x)−h(x)(ut) ≥ G(x, t) for all x ∈ X and t > 0

}.

By Lemma 1.5, (E, dG) is a complete generalized metric space. Now, let usconsider the linear mapping J : E → E,

Jg(x) :=1

ag(bx).

We show that J is a strictly contractive self-mapping of E with the Lipschitzconstant k = α/ϕ(a).

Indeed, let g, h in E be such that dG(g, h) < ε. Then

∀x ∈ X ∀t ∈ (0,∞) : νg(x)−h(x)(εt) ≥ G(x, t),

whence

νJg(x)−Jh(x)

(α

ϕ(a)εt

)= ν 1

a (g(bx)−h(bx))

(α

ϕ(a)εt

)= νg(bx)−h(bx)(αεt) ≥ G(bx, αt) (x ∈ X, t > 0).

Since G(bx, αt) ≥ G(x, t), then νJg(x)−Jh(x)(α

ϕ(a) εt) ≥ G(x, t), that is,

dG(g, h) < ε =⇒ dG(Jg, Jh) ≤ α

ϕ(a)ε.

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STABILITY OF SOME FUNCTIONAL EQUATIONS IN MENGER ϕ-NORMED SPACES

This means that

dG(Jg, Jh) ≤ α

ϕ(a)dG(g, h),

for all g, h in E.

Next, from

νf(x)− 1a f(bx)(t) ≥ G(x, t)

it follows that dG(f, Jf) ≤ 1. Using the Luxemburg-Jung theorem (Lemma 1.4),we deduce the existence of a fixed point of J , that is, the existence of a mappingg : X → Y such that g(bx) = ag(x), for all x ∈ X.

Since for any x ∈ X and t > 0,

dG(u, v) < ε =⇒ νu(x)−v(x)(t) ≥ G

(x,

t

ε

),

from dG(Jnf, g) → 0, it follows that lim

n→∞f(bnx)

an = g(x), for any x ∈ X.

Also, dG(f, g) ≤ 11−kd(f, Jf) implies the inequality dG(f, g) ≤ 1

1− αϕ(a)

from

which it immediately follows νg(x)−f(x)(ϕ(a)

ϕ(a)−α t) ≥ G(x, t) for all t > 0 and

x ∈ X. This means that

∀x ∈ X ∀t ∈ (0,∞) : νg(x)−f(x)(t) ≥ G

(x,

ϕ(a)− α

ϕ(a)t

).

The uniqueness of g follows from the fact that g is the unique fixed point ofJ with the property: there is C ∈ (0,∞) such that νg(x)−f(x)(Ct) ≥ G(x, t) forall x ∈ X and t > 0. �

Remark 2.2� For a similar result in the setting of Menger spaces, the reader isreferred to the paper [26].

3. Probabilistic stability for the Jensen, quadratic, cubicand quartic functional equations

In the following we will appeal to Theorem 2.1 in order to prove the prob-abilistic stability for the Jensen, the quadratic, cubic and quartic functionalequations.

Recall that the functional equation

f

(1

2(x+ y)

)=

1

2

(f(x) + f(y)

)(3.1)

is called Jensen functional equation. A mapping f between two linear spaceswith f(0) = 0 satisfies the Jensen equation if and only if it is additive; cf. [32].

213

DOREL MIHET — REZA SAADATI

Stability of Jensen equation has been studied at first by Kominek [21] and thenby several other mathematicians, see [4], [16] and references therein.

The functional equation

f(2x+ y) + f(2x− y) = 2f(x+ y) + 2f(x− y) + 12f(x) (3.2)

is called a cubic functional equation since the function f(x) = cx3 is a solution.Every solution of the cubic functional equation is said to be a cubic mapping.The stability problem for the cubic functional equation was firstly considered in[12] for mappings f : X → Y , where X is a real normed space and Y is a Banachspace. Later, a number of mathematicians have worked on the stability of sometypes of the cubic equation [14,34].

