Fixed Points and Stability for Functional Equations in Probabilistic Metric and Random Normed Spaces

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Volume 2009, Article ID 589143,18pages doi:10.1155/2009/589143

Research Article

Fixed Points and Stability for Functional Equations

in Probabilistic Metric and Random Normed Spaces

Liviu C ˘adariu

1

_{and Viorel Radu}

2

1_{Department of Mathematics, “Politehnica” University of Timis¸oara, Piat¸a Victoriei number 2,}

300006 Timis¸oara, Romania

2_{Department of Mathematics, Faculty of Mathematics and Computer Science, West University of Timis¸oara,}

Vasile Pˆarvan 4, 300223 Timis¸oara, Romania

Correspondence should be addressed to Liviu C˘adariu,[email protected]

Received 24 April 2009; Accepted 19 October 2009

Recommended by Tomas Dominguez Benavides

We prove a general Ulam-Hyers stability theorem for a nonlinear equation in probabilistic metric spaces, which is then used to obtain stability properties for diﬀerent kinds of functional equations

linear functional equations, generalized equation of the square root, spiral generalized gamma equationsin random normed spaces. As direct and natural consequences of our results, we obtain general stability properties for the corresponding functional equations indeterministicmetric and normed spaces.

Copyrightq2009 L. C˘adariu and V. Radu. 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.

1. Introduction

Hyers 1, 1941 has given an aﬃrmative answer to a question of Ulam by proving the stability of additive Cauchy equations in Banach spaces. Then Aoki, Bourgin, Rassias, Forti and Gajda considered the stability problem with unbounded Cauchy diﬀerences. Their results include the following theorem.

Theorem 1.1see Hyers1, Aoki2, and Gajda3. Suppose thatEis a real normed space,F is a real Banach space andf:E → Fis a given function, such that the following condition holds:

fxy−fx−fy_F ≤θxp_Ey_Ep, ∀x, y∈E, 1.1

for somep∈0,∞\ {1}. Then there exists a unique additive functiona:E → Fsuch that

_f_x₋_a_x

F ≤

2θ

|2−2p_|x p

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This phenomenon is calledgeneralized Ulam-Hyers stabilityand has been extensively investigated for diﬀerent functional equations. Almost all proofs used the idea conceived by Hyers. Namely, the additive functiona: E → F is constructed, starting from the given functionf, by the following formulae:

ax lim

n→ ∞

1 2nf2

n_x_, _if_{p <}₁_,

ax lim

n→ ∞2

n_fx

2n

, ifp >1.

1.3

This method is calledthe direct methodorHyers method. It is often used to construct a solution of a given functional equation and is seen to be a powerful tool for studying the stability of many functional equationssee4–6for details. It is worth noting that in 1978 Rassias 7proved that the additive mappinga, obtained by Hyers or Aoki, is linear if, in addition, for eachx ∈ Ethe mapping ftx is continous in t ∈ R. Subsequently, the above results were extended by replacing the control mappingθxp_Eyp_Ein1.1by functionsϕwith suitable propertiessee8–10.

On the other hand, in11–13afixed point methodwas proposed, by showing that many theorems concerning the stability of Cauchy and Jensen equations are consequences of the fixed point alternative. The method has also been used in14–16or17.

Our aim is to highlight generalized Ulam-Hyers stability results for functional equations defined by mappings with values in probabilistic metric spaces and in random normed spaces, obtained by using the fixed point alternative, which is now recalled, for the sake of conveniencecf.18, see also19, chapter 5:

Theorem 1.2. Suppose that one is given a complete generalized metric spaceE, d-that is, one for which the metricdmay assume infinite values—and a strictly contractive mappingA:E → E,with the Lipschitz constantL. Then, for each given elementf∈ E, either

A1dAnf, An1f ∞,for alln≥0,or;

A2there exists a natural numbern0such that;

A20dAnf, An1f<∞, for alln≥n0;

A21the sequenceAnfis convergent to a fixed pointf∗ofA;

A22f∗is the unique fixed point ofAin the setE {g∈ E, dAn0f, g<∞};

A23dg, f∗≤1/1−Ldg, Ag, for allg∈ E.

