ON APPROXIMATELY（p，q）-WRIGHT AFFINE FUNCTIONS AND INNER PRODUCT SPACES?

Anna BAHYRYCZMagdalena PISZCZEK

Institute of Mathematics，Pedagogical University，Podchor?a˙zych 2，30-084 Krak′ow，Poland

E-mail∶bah@up.krakow.pl；magdap@up.krakow.pl

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ON APPROXIMATELY（p，q）-WRIGHT AFFINE FUNCTIONS AND INNER PRODUCT SPACES?

Anna BAHYRYCZMagdalena PISZCZEK

Institute of Mathematics，Pedagogical University，Podchor?a˙zych 2，30-084 Krak′ow，Poland

E-mail∶bah@up.krakow.pl；magdap@up.krakow.pl

We prove，using the fixed point approach，some results on hyperstability（in normed spaces）of the equation that defines the generalization of p-Wright affine functions and show that they yield a simple characterization of the complex inner product spaces.

Hyperstability；p-Wright affine function；complex inner product space；fixed point theorem

2010 MR Subject Classification39B52；39B55；47H10

1　Introduction

Let X and Y be linear spaces over fields F∈｛R，C｝and K，respectively，and p，q∈F be fixed.The functional equation

for function g：X→Y，generalizes the equation

For X=Y=R and p∈（0，1），the solutions of equation（1.2）are called the p-Wright affine functions.The functions satisfying equation（1.1）are called the（p，q）-Wright affine functions.

The definition of p-Wright affine functions is connected to the notions of p-Wright convexity and p-Wright concavity（see，for example，［9，10，12，14，17］）.Clearly，for p=1/2，equation（1.2）is just the well known Jensen's equation

For p=1/3，equation（1.2）takes the form

which was investigated by Najati and Park［13］；in particular，they proved some results on its stability and applied them in the investigation of the generalized（σ，τ）-Jordan derivations on Banach algebras.

In this article，we present some hyperstability results for equation（1.1）.Namely，we prove that for some particular forms of θ，the functional equation（1.1）is θ-hyperstable in the class of functions g：X→Y，that is，each g：X→Y satisfying the inequality

‖g（px+qy）+g（qx+py）?g（x）?g（y）‖≤θ（x，y），x，y∈X，

must be（p，q）-Wright affine function（see Theorems 3.1 and 3.2）.We show that they yield a simple characterization of complex inner product spaces（see Corollary 4.1）.For

with some α∈R，equation（1.1）characterizes norms in the complex inner product spaces.

It is known（see［16］）that every solution g of（1.1）has the form

with some c∈Y，an additive A：X→Y，and a quadratic B：X→Y.

For more information and numerous on the equation of the p-Wright affine functions and its stability，please refer to，for example，［2-4，9，11，15］.

2　Auxiliary Results

To present an auxiliary（fixed point）result，we need to introduce some necessary hypotheses（R+stands for the set of nonnegative reals and ABdenotes the family of all functions mapping a set B 6=?into a set A 6=?）.

（H1）X is a nonempty set，E is a Banach space，f1，···，fk：X→X and L1，···，Lk：X→R+are given，and T：EX→EXis an operator satisfying the inequality

Now，we can present the above mentioned fixed point theorem proved in［6，Theorem 1］（see also［7，Theorem 2］and［5，8］）.

Theorem 2.1Let hypotheses（H1），（H2）be valid and functions ε：X→R+and ?：X→E fulfil the following two conditions：

Then，there exists a unique fixed point ψ of T with

Moreover，

First，we prove an auxiliary lemma，which shows that（p，q）-Wright affine function on X｛0｝is（p，q）-Wright affine function on the whole X.

Lemma 2.2Let p，q∈F｛0｝.If a function g：X→Y satisfies

then g is（p，q）-Wright affine function.

