International Journal of Analysis and Applications ISSN 2291-8639 Volume 4, Number 2 (2014), 130-147 http://www.etamaths.com GENERAL STABILITY OF A RECIPROCAL TYPE FUNCTIONAL EQUATION IN THREE VARIABLES K. RAVI1,∗, J.M. RASSIAS2 AND B.V. SENTHIL KUMAR3 Abstract. In this paper, we obtain the solution of a reciprocal type functional equation in three variables of the form g (2(k− 1)x1 + x2 + x3) = g((k− 1)x1 + x2)g((k− 1)x1 + x3) g((k− 1)x1 + x2) + g((k− 1)x1 + x3) and investigate its generalized Hyers-Ulam stability where k ≥ 2 is a positive integer, g : X → R is a mapping with X as the space of non-zero real numbers and 2(k − 1)x1 + x2 + x3 6= 0, g((k − 1)x1 + x2) + g((k − 1)x1 + x2) 6= 0, for all x1,x2,x3 ∈ X. We also provide counter-examples for non-stability. 1. INTRODUCTION An inquisitive question that was given a serious thought by S.M. Ulam [42] concerning the stability of group homomorphisms gave rise to the stability problem of functional equations. The laborious intellectual strivings of D.H. Hyers [15] did not go in vain because he was the first to come out with a partial answer to solve the question posed by Ulam on Banach spaces. In course of time, the theorem formulated by Hyers was generalized by T. Aoki [4] for additive mappings and by Th.M. Rassias [40] for linear mappings by taking into consideration an unbounded Cauchy difference. The findings of Th.M. Rassias have exercised a delectable influence on the development of what is addressed as the generalized Hyers-Ulam-Rassias stability of functional equations. A generalized and modified form of the theorem evolved by Th.M. Rassias was advocated by P. Gavruta [13] who replaced the unbounded Cauchy difference by driving into study a general control function within the viable approach designed by Th.M. Rassias. In 1982-1989, a generalization of the result of D.H. Hyers was proved by J.M. Rassias using weaker conditions controlled by a product of different powers of norms ([31], [32], [33]). The investigation of stability of functional equations involving with the mixed type product-sum of powers of norms is introduced by J.M. Rassias [34]. A further research materialized by F. Skof [41] found solution to Hyers-Ulam- Rassias stability problem for quadratic functional equation (1.1) f(x + y) + f(x−y) = 2f(x) + 2f(y) 2010 Mathematics Subject Classification. 39B22, 39B52, 39B72. Key words and phrases. Reciprocal function; Reciprocal type functional equation; Generalized Hyers-Ulam stability. c©2014 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 130 STABILITY OF A RECIPROCAL FUNCTIONAL EQUATION IN THREE VARIABLES 131 for a class of functions f : A → B, where A is a normed space and B is a Banach space. The functional equation (1.1) is used to characterize inner product spaces ([1], [2], [24]). The stability problems of several functional equations have been ex- tensively investigated by a number of mathematicians, posed with creative thinking and critical dissent who have arrived at interesting results (see [5], [8], [9], [11], [14], [16], [25], [35]). Functional equations find a lot of application in information