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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 878543, 6 pages
Remarks on the Conditional Quadratic and the Conditional D’Alembert Functional Equations on Particular Normed Spaces
Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, I-10123 Torino, Italy
Received 20 March 2013; Accepted 6 June 2013
Academic Editor: Pedro M. Lima
Copyright © 2013 Margherita Fochi and Gabriella Viola. 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.
Let be a real normed space with dimension greater than 2 and let be a real functional defined on . Applying some ideas from the studies made on the conditional Cauchy functional equation on the restricted domain of the vectors of equal norm and the isosceles orthogonal vectors, the conditional quadratic equation and the D’Alembert one, namely, and , have been studied in this paper, in order to describe their solutions. Particular normed spaces are introduced for this aim.
The conditional functional equation where is a continuous mapping from an inner product space of dimension into a real topological vector space , was first studied by Alsina and Garcia Roig in . They recognized the connection between this equation and the orthogonally additive functional equation that is which is a conditional functional equation studied by several authors and for long time (see among others [2–5]). In particular we mention the general results obtained by Rätz : he described the solutions of in the so-called orthogonality spaces in which the orthogonality relation is defined in axiomatic way and it is homogeneous.
Then Szabó  studied the previous for functionals defined in a real normed space with values in an abelian group. In this paper he considered also the following conditional equation: which is the additive Cauchy functional equation on the restricted domain of the isosceles orthogonal vectors in the sense of James ; that is, are isosceles orthogonal if .
This study of functional equations on the restricted domain of the vectors of equal norm or the isosceles orthogonal vectors in linear normed spaces, which are not inner product spaces, is quite difficult.
In particular, the lack of homogeneity for the isosceles orthogonality leads to the search for a new different approach for the proof, since the general theory of the orthogonality spaces in the sense of Rätz does not cover this case. The method of the proof of those theorems is by no means elementary. In particular, some sophisticated connectivity results on intersection of spheres of equal radii in normed linear spaces are involved. The proofs are based on the existence of particular auxiliary vectors of equal norm in normed spaces. Moreover, the method spoken of requires the dimension of the space considered to be greater than or equal to 3.
The aim of our paper is to consider two other functional equations: the quadratic functional equation and the D’Alembert functional equation and to study the solutions of their conditional version on the restricted domain of vectors of equal norm in the class of functionals defined on a normed space; this is the starting point. In the future we purpose to study the above conditional equations on the restricted domain of the isosceles orthogonal vectors.
Let . We study the following conditional equations: taking hint from the techniques introduced in the previous papers, based on suitable vectors in normed spaces.
We notice that the additive equation involves the values of on the vectors , , , and the auxiliary vectors introduced by Szabó can perform in this case. For (3) and (4) we have to deal with the values of on , and both vectors and . So we should assume the existence of useful vectors, defining particular normed spaces in which such vectors exist. We describe those spaces in Section 2.
Then, in Section 5, we consider the even isosceles orthogonal additive functionals in the particular class of the functional defined on the normed spaces introduced in Section 2, and we give their representation in order to characterize the particular normed spaces introduced in this paper; unfortunately this characterization remains since now an open problem.
2. Particular Normed Spaces
Let be a normed space with . Following the ideas of Szabó, let us introduce in some auxiliary vectors by the following conditions.
Let be vectors with ; then there exist such that
Let be vectors with ; then there exist such that
Let be vectors with ; then there exist such that
Let be vectors with ; then there exists such that
Condition has been introduced by Szabó  and the existence of such vectors is proved in every normed spaces, as we can see from the following.
Proof. Let be vectors with . We put ; consider the connected intersection of spheres and , with the continuous function defined by ; we easily prove that so that the function changes its sign on which is connected. Hence there is a vector with , that is , so the assertion is true if we put and .
We remark that from this condition we deduce that for every nonnull vector, there exists a vector of equal norm which is also isosceles orthogonal to the given one; this is easily proved setting in condition .
Condition is easily proved from Proposition 1, using the auxiliary functional . For , condition leads us to the same conclusion as in the case .
We remark that the pair of vectors and the pair of vectors in conditions and may be different from each other.
Condition requires that there exists a pair of vectors satisfying both conditions; this case is not satisfied in every normed space as we can see from the following counter example.
Example 2. Let be the normed space in which we define the norm
In this space we prove that there exist with for which it is impossible to find vectors satisfying . For this aim, let us consider and . Those vectors satisfy since and . We use now condition .
The idea of the proof that there are not vectors satisfying condition is as follows: we suppose by an absurd that there exists a vector with satisfying condition . We compute ; since , we are forced to consider only particular values for . In fact from we remark that conditions: , , must be satisfied in a suitable manner.
First suppose that ; hence or in order to satisfy (6). The following three cases hold: , , and , with and .
In the first case compute and , so that this case is impossible.
In the second case compute and , so also this case is impossible.
In the third case compute and . We have only if , so that we have the vector . We compute in this case and , but and ; hence this vector does not satisfy the two conditions required.
All the remaining cases are similar, so we prove that in the space , there exist no vectors satisfying condition .
Since the normed space of the previous example is a noninner product space, we purpose to find if condition characterizes or not the inner product spaces among normed spaces. First we prove the sufficient condition.
In fact, using the definition of the norm in inner product spaces and the basic property of the inner product this theorem may be easily proved.