Remark 1.3. The fixed points ofA,if any, need not be uniquely determinedin the whole space

Eand do depend on the initial guessf.

We recallsee, e.g., Schweizer and Sklar20that adistance distribution functionF is a mapping from−∞,∞into0,1, nondecreasing, and left-continuous, withF0 0. The class of all distribution functionsF, with limx→ ∞Fx 1, is denoted byD. For any

a∈0,∞, εat

⎧ ⎨ ⎩

0, ift≤a,

1, ift > a

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is an element ofD. For anyλ >0 andF∈D, the distribution functionλ◦Fis defined by

λ◦Ft: F

t λ

, for allt >0. 1.5

Let us considerX a real vector space,Fa mapping fromX intoD for anyxinX, Fxis denoted byFxandT at-norm. The tripleX,F, Tis calledarandom normed space

briefly RN-spaceiﬀthe following conditions are satisfied: RN-1:Fx ε0if and only ifx θ, the null vector;

RN-2:Fax |a| ◦Fx,∀t >0,∀a∈R, a /0;

RN-3:Fxyt1t2≥TFxt1, Fyt2,∀x, y∈Xandt1, t2>0.

Notice that atriangular norm (t-norm)is a mappingT :0,1×0,1 → 0,1, which is associative, commutative and increasing in each variable, withTa,1 1,for all a∈0,1. The most importantt-norms areTMa, b min{a, b}, Proda, b a·b,T1a, b max{a

b−1,0}.

Every normed spaceX,||·||defines a random normed spaceX,F, TM, withFx∈D and

Fxt

t

tx, ∀t >0. 1.6

If the t-norm T is such that sup₀_<a<₁Ta, a 1, then every random normed space X,F, Tis ametrizable linear topological spacewith theε, λ-topology, induced by the base of neighborhoodsUε, λ {x∈ X, Fxε >1−λ}.IfT TM, thenX,F, Tis locally convex

20, Theorems 12.1.6 and 15.1.2.

A sequencexnin a random normed spaceX,F, Tconvergestox∈Xin theε, λ

-topology if limn→ ∞Fxn−xt 1,for allt > 0.A sequence xn is called Cauchy sequence if limm,n→ ∞Fxn−xmt 1,for allt > 0.The random normed spaceX,F, Tiscompleteif every Cauchy sequence inXis convergent.

It is worth notice that a random norm induces a probabilistic metric by the formula Fuvt: Fu−vt,∀tsee, e.g.,20, Theorem 15.1.2. For more details on probabilistic metric

spaces and random normed spaces, seeSchweizer and Sklar20, Chapters 8, 12 and 15, Radu21or Chang et al.22.

In the present paper we prove a very general Ulam-Hyers stability theorem for a nonlinear equation in probabilistic metric spaces, which is then used to obtain stability properties for diﬀerent kinds of functional equationslinear functional equations, generalized equation of the square root spiral, generalized gamma equations in random normed spaces. As direct and natural consequences of our results, we obtain general stability properties for the corresponding functional equations indeterministicmetric and normed spaces.

2. The Generalized Ulam-Hyers Stability of a Nonlinear Equation

The Ulam-Hyers stability for the nonlinear equation

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was discussed by Baker 23. Recently, we extended his result in 15, by proving the generalized stability in Ulam-Hyers sense for2.1, by means of the fixed point alternative. In24the stability of additive Cauchy equation in random normed spaces is proved.

In the next theorem we prove a generalized Ulam-Hyers stability result for the nonlinear equation2.1, where the unknownfis mapping a nonempty setXinto acomplete probabilistic metric spaceY,F, TM.

Theorem 2.1. LetX be a nonempty set and letY,F, TMbe a complete probabilistic metric space.