The functional equation

f(x+ y) + f(x− y) = 2f(x) + 2f(y) (3.3)

is called a quadratic functional equation. F. Skof [39] was the first one whoinvestigated the stability of this equation. She showed that, if f is a mappingfrom a normed space X into a Banach space Y satisfying ‖f(x+ y)+ f(x− y)−2f(x) − 2f(y)‖ ≤ ε for some ε > 0, then there is a unique quadratic functiong : X → Y such that ‖f(x) − g(x)‖ ≤ ε

2 . P. W. Cholewa [5] extended Skof’stheorem by replacing X by an abelian group. Skof’s result was later generalizedby S. Czerwik [6] in the spirit of Hyers-Ulam-Rassias. The stability problem ofthe quadratic equation has been extensively investigated by a number of othermathematicians, see e.g. [3,7,15,18,20,31,35] and references therein.

The functional equation

f(2x+ y) + f(2x− y) = 4f(x+ y) + 4f(x− y) + 24f(x)− 6f(y) (3.4)

is called a quartic functional equation, since the function f(x) = cx4 is a solution.Every solution of the quartic functional equation is called a quartic mapping.The stability problem for the quartic functional equation first was considered byTh. M. Rassias [36] for mappings f : X → Y , where X is a real normed spaceand Y is a Banach space.

Our first theorem in this section is a stability result for the Jensen functionalequation.

���� � 3.1� Let X be a real linear space, (Y, ν, TM) be a complete Mengerprobabilistic ϕ-normed space and Φ be a mapping from X2 to D+ (Φ(x, y) isdenoted by Φx,y) such that, for some 0 < α < ϕ(1/2),

Φ 12x,

12y(αt) ≥ Φx,y(t) (x, y, z ∈ X, t > 0) (3.5)

and

limn→∞αnϕ(2n) = 0. (3.6)

214

STABILITY OF SOME FUNCTIONAL EQUATIONS IN MENGER ϕ-NORMED SPACES

Suppose that f : X → Y has the properties: f(0) = 0 and

νf( 12 (x+y))−( 1

2 (f(x)+f(y))(t) ≥ Φx,y(t) (x, y ∈ X, t > 0). (3.7)

Then there exists a unique additive mapping g : X → Y such that

νf(x)−g(x)(t) ≥ Φx,0(Mt) (x ∈ X, t > 0), (3.8)

where M = ϕ(1/2)−αϕ(1/2)ϕ(2) .

P r o o f. By setting y = 0 in (3.7) we obtain that

νf( 12x)− 1

2 f(x)(t) ≥ Φx,0(t) (x ∈ X, t > 0),

that is,

ν2f( 12x)−f(x)(ϕ(2)t) = νf( 1

2x)− 12 f(x)

(t) ≥ Φx,0(t) (x ∈ X, t > 0).

It follows that

ν2f( 12x)−f(x)(t) ≥ G(x, t),

where G(x, t) := Φx,0

(t

ϕ(2)

).

From Theorem 2.1 we deduce that g(x) = limn→∞ 2nf( 1

2nx) (x ∈ X) is the

unique mapping g : X → Y such that g(12x

)= 1

2g(x), for all x ∈ X and

νg(x)−f(x)(t) ≥ Φx,0

(ϕ(1/2)− α

ϕ(1/2)ϕ(2)t

)(x ∈ X, t > 0).

To show that g is a solution for the Jensen equation, it suffices to prove thatthe mapping g is Cauchy additive. We have:

νg(x)+g(y)−g(x+y)(t)

≥ Min{νg(x)−2nf( 1

2n x)(t4

), ν

g(y)−2nf(

12n y

)( t4

), ν

g(x+y)−2nf(

12n (x+y)

) ( t4

),

ν2nf( 12n (x+y))−2nf( 1

2n x)−2nf( 12n y)

(t4

) }(x, y ∈ X, t > 0, n ∈ N).