The necessary condition remains an open problem, but in Section 5 we describe an interesting partial result.
3. The Conditional Quadratic Functional Equation
We give now a characterization of the quadratic equation on the restricted domain of equal-norm vectors in a normed space introduced in the previous section, that is a space in which condition is fulfilled.
Proof. It is easily seen that (1) implies (3).
Suppose now that (3) is satisfied. First we prove the following relations: It is sufficient to put , and choose the pair in (3), respectively.
Let be arbitrarily given vectors with . For property , there exist the vectors such that and Substituting in (3) in place of , we obtain Since is even, for (3), (8), and (11), we get The proof is similar in case of : it is sufficient to change the role of and , since is even.
4. The D’Alembert Conditional Functional Equation
We give now a characterization of the D’Alembert equation on the restricted domain of equal-norm vectors in a normed space introduced in the previous section, that is, a space in which condition is fulfilled.
Proof. It is easily seen that (2) implies (4).
Suppose now that (4) is satisfied. First we prove that or , putting in (4) and obtaining .
(a) Let ; we shall prove that, in this case, for all . Setting in (4) , we get Then, substituting in (4) and in place of , we get for , and using (13) we have On the other hand, squaring both sides of (4), we get Comparing (14) with (15), we have Now, let be an arbitrary vector. Then there exists a vector such that and (see Szabó, [6, Page 270]). Substituting and for and , respectively, in (4), we get , and then by (16), . Hence, by (13), we get ; that is, .
(b) Let , we put in (4) and obtain We now prove that is even. Substitution in (4) of and for and leads to , for all . Then, putting in this last equation, we get ; hence .
Using (17) and (4), we prove the following useful equations:
Proof of (18). Substituting in (4) and in place of , we get for , and using (17) we have On the other hand, squaring both sides of (4), we get (15). Then, comparing (20) and (15) we prove (18).
Proof of (19). Substituting in (18) and in place of , we get ; that is, , and using (17) in the right hand side, we have (19).
Now we have to prove that satisfies the D’Alembert functional equation for all .
Let be arbitrarily given vectors; we may assume that . We distinguish two cases: first and then .
(a) . Using (4) with and in place of and , respectively, we have , and then by (17) we get Moreover, since , substituting and in place of and , respectively, in (4) we obtain . Squaring both sides of this last equation and using (21) we get From (17) we have , so that substitution of in the right hand side of (22) leads to , which is the D’Alembert functional equation for this values of .
(b) Let us now suppose that and use the auxiliary vector of condition .
Since , from (4) we get and then by we obtain in the same way Addition of both sides of (23) and (24) leads to Now, writing in (19) and in place of and , respectively, since , and from the evenness of , we arrive at In the same way, by and , we obtain from (19) Adding both sides of (27) and then using (25) and (26), we get We have to deal with the following possibilities: or .
In the first case, that is, , we prove that the D’Alembert functional equation is satisfied for these values of .
In the second case we suppose that . From (17) we deduce that and by a simple calculation. Hence, we can easily show that (28) can be reformulated as follows: so that we obtain Squaring both side of (30) and we get and by elementary calculations in order to obtain, using (17), Now let us consider the following identity: by (17) we get . We substitute the right hand side of this last equation in the right hand side of (34) and have We purpose now to prove the following identity: in order to show that the D’Alembert functional equation is satisfied in this case too.
Squaring both sides of (31) we have , and by the suitable calculation above used, we get .
Now let us use (17) in order to have .
We now consider (30) and we can prove that , by the same techniques used in the proof of (37). Squaring both sides of this last equation we have and by suitable calculations using (17) and (30) we prove (37).
Hence the theorem is proved.
5. The Isosceles Orthogonal Additive Functional Equation
In order to study the problem of the characterization of the normed spaces satisfying condition , we recall the following result of Szabó  concerning a characterization of the inner product spaces among the normed spaces; its characterization involves a particular conditional functional equation.
Theorem 6 (Szabó). There exists a nontrivial, even, isosceles orthogonally additive mapping from X to Y if and only if X is an inner product space.
Until now we can prove a partial result: we describe two properties of the isosceles orthogonally additive mappings in the normed spaces .
Theorem 7. Let be an even isosceles orthogonally additive mapping, that is, an even solution of the conditional equation then satisfies everywhere the quadratic equation, that is, (1), and it is constant on each sphere centred at zero.
Proof. Let be an even isosceles orthogonally additive mapping. From , we may show that ; hence from (38), in which we substitute and in place of and , we get , using the oddness of . Adding both sides of (38) and this last equation, we have
Now, consider an arbitrary . We prove that for all
We observe that for all , there exists such that and ; it is sufficient to consider the continuous functional defined by . This functional has a connected domain and satisfies ; hence there exists a vector such that .
From (38), using the evenness of the function , we get On the other hand, since , from (38) and the evenness of again, we have Thus, substituting in place of in (41) we obtain (40).
Let us now consider with . Putting we have , , that is, . From (39) we obtain ; hence, . From (40) we deduce that (3) is true in this case. So Theorem 4 gives us that is everywhere quadratic.
Let us now prove that is constant on each sphere centred at the origin. Let be such that . By (43), substituting in place of in (38), we have ; hence . We now remember that satisfies the quadratic equation everywhere, in order to show that ; then, for (40) we obtain that for all with . Consequently, there exists a function such that for all . The theorem is proved.
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