On the Stability of Quadratic Functional Equations in -Spaces

Abstract

The Hyers-Ulam-Rassias stability of quadratic functional equation and orthogonal stability of the Pexiderized quadratic functional equation in -spaces are proved.

1. Introduction

In 1940, Ulam [1] proposed the following stability problem: given metric group , number and mapping which satisfies inequality for all in , do automorphism of and constant , depending only on , such that for all in , exist? If the answer is affirmative, we call equation of automorphism stable. One year later, Hyers [2] provided a positive partial answer to Ulam’s problem. In 1978, a generalized version of Hyers’ result was proved by Rassias in [3]. Since then, the stability problems of several functional equations have been extensively investigated by a number of authors [4–17]. In fact, we also refer the readers to the paper [18] for recent developments in Ulam’s type stability, [19] for recent developments of the conditional stability of the homomorphism equation and books, and [8, 20] for the general understanding of the stability theory.

Another important stability problem is orthogonal stability, which is closely related to the notion of orthogonality spaces; we know that a number of definitions of orthogonality in vector spaces, in addition to the usual one for inner product spaces, have appeared in the literature during the past half century. Many of these are mentioned in an article by Drljević [21]. Perhaps the best known of these is the “Birkhoff-James” orthogonality (see James [22]) for real normed vector spaces, where is orthogonal to meaning that for all . In giving his axiomatic definition of orthogonality, Rätz in a 1985 paper [23] modified the definition given on pp. 427-428 of Gudder and Strawther [24] and arrived at the following.

Definition 1. Suppose that is a real vector space with and is a binary relation on with the following properties:(O1)Totality of for zero: ,   for all .(O2)Independence: if ,  , then are linearly independent.(O3)Homogeneity: if ,  , then for all .(O4)The Thalesian property: Let be a 2-dimensional subspace of . If and , then there exists such that and .The pair is called an orthogonality space.

Rätz points out that this definition is more restrictive than that given by Gudder and Strawther [24], but he showed that his definition includes the following basic examples.

Example 2. The trivial orthogonality on vector space is defined by (O1), and for nonzero elements ,  , if and only if are linearly independent.

Example 3. The ordinary orthogonality on inner product space is given by if and only if .

Example 4. The Birkhoff-James orthogonality on a normed space is defined by if and only if for all .

It is well-known that two orthogonal vectors can not be commutated, and it is necessary to introduce the following definition.

Definition 5. Relation is called symmetric if implies that for all .

Clearly, Examples 2 and 3 are symmetric, but Example 4 is not. It is remarkable to note, however, that a real normed space of dimension greater than or equal to 3 is an inner product space if and only if the Birkhoff-James orthogonality is symmetric. Some notions of orthogonally additive or orthogonally quadratic function equation are given as follows.

Definition 6. Let be a vector space (an orthogonality space) and be an abelian group. Mapping is called (orthogonally) additive if it satisfies the so-called (orthogonal) additive functional equation:for all with .

Definition 7. Mapping is said to be (orthogonally) quadratic if it satisfies the so-called (orthogonally) Jordan-von Neumann quadratic function equation: for all with .

The orthogonal quadratic equation (2) was first investigated by Vajzović [25] when is a Hilbert space, is the scalar field, is continuous, and means the Hilbert space orthogonality. Later Drljević [21], Fochi [26], and Szabó [27] generalized this result. One of the significant conditional equations is the so-called orthogonally quadratic functional equation of Pexider type: Moslehian [28] considered the stability of this equation in the spirit of Hyers-Ulam under certain conditions. We will examine the stability of this equation in more general setting such that the target spaces are -spaces or complete -normed spaces. We have obtained some results which generalize work of Moslehian [28] and will be presented in Section 3. First, we recall some notions of -spaces and -normed spaces; for detailed understanding of the properties of the above spaces, the readers are required to read the book [29].

Definition 8. Let be a linear space over that denotes either complex or real numbers. A nonnegative valued function defined on is called -norm (or briefly a norm) if it satisfies the following conditions:(n1) if and only if .(n2) for all , .(n3).(n4) provided .(n5) provided .(n6) provided ,  .A linear space equipped with -norm is called -space which will be denoted by or . A complete -space is called -space.

Definition 9. Let be a linear space over that denotes either complex or real numbers and . A nonnegative valued function defined on is called -norm if it satisfies the following conditions:(n1) if and only if .(n2) for all .(n3).A linear space equipped with -norm is called -normed space, which will be denoted by or briefly .

Remark 10. It is clear that -normed space is a special -space, and when , -normed space become normed space. Comparing with the normed space, -space does not possess good metric properties, and the study of the stability of functional equations becomes more difficult.

There are many forms of the quadratic functional equation among them of great interest to us is the following:

The purpose of this paper is to study the stability or orthogonal stability of equations in -space or -normed space. Section 2 is devoted to the study of stability of (4) in more general setting such that the target spaces are -spaces or complete -normed spaces. In Section 3, we focus on the study of orthogonal stability of (3) under the condition that the target spaces are -spaces or complete -normed space; some open problems are also proposed.

Throughout this paper, let be a real -normed space or orthogonality space, and be complete -spaces or -normed space. Also , , and stand for the set of all real numbers, real numbers, or complex numbers and natural numbers, respectively.

