Anna BAHYRYCZKrzysztof CIEPLI′NSKI2,?Jolanta OLKO

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

2.AGH University of Science and Technology,Faculty of Applied Mathematics, Mickiewicza 30,30-059 Krakow,Poland

E-mail:bah@up.krakow.pl;cieplin@agh.edu.pl;jolko@up.krakow.pl

ON AN EQUATION CHARACTERIZING MULTI-CAUCHY-JENSEN MAPPINGS AND
ITS HYERS-ULAM STABILITY?

Anna BAHYRYCZ1Krzysztof CIEPLI′NSKI2,?Jolanta OLKO1

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

2.AGH University of Science and Technology,Faculty of Applied Mathematics, Mickiewicza 30,30-059 Krakow,Poland

E-mail:bah@up.krakow.pl;cieplin@agh.edu.pl;jolko@up.krakow.pl

In this paper,we give two characterizations of multi-Cauchy-Jensen mappings. One of them reduces the system of n equations defning these mappings to a single functional equation.We also prove,using the fxed point method,the generalized Hyers-Ulam stability of this equation.Our results generalize some known outcomes.

multi-Cauchy-Jensen mapping;(generalized)Hyers-Ulam stability;Cauchy’s functional equation;Jensen’s functional equation;fxed point method2010 MR Subject Classifcation39B82;39B72;38B52

1 Introduction

Throughout this paper N stands for the set of all positive integers,N0:=N∪{0}, R+:=[0,∞),n∈N and k∈{0,···,n}.Moreover,given a nonempty set V,we identify x=(x1,···,xn)∈Vnwith(x1,x2)∈Vk×Vn-k,where x1:=(x1,···,xk)and x2:=(xk+1,···,xn).

It is well-known that among functional equations the Cauchy equation

and the Jensen equation

(which is closely connected with the notion of convex function)play a prominent role.A lot of information about them and their applications can be found for instance in[1,2].

Let us note that for k=n the above defnition leads to the so-called multi-additive mappings (some basic facts on such mappings can be found for instance in[2],where their application to the representation of polynomial functions is also presented);for k=0 we obtain the notion of multi-Jensen function(which was introduced in 2005 by W.Prager and J.Schwaiger(see[4]) with the connection with generalized polynomials);a 1-Cauchy and 1-Jensen mapping is just a Cauchy-Jensen mapping defned by W.-G.Park and J.-H.Bae in[5].

In this paper,we give two characterizations of multi-Cauchy-Jensen mappings.One of them (Theorem 2.2)reduces the system of n equations defning these mappings to a single functional equation.Next,we prove the generalized Hyers-Ulam stability of this equation.Our results are signifcant supplements and/or generalizations of some classical outcomes from[6-10]and recent results from[5,11-23].

Let us also recall that speaking of the stability of a functional equation we follow the question raised in 1940 by S.M.Ulam:“when is it true that the solution of an equation difering slightly from a given one,must of necessity be close to the solution of the given equation?”.The frst partial answer(in the case of Cauchy’s equation(1.1)in Banach spaces)to Ulam’s question was given by D.H.Hyers(see[9]).After his result a great number of papers(see for instance [1,24-29]and the references given there)on the subject has been published,generalizing Ulam’s problem and Hyers’s theorem in various directions and to other(not only functional)equations.

In the proof of our stability result(Theorem 3.2)we use the fxed point method,which was used for the investigation of the Hyers-Ulam stability of functional equations for the frst time by J.A.Baker in[30].For more information about this method we refer the reader to recent papers[25,27,31,32].

To fnish this introductory section let us fnally mention that some results on the stability of Cauchy-Jensen mappings can be found in[5,17-21,33,34].

2 Some Characterizations of Multi-Cauchy-Jensen Mappings

In this section,we present two characterizations of multi-Cauchy-Jensen mappings.

First,we reduce the system of n equations defning the k-Cauchy and n-k-Jensen mapping to obtain a single functional equation for f.In order to do this,we use the following lemma generalizing Lemma 1.1 from[22].

Lemma 2.1Assume that G and H are semigroups uniquely divisible by 2,H is commutative and n∈N.Then f:Gn→H is an n-Jensen mapping if and only if for any xi=(x1i,···,xni)∈Gn,i∈{1,2}we have

ProofAssume that f:Gn→H satisfes equation(2.1).Fix j∈{1,···,n}and xj1,xj2,yi∈G,i∈{1,···,n}{j}.Set xi:=(x1i,···,xni),where xk1=xk2=ykfork∈{1,···,n}{j},i∈{1,2}.It is easily seen that for any i1,···,in∈{1,2}we have

Consequently,by(2.1),we get

which shows that f satisfes Jensen’s equation in the j-th variable.

Necessity of(2.1)is proved by induction on n.The case n=1 is obvious.