The first three terms on the right-hand side of the above inequality tend to 1 asn → ∞. Also, let us note from (3.5) it immediately follows by induction on nthat

Φ 12n x, 1

2n y (αnt) ≥ Φx,y(t) (x, y ∈ X, t > 0),

hence

Φ 12n x, 1

2n y(t) ≥ Φx,y

(t

αnt

)(x, y ∈ X, t > 0). (3.9)

215

DOREL MIHET — REZA SAADATI

Then, by using (3.7), we obtain

ν2nf( 12n (x+y))−2nf( 1

2nx )−2nf( 12n y)

(t

4

)= νf( 1

2n (x+y))−f( 12n x)−f( 1

2n y)

(t

4ϕ(2n)

)

≥ Φ 12n x, 1

2n y

(t

4ϕ(2n)

)

≥ Φx,y

(t

4αnϕ(2n)

).

From (3.9) we deduce that the fourth term also tends to 1 when n tends to ∞,concluding that g is additive. �

In what follows we obtain similar results on the stability of cubic, quadraticand quartic functional equations.

���� � 3.2� Let X be a real linear space, let f be a mapping from X into acomplete Menger probabilistic ϕ-normed space (Y, ν, TM) with f(0) = 0 and letΦ: X2 → D+ be a mapping with the property

∃α ∈ (0, ϕ(8)) ∀x, y ∈ X ∀t ∈ (0,∞) : Φ2x,2y(αt) ≥ Φx,y(t). (3.10)

If

νf(2x+y)+f(2x−y)−2f(x+y)−2f(x−y)−12f(x)(t) ≥ Φx,y(t) for all x, y ∈ X,(3.11)

and

limn→∞

αnϕ

(1

23n

)= 0, (3.12)

then there is a unique cubic mapping g : X → Y such that

νg(x)−f(x)(t) ≥ Φx,0 (Mt) for all x ∈ X, t > 0, (3.13)

where

M =ϕ(8)− α

ϕ(8)ϕ(1/16).

Moreover,

g(x) = limn→∞

f(2nx)

23n.

P r o o f. By setting y = 0 in (3.11), we obtain

ν2f(2x)−16f(x)(t) ≥ Φx,0(t),

for all x ∈ X, whence

ν 18 f(2x)−f(x)(t) = ν 1

16 (2f(2x)−16f(x))(t) = ν2f(2x)−16f(x)

(t/ϕ(

1

16)

)

≥ Φx,0

(t/ϕ(

1

16)

), for all x ∈ X, t > 0.

216

STABILITY OF SOME FUNCTIONAL EQUATIONS IN MENGER ϕ-NORMED SPACES

Let

G(x, t) := Φx,0

(t

ϕ( 116 )

).

By Theorem 2.1 it follows the existence of a unique mapping g : X → Y suchthat g(2x) = 8g(x), for all x ∈ X and

νg(x)−f(x)(t) ≥ Φx,0

(ϕ(8)− α

ϕ(8)ϕ(1/16)t

)for all x ∈ X, t > 0.

Moreover, limn→∞

f(2nx)23n = g(x), for any x ∈ X.

The proof of the fact that g is a cubic mapping is similar to the proof oflinearity in the preceding theorem. �

In a similar way we can obtain stability results for the quadratic and thequartic functional equations, which are stated in the following.

���� � 3.3� Let X be a real linear space, let f be a mapping from X into acomplete Menger probabilistic ϕ-normed space (Y, ν, TM) with f(0) = 0 and letΦ: X2 → D+ be a mapping with the property

∃α ∈ (0, ϕ(4)) ∀x, y ∈ X ∀t ∈ (0,∞) : Φ2x,2y(αt) ≥ Φx,y(t).

If

νf(x+y)+f(x−y)−2f(x)−2f(y)(t) ≥ Φx,y(t) for all x, y ∈ X,

and

limn→∞αnϕ

(1

22n

)= 0,

then the formula g(x) = limn→∞

f(2nx)22n defines a unique quadratic mapping g : X→ Y

such that

νg(x)−f(x)(t) ≥ Φx,x (Mt) , for all x ∈ X, t > 0,

where

M =ϕ(4)− α

ϕ(4)ϕ(1/4).