theory, infor- mation science, measure of information, coding theory, computer graphics, spatial filtering in image processing, behavioral and social sciences, astronomy, number theory, fuzzy system models, economics, stochastic processes, mechanics, cryptog- raphy and physics. In 1998, S.M. Jung [18] investigated the Hyers-Ulam-Rassias stability for the Jensen functional equation (1.2) 2f ( x + y 2 ) = f(x) + f(y) and applied the stability result to the study of an asymptotic behaviour of the additive mappings. In 2008, W.G. Park and J.H. Bae [29] obtained the general solution and the stability of the functional equation f(x + y + z,u + v + w) + f(x + y −z,u + v + w) + 2f(x,u,−w) + 2f(y,v,−w) = f(x + y,u + w) + f(x + y,v + w) + f(x + z,u + w) + f(x−z,u + v −w) + f(y + z,v + w) + f(y −z,u + v −w).(1.3) The function f : R × R → R given by f(x,y) = x3 + ax + b − y2 having level curves as elliptic curves is a solution of (1.3). The stability result of the functional equation (1.3) is related with the canonical height function of the elliptic curves. In the year 1996, Isac and Th.M. Rassias [17] were the first to provide appli- cations of stability theorem of functional equations for the proof of new fixed point theorems with applications. Usually, the stability problem for functional equations is solved by direct method in which the exact solution of the functional equation is explicitly con- structed as a limit of a (Hyers) sequence, starting from the given approximate solution of f (see [3], [10], [16], [18], [19]). In the year 2003, Radu [30] proposed a new method for obtaining the existence of exact solutions and error estimations, based on the fixed point alternative. This method was recently been used by many authors (see [20], [21], [22], [23], [27], [28]). Cadariu and Radu ([6], [7]) applied a fixed point method to investigate the Jensen’s and Cauchy additive functional equations. In the year 2010, K. Ravi and B.V. Senthil Kumar [36] investigated the gen- eralized Hyers-Ulam stability for the reciprocal functional equation (1.4) g(x + y) = g(x)g(y) g(x) + g(y) where g : X → Y is a mapping on the spaces of non-zero real numbers. The reciprocal function g(x) = c x is a solution of the functional equation (1.4). The functional equation (1.4) holds good for the “Reciprocal formula” of any electric 132 RAVI, RASSIAS AND KUMAR circuit with two resistors connected in parallel. S.M. Jung [22] applied fixed point method for proving Hyers-Ulam stability for the reciprocal functional equation (1.4). K. Ravi, J.M. Rassias and B.V. Senthil Kumar [37] disucssed the generalized Hyers-Ulam stability for the reciprocal functional equation in several variables of the form (1.5) g ( m∑ i=1 αixi ) = ∏m i=1 g(xi)∑m i=1 [ αi (∏m j=1,j 6=i g(xj) )] for arbitrary but fixed real numbers (α1,α2, . . . ,αm) 6= (0, 0, . . . , 0), so that 0 < α = α1 + α2 + · · · + αm = ∑m i=1 αi 6= 1 and g : X → Y with X and Y are the spaces of non-zero real numbers. Later, J.M. Rassias and et.al., [38] introduced the Reciprocal Difference Func- tional equation (1.6) r ( x + y 2 ) −r(x + y) = r(x)r(y) r(x) + r(y) and the