Advances on Multivalued Operators and Related Fixed Point ProblemsView this Special Issue
Solution and Stability of the Multiquadratic Functional Equation
We consider the multiquadratic functional equation. We establish its general solution and provide a characterization for this functional equation. Finally, we prove the Hyers-Ulam-Rassias stability of this functional equation.
In 1940, Ulam  gave a talk before the Mathematics Club of the University of Wisconsin, in which he discussed a number of unsolved problems. The stability of a functional equation originated from a question raised by Ulam: “when is it true that the solution of an equation differing slightly from a given one must of necessity be close to the solution of the given equation?” This question was solved by Hyers  in the case of the approximately additive functions between Banach spaces. In 1978, Rassias  provided a generalized version of Hyers’ result by allowing the Cauchy difference to be unbounded. The paper of Rassias  has provided a lot of influence in the development of the stability of functional equations, and this new concept is known as generalized Hyers-Ulam-Rassias stability or Hyers-Ulam-Rassias stability. Since then, the stability problems have been widely studied and extensively developed by many authors for a number of functional equations; see, for example, [4–10] and the books [11–14].
The functional equation is called the quadratic functional equation, and every solution of the quadratic functional equation is said to be a quadratic function. It is well known that a quadratic function between vector spaces can be expressed by a symmetric biadditive (i.e., additive for each fixed one variable) function . On the other hand, the stability problem for the quadratic functional equation has been studied by many mathematicians under various degrees of generality imposed on the equation or on the underlying space; see, for example, [15–20] and the references therein.
In , Park and Bae obtained the general solution and the generalized Hyers-Ulam-Rassias stability of the biquadratic functional equation. Let and be vector spaces. Recall from  that a mapping is called biquadratic if satisfies the system of equations for all ; that is, is quadratic for each fixed one variable.
A general version of the biquadratic functional equation is the multiquadratic functional equation. Recall from  that a mapping , where is a commutative group, is a linear space, and is an integer, is called multiquadratic if it is quadratic in each variable. On the other hand, for more details about the multiadditive (resp., the multi-Jensen mappings) (i.e., mappings satisfying Cauchy’s (resp., Jensen’s) functional equation in each variable) and the stability for them, one can see [23–28] and the references given there.
The stability of the multiquadratic functional equation was also studied by some authors. For example, Park  proved the stability of the multiquadratic functional equation in Banach spaces. Ciepliński  proved the stability of this functional equation in complete non-Archimedean spaces as well as in Banach spaces but using the fixed point method. However, to our knowledge, not many results are known about the solution of this functional equation.
In the present paper, we establish the general solution of the multiquadratic functional equation and provide a sufficient and necessary condition for a mapping to be multiquadratic. Finally, we prove its Hyers-Ulam-Rassias stability.
2. General Solution
Throughout this section, let and be vector spaces, and let be a positive integer. We begin with the following useful proposition.
Proposition 1 (see ). A function is quadratic if and only if there exists a unique symmetric biadditive function such that for any . The biadditive function is given by
In the following, we give the general solution of the multiquadratic functional equation.
Theorem 2. A mapping is multiquadratic if and only if there exists a multiadditive mapping such that for all , and satisfies the following symmetric condition for all , where and . Moreover, the mapping is given by where , , .
Proof. We prove this theorem by using induction on . Clearly, Theorem 2 is true for thanks to Proposition 1. Now, we assume that the present theorem is true for some , and we consider the case for .
We first assume that there exists a multiadditive mapping such that for all , and satisfies the following symmetric condition: for all , where and . Then, for each , we have that
for all . Thus, is multiquadratic.
Conversely, we assume that is a multiquadratic function. We need to find the desired multiadditive function . For this, we give the following notations.
For each fixed , define the mapping by
Then is a multiquadratic mapping (as is multiquadratic). By induction, we let denote the corresponding multiadditive mapping for ; that is, satisfies the symmetric condition (5) and for all . Moreover, the mapping is given by for all , where and .