P r o o f. (Sketch) By setting x = y in (3.4), we obtain

νf(2x)−4f(x)(t) ≥ Φx,x(t),

for all x ∈ X, whence

ν 14 f(2x)−f(x)(t) ≥ G(x, t), for all x ∈ X, t > 0,

where G(x, t) := Φx,x

(t/ϕ(14 )

). �

217

DOREL MIHET — REZA SAADATI

���� � 3.4� Let X be a real linear space, let f be a mapping from X into acomplete Menger probabilistic ϕ-normed space (Y, ν, TM) with f(0) = 0 and letΦ: X2 → D+ be a mapping with the property

∃α ∈ (0, ϕ(16)) ∀x, y ∈ X ∀t ∈ (0,∞) : Φ2x,2y(αt) ≥ Φx,y(t).

If

νf(2x+y)+f(2x−y)−4f(x+y)−4f(x−y)−24f(x)+6f(y)(t) ≥ Φx,y(t) for all x, y ∈ X,

and

limn→∞αnϕ

(1

24n

)= 0,

then there is a unique quartic mapping g : X → Y , g(x) = limn→∞

f(2nx)24n , such that

νg(x)−f(x)(t) ≥ Φx,0 (Mt) for all x ∈ X, t > 0,

where

M =ϕ(16)− α

ϕ(16)ϕ(1/32).

4. Particular cases

For concrete choices of ϕ, Φ or ν, one can obtain stability theorems for differ-ent functional equations in random normed spaces or in linear normed spaces.

���� � 4.1� Let X be a real linear space, (Y, ν, TM) be a complete probabilisticnormed space and Φ be a mapping from X2 to D+ such that, for some 0 < α< 1/2,

Φ 12x,

12y(αt) ≥ Φx,y(t) (x, y ∈ X, t > 0).

Suppose that f : X → Y has the properties: f(0) = 0 and

νf( 12 (x+y))−( 1

2 (f(x)+f(y))(t) ≥ Φx,y(t) (x, y ∈ X, t > 0).

Then there exists a unique additive mapping g : X → Y such that

νf(x)−g(x)(t) ≥ Φx,0((1/2− α)t) (x ∈ X, t > 0).

P r o o f. The conclusion follows by considering ϕ(α) = |α| in Theorem 3.1 (we

note that in this case ϕ(1/2)−αϕ(1/2)ϕ(2) = ϕ(1/2− α).

The conditionlim

n→∞αnϕ(2n) = 0

is satisfied, as it reduces tolimn→∞

(2α)n = 0.

�

218

STABILITY OF SOME FUNCTIONAL EQUATIONS IN MENGER ϕ-NORMED SPACES

���� � 4.2� Let (X, ‖ · ‖) be a real normed linear space, (Y, ν, TM) be acomplete RN-space and p be nonnegative real number. If f : X → Y is a mappingsuch that

ν f(x)+f(y)2 −f( x+y

2 )(t) ≥ t

t+ ‖x‖p + ‖y‖p (x, y ∈ X, t > 0), (4.1)

and p < 1, then there exists a unique additive mapping g : X → Y such that

νf(x)−g(x)(t) ≥ (1− 2p−1)t

(1− 2p−1)t+ 2p‖x‖p . (4.2)

for all x ∈ X and t > 0.

P r o o f. Consider the mapping Φ: X2 → D+ defined through

Φx,y(t) =t

t+ ‖x‖p + ‖y‖pand let ϕ(t) = |t|p (t ∈ R), where 0 < p < 1, α = 1/2.

It is immediate that0 < 1/2 < ϕ(1/2),

Φ 12x,

12y(αt) ≥ Φx,y(t) (x, y, z ∈ X, t > 0)

andlim

n→∞αnϕ(2n) = lim

n→∞2(p−1)n = 0.