Reciprocal Adjoint Functional equation (1.7) r ( x + y 2 ) + r(x + y) = 3r(x)r(y) r(x) + r(y) and investigated the generalized Hyers-Ulam stability of the equations (1.6) and (1.7). Soon after, J.M. Rassias and et.al., [39] applied fixed point method to investi- gate the generalized Hyers-Ulam stability of the equations (1.6) and (1.7). G.L. Forti [12] obtained the stability of the functional equation Ψ ◦f ◦a = f, by proving the following theorem. Theorem 1.1. Assume that (Y,d) is a complete metric space, K is a nonempty set, f : K → Y , Ψ : Y → Y , a : K → K, h : K → [0,∞), λ ∈ [0,∞), d(Ψ ◦ f ◦ a(x),f(x)) ≤ h(x) for x ∈ K, d(Ψ(x), Ψ(y)) ≤ λd(x,y) for x,y ∈ Y, and H(x) = ∞∑ i=0 λih ( ai(x) ) < ∞ for x ∈ K. Then, for every x ∈ K, the limit r(x) = lim n→∞ Ψn ◦f ◦an(x) exists and r : K → Y is a unique function such that Ψ ◦r ◦a = r and d(f(x),r(x)) ≤ H(x), for x ∈ K. In this paper, we obtain the general solution and investigate the generalized Hyers-Ulam stability of the reciprocal type functional equation in three variables of the form (1.8) g (2(k − 1)x1 + x2 + x3) = g((k − 1)x1 + x2)g((k − 1)x1 + x3) g((k − 1)x1 + x2) + g((k − 1)x1 + x3) where k ≥ 2 is a positive integer, g : X → R is a mapping with X as the space of non-zero real numbers and 2(k − 1)x1 + x2 + x3 6= 0, g((k − 1)x1 + x2) + g((k − 1)x1 + x2) 6= 0, for all x1,x2,x3 ∈ X using direct method, fixed point method and Theorem 1.1. We also provide counter-examples for non-stability. Throughout this paper, let X be the space of non-zero real numbers. We also assume that 2(k − 1)x1 + x2 + x3 6= 0, g(x) 6= 0, g((k − 1)x1 + x2) 6= 0, STABILITY OF A RECIPROCAL FUNCTIONAL EQUATION IN THREE VARIABLES 133 g((k − 1)x1 + x3) 6= 0 and g((k − 1)x1 + x2) + g((k − 1)x1 + x3) 6= 0 for all x,x1,x2,x3 ∈ X to prove our main results. For the sake of convenience, let us define D3g(x1,x2,x3) = g (2(k − 1)x1 + x2 + x3) − g((k − 1)x1 + x2)g((k − 1)x1 + x3) g((k − 1)x1 + x2) + g((k − 1)x1 + x3) (1.9) for all x1,x2,x3 ∈ X. The paper is organised as follows. In Section 2, we present preliminaries required for proving our main results. In Section 3, we obtain the general solution of the functional equation (1.8). In Section 4, we investigate the generalized Hyers- Ulam stability of the equation (1.8) using direct method. In Section 5, we apply fixed point method to gain the generalized Hyers-Ulam stability of the equation (1.8) and in Section 6, we find the generalized Hyers-Ulam stability of the equation (1.8) in the sense of G.L. Forti. In Section 7, we illustrate counter-examples for singular cases. 2. PRELIMINARIES In this section, we present the definition of generalized metric and fundamental result of fixed point theory. Let A be a set. A function d : A×A → [0,∞] is called a generalized metric on A if d satisfies the following conditions: 1. d(x,y) = 0 if and only if x = y; 2. d(x,y) = d(y,x) for all x,y ∈ A; 3. d(x,z) ≤ d(x,y) + d(y,z) for al x,y,z ∈ A. We note that the only one difference of the generalized metric from the usual metric is that the range of the former is permitted to include infinity. The following theorem is very useful for