2. On the Hyers-Ulam-Rassias Stability of (4)

From now on, let be a real vector space and let be -space in which there exists such that for all , unless we give any specific reference. We will investigate the Hyers-Ulam-Rassias stability problem for functional equation (4). Thus, we find the condition that there exists a true quadratic function near an approximately quadratic function.

Theorem 11. Let be a real vector space and be -space in which there exists such that for all , and let be a function such that converges and for all . Suppose that satisfies for all Then, there exists unique quadratic function which satisfies (4) and the inequality for all . Function is given by for all .

Proof. Putting in (7), we haveIt follows that for all . Replacing by in (11) and by the assumption on norm, we get Hence, for all . Using the induction on positive integer , we obtain that for all . In order to prove convergence of sequence , replace by to find that, for ,Since the right hand side of the inequality tends to 0 as tends to infinity, sequence is a Cauchy sequence. Therefore, we may define for all . By letting in (14), we arrive at formula (8). To show that satisfies (4), replace by , respectively; then, it follows thatTaking the limit as , we find that satisfies (4) for all . To prove the uniqueness of quadratic function subject to (8), let us assume that there exists quadratic function which satisfies (4) and inequality (8). Obviously, we have and for all and . Hence, it follows from (8) that for all . By letting in the preceding inequality, we immediately find the uniqueness of . This completes the proof of the theorem.

Corollary 12. Let be a real vector space and be a complete -normed space , and let be a function such that converges andfor all . Suppose that satisfies for all Then, there exists unique quadratic function which satisfies (4) and inequality for all . Function is given by for all .

3. On the Orthogonal Stability of (3)

Applying some ideas from [30], we deal with the conditional stability problem of the following equation:where is odd and is symmetric. Throughout this section, denotes an orthogonality space in the sense of Rätz and is a real -space or -Banach space . First, we give a technical lemma.

Lemma 13. If fulfills for all with and is symmetric, then is orthogonally additive.

Proof. Assume that for all with Putting , we get ,  . Let . Then, and so . Hence, Thus,Therefore, is orthogonally additive.

Remark 14. Rätz gave example to demonstrate that there exists odd mapping from an orthogonality space into a uniquely 2-divisible group (i.e., an abelian group in which map ,   is bijective) satisfying ,  , such that . He considered and ,  

Theorem 15. Let be an orthogonality space, and be -space in which there exists such that for all . Suppose that is symmetric on and are mappings fulfilling for some and for all with . Assume that is odd, and . Then, there exist one additive mapping and one quadratic mapping such that for all .

Proof. Put in (26). We can do this because of (O1). Then, Therefore, Similarly, by putting in (26), we get for all . Hence, for all with . Fix . By (O4), there exists such that and . Since is symmetric, too. Using inequality (31) and the oddness of , we get So that It is not hard to see that for all . In fact, It follows from that (34) holds for . Assume that (34) holds for , when . Replacing by in (34), we getHence, So, formula (34) is proved. Replacing by in inequality (34), we have which implies that is a Cauchy sequence in -space , and exists and map is well defined odd map from into satisfying For all with , by applying inequality (31) and (O3), we obtain If , then we deduce that for all with . Moreover, . Using Lemma 13, we conclude that is an orthogonally additive mapping. By Corollary of [23], is of form with additive and quadratic. If there are another quadratic mapping and another additive mapping satisfying the required inequalities in our theorem and , then for all . Using the fact that additive mappings are odd and quadratic mappings are even, we obtainHence, Letting tends to we infer that . Similarly, for all . Hence,for all . Taking the limit as , we conclude that . Using (30) and (39), we infer that, for all ,

Corollary 16. Let be an orthogonality space, and be -Banach space. Suppose that is symmetric on and are mappings fulfilling for some and for all with . Assume that is odd, and . Then, there exist one additive mapping and one quadratic mapping , such that for all .

Remark 17. (i) If for some number , then inequality (30) implies that , . Hence, ,  . So ,  .
(ii) Similarly, if for some number , then it follows from (29) that for all .

As far as the author knows, unlike orthogonally additive maps (see Corollary of [23]), there is no characterization for orthogonally quadratic maps. Every orthogonally quadratic mapping into a uniquely 2-divisible abelian group is even. In fact, , so = . Therefore, . For all , we have and hence .

Thus, . There are some characterizations of orthogonally quadratic maps in various notions of orthogonality. For example, if -orthogonality on Hilbert space is defined by , where is a bounded self-adjoint operator on , then, as shown by Fochi, every -orthogonally quadratic functional is quadratic if (see [26, 27]).

To conclude this paper, we propose the following problem.

Problem. Let be an orthogonality space, and be -space in which there exists such that for all . Suppose that is symmetric on and are mappings fulfillingfor some and for all with . Assume that is even, and . Does there exist an orthogonally quadratic mapping , under certain conditions such that for some scalars and for all ?

Competing Interests

The author declares that there are no competing interests regarding the publication of this paper.

Acknowledgments

The paper is supported by the National Natural Science Foundation of China (Grant no. 11371119), the Key Foundation of Education Department of Hebei Province (Grant no. ZD2016023), and by Natural Science Foundation of Education Department of Hebei Province (Grant no. Z2014031).