On the other hand, for any fixed elements , define by
for all . It can be verified that is a quadratic mapping. Thus, it follows from Proposition 1 that there exists a symmetric biadditive mapping such that
for all . The mapping is given by
for all .
Now, we define the mapping by for all , , . In the following, we will show that is the desired function for . First, we show that is multiadditive. Indeed, by the definition of (see (16)) and noting that for any the function is multiadditive, one can obtain that for each , for all . Moreover, by the definition of in (16) and the notations we gave in (13) and (15), we have that for all . Similarly, we can see that is additive in the other variables. Thus, we have shown that is multiadditive.
Furthermore, since is multiquadratic, we obtain that for all . Thus, by the definition of in (16) and the notations we gave in (10) and (11), one has for all .
Now, we verify the expression of the mapping . By the definition of again and the notations we gave in (10) and (12), also noting that is multiquadratic, one can obtain that for all , , .
Finally, we check the symmetric property of . Fix any , where and . Since is multiquadratic, it follows that is an even mapping in each variable. Then by (21), it is easy to verify that Moreover, due to the symmetric property of and and from the definition of (see (16)) we can get for each . So the desired symmetric property of is proved. Thus, we have shown that is the desired multiadditive mapping for the multiquadratic mapping . The proof is complete.
3. A Characterization for the Multiquadratic Functional Equation
The following theorem provides a sufficient and necessary condition for a mapping to be multiquadratic.
Theorem 3. Let be a commutative semigroup with the identity element , and let be a linear space. A mapping is multiquadratic if and only if for all , .
Proof. Assume that satisfies (24). Putting
in (24) we get , and consequently, we have . Next, fix , , and put , where , for . Then, by (24),
and thus . Continuing in this fashion, we obtain that for any with at least one component which is equal to .
Now, fix , and put for in (24). Then and thus which proves that is multiquadratic.
Conversely, we assume that is multiquadratic, and we prove (24) by mathematical induction. If is quadratic, then for all . So, (24) holds for . It is easy to verify that (24) holds for . Indeed, for all . Assume that (24) holds for some positive integer . Then, Thus, (24) holds for , and this completes the proof.
In this section, we give two results on the stability of the multiquadratic functional equation. Throughout this section, let be a commutative semigroup with the identity element , and let be a Banach space.
Theorem 4. Assume that for every , is a mapping such that for any If is a function satisfying for all , , then for every there exists a multiquadratic mapping such that for any one has For every the function is given by for all .
Proof. Fix , (where denotes the set of the positive integers) and . Putting in (32), we get
Dividing both sides of the above inequality by and replacing by , we obtain
and consequently for any nonnegative integers and with , we obtain
Therefore, it follows from (31) that is a Cauchy sequence. Since the space is complete, this sequence is convergent, and we define by (34). Putting , letting in (38), and using (31), we see that (33) holds.
Finally, fix also , , and notice that according to (32) we have Next, fix , , and assume that (the same arguments apply to the case where ). Then, it follows from (32) that Letting in the above two inequalities and using (31), we see that the mapping is multiquadratic.
Theorem 5. Assume that is a mapping such that for all . If is a function satisfying for all and letting for any with one component which is equal to , then there exists a unique multiquadratic mapping such that for all . The function is given by for all .
Proof. Fix and . Putting for in (42), we get
Dividing both sides of the above inequality by and replacing by for , we see that
and consequently for any nonnegative integers and with we obtain
Therefore, it follows from (41) that is a Cauchy sequence. Since the space is complete, this sequence is convergent, and we define by (44). Putting , taking in (47), and using (41), we can see that the inequality (43) holds.
Next, fix also , and note that according to (42) we have Letting in the above inequality and using (41), we see that satisfies (24). By Theorem 3, we obtain that is multiquadratic.
Finally, assume that is another multiquadratic mapping satisfying (43). Fix . Since and are multiquadratic mappings, it is easy to verify that Then, using (41) and (43), we have hence letting we obtain .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, NY, USA, 1960.View at: MathSciNet