proving our results in Section 5, which is due to Margolis and Diaz [26]. Theorem 2.1. Let (A,d) be a complete generalized metric space and let J : A → A be a strictly contractive mapping with Lipschitz constant L < 1. Then for each given element x ∈ A, either (2.1) d ( Jnx,Jn+1x ) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that 1. d ( Jnx,Jn+1x ) < ∞ for all n ≥ n0; 2. the sequence {Jnx} converges to a fixed point y∗ of J; 3. y∗ is the unique fixed point of J in the set Y = {y ∈ A|d (Jn0x,y) < ∞}; 4. d(y,y∗) ≤ 1 1−Ld(y,Jy) for all y ∈ Y . 134 RAVI, RASSIAS AND KUMAR 3. GENERAL SOLUTION OF FUNCTIONAL EQUATION (1.8) Theorem 3.1. A mapping g : X → R satisfies the functional equation (1.8) for all x1,x2,x3 ∈ X if and only if there exists a reciprocal mapping g : X → R satisfying the reciprocal functional equation (1.4) for all x,y ∈ X. Hence, every solution of functional equation (1.8) is also a reciprocal function. Proof. Let the mapping g : X → R satisfy the functional equation (1.8). Replacing (x1,x2,x3) by ( x+y k−1 ,−y,−x ) in (1.8), we obtain the equation (1.4). Conversely, let the mapping g : X → R satisfy the functional equation (1.4). Replacing (x,y) by ((k−1)x1 + x2, (k−1)x1 + x3) in (1.4), we obtain the equation (1.8), which completes the proof of Theorem 3.1. � 4. GENERALIZED HYERS-ULAM STABILITY OF EQUATION (1.8) USING DIRECT METHOD Theorem 4.1. Let g : X → R be a mapping satisfying (1.9) and (4.1) |D3g(x1,x2,x3)| ≤ φ(x1,x2,x3) for all x1,x2,x3 ∈ X, where φ : X ×X ×X → R be a given function such that (4.2) Φ(x) = ∞∑ i=0 2i+1φ ( 2ix k , 2ix k , 2ix k ) with the condition (4.3) lim n→∞ 2nφ ( 2nx k , 2nx k , 2nx k ) = 0 holds for every x ∈ X. Then there exists a unique reciprocal mapping r : X → R which satisfies (1.8) and the inequality (4.4) |g(x) −r(x)| ≤ Φ(x) for all x ∈ X. Proof. Replacing (x1,x2,x3) by ( x k , x k , x k ) in (4.1) and multiplying by 2, we get (4.5) |2g(2x) −g(x)| ≤ 2φ (x k , x k , x k ) for all x ∈ X. Again replacing x by 2x in (4.5), multiplying by 2 and summing the resulting inequality with (4.5), we get ∣∣22g(22x) −g(x)∣∣ ≤ ∑1i=0 2i+1φ(2ixk , 2ixk , 2ixk ) for all x ∈ X. Proceeding further and using induction on a positive integer n, we STABILITY OF A RECIPROCAL FUNCTIONAL EQUATION IN THREE VARIABLES 135 obtain |2ng(2nx) −g(x)| ≤ n−1∑ i=0 2i+1φ ( 2ix k , 2ix k , 2ix k ) ≤ ∞∑ i=0 2i+1φ ( 2ix k , 2ix k , 2ix k ) (4.6) for all x ∈ X. In order to prove the convergence of the sequence {2ng (2nx)}, replace x by 2px in (4.6) and multiply by 2p, we find that for n > p > 0∣∣2pg (2px) − 2n+pg(2n+px)∣∣ = 2p ∣∣g (2px) − 2ng(2n+px)∣∣ ≤ ∞∑ i=0 2p+i+1φ ( 2p+ix k , 2p+ix k , 2p+ix k ) → 0 as p →∞. Thus the sequence {2ng (2nx)} is a Cauchy sequence. Allowing n →∞ in (4.6), we arrive (4.4) with r(x) = lim n→∞ 2ng (2nx) . To show that r satisfies (1.8), replacing (x,y) by (2nx, 2ny) in (4.1) and multiplying by 2n, we obtain (4.7) 2n |D3g (2nx1, 2nx2, 2nx3)| ≤ 2nφ (2nx1, 2nx2, 2nx3) . Allowing n → ∞ in (4.7), we see that r satisfies (1.8) for all x1,x2,x3 ∈ X. To prove r is a unique reciprocal mapping satisfying (1.8), let r1 : X → R be another reciprocal mapping