Now the conclusion follows from Theorem 3.1. �

�� ��� � 4.3� ([16: Theorem 1], with δ = 0, θ = 2) Let f : X → Y be amapping between real Banach spaces and let 0 ≤ p < 1 be fixed. If f satisfies theinequality ∥∥∥∥f

(x+ y

2

)− f(x) + f(y)

2

∥∥∥∥ ≤ ‖x‖p + ‖y‖p

for all x, y ∈ X, then there exists a unique additive mapping F : X → Y suchthat

‖f(x)− F (x)‖ ≤ 2p

1− 2p−1‖x‖

for all x ∈ X.

P r o o f. Consider the induced RN space (X, ν, TM), where νx(t) =t

t+‖x‖p . Then

(4.1) is equivalent to∥∥∥∥f(x+ y

2

)− f(x) + f(y)

2

∥∥∥∥ ≤ ‖x‖p + ‖y‖p,

while (4.2) is equivalent to

‖f(x)− g(x)‖ ≤ 2p

1− 2p−1‖x‖. �

219

DOREL MIHET — REZA SAADATI

���� � 4.4� Let X be a real linear space, (Y, ν, TM) be a complete probabilisticnormed space and Φ be a mapping from X2 to D+ such that, for some 0 < α < 8,

Φ2x,2y(αt) ≥ Φx,y(t) (x, y ∈ X, t > 0).

Suppose that f : X → Y has the properties: f(0) = 0 and

νf(2x+y)+f(2x−y)−2f(x+y)−2f(x−y)−12f(x)(t) ≥ Φx,y(t) (x, y ∈ X, t > 0).

Then there exists a unique cubic mapping g : X → Y such that

νf(x)−g(x)(t) ≥ Φx,0(2(8− α)t) (x ∈ X, t > 0).

P r o o f. The conclusion follows by considering ϕ(α) = |α| in Theorem 3.2 (we

note that in this case ϕ(8)−αϕ(8)ϕ(1/16) = ϕ(2(8− α)).

The condition

limn→∞αnϕ

(1

23n

)= 0

is satisfied, as it reduces to

limn→∞

(α8

)n

= 0. �

���� � 4.5� Let (X, ‖ · ‖) be a real normed linear space, (Y, ν, TM) be acomplete RN-space and p be nonnegative real number. If f : X → Y is a mappingsuch that

νf(2x+y)+f(2x−y)−2f(x+y)−2f(x−y)−12f(x)(t) ≥ t

t+ ‖x‖p + ‖y‖p(x, y ∈ X, t > 0),

(4.3)

and 1 < p < 3, then there exists a unique cubic mapping g : X → Y such that

νf(x)−g(x)(t) ≥ (8p − 8)t

(8p − 8)t+ 2−p‖x‖p . (4.4)

for all x ∈ X and t > 0.

P r o o f. Consider the mapping Φ: X2 → D+ defined through

Φx,y(t) =t

t+ ‖x‖p + ‖y‖pand let ϕ(t) = |t|p (t ∈ R), where 1 < p < 3, α = 8.

It is immediate that0 < 8 < ϕ(8),

Φ2x,2y(αt) ≥ Φx,y(t) (x, y, z ∈ X, t > 0)

and

limn→∞αnϕ

( 1

8n

)= lim

n→∞ 8(1−p)n = 0.

Now the conclusion follows from Theorem 3.2. �

220

STABILITY OF SOME FUNCTIONAL EQUATIONS IN MENGER ϕ-NORMED SPACES

���� � 4.6� Let (X, ‖ · ‖) be a real normed linear space, (Y, ν, TM) be acomplete RN-space and p be nonnegative real number. If f : X → Y is a mappingsuch that

νf(2x+y)+f(2x−y)−2f(x+y)−2f(x−y)−12f(x)(t) ≥ t

t+ ‖x‖p + ‖y‖p(x, y ∈ X, t > 0),

(4.5)

and 13 < p < 1, then there exists a unique cubic mapping g : X → Y such that

νf(x)−g(x)(t) ≥ (8p − 2)t

(8p − 2)t+ 2−p‖x‖p . (4.6)

for all x ∈ X and t > 0.