Let us consider the mappingsη:X → X,g :X → R\ {0}andE:X×Y → Y. Suppose that

FEx,uEx,v≥gx◦Fuv, ∀x∈X, ∀u, v∈Y. 2.2

Iff:X → Y satisfies

F_f_x_E_x,f_η_x≥ψx, ∀x∈X, Cψ

whereψ:X → Dis a mapping for which there existsL <1such that

gx◦ψηxLt≥ψxt, ∀x∈X, ∀t >0, Hψ

then there exists a unique mappingc:X → Ywhich satisfies both the equation

cx Ex, cηx, ∀x∈X, 2.3

and the estimation

Fcxfxt≥ψx1−Lt, ∀x∈X, Estψ

for almost allt >0. Moreover, the solution mapping chas the form

cx lim

n→ ∞F

x, FFηx, . . . , Fηx,f◦ηnηx, ∀x∈X. 2.4

Proof. Let us consider the mappingGx, t: ψxt, and the setE: {h: X → Y, h0 0}.

We introduce a generalized metric onEas usual, inf∅ ∞:

dh1, h2 dGh1, h2 inf

K∈R, Fh1xh2xKt≥Gx, t, ∀x∈X, ∀t >0

. 2.5

The proof of the fact thatE, dGis acomplete generalized metric spacecan be found e.g. inRadu

25, Hadˇzi´c et al.26or Mihet¸ and Radu27. Now, define the operator

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Step 1. Using our hypotheses, it follows that J is strictly contractive onE. Indeed, for any h1, h2∈ Ewe have:

dh1, h2< K ⇒Fh1xh2xKt≥Gx, t ψxt, ∀x∈X, ∀t >0,

FJh1xJh2xLKt FEx,h1ηxEx,h2ηxLKt≥Fh1ηxh2ηx

LKt _g_x

≥ψηx

Lt _g_x

≥ψxt.

2.6

ThereforedJh1, Jh2≤LK,which implies

dJh1, Jh2≤Ldh1, h2, ∀g, h∈ E. CCL

This says thatJis astrictly contractiveself-mapping ofE,with the Lipschitz constantL <1.

Step 2. Obviously,df, Jf<∞. In fact, using the relation1.6it results thatdf, Jf≤1.

Step 3. Now we can apply the fixed point alternative to obtain the existence of a mapping c:X → Ysuch that,

icis a fixed point ofJ, that is

cx Ex, cηx, ∀x∈X. 2.7

The mappingcis the unique fixed point ofJin the set

F h∈ E, dG

f, h<∞. 2.8

This says thatcis the unique mapping verifyingboththe above equation2.7and the next condition:

∃K <∞ such thatFcx−fxKt≥ψxt, ∀x∈X, ∀t >0. 2.9

Moreover,

iidJn_{f, c} _→

n→ ∞0, which implies limn→ ∞FcxJnfxt 1,∀t >0 and∀x∈X,whence

cx lim

n→ ∞F

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iiidf, c≤1/1−Ldf, Jf,which implies the inequality

df, c≤ 1

1−L, 2.11

hence

Fcxfx

t 1−L

≥ψxt, ∀x∈X, 2.12

for almost allt >0. It results that

Fcxfxt≥ψx1−Lt, ∀x∈X, 2.13

for almost allt >0, that is2.3is seen to be true.

As a direct consequence ofTheorem 2.1, the following Ulam-Hyers stability result for the nonlinear equation2.1is obtained.

Corollary 2.2. LetXbe a nonempty set and letY,F, TMbe a complete probabilistic metric space.

Consider the mappingsη:X → X,E:X×Y → Yand0≤L <1. Suppose that

FEx,uEx,vLt≥Fuvt, ∀x∈X, ∀u, v∈Y, ∀t >0. 2.14

F_f_x_E_x,f_η_x≥εδ, ∀x∈X, 2.15

where

εδt

⎧ ⎨ ⎩

0, ift≤δ,

1, ift > δ

2.16

(an element ofDforδ >0), then there exists a unique mappingc:X → Y which satisfies both the

equation

cx Ex, cηx, ∀x∈X, 2.17

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for almost allt >0. Moreover, the solution mapping chas the form

cx lim

n→ ∞F

x, FFηx, . . . , Fηx,f◦ηnηx, ∀x∈X. _2.19

Proof. It follows fromTheorem 2.1, by choosingψxt: εδt,∀t >0, whereδ >0 is fixed.