which satisfies (1.8) and the inequality (4.4). Clearly r1 (2 nx) = 2−nr1(x), r (2 nx) = 2−nr(x) and using (4.4), we arrive |r1(x) −r(x)| ≤ 2n |r1 (2nx) −r (2nx)| ≤ 2n (|r1 (2nx) −g (2nx)| + |g (2nx) −r (2nx)|) ≤ 2 ∞∑ i=0 2n+i+1φ ( 2n+ix k , 2n+ix k , 2n+ix k ) → 0 as n →∞ which implies that r is unique. This completes the proof of Theorem 4.1. � Theorem 4.2. Let g : X → R be a mapping satisfying (1.9) and (4.1), for all x1,x2,x3 ∈ X, where φ : X ×X ×X → R be a given function such that (4.8) Φ(x) = ∞∑ i=0 1 2i φ ( x 2i+1k , x 2i+1k , x 2i+1k ) with the condition (4.9) lim n→∞ 1 2n φ ( x 2n+1k , x 2n+1k , x 2n+1k ) = 0 136 RAVI, RASSIAS AND KUMAR holds for every x ∈ X. Then there exists a unique reciprocal mapping r : X → R which satisfies (1.8) and the inequality (4.10) |g(x) −r(x)| ≤ Φ(x) for all x ∈ X. Proof. The proof is obtained by replacing (x1,x2,x3) by ( x 2k , x 2k , x 2k ) in (4.1) and proceeding by similar arguments as in Theorem 4.1. � Corollary 4.3. Let g : X → R be a mapping and let there exist real numbers q 6= −1 and θ1 ≥ 0 such that (4.11) |D3g(x1,x2,x3)| ≤ θ1 ( 3∑ i=1 |xi|q ) for all x1,x2,x3 ∈ X. Then there exists a unique reciprocal mapping r : X → R satisfying (1.8) and (4.12) |g(x) −r(x)| ≤   6θ1 kq(1−2q+1)|x| q, for q < −1 6θ1 kq(2q+1−1)|x| q, for q > −1 for every x ∈ X. Proof. The proof follows immediately by taking φ(x1,x2,x3) = θ1 (∑3 i=1 |xi| q ) , for all x1,x2,x3 ∈ X in Theorems 4.1 and 4.2 respectively. � Corollary 4.4. Let g : X → R be a mapping and let there exist a real number q 6= −1. Let there exist θ2 ≥ 0 such that (4.13) |D3g(x1,x2,x3)| ≤ θ2 ( 3∏ i=1 |xi| q 3 ) for all x1,x2,x3 ∈ X. Then there exists a unique reciprocal mapping r : X → R satisfying (1.8) and (4.14) |g(x) −r(x)| ≤   2θ2 kq(1−2q+1)|x| q, for q < −1 2θ2 kq(2q+1−1)|x| q, for q > −1 for every x ∈ X. Proof. The required results in Corollary 4.4 can be easily derived by considering φ(x1,x2,x3) = θ2 (∏3 i=1 |xi| q 3 ) , for all x1,x2,x3 ∈ X in Theorems 4.1 and 4.2 respectively. � STABILITY OF A RECIPROCAL FUNCTIONAL EQUATION IN THREE VARIABLES 137 Corollary 4.5. Let θ3 ≥ 0 and q 6= −1 be real numbers, and g : X → R be a mapping satisfying the functional inequality (4.15) |D3g(x1,x2,x3)| ≤ θ3 ( 3∏ i=1 |xi| q 3 + ( 3∑ i=1 |xi|q )) for all x1,x2,x3 ∈ X. Then there exists a unique reciprocal mapping r : X → R satisfying (1.8) and (4.16) |g(x) −r(x)| ≤   8θ3 kq(1−2q+1)|x| q, for q < −1 8θ3 kq(2q+1−1)|x| q, for q > −1 for every x ∈ X. Proof. By choosing φ(x1,x2,x3) = θ3 (∏3 i=1 |xi| q 3 + (∑3 i=1 |xi| q )) , for all x1,x2,x3 ∈ X in Theorems 4.1 and 4.2 respectively, the proof of Corollary 4.5 is complete. � 5. GENERALIZED HYERS-ULAM STABILITY OF EQUATION (1.8) USING FIXED POINT METHOD Theorem 5.1. Suppose that the mapping g : X → R satisfies the inequality (5.1) |D3g(x1,x2,x3)| ≤ φ(x1,x2,x3) for all x1,x2,x3 ∈ X, where φ : X×X×X → R is a given function. If there exists L < 1 such that the mapping x → Φ(x) = 2φ (x k , x k , x k ) has the property Φ(2x) ≤ 1 2 LΦ(x), for all x ∈ X and the mapping φ has the property (5.2) lim n→∞ 2nφ (2nx1, 2 nx2, 2 nx3) = 0 for all x1,x2,x3 ∈ X, then there exists