1 Introduction

Let G be an abelian group,R,C the set of real numbers,complex numbers,respectively.Hom(G,F)denotes the group of homomorphisms from the group G to the multiplicative field F.Every element in Hom(G,F)is called a(group)character and denoted by χ.Thus a character is a nonzero multiplicative homomorphism from group G into the multiplicative group of nonzero complex numbers.If χ is a group character of G,then we denote χ(x−1)by(x).Clearly,(x)is also a character of G.Denote σ:G→G be an involution,provided that σ(x+y)= σ(x)+ σ(y)and σ(σ(x))=x for all x,y ∈ G.For convenience we always denote σ(x)as simply σx.A function m:G → F is called an exponential function provided that m(x+y)=m(x)m(y)for all x,y∈G.A function f:G→F is said to be an abelian function if and only if f(xy)=f(yx)for all x,y∈G.And a function f:G→F is said to be σ-odd with respect to an involution σ:G → G if and only if f(σx)= −f(x)for all x ∈ G.It is easy to see that if f is a σ-odd function,then f(e)=0,where e is a unit element in G.Similarly,a function f:G → F is said to be σ-even with respect to an involution σ:G → G if and only if f(σx)=f(x)for all x ∈ G.A function m:G → F is called σ-exponential if m satis fies m(xy)=m(x)m(y)and m(σx)=m(x)for x ∈ G,and denoted by mσ.

In 1910,Vleck[1]considered the functional equation

where a＞0 is fixed.He proved that f is a periodic function with period 4a and(1.1)implies the cosine functional equation.

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In 2009,Kannappan[2]considered the functional equation

and proved the following result:the general solution f:R→C of(1.2)is either f=0 or f(x)=g(x−a),where g is an arbitrary solution of the cosine functional equation

with period 2a.

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In 2002,Czerwi[3]studied the functional equation

on a locally compact abelian group G.And this work was extended by Fechner[4]in 2009.

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Serval years later,Perkins and Sahoo[5]generalized the above functional equation

Generally,these above functional equations are called trigonometric functions.More stability and solution of trigonometric functions which was solved in the reference[6]–[12].

On the other hand,Hyers[10]proposed the Hyers-Ulam stability problems of functional equations which were concerning the approximate homomorphisms from a group to a metric group.Later,Szekelyhidi[11]investigated the Hyers-Ulam stability question of the following trigonometric functional equations

As a particular case of the result in[11],he obtained the stability of the functional inequalities

当TBR>1时，表明实际完成时间小于其交货周期，其时间效益值大；当TBR=1时，表明实际完成时间与交货周期一致。上述两种情况说明目前的制造能力从时间效益来说是满足制造需求的，具有时间效益增益。而当TBR<1时，表明实际完成时间大于交货周期，没有产生时间效益，说明不能满足制造需求，因此在制造过程需要避免这种情况出现。

for all x,y∈G,where f:G→C and φ:G→R+.

From above reference,we found that though some researchers investigated solutions and stability of functional equations with sine functions and cosine functions(see[10]–[11]),there were some pity needing to be completed.Secondly,there were similar ways to obtain their solutions.

Putting(3.17)into(1.11),we have

And this method is different from above reference,above all,we give the stability of the following functional inequalities(1.11)and(1.12):

for all x,y∈G and f:G→F,where φ:G→R+is arbitrary.

Furthermore,it is important that this may be a partial answer of the question in[10]and[11].And we correct an important condition that f and g must be independent which may be forgotten in[11].

2 Solutions of(1.9)and(1.10)

In this section,we determine the solution of(1.9)with involution on arbitrary groups.The solution of(1.10)can be solved by the similar ways.We manipulate the variables to find properties of functions f.Using these properties we can get some relations between different functional equations.

Substituting(x,y)=(y,x)in(1.9).Then we get

where χ(x)is an arbitrary character function of G.

Proof. It is clear that f≡0,g≡0 is a solution to(2.1).Next,we are only concerned with the nonzero solutions of(2.1).

Theorem 2.1 Let G be a group,F be the field of complex numbers,and σ:G → G be an involution.If f,g:G→F satisfy(1.9)for all x,y∈G,then either f≡0 or f is given by

Combining with(1.9),it yields that

That is,f is a σ-even function.

Fix y0such that g(y0)0.Then by(1.9),we have

That is,g is a σ-odd function.

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Setting y= σy,x= σx in(1.9),respectively,we can get

By Proposition 1 in[11],we have

where χ(x)is an arbitrary character function of G.The proof is completed.

By this Theorem,we can get the following remark.

Denote

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where χ(x)is an arbitrary character function of G.

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3 Ulam-Hyers Stability of(1.11)and(1.12)

Since F(x,y)is bounded,when y,z are fixed andevery term on the left hand side of(3.8)is bounded by a constant M,where

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Lemma 3.1 Let G be a group,F be a commutative field on C.Suppose that f,g:G→F satisfy the inequality(1.11)for all x,y ∈ G.Then either there existµ1,µ2∈ F withand positive number M such that

or else

Proof.Define F:G×G→F by

Then F(x,y)is bounded by(1.11).Choose y0satisfying g(y0)0.Then we get

Let wThen(3.4)turns into(3.5).