P r o o f. Consider the mapping Φ: X2 → D+ defined through

Φx,y(t) =t

t+ ‖x‖p + ‖y‖pand let ϕ(t) = |t|p (t ∈ R), where 1

3 < p < 1, α = 2.

It is immediate that0 < 2 < ϕ(8),

Φ2x,2y(αt) ≥ Φx,y(t) (x, y, z ∈ X, t > 0)

and

limn→∞αnϕ

(1

8n

)= lim

n→∞ 2(1−3p)n = 0.

Now the conclusion follows from Theorem 3.2. ��� ��� � 4.7� ([12: Theorem 1], with δ = 0, θ = 2) Let f : X → Y be amapping between real Banach spaces and let 1/3 ≤ p < 1 be fixed. If f satisfiesthe inequality

‖f(2x+ y) + f(2x− y)− 2f(x+ y)− 2f(x− y)− 12f(x)‖ ≤ ‖x‖p + ‖y‖pfor all x, y ∈ X, then there exists a unique cubic mapping g : X → Y such that

‖f(x)− g(x)‖ ≤ 2−p

8p − 2‖x‖

for all x ∈ X.

P r o o f. Consider the induced RN space (X, ν, TM), where νx(t) =t

t+‖x‖p . Then

(4.5) is equivalent to

‖f(2x+ y) + f(2x− y)− 2f(x+ y)− 2f(x− y)− 12f(x)‖ ≤ ‖x‖p + ‖y‖p,while (4.6) is equivalent to

‖f(x)− g(x)‖ ≤ 2−p

8p − 2‖x‖. �

221

DOREL MIHET — REZA SAADATI

���� � 4.8� Let X be a real linear space, (Y, ν, TM) be a complete probabilisticnormed space and Φ be a mapping from X2 to D+ such that, for some 0 < α < 4,

Φ2x,2y(αt) ≥ Φx,y(t) (x, y, z ∈ X, t > 0).

Suppose that f : X → Y has the properties: f(0) = 0 and

νf(x+y)+f(x−y)−2f(x)−2f(y)(t) ≥ Φx,y(t) (x, y ∈ X, t > 0).

Then there exists a unique quadratic mapping g : X → Y such that

νf(x)−g(x)(t) ≥ Φx,x((4− α)t) (x ∈ X, t > 0).

P r o o f. The conclusion follows by considering ϕ(α) = |α| in Theorem 3.3 (we

note that in this case ϕ(4)−αϕ(4)ϕ(1/4) = ϕ(4− α).

The condition

limn→∞αnϕ

(1

22n

)= 0

is satisfied, as it reduces to

limn→∞

(α4

)n

= 0. �

���� � 4.9� Let (X, ‖ · ‖) be a real normed linear space, (Y, ν, TM) be acomplete RN-space and p be nonnegative real number. If f : X → Y is a mappingsuch that

νf(x+y)+f(x−y)−2f(x)−2f(y)(t) ≥ t

t+ ‖x‖p + ‖y‖p (x, y ∈ X, t > 0), (4.7)

and 1 < p < 2, then there exists a unique quadratic mapping g : X → Y suchthat

νf(x)−g(x)(t) ≥ (4p − 4)t

(4p − 4)t+ 2‖x‖p . (4.8)

for all x ∈ X and t > 0.

P r o o f. Consider the mapping Φ: X2 → D+ defined through

Φx,y(t) =t

t+ ‖x‖p + ‖y‖pand let ϕ(t) = |t|p (t ∈ R), where 1 < p < 2, α = 4.

It is immediate that

0 < 4 < ϕ(4),

Φ2x,2y(αt) ≥ Φx,y(t) (x, y, z ∈ X, t > 0)

and

limn→∞αnϕ

(1

4n

)= lim

n→∞ 4(1−p)n = 0.