a unique reciprocal mapping r : X → R such that (5.3) |g(x) −r(x)| ≤ 1 1 −L Φ(x) for all x ∈ X. Proof. Define a set S by S = {h : X → R|h is a function} 138 RAVI, RASSIAS AND KUMAR and introduce the generalized metric d on S as follows: (5.4) d(g,h) = dΦ(g,h) = inf{C ∈ R+ : |g(x) −h(x)| ≤ CΦ(x), for all x ∈ X}. Now, we show that (S,d) is complete. Using the idea from [21], we prove the completeness of (S,d). Let {hn} be a Cauchy sequence in (S,d). Then for any � > 0, there exists an integer N� > 0 such that d(hm,hn) ≤ �, for all m,n ≥ N�. From (5.4), we arrive (5.5) ∀� > 0,∃N� ∈ N,∀m,n ≥ N�,∀x ∈ X : |hm(x) −hn(x)| ≤ �Φ(x). If x is a fixed number, (5.5) implies that {hn(x)} is a Cauchy sequence in (R, |.|). Since (R, |.|) is complete, {hn(x)} converges for all x ∈ X. Therefore, we can define a function h : X → R by h(x) = lim n→∞ hn(x) and hence h ∈ S. Letting m →∞ in (5.5), we have ∀� > 0,∃N� ∈ N,∀n ≥ N�,∀x ∈ X : |h(x) −hn(x)| ≤ �Φ(x). By considering (5.4), we arrive ∀� > 0,∃N� ∈ N,∀n ≥ N� : d(h,hn) ≤ �, which implies that the Cauchy sequence {hn} converges to h in (S,d). Hence (S,d) is complete. Define a mapping σ : S → S by (5.6) σh(x) = 2h(2x) (x ∈ X) for all h ∈ S. We claim that σ is strictly contractive on S. For any given g,h ∈ S, let Cgh ∈ [0,∞] be an arbitrary constant with d(g,h) ≤ Cgh. Hence d(g,h) < Cgh ⇒|g(x) −h(x)| ≤ CghΦ(x), for all x ∈ X ⇒|2g(2x) − 2h(2x)| ≤ 2CghΦ(2x), for all x ∈ X ⇒|2g(2x) − 2h(2x)| ≤ LCghΦ(x), for all x ∈ X ⇒ d(σg,σh) ≤ LCgh. Therefore, we see that d(σg,σh) ≤ Ld(g,h), for all g,h ∈ S. That is, σ is strictly contractive mapping of S, with the Lipschitz constant L. Now, replacing (x1,x2,x3) by ( x k , x k , x k ) in (5.1) and multiplying by 2, we get |2g(2x) −g(x)| ≤ 2φ (x k , x k , x k ) = Φ(x) STABILITY OF A RECIPROCAL FUNCTIONAL EQUATION IN THREE VARIABLES 139 for all x ∈ X. Hence (5.4) implies that d(σf,f) ≤ 1. Hence by applying the fixed point alternative Theorem 2.1, there exists a function r : X → R satisfying the following: (1) r is a fixed point of σ, that is (5.7) r(2x) = 1 2 r(x) for all x ∈ X. The mapping r is the unique fixed point of σ in the set µ = {f ∈ S : d(f,g) < ∞}. This implies that r is the unique mapping satisfying (5.7) such that there exists C ∈ (0,∞) satisfying |r(x) −g(x)| ≤ CΦ(x), ∀x ∈ X. (2) d(σng,r) → 0 as n →∞. Thus, we have (5.8) lim n→∞ 2ng (2nx) = r(x) for all x ∈ X. (3) d(g,r) ≤ 1 1−Ld(σg,r), which implies d(g,r) ≤ 1 1 −L . Thus the inequality (5.3) holds. Hence from (5.1), (5.2) and (5.8), we have |D3g(x1,x2,x3)| = lim n→∞ 2n|D3g(2nx1, 2nx2, 2nx3)| ≤ lim n→∞ 2nφ(2nx1, 2 nx2, 2 nx3) = 0 for all x1,x2,x3 ∈ X. Hence r is a solution of the functional equation (1.8). By Theorem 3.1, r : X → R is a reciprocal mapping. Next, we show that r is the unique reciprocal mapping satisfying (1.8) and (5.3). Suppose, let r1 : X → R be another reciprocal mapping satisfying (1.8) and (5.3). Then from (1.8), we have that r1 is a fixed point of σ. Since d(g,r1) < ∞, we have r1 ∈ S∗ = {f ∈ S|d(f,g) < ∞}. From Theorem 2.1 (3) and since both r and r1 are fixed points of σ, we have r = r1. Therefore, r is unique. Hence, there exists