2.1. A Consequence for Metric Spaces

By usingTheorem 2.1, we can immediately obtain the following generalized stability result of Ulam-Hyers type for the nonlinear equation

fx Fx, fηx 2.20

incomplete metric spaces, firstly proved by us in15, Theorem 4.1.

Corollary 2.3. Consider a nonempty setX, a complete metric spaceY, dand the mappingsη:X → X,g:X → R\ {0}andF:X×Y → Y. Suppose that

dFx, u, Fx, v≤gx·du, v, ∀x∈X, ∀u, v∈Y. 2.21

dfx, Fx, fηx≤ϕx, ∀x∈X, 2.22

with a mappingϕ:X → 0,∞for which there existsL <1such that

gxϕ◦ηx≤Lϕx, ∀x∈X, 2.23

cx Fx, cηx, ∀x∈X 2.24

dfx, cx≤ ϕx

1−L, ∀x∈X. 2.25

Moreover,

cx lim

n→ ∞F

x, FFηx, . . . , Fηx,f◦ηnηx, ∀x∈X. _2.26

Proof. Let us consider the probabilistic metricY×Y u, v → Fuv∈D, where

Fuvt 1−exp

− t

du, v

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and the mapping

ψ:X → D, ψxt 1−exp

− t

ϕx

. 2.28

ThenY is a complete probabilistic metric space underTM Minsee20, Theorem 8.4.2.

The hypothesis relations 2.21, 2.22 and 2.23 are equivalent to 2.2, 1.6 and Cψ, respectively. Therefore we can apply ourTheorem 2.1to obtain the above result.

Remark 2.4. Corollary 2.3can also be obtained by using the probabilistic metric

Y ×Y u, v−→Fuv∈D, Fuvt

t

tdu, v, ∀t >0, 2.29

and the mapping

ψ :X−→D, ψxt

t

tϕx, ∀t >0. 2.30

Again by usingTheorem 2.1, the following Ulam-Hyers stability resultcf.see23, Theorem 2or28, Theorem 13for the nonlinear equation2.1incomplete metric spaces

can be proved.

Corollary 2.5. LetX be a nonempty set andY, dbe a complete metric space. Letη : X → X, F:X×Y → Y and0≤L <1. Suppose that

dFx, u, Fx, v≤L·du, v, ∀x∈X, ∀u, v∈Y. 2.31

Iff:X → Ysatisfies

dfx, Fx, fηx≤δ, ∀x∈X, 2.32

with a fixed constantδ >0, then there exists a unique mappingc:X → Y which satisfies both the equation

cx Fx, cηx, ∀x∈X 2.33

dfx, cx≤ δ

1−L, ∀x∈X. 2.34

Moreover,

cx lim

n→ ∞F

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Proof. It follows, by choosing inTheorem 2.1the probabilistic metric

Y ×Y u, v−→Fuv∈D, Fuvt

t

tdu, v, ∀t >0, 2.36

and the mapping

ψ :X −→D, ψxt

t

tδ, ∀t >0. 2.37

Remark 2.6. The above result can also be obtained byCorollary 2.2and the probabilistic metric

Y ×Y u, v−→Fuv∈D, Fuvt ε1

t du, v

, ∀t >0, du, v>0. 2.38

3. The Generalized Ulam-Hyers Stability of a Linear

Functional Equation

If we consider

Ex, fηx gx·fηxhx, 3.1

then2.1becomes

fx gx·fηxhx, 3.2

whereg, η, hare given mappings andfis an unknown function. The above equation is called

linear functional equation. A lot of results concerning monotonic solutions, regular solutions and convex solutions of3.2were given by Kuczma et al.29.