Now the conclusion follows from Theorem 3.3. �

222

STABILITY OF SOME FUNCTIONAL EQUATIONS IN MENGER ϕ-NORMED SPACES

���� � 4.10� Let (X, ‖ · ‖) be a real normed linear space, (Y, ν, TM) be acomplete RN-space and p be nonnegative real number. If f : X → Y is a mappingsuch that

νf(x+y)+f(x−y)−2f(x)−2f(y)(t) ≥ t

t+ ‖x‖p + ‖y‖p (x, y ∈ X, t > 0), (4.9)

and 12 < p < 1, then there exists a unique quadratic mapping g : X → Y such

that

νf(x)−g(x)(t) ≥ (4p − 2)t

(4p − 2)t+ 2‖x‖p . (4.10)

for all x ∈ X and t > 0.

P r o o f. Consider the mapping Φ: X2 → D+ defined through

Φx,y(t) =t

t+ ‖x‖p + ‖y‖pand let ϕ(t) = |t|p (t ∈ R), where 1

2 < p < 1, α = 2.

It is immediate that

0 < 2 < ϕ(4),

Φ2x,2y(αt) ≥ Φx,y(t) (x, y, z ∈ X, t > 0)

and

limn→∞

αnϕ

(1

4n

)= lim

n→∞2(1−2p)n = 0.

Now the conclusion follows from Theorem 3.3. �

�� ��� � 4.11� ([39: Theorem 1], with δ = 0, θ = 2) Let f : X → Y be amapping between real Banach spaces and let 1/2 ≤ p < 1 be fixed. If f satisfiesthe inequality

‖f(x+ y) + f(x− y)− 2f(x)− 2f(y)‖ ≤ ‖x‖p + ‖y‖pfor all x, y ∈ X, then there exists a unique quadratic mapping g : X → Y suchthat

‖f(x)− g(x)‖ ≤ 2

4p − 2‖x‖

for all x ∈ X.

P r o o f. Consider the induced RN space (X, ν, TM), where νx(t) =t

t+‖x‖p . Then

(4.9) is equivalent to

‖f(x+ y) + f(x− y)− 2f(x)− 2f(y)‖ ≤ ‖x‖p + ‖y‖p,while (4.10) is equivalent to

‖f(x)− g(x)‖ ≤ 2

4p − 2‖x‖. �

223

DOREL MIHET — REZA SAADATI

���� � 4.12� Let X be a real linear space, (Y, ν, TM) be a complete proba-bilistic normed space and Φ be a mapping from X2 to D+ such that, for some0 < α < 16,

Φ2x,2y(αt) ≥ Φx,y(t) (x, y, z ∈ X, t > 0).

Suppose that f : X → Y has the properties: f(0) = 0 and

νf(2x+y)+f(2x−y)−4f(x+y)−4f(x−y)−24f(x)+6f(y)(t) ≥ Φx,y(t)

(x, y ∈ X, t > 0).

Then there exists a unique quadratic mapping g : X → Y such that

νf(x)−g(x)(t) ≥ Φx,0(2(16− α)t) (x ∈ X, t > 0).

P r o o f. The conclusion follows by considering ϕ(α) = |α| in Theorem 3.4 (we

note that in this case ϕ(16)−αϕ(16)ϕ(1/32) = ϕ(2(16− α)).

The condition

limn→∞αnϕ

(1

24n

)= 0

is satisfied, as it reduces to

limn→∞

( α

16

)n

= 0. �

���� � 4.13� Let (X, ‖ · ‖) be a real normed linear space, (Y, ν, TM) be acomplete RN-space and p be nonnegative real number. If f : X → Y is a mappingsuch that

νf(2x+y)+f(2x−y)−4f(x+y)−4f(x−y)−24f(x)+6f(y)(t) ≥ t

t+ ‖x‖p + ‖y‖p(x, y ∈ X, t > 0),

(4.11)

and 1 < p < 4, then there exists a unique quartic mapping g : X → Y such that

νf(x)−g(x)(t) ≥ (16p − 16)t

(16p − 16)t+ 2−p‖x‖p . (4.12)

for all x ∈ X and t > 0.