Fix an n∈N{1},assume that our assertion is true for n-1 and let f:Gn→H be an n-Jensen function.Fix also xi=(x1i,···,xni)∈Gn,i∈{1,2},put?xi:=(x1i,···,xn-1i)and defne fxni:Gn-1→H by

Since the functions fxniare n-1-Jensen,from the inductive hypothesis it follows that

On the other hand,f satisfes Jensen’s equation in the n-th variable,and therefore

which,together with(2.2),completes the proof.

ProofLet us frst assume that f:Vn→W is a k-Cauchy and n-k-Jensen mapping. Since then for any x2∈Vn-kthe mapping gx2:Vk→W given by

is k-Cauchy,Theorem 2 in[16](in which the commutativity of V can be omitted)shows that for any x1i=(x1i,···,xki)∈Vk,i∈{1,2}we have

which means that

is n-k-Jensen,and therefore from Lemma 2.1 it follows that

which proves that f satisfes equation(2.3).

To get our second characterization of multi-Cauchy-Jensen mappings,we need two lemmas. It seems that they are interesting in themselves,and will be useful for proving the hyperstability of some related functional equations(for a survey on this issue see[26]).

Lemma 2.3Let G and H be groups,and assume that there exists an x0∈G with 2x0:=x0+x0/=0.Then a mapping f:G→H satisfes Cauchy’s functional equation(1.1) for x,y∈G if and only if it satisfes this equation for x,y∈G{0}.

ProofAssume that f:G→H satisfes equation(1.1)for x,y∈G{0}.In order to see that it fulfls(1.1)for any x,y∈G,we need only to show that f(0)=0.

Take an x0∈G with 2x0/=0.Since x0/=0,we have

and consequently f(-x0)=-f(x0).Therefore,

Let us note that if we drop the assumption on x0from Lemma 2.3,then its assertion is not true.Indeed,the function f:Z2→Z3with f(0)=2,f(1)=1 satisfes equation(1.1)on Z2{0},but f does not satisfy it on the whole Z2.

Lemma 2.4Let G and H be commutative groups uniquely divisible by 2,and assume that there exists an x0∈G with 3x0/=0.Then a mapping f:G→H satisfes Jensen’s functional equation(1.2)for x,y∈G if and only if it satisfes this equation for x,y∈G{0}.

ProofAssume that f:G→H satisfes equation(1.2)for x,y∈G{0}and fx an x0∈G such that 3x0/=0.Then x0/=0 and G{0,-x0}/=?.

for x∈G{0}.Obviously,

Then for every y∈G{0,-x0}we have

This yields

which with(2.6)applied for x0completes the proof.?

The condition concerning x0is essential in Lemma 2.4,because f:Z3→Z5given by

satisfes Jensen’s equation only on Z3{0}.

As an easy consequence of the above two lemmas we obtain the mention characterization of k-Cauchy and n-k-Jensen mappings.Namely,we have the following.

Corollary 2.5Let G and H be commutative groups uniquely divisible by 2,and assume that there exists an x0∈G with 3x0/=0.Then f:Gn→H is a k-Cauchy and n-k-Jensen mapping if and only if f satisfes on G{0}Cauchy’s equation in each of the frst k variables and Jensen’s equation in each of the other variables.

3 Stability of Equation(2.3)

In this section,we prove the generalized(in the spirit of D.G.Bourgin and P.G?avrut?a) Hyers-Ulam stability of equation(2.3).The proof is based on a fxed point result that can be derived from[35,Theorem 1].To present it,we introduce the following three hypotheses:

(H1)E is a nonempty set,Y is a Banach space,j∈N,f1,···,fj:E→E and L1,···,Lj: E→R+.

(H2)T:YE→YEis an operator satisfying the inequality

Now,we are in a position to present the mentioned fxed point theorem.

Theorem 3.1Let hypotheses(H1)-(H3)hold and functions ε:E→R+and φ:E→Y fulfll the following two conditions

Then there exists a unique fxed point ψ of T such that

With this notation,we have the following result.

Theorem 3.2Suppose that V is a linear spaces over the rationals and W is a Banach space.Let f:Vn→W and θ:Vn×Vn→R+be mappings satisfying the inequality

for x=(x1,x2)∈Vnand

and consequently

Fix an x∈Vnand write

Then,by(3.5)and(3.6),we obtain

Next,put

so hypothesis(H2)is also valid.

Finally,using induction,one can check that for any l∈N0and x∈Vnwe have

which,together with(3.2),shows that all assumptions of Theorem 3.1 are satisfed.Therefore, there exists a unique function F:Vn→W such that

and(3.4)holds.Moreover,

Now,we show that

Letting l→∞in(3.7)and using(3.3)we obtain

which means the function F satisfes equation(2.3).?

Theorem 3.2 yields immediately the following stability result corresponding to[20,Corollary 2.3].

Corollary 3.3Assume that V is a linear spaces over the rationals,W is a Banach space, k≥1 and ε＞0.If f:Vn→W satisfes the inequality

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?Received August 19,2014;revised April 27,2015.?

Krzysztof CIEPLI′NSKI.