Abstract
We introduce the notion of multi--normed space () and study the stability of the alternative additive functional equation of two forms in this type of space.
1. Introduction
In 1940, Ulam [1] proposed the following stability problem: given a metric group , a number , and mapping which satisfies the inequality for all in , does there exist an automorphism of and a constant , depending only on , such that for all in ? If the answer is affirmative, we call the 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 [4–12]. In particular, we also refer the readers to the survey paper [13] for recent developments in Ulam’s type stability, [14] for recent developments of the conditional stability of the homomorphism equation, and books [15–18] for the general understanding of the stability theory.
The notion of multinormed space was introduced by Dales and Polyakov [19]. This concept is somewhat similar to operator sequence space and has some connections with operator spaces. Because of its applications in and outside of mathematics, the study on the stability of various functional equations has become one of the most important research subjects in the field of functional equations and attracts much attention from many researchers worldwide. Many examples of multinormed spaces can be found in [19], and further development of the stability in multinormed spaces can be found in papers [20–24].
In order to study the stability problem in more general setting, in this paper we introduce the notion of multi--normed spaces which are the combination of multinormed spaces and -normed spaces, and the definition is given as follows.
In this paper we will use the following notations. Let be a complex -normed space with , and let . We denote by the linear space consisting of -tuples , where . The linear operations on are defined coordinatewise. The zero element of either or is denoted by . We denote the set and denote by the group of permutation on .
Definition 1. A multi--norm on is a sequence such that is -norm on for each , for each , and the following axioms are satisfied for each with : (), ;(), ;(), ;(), .In this case, we say that is a multi--normed space.
The following two properties of multi--normed spaces are easily obtained:
It follows from (2) that if is a complete -normed space, then is complete -normed space for each ; in this case is a complete multi--normed space. In particular, if is Banach space, then the space is multi-Banach space. Now we give one example of multi--normed space.
Example 2. Let be an arbitrary -normed space. The sequence on defined byis a multi--norm.
Lemma 3. Let and . For each , let be a sequence in such that . Then for each one has
Definition 4. Let be a multi--normed space. A sequence in is a multinull sequence if, for each , there exists such that for all . Let ; we say that the sequence is multiconvergent to in if is a multinull sequence. In this case, is called the limit of the sequence and we denote it by
In this paper we will study the stability in the multi--normed space of alternative additive equation of the two forms, which were further studied in the normed spaces in paper [25], and their definitions are presented as follows.
Definition 5 (see [25]). Let , be linear spaces and let be mapping from to . The equation is called alternative additive of the first form if satisfies the functional equation Obviously (6) is equivalent to the alternative Jensen equation
Definition 6 (see [25]). Let , be linear spaces and let be mapping from to . The equation is called alternative additive of the second form if satisfies the functional equation Obviously (6) is equivalent to the alternative Jensen equation
2. Stability of Alternative Additive Equation of the First Form
In this section we will study the stability of the alternative additive equation of the first form in multi--normed space and on the restricted domain. First, we investigate the general case where the domain of the mapping is the whole space. The following theorem is obtained.
Theorem 7. Let be a real normed space, and let be a complete real multi--normed space. Suppose that ; mapping satisfies for all Then there exists unique alternative additive mapping of the first form satisfying for all .
Proof. Letting in (10) yields Setting yields It follows from (11), (13), and (14) that Therefore, we have for all , , .
It follows from () and (17) that Hence is Cauchy sequence, which must be convergent in complete real multi--normed space; that is, there exists mapping such that Hence, for arbitrary , there exists ; if , then we have Considering (2), we obtain If we let in (17), then we have Letting and making use of Lemma 3 and (20), we know that mapping satisfies (12).
Let . Setting , in (10) and dividing both sides by yield which together with (1) implies Taking limit as , we haveSo is the alternative additive mapping of the first form. It remains to show that is uniquely determined. Let be another alternative additive mapping of the first form that satisfies (12). It follows from (24) that some properties of mapping are obtained:(1)If we let , we get , so is odd mapping.(2)If we let , we have .(3)Putting yields ; that is, .(4)Replacing with , respectively, yields ; hence .(5)Replacing with , respectively, yields ; that is, .Proceeding in an obvious fashion yields . Similarly, we have . Letting in (12) and in view of (1) we obtain Similarly we have Therefore, Taking limit as , we have .
It is a time to study the stability of this type mapping on the local domain. We only prove the stability result when the target spaces are real multi-Banach spaces, that is, the special case of real multi--normed space when . For , it is an interesting open problem. The following are our results.
Theorem 8. Let be a real normed space, let be a real multi-Banach space, and let , . Suppose that mapping satisfies for all that satisfy and . Then there exists unique alternative additive mapping of the first form such that for all .
Proof. Fix . Let and satisfy . If , then let and . If or , letIf , we get . If , we have . Therefore, It follows from (28) that It follows from Theorem 7 that there exists unique alternative additive mapping of the first form satisfying (29) for all .
Corollary 9. Let be a real multinormed space, and let be a multi-Banach space. Mapping satisfies alternative additive equation of the first form if and only if, for each , if and , one has
3. Stability of Alternative Additive Equation of the Second Form
In this section we will study the stability of the alternative additive equation of the second form in multi--normed space and on the restricted domain. First, we investigate the general case where the domain of the mapping is the whole space. The following theorem is obtained.
Theorem 10. Let be a real normed space, and let be a complete real multi--normed space. Suppose that ; mapping satisfies for all . Then there exists unique alternative additive mapping of the second form such that for all .
Proof. Let in (34); we get Replacing with , we obtain Let and replace with ; we obtain Hence for all we have Therefore, for all , , , we have We omit the following arguments because they are similar to that of Theorem 7.
Theorem 11. Let be a real normed space, let be a complete real multi-Banach space, and let , . Suppose that satisfies for that satisfy and . Then there exists unique alternative additive mapping of the second form such that for all .
Proof. Fix ; choose and with . If , then let and . If or , then letIf , then . If , then . Therefore,It follows from (41) that According to Theorem 10, there exists unique alternative additive mapping of the second form such that (42) holds true.
Corollary 12. Let be a real multinormed space and let be a multi-Banach space. Mapping satisfies alternative additive equation of the second form if and only if, for each if and , one has
Competing Interests
The authors declare that they have no competing interests.
Authors’ Contributions
All authors conceived of the study, participated its design and coordination, drafted the paper, participated in the sequence alignment, and read and approved the final paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant 11371119), the Key Foundation of Education Department of Hebei Province (Grant ZD2016023), and Natural Science Foundation of Education Department of Hebei Province (Grant Z2014031).