Approximation on the Quadratic Reciprocal Functional Equation

In [1],Ulamproposed thewell-knownUlam 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 φ 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


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 Gȃvrut ¸a [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][11][12][13][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 () = 1/ 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][17][18][19][20][21][22].
In the following theorem, we obtain an approximation for approximate quadratic reciprocal mappings on nonzero real numbers.
Theorem 2. Let  : R * → R * be a mapping for which there exists a constant  (independent of  and ) such that the functional inequality holds for all ,  ∈ R * .Then the limit exists for all  ∈ R * ,  ∈ N and  : R * → R * is the unique mapping satisfying the Rassias quadratic reciprocal functional equation (2), such that for all  ∈ R * .Moreover, the functional identity holds for all  ∈ R * and  ∈ N.
Proof.Putting  =  in (4), we get for all  ∈ R * .Thus we have for all  ∈ R * .Substituting  by /3 in (9) and then dividing both sides by 3 2 , we obtain for all  ∈ R * .It follows from ( 9) and ( 10) that for all  ∈ R * .The above process can be repeated to obtain for all  ∈ R * and all  ∈ N. In order to prove the convergence of the sequence {(1/3 2 )(/3  )}, we have if  >  > 0, then by ( 12) for all  ∈ R * in which  = /3  .The above relation shows that the mentioned sequence is a Cauchy sequence and thus limit ( 5) exists for all  ∈ R * .Taking that  tends to infinity in (12), we can see that inequality (6) holds for all  ∈ R * .Replacing ,  by /3  , /3  , respectively, in (4) and dividing both sides by 3 2 , we deduce that holds for all ,  ∈ R * .Allowing  → ∞ in ( 14), we see that  satisfies (2) for all ,  ∈ R * .To prove that  is a unique quadratic reciprocal function satisfying (2) subject to (6), let us consider a Q : R * → R * to be another quadratic reciprocal function which satisfies (2) and inequality (6).Clearly  and Q satisfy (7) and using (6), we get for all  ∈ R * .This shows the uniqueness of .
Proof.We prove the result only in the case that  = 1.Another case is similar.Putting  =  in ( 17), we have for all  ∈ R * .Replacing  by /3 in the above inequality, we get for all  ∈ R * .Replacing  by /3  in (20) and then dividing both sides by 3 2 , we have for all  ∈ R * and all nonnegative integers .Thus the sequence {(1/3 2 )(/3  )} is Cauchy by (16) and so this sequence is convergent.Indeed, for all  ∈ R * .On the other hand, by using (20)   for all  ∈ R * .