Combining(3.4)and(3.5),we obtain

On the other hand,by definition we have

So comparing two equations(3.6)and(3.7),transposing all terms which contain F to the left hand side,we have

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Remark 2.1 Let G be a group,F be the field of complex numbers,and σ:G → G be an involution.If f,g:G→F satisfy(1.9)for all x,y∈G,then g is given by

In this section we consider the stability of functional equation(1.11).And the method is suit for the stability of functional equation(1.12).

Hence it yields the first conclusion.

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Whenµ1= µ2=0,

So we get

F(xy,z)−F(x,yz)=(w2F(x,y)+w1F(xy,y0)−w1F(x,yσy))g(σz)−F(x,y)f(σz).(3.9)

On the other hand,

Next,if g is unbounded,then by Lemma 3.1,since

Next,we give the main Theorem of this section.

Theorem 3.1 Let G be a group,F be a commutative field on C.If the functions f,g satisfy the inequality(1.11)for all x,y∈G,then(f,g)satis fies one of the following statements:

(1)f,g are both bounded functions;

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(2)f,g are both exponential functions,and given by

where m is an exponential function;

(3)g=±i(f−m)for a bounded exponential function m,and g satis fies

In particular,if φ(x)=0,then f(0)=1,g(0)=0.

Proof. First,if g is bounded,then by using Lemma 3.1,from triangle inequality we obtain

that is,there exists a nonnegative number N such that

So we can get from Lemma 2.1 in[10]f is bounded or is a nonzero exponential function.

Now,if f is bounded,then the case(1)follows.

If f is a nonzero exponential function,replacing(x,y)by(y,x),and by using triangle inequality again,we obtain

This follows that

and this implies f≡1,so the case(1)holds.

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Again the right hand side of(3.10)is bounded by φ(σx)+ φ(σ(xy)),and the right hand side of(3.9)in fact is a function of variable z when x,y are fixed.So there only have one probability which F(x,y)≡0.This completes the proof.

it is easy to see that f is also unbounded.Suppose that there exists a nonnegative number λ∈F and a bounded function r:G→F such that

Based on this,we determine the general solution of the following functional equations(1.9)and(1.10)with involution on groups.

Hence,

Since the right hand side of(3.19)is a positive constant for variable y,by using Lemma 2.1 in[10]again and from the above analysis,there exists an exponential function m(y)such that

If λ2 −1,then

Putting(3.20)into(1.11),multiplying|1+λ2|,and using the triangle inequality,we have for some d≥0,

Since m is unbounded,we get

and this implies m(x)≡1.But this is contradicting to the fact that m is unbounded.So

Therefore,by(3.17)and(3.20),we have

where m is an exponential function.Putting(3.23)in(1.11),we get

Replacing y by x in(3.24),dividing the result by constant,we have

From(3.23)and(3.25)we get(3).

When f,g satisfy(3.2),in view of Lemma 3.1,the solution is contained in(1)or given by(2).Furthermore,if φ(0)=0 in(3.25),Then f(0)=1.From(1.11)we get g(0)=0.This completes the proof.

References

[1]Vleck E B V.A functional equation for the sine.Ann.Math.,1910,11(4):161–165.

[2]Kannappan P.Functional Equations and Inequalities with Applications.New York:Springer,2009.

[3]Czerwi S.Functional Equations and Inequalities in Several Variables.Singapore:World Scientific,2002.

[4]Fechner Z.A generalization of Gajda’s equation.J.Math.Anal.Appl.,2009,354(2):584–593.[5]Perkins A M,Sahoo P K.On two functional equations with involution on groups related to sine and cosine functions.Aequat.Math.,2015,89(5):1251–1263.

[6]Perkins A M,Sahoo P K.A functional equation with involution related to the cosine function.Aequat.Math.,2016,90(1):123–131.

[7]Chung J,Sahoo P K.Stability of Wilson’s functional equations with involutions.Aequat.Math.,2015,89(3):1–15.

[8]Perkins A M,Sahoo P K.On two functional equations with involution on groups related to sine and cosine functions.Aequat.Math.,2015,89(5):1251–1263.

[9]Cheng L H,Zhang J M.Fixed point approach to the fuzzy stability of a mixed type functional equation.Comm.Math.Res.,2016,32(2):122–130.

[10]Hyers D H.On the stability of the linear functional equation.Proc.Nat.Acad.Sci.USA,1941,27(4):222–224.

[11]Szekelyhidi L.The stability of the sine.1990,110(1):109–115.

[12]Chung J,Choi C K,Kim J.Ulam-Hyers stability of trigonometric functional equation with involution.J.Funct.Spaces Appl.,2015,2015(6):1–7.