ProofAssume that g fulfils（2.1）.Let ge，go：X→Y denote the even and the odd parts of g，respectively，that is，

Putting y=x in（2.1），we have

Setting y=?x in（2.1），we obtain

Thus，the even part of g satisfies

Replacing in（2.1）x by px，y by?qx，we get

and hence using（2.2）and（2.3），we obtain

Replacing in（2.1）x by px，y by qy，we have

Setting y=x in（2.6），we get

Adding equations（2.4）and（2.7），we obtain，for the odd part of g，

Whence using（2.1），we have，for x∈X｛0｝，

thus

Next，applying（2.6）and（2.2），we obtain

Replacing in the above equation x bywe obtain

which together with（2.5）gives

and finishes the proof.

3　Hyperstability Results

In this section we present the results concerning hyperstability of equation（1.1）.

Theorem 3.1Let X be a normed space over a field F∈｛R，C｝，Y be a normed space，p，q∈F｛0｝，c≥0，k＞0，and l＜0.Then，every function g：X→Y with

satisfies equation（1.1）.

ProofFirst，without loss of generality，we assume that Y is a Banach space，because otherwise we can replace it by its completion.

Replacing y by mx in（3.1），for m∈N，we get

Write

and

Define

so（H1）is valid.

Next，we can find m0∈N，such that

Therefore，

for m≥m0and x∈X｛0｝.

Thus，according to Theorem 2.1，for each m≥m0，there exists a unique solution Gm：X｛0｝→Y of the equation

such that

for x∈X｛0｝.Moreover，

Now，we show that

for every x，y∈X｛0｝，n∈N0（nonnegative integers）.

If n=0，then（3.3）is simply（3.1）.So，take i∈N0and suppose that（3.3）holds for n=i and x，y∈X｛0｝.Then，

Letting n→∞in（3.3），we obtain

for x∈X｛0｝.Hence，letting m→∞，（2.1）holds.So，according to Lemma 2.2 the function g is the（p，q）-Wright affine function，which completes the proof.

In the analogous way，we can prove the following theorem.

Theorem 3.2Let X be a normed space over a field F∈｛R，C｝，Y be a normed space，c≥0，k＞0，l＜0，and p，q∈F｛0｝，with q 6=?p.Then，every function g：X→Y with

satisfies equation（1.1）.

We notice that for p=q，from（3.5）we obtain

For x∈X｛0｝，we define

Then，（3.6）（if p=q）and（3.5）（if p 6=q）take form

and it is easily seen that Λmthat has the form described in（H2）and（H1）is valid.Write

Next，we can find m0∈N，such that αm＜1 for all m≥m0.Therefore，after simple calculations，we have

Hence，according to Theorem 2.1，for each m∈Nm0，there exists a unique fixed point Gm：X｛0｝→Y of Tmwith

Moreover，

It is easy to prove that for every x，y∈X｛0｝，m∈Nm0，and n∈N0，

Letting n→∞in（3.8），we obtain

Remark 3.3We notice that in the particular case when q=1?p，we obtain some results for hyperstability of p-Wright affine functions. The following example show that the assumption q 6=?p in the above theorem is essential. Example 3.4Let g：R→R be defined as g（x）=x2k+1for x 6=0 and g（0）=0，where k is a negative integer number.Then，g satisfies

but g does not satisfy（p，?p）-Wright affine equation，because g is not a constant.

4　Characterization of the Complex Inner Product Spaces

In this part，we show that Theorem 3.1 and Theorem 3.2 yield a characterization of the complex inner product spaces.

First，we define the function θj（x，y）：（X｛0｝）×（X｛0｝）→R+，for j∈｛1，2｝in the following way

where d≥0，k＞0，l＜0.

Corollary 4.1The following three statements are valid.

（i）Let X be a normed space over F∈｛R，C｝.Then，for every s＞0，j∈｛1，2｝and p，q∈F｛0｝with|p+q|6=1 or|p?q|6=1，and additionaly q 6=?p when j=2，we have

（ii）Assume that X is the complex normed space and there exist j∈｛1，2｝，p，q∈CR with q 6=?p when j=2，such that

Then，X is an inner product space and|p+q|=|p?q|=1.