P r o o f. Consider the mapping Φ: X2 → D+ defined through

Φx,y(t) =t

t+ ‖x‖p + ‖y‖pand let ϕ(t) = |t|p (t ∈ R), where 1 < p < 4, α = 16.

224

STABILITY OF SOME FUNCTIONAL EQUATIONS IN MENGER ϕ-NORMED SPACES

It is immediate that0 < 16 < ϕ(16),

Φ2x,2y(αt) ≥ Φx,y(t) (x, y, z ∈ X, t > 0)

and

limn→∞αnϕ

( 1

16n

)= lim

n→∞ 16(1−p)n = 0.

Now the conclusion follows from Theorem 3.4. �

���� � 4.14� Let (X, ‖ · ‖) be a real normed linear space, (Y, ν, TM) be acomplete RN-space and p be nonnegative real number. If f : X → Y is a mappingsuch that

νf(2x+y)+f(2x−y)−4f(x+y)−4f(x−y)−24f(x)+6f(y)(t) ≥ t

t+ ‖x‖p + ‖y‖p(x, y ∈ X, t > 0),

(4.13)

and 14 < p < 1, then there exists a unique quartic mapping g : X → Y such that

νf(x)−g(x)(t) ≥ (16p − 2)t

(16p − 2)t+ 2−p‖x‖p . (4.14)

for all x ∈ X and t > 0.

P r o o f. Consider the mapping Φ: X2 → D+ defined through

Φx,y(t) =t

t+ ‖x‖p + ‖y‖pand let ϕ(t) = |t|p (t ∈ R), where 1

4 < p < 1, α = 2.

It is immediate that0 < 2 < ϕ(16),

Φ2x,2y(αt) ≥ Φx,y(t) (x, y, z ∈ X, t > 0)

and

limn→∞αnϕ

(1

16n

)= lim

n→∞ 2(1−4p)n = 0.

Now the conclusion follows from Theorem 3.4. �

�� ��� � 4.15� ([36: Theorem 1], with δ = 0, θ = 2) Let f : X → Y be amapping between real Banach spaces and let 1/4 ≤ p < 1 be fixed. If f satisfiesthe inequality

‖f(2x+ y)+ f(2x− y)− 4f(x+ y)− 4f(x− y)− 24f(x)+ 6f(y)‖ ≤ ‖x‖p+ ‖y‖pfor all x, y ∈ X, then there exists a unique quartic mapping g : X → Y such that

‖f(x)− g(x)‖ ≤ 2−p

16p − 2‖x‖

for all x ∈ X.

225

DOREL MIHET — REZA SAADATI

P r o o f. Consider the induced RN space (X, ν, TM), where νx(t) =t

t+‖x‖p . Then

(4.13) is equivalent to

‖f(2x+ y)+ f(2x− y)− 4f(x+ y)− 4f(x− y)− 24f(x)+6f(y)‖ ≤ ‖x‖p+ ‖y‖p,while (4.14) is equivalent to

‖f(x)− g(x)‖ ≤ 2−p

16p − 2‖x‖.

�

Recently (see [41]) it was shown that if (X,µ, T ) is a Menger probabilisticϕ-normed space with T = TM (as in our paper), then ϕ(t) = |t|p, with p ∈ (0, 1].Nevertheless, Menger probabilistic ϕ-normed spaces still represent a rich enoughstructure, allowing one to unify many known stability results, as it have seen inSections 2 and 3. We also mention here the paper [42], where complementaryresults in the case α > ϕ(a) are provided.

Acknowledgement� The authors would like to thank the referee and man-aging editor Professor Anatolij Dvurecenskij for giving useful suggestions andcomments for the improvement of this paper.

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Received 1. 9. 2011Accepted 10. 10. 2011

*Faculty of Mathematics and Computer ScienceWest University of TimisoaraBv. V. Parvan 4

RO–300223 TimisoaraROMANIA

E-mail : mihet@math.uvt.ro

**Department of Mathematics

Iran University of Science and TechnologyTehranIRAN

E-mail : rsaadati@iust.ac.ir

228