In this section we prove a generalized Ulam-Hyers stability result for the linear functional equation3.2, as a particular case ofTheorem 2.1. The unknown is a mapping ffrom the nonempty setXinto acomplete random normed spaceY,F, TM.

Theorem 3.1. Let X be a nonempty set and letY,F, TM be a complete random normed space.

Suppose thatη:X → X,g :X → R\ {0}. Iff:X → Ysatisfies

F_g_x_·_f_η_x_h_x₋_f_x≥ψx, ∀x∈X, 3.3

whereψ:X → Dis a mapping for which there existsL <1such that

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then there exists a unique mappingc:X → Y,

cx hx lim

n→ ∞

⎛

⎝_f_ηn_x_·n−1 i 0

gηix

n−2

j 0

hηj1x·

j

i 0

gηix ⎞

⎠_, _3.5

for allx∈X,which satisfies both the equation

cx gx·cηxhx, ∀x∈X, 3.6

Fcx−fxt≥ψx1−Lt, ∀x∈X, 3.7

for almost allt >0.

Proof. We consider in Theorem 2.1 the probabilistic metric Fuv on Y × Y, induced by

the random norm, namely, Fuvt : Fu−vt,∀t > 0 and the function Ex, fηx :

gxfηx hx,∀x ∈ X,withg, η, has in hypothesis ofTheorem 3.1. The relation2.2

holds with equality. ApplyingTheorem 2.1, there exists a unique mappingcwhich satisfies the equation3.2and the estimation3.7. Moreover,

cx lim

n→ ∞J

n_f_x_, _∀_x_∈_X,

3.8

where

Jnfx gx·Jn−1fηxhx

gx·gηx·Jn−2fη2xgx·hηxhx, ∀x∈X,

3.9

whence, for allx∈X,

Jnfx: hx fηnx·

n−1

i 0

gηix

n−2

j 0

hηj1x·

j

i 0

gηix. 3.10

As in Section 2.1, we can obtain Ulam-Hyers stability result for 3.2 in complete random normed space, by takingψxt: εδt,∀t >0 inTheorem 3.1.

3.1. A Consequence for Normed Spaces

As a particular case ofTheorem 3.1, we can prove immediately the generalized stability result of Ulam-Hyers type for the linear functional equation

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in Banach spaces, obtained by us in15, Theorem 5.1, by using the fixed point alternative.

Corollary 3.2. ConsiderX a nonempty set andY a real (or complex) Banach space. Suppose that η:X → X,g:X → R\ {0}. Iff :X → Y satisfies

fx−gxfηx−hx_Y ≤ϕx, ∀x∈X, 3.12

with some fixed mappingϕ:X → 0,∞and there existsL <1such that

gxϕ◦ηx≤Lϕx, ∀x∈X, 3.13

then there exists a unique mappingc:X → Y

cx hx lim

n→ ∞

⎛

⎝_f_ηn_x_·n−1 i 0

gηix

n−2

j 0

hηj1x·

j

i 0

gηix ⎞

⎠_, _3.14

cx gx·cηxhx, ∀x∈X 3.15

fx−cx_Y ≤ ϕx

1−L, ∀x∈X. 3.16

Proof. Let us consider the random normY u → Fu∈D, where

Fut 1−exp

− t u

, ∀t >0, 3.17

and the mapping

ψ:X → D, ψxt 1−exp

− t

ϕx

. 3.18

ThenYis a complete random normed space underTM Min. The hypothesis relations3.12,

3.13and3.16are equivalent to3.3,3.4and3.7, respectively. Therefore we can apply ourTheorem 3.1to obtain the above result.

Remark 3.3. Corollary 3.2can also be obtained by using the random norm

Y u → Fu∈D, Fut t

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and the mapping

ψ:X → D, ψxt t

tϕx. 3.20

4. Applications to the Generalized Equation of the Square Root Spiral

Wang et al. proved in30, by the direct method, a generalized Ulam-Hyers stability result for thegeneralized equation of the square root spiral

fp−1px k fx h1x, 4.1

in Banach spaces.