（iii）Let X be an inner product space over F∈｛R，C｝.Then，

ProofTake s＞0，j∈｛1，2｝and p，q∈F｛0｝with|p+q|6=1 or|p?q|6=1，additionaly q 6=?p if j=2 and suppose that

This means that a function g：X→R，g（x）=‖x‖s，satisfies

with some M≥0.Consequently，in view of Theorem 3.1 if j=1，and Theorem 3.2 if j=2，we have

Setting in（4.4）y=x and then y=?x，we get

and

respectively.Hence，|p+q|=|p?q|=1，which is a contraction.

For the proof of（ii），observe that（4.1）is just condition（4.2）with s=2.Hence，

and|p+q|=|p?q|=1，and therefore（1.4）holds with some c∈R，an additive A：X→R，and a quadratic B：X→R.From the fact that g is even and g（0）=0，we obtain

which means that for every x，y∈X，we have the parallelogram equality

and consequently，X is an inner product space.

It remains to show（iii）.So，fix p，q∈F with|p+q|=|p?q|=1.Note that the case F=R is trivial，because then|p|=1 and q=0 or p=0 and|q|=1.So，assume that F=C.Let〈x，y〉denote the inner product of vectors x，y∈X.Write

Then，

whence（with x replaced by px and y by qx）we getNext，by simple calculations，we get Consequently，we obtain

Remark 4.2The condition|p+q|=|p?q|=1 implies that|p|2+|q|2=1.If p，q∈R，then|p|=1 and q=0 or p=0 and q=|1|.Moreover，every pair（p，q）∈C2｛（0，0）｝satisfying the condition|p+q|=|p?q|=1 is of the form（1.3）with some α∈R，and for such pairs（p，q），equation（1.1）characterizes norms in the complex inner product spaces.

References

［1］Acz′el J，Dhombres J.Functional Equations in Several Variables.Cambridge University Press，1989

［2］Bahyrycz A，Piszczek M.Hyperstability of the Jensen functional equation.Acta Math Hungar，2014，142：353-365

［3］Bahyrycz A，Brzd?ek J，Piszczek M.On approximately p-Wright affine functions in ultrametric spaces.J Funct Space Appl，2013.Art ID 723545

［4］Brzd?ek J.Stability of the equation of the p-Wright affine functions.Aequationes Math，2013，85：497-503

［5］Brzd?ek J，C?adariu L，Ciepli′nski K.Fixed point theory and the Ulam stability.J Funct Spaces，2014.Art ID 829419

［6］Brzd?ek J，Chudziak J，P′ales Zs.A fixed point approach to stability of functional equations.Nonlinear Anal，2011，74：6728-6732

［7］Brzd?ek J，Ciepli′nski K.A fixed point approach to the stability of functional equations in non-Archimedean metric spaces.Nonlinear Anal，2011，74：6861-6867

［8］C?adariu L，G?avrut?a L，G?avrut?a P.Fixed points and generalized Hyers-Ulam stability.Abstr Appl Anal，2012.Art ID 712743

［9］Dar′oczy Z，Lajk′o K，Lovas R L，Maksa G，P′ales Zs.Functional equations involving means.Acta Math Hung，2007，116：79-87

［10］Gil′anyi A，P′ales Zs.On Dinghas-type derivatives and convex functions of higher order.Real Anal Exchange，2001/2002，27：485-493

［11］Lajk′o K.On a functional equation of Alsina and Garc′?a-Roig.Publ Math Debrecen，1998，52：507-515

［12］Maksa G，Nikodem K，P′ales Zs.Results on t-Wright convexity.C R Math Rep Acad Sci Canada，1991，13：274-278

［13］Najati A，Park C.Stability of homomorphisms and generalized derivations on Banach algebras.J Inequal Appl，2009，2009：1-12

［14］Nikodem K，P′ales Zs.On approximately Jensen-convex and Wright-convex functions.C R Math Rep Acad Sci Canada，2001，23：141-147

［15］Sz′ekelyhidi L.The stability of linear functional equations.C R Math Rep Acad Sci Canada，1981，3（2）：63-67

［16］Sz′ekelyhidi L.Convolution type functional equations on topological abelian groups.World Scientific Publishing Co Inc Teaneck NJ，1991

［17］Wright E M.An inequality for convex functions.Amer Math Monthly，1954，61：620-622

December 4，2014；revised January 8，2015.