Approximation on the Quadratic Reciprocal Functional Equation

Abstract

The quadratic reciprocal functional equation is introduced. The Ulam stability problem for an -quadratic reciprocal mapping between nonzero real numbers is solved. The Găvruţa stability for the quadratic reciprocal functional equations is established as well.

1. Introduction

In [1], Ulam proposed the well-known Ulam stability problem and one year later, the problem for linear mappings was solved by Hyers [2]. Bourgin [3] also studied the Ulam problem for additive mappings. Gruber [4] claimed that this kind of stability problem is of particular interest in the probability theory and in the case of functional equations of different types. The result of Hyers was generalized for approximately additive mappings by Aoki [5] and for approximately linear mappings, by considering the unbounded Cauchy differences by Rassias [6]. A further generalization was obtained by Găvruţa [7] by replacing the Cauchy differences with a control function satisfying a very simple condition of convergence. Skof [8] was the first author to solve the Ulam problem for quadratic mappings on Banach algebras. Cholewa [9] demonstrated that the theorem of Skof is still true if relevant domain is replaced with an abelian group (see also [10–14]).

Ravi and Senthil Kumar [15] studied the Hyers-Ulam stability for the reciprocal functional equation where is a mapping in the space of nonzero real numbers. It is easy to check that the reciprocal function is a solution of the functional equation (1). Other results regarding the stability of various forms of the reciprocal functional equation can be found in [16–22].

In this paper, we study the Ulam-Găvruţa-Rassias stability for a new 2-dimensional quadratic reciprocal functional mapping satisfying the Rassias quadratic reciprocal functional equation It is easily verified that the quadratic reciprocal function is a solution of the functional equation (2). As some corollaries, we investigate the pertinent stability of the Rassias equation (2) controlled by the “sum, product, and the mixed product-sum of powers of norms.”

2. -Stability of (2)

Throughout this paper, we denote the space of nonzero real numbers by .

Definition 1. A mapping is called Rassias quadratic reciprocal, if the Rassias quadratic reciprocal functional equation (2) holds for all .
Discussion on the above Definition and (2). We firstly note that, in the above definition, the equalities and can not occur because and do not belong to . On the other hand, if , we consider (2) which is equivalent to Since , we have . If , then is not defined. This is a contradiction. So, . Hence, it follows that . However, in the case , there is no problem in the definition of (2).
In the following theorem, we obtain an approximation for approximate quadratic reciprocal mappings on nonzero real numbers.

Theorem 2. Let be a mapping for which there exists a constant (independent of and ) such that the functional inequality holds for all . Then the limit exists for all , and is the unique mapping satisfying the Rassias quadratic reciprocal functional equation (2), such that for all . Moreover, the functional identity holds for all and .

Proof. Putting in (4), we get for all . Thus we have for all . Substituting by in (9) and then dividing both sides by , we obtain for all . It follows from (9) and (10) that for all . The above process can be repeated to obtain for all and all . In order to prove the convergence of the sequence , we have if , then by (12) for all in which . The above relation shows that the mentioned sequence is a Cauchy sequence and thus limit (5) exists for all . Taking that tends to infinity in (12), we can see that inequality (6) holds for all . Replacing , by , , respectively, in (4) and dividing both sides by , we deduce that holds for all . Allowing in (14), we see that satisfies (2) for all . To prove that is a unique quadratic reciprocal function satisfying (2) subject to (6), let us consider a to be another quadratic reciprocal function which satisfies (2) and inequality (6). Clearly and satisfy (7) and using (6), we get for all . This shows the uniqueness of .

3. Găvruţa Stability of (2)

Theorem 3. Let be fixed. Suppose that is a function such that for all . Assume in addition that is a function which satisfies the functional inequality holds for all . Then there exists a unique quadratic reciprocal function which satisfies the Rassias equation (2) and the inequality for all .

Proof. We prove the result only in the case that . Another case is similar. Putting in (17), we have for all . Replacing by in the above inequality, we get for all . Replacing by in (20) and then dividing both sides by , we have for all and all nonnegative integers . Thus the sequence is Cauchy by (16) and so this sequence is convergent. Indeed, for all . On the other hand, by using (20) and applying mathematical induction to a positive integer , we obtain for all and all . Letting in (23) and using (22), one sees that inequality (18) holds for all . Replacing , by , in (17) and dividing both sides by , we deduce that holds for all . Taking in (24), we see that satisfies (2) for all . Now, let be another quadratic reciprocal function which satisfies (2) and inequality (18). Obviously and satisfy (7). Using (18), we get for all . Taking in the preceding inequality, we immediately find the uniqueness of . For , we obtain from which one can prove the result by a similar technique. This completes the proof.

Corollary 4. Let , be nonnegative real numbers with . Suppose that is a function which satisfies the functional inequality for all . Then there exists a unique quadratic reciprocal function that satisfies the Rassias equation (2) and the inequality for all .

Proof. Letting in Theorem 3, we can get the result.

Corollary 5. Let , , be nonnegative real numbers such that . Suppose that is a function which satisfies the functional inequality for all . Then there exists a unique quadratic reciprocal function that satisfies the Rassias equation (2) and the inequality for all .

Proof. Defining and applying Theorem 3, one can obtain the desired result.

The proof of the following corollary is similar to the above results, so it is omitted.

Corollary 6. Let , be nonnegative real numbers with . Suppose that is a function which satisfies the functional inequality for all . Then there exists a unique quadratic reciprocal function that satisfies the Rassias equation (2) and the inequality for all .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors sincerely thank the anonymous reviewers for their careful reading, constructive comments, and fruitful suggestions to improve the quality of the first draft.