We consider this equation for mappingsffrom an Abelian semigroupXinto acomplete random normed spaceY,F, TM, withp:X → Xandh1 :X → Y some given functionsp−1

is the inverse ofpandkis a fixed constant, which is not an identity element inX.

We can prove, as a consequence ofTheorem 3.1, the following generalized Ulam-Hyers stability result for4.1:

Theorem 4.1. LetXbe an Abelian semigroup and letY,F, TMbe a complete random normed space.

Ffp−1_p_x_k₋_h

1x−fx≥ψx, ∀x∈X, 4.2

ψp−1_p_x_kLt≥ψ_xt, ∀x∈X, ∀t >0, 4.3

cx lim

n→ ∞

fp−1px nk−

n−1

i 0

h1

p−1px ik

, ∀x∈X, 4.4

which satisfies both the equation

cp−1px k cx h1x, ∀x∈X, 4.5

Fcx−fxt≥ψx1−Lt, ∀x∈X, 4.6

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Proof. It is easy to see that4.1is a particular case of3.2. In fact, we can considerg ≡ 1, h: −h1,ηx: p−1px k,∀x∈X, withpbijective onXandka fixed constant, which is

not an identity element inX. By using the above notations,Theorem 4.1can be obtained as a consequence ofTheorem 3.1, with

cx hx lim

n→ ∞

⎛

⎝_f_ηn_x_·n−1 i 0

gηix

n−2

j 0

hηj1x·

j

i 0

gηix ⎞ ⎠

−h1x lim

n→ ∞

fp−1px nk−

n−1

i 1

h1

p−1px ik

lim

n→ ∞

fp−1px nk−

n−1

i 0

h1

p−1px ik

, ∀x∈X.

4.7

As a consequence of Theorem 4.1, we can prove the generalized stability result of Ulam-Hyers type for the generalized equation of the square root spiral4.1in Banach spaces, obtained by us in15, Theorem 3.1, by using the fixed point alternative.

Corollary 4.2. LetX be an Abelian semigroup, letY be a Banach space and let p : X → X and h1 :X → Y some given functions, withpbijective. Ifkis a fixed constant, which is not an identity

element inXandf :X → Y satisfies

fp−1px k−fx−h1x

Y ≤ψx, ∀x∈X 4.8

with a mappingψ :X → 0,∞for which there existsL <1such that

ψp−1px kx≤Lψx, ∀x∈X, 4.9

cp−1px k cx h1x, ∀x∈X 4.10

fx−cx_Y ≤ ψx

1−L, ∀x∈X. 4.11

Moreover,

cx lim

n→ ∞

fp−1px nk−

n−1

i 0

h1

p−1px ik

, ∀x∈X. 4.12

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Aspecial case of4.1is obtained on0,∞, fork 1,px xn_,_n _≥ _{2 and}_h

1x

arctan1/x. It is the so-called “nth root spiral equation”

f√n

xn₁ _f_x _arctan1

x. 4.13

If we take in Theorem 4.1X R 0,∞ and Y R, we obtain the following generalized stability result for the functional equation4.13:

Corollary 4.3. LetR,F, TMbe a complete random normed space. Iff :R → Rsatisfies

F_f√n_xn₁_−arctan₁_/x₋_f_x≥ψx, ∀x∈R, 4.14

whereψ:R → 0,∞is a mapping for which there existsL <1such that

ψ√n_xn₁Lt≥ψxt, ∀x∈R, ∀t >0, 4.15

then there exists a unique mappingc:R → R,

cx lim

m→ ∞

n √

xn_m₋ m−1

i 0

arctan√_n 1

xn_i

, ∀x∈R, 4.16

which satisfies both4.13and the estimation

Fcx−fxt≥ψx1−Lt, ∀x∈R, 4.17

Moreover, as inCorollary 3.2, it is obtained a generalized Ulam-Hyers stability result for4.13, inR:

Corollary 4.4see15, Theorem 3.2. Iff:R → Rsatisfies

f√nxn₁₋_f_x₋_arctan1

x

≤ψx, ∀x∈R, 4.18

with some fixed mappingψ:R → 0,∞and there existsL <1such that

ψ√nxn₁_≤_Lψ_x_, _∀_x_∈_R

, 4.19

then there exists a unique mappingc:R → R,

cx lim

m→ ∞

n √

xn_m₋ m−1

i 0

arctan√_n 1 xn_i

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which satisfies both4.13and the estimation

_f_x₋_c_x_≤ ψx

1−L, ∀x∈R. 4.21

5. Applications to the Generalized Gamma Functional Equation

As a consequence ofTheorem 3.1, we obtain the following slight extension of the result in Kim31for thegeneralized gamma functional equation.

fxk gxfx. 5.1

Theorem 5.1. LetX be an Abelian group and letY,F, TMbe a complete random normed space.

Suppose thatg :X → R\ {0}and kis a fixed constant inX, diﬀerent from the identity element. If f:X → Ysatisfies

Ffxk−gx·fx≥ψxk, ∀x∈X, 5.2

gx◦ψxLt≥ψxkt, ∀x∈X,∀t >0, 5.3

cx lim

n→ ∞fx−nk·

n

i 1

gx−ik, 5.4

cxk gx·cx, ∀x∈X, 5.5

Fcx−fxt≥ψx1−Lt, ∀x∈X, 5.6

Proof. It is easy to see that 5.1is a particular case of the equation 3.2. In fact, we can considerh≡0,gx: gx−k,ηx: x−k,∀x∈X, with a fixed constantkinX, diﬀerent from the identity element. By using the above notations,Theorem 5.1can be obtained as a consequence ofTheorem 3.1.

Another generalized Ulam-Hyers stability result for5.1can be obtained by using our

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Theorem 5.2. LetXbe an Abelian semigroup and letY,F, TMbe a complete random normed space.

Suppose thatg :X → R\ {0}andkis a fixed constant, which is not an identity element inX. If f:X → Ysatisfies

Ffxk−gx·fx≥gx◦ψx, ∀x∈X, 5.7

whereψ:X → Dis a mapping for which there existsL <1such that

ψxkLt≥gx◦ψxt, ∀x∈X, ∀t >0, 5.8

cx lim

n→ ∞

fxnk·

n−1

i 0

1

gxik

, 5.9

cxk gx·cx, ∀x∈X, 5.10

Fcx−fxt≥ψx1−Lt, ∀x∈X, 5.11

Proof. Obviously,5.1is a particular case of the equation3.2. In fact, we can considerh≡0, gx 1/gx,ηx: xk,∀x∈X, with a fixed constantkinX, diﬀerent from the identity element. With the above notations,Theorem 5.2can be obtained by usingTheorem 3.1.

As a particular case of5.1we have, fork 1 andgx : x,thegamma functional equation

fx1 xfx, ∀x∈0,∞. 5.12

If we consider inTheorem 5.2:X 0,∞,Y Randk and g as in the above, we obtain the following generalized Ulam-Hyers stability for the functional equation5.12in random normed spaces.

Corollary 5.3. LetR,F, TMbe a complete random normed space. Iff :R → Rsatisfies

Ffx1−x·fx≥x◦ψx, ∀x∈0,∞, 5.13

whereψ:0,∞ → Dis a mapping for which there existsL <1such that

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then there exists a unique mappingc:0,∞ → R,

cx lim

n→ ∞fxnk·

n−1

i 0

1

xi, ∀x∈0,∞, 5.15

which satisfies both the equation

cx1 xcx, ∀x∈0,∞, 5.16

Fcx−fxt≥ψx1−Lt, ∀x∈0,∞, 5.17

Remark 5.4. The corresponding results of generalized Ulam-Hyers stability for 5.1 and 5.12can be obtained, in the deterministic case, as inCorollary 3.2.

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