a unique reciprocal mapping r : X → R satisfying (1.8) and (5.3), which completes the proof of Theorem 5.1. � 140 RAVI, RASSIAS AND KUMAR Theorem 5.2. Suppose that the mapping g : X → R satisfies the inequality (5.1) for all x1,x2,x3 ∈ X, where φ : X×X×X → R is a given function. If there exists L < 1 such that the mapping x → Φ(x) = φ ( x 2k , x 2k , x 2k ) has the property Φ (x 2 ) ≤ 2LΦ(x), for all x ∈ X and the mapping φ has the property (5.9) lim n→∞ 2−nφ ( 2−nx1, 2 −nx2, 2 −nx3 ) = 0 for all x1,x2,x3 ∈ X, then there exists a unique reciprocal mapping r : X → R such that (5.10) |g(x) −r(x)| ≤ 1 1 −L Φ(x) for all x ∈ X. Proof. The proof of Theorem 5.2 goes through the same way as in Theorem 5.1. � Corollary 5.3. Let g : X → R be a mapping and let there exist real numbers q 6= −1 and θ1 ≥ 0 such that (4.11) holds for all x1,x2,x3 ∈ X. Then there exists a unique reciprocal mapping r : X → R satisfying (1.8) and (4.12) for every x ∈ X. Proof. The proof is obtained by assuming φ(x1,x2,x3) = θ1 (∑3 i=1 |xi| q ) , for all x1,x2,x3 ∈ X and L = 2−q−1,L = 2q+1 in Theorems 5.1 and 5.2 respectively. � Corollary 5.4. Let g : X → R be a mapping and let there exist a real number q 6= −1. Let there exist θ2 ≥ 0 such that (4.13) holds for all x1,x2,x3 ∈ X. Then there exists a unique reciprocal mapping r : X → R satisfying (1.8) and (4.14) for every x ∈ X. Proof. It is easy to derive the required results in Corollary 5.4 by considering φ(x1,x2,x3) = θ2 (∏3 i=1 |xi| q 3 ) , for all x1,x2,x3 ∈ X and L = 2−q−1,L = 2q+1 in Theorems 5.1 and 5.2 respectively. � Corollary 5.5. Let θ3 ≥ 0 and q 6= −1 be real numbers, and g : X → R be a mapping satisfying the functional inequality (4.15) for all x1,x2,x3 ∈ X. Then there exists a unique reciprocal mapping r : X → R satisfying (1.8) and (4.16) for every x ∈ X. STABILITY OF A RECIPROCAL FUNCTIONAL EQUATION IN THREE VARIABLES 141 Proof. The proof is complete by choosing φ(x1,x2,x3) = θ3 (∏3 i=1 |xi| q e + (∑3 i=1 |xi| q )) , for all x1,x2,x3 ∈ X and L = 2−q−1,L = 2q+1 in Theorems 5.1 and 5.2 respective- ly. � 6. GENERALIZED HYERS-ULAM STABILITY OF EQUATION (1.8) IN THE SENSE OF G.L. FORTI Theorem 6.1. Suppose that the mapping g : X → R satisfies the inequality (6.1) |D3g(x1,x2,x3)| ≤ φ(x1,x2,x3) for all x1,x2,x3 ∈ X, where φ : X×X×X → R is a given function. Suppose there exists β ∈ (0,∞) such that 2β < 1, (6.2) φ ( 2x1 k , 2x2 k , 2x3 k ) ≤ βφ (x1 k , x2 k , x3 k ) for all x1,x2,x3 ∈ X and k(> 1) ∈ Z+. Then there exists a unique reciprocal mapping r : X → R such that (6.3) |r(x) −g(x)| ≤ 2 1 − 2β φ (x k , x k , x k ) for all x ∈ X. Proof. Replacing (x1,x2,x3) by ( x k , x k , x k ) in (6.1), we get |g(2x) − 1 2 g(x)| ≤ φ (x k , x k , x k ) , for all x ∈ X. Hence, we obtain |2g(2x) −g(x)| = 2 ∣∣∣∣g(2x) − 12g(x) ∣∣∣∣ ≤ 2φ (x k , x k , x k ) , for x ∈ X. Considering f = 1 2 g, Ψ(z) = 2z, λ = 2, h(x) = 2φ ( x k , x k , x k ) , a(x) = 2x and d(x,y) = |x−y|, for all x,y ∈ X in Theorem 1.1, we see that the limit r(x) exists and |r(x) −g(x)| ≤ H(x), for all x ∈ X. Using (6.1), we obtain (6.4) 2n|D3g (2nx1, 2nx2, 2nx3) | ≤ (2β)nφ(x1,x2,x3) for all x1,x2,x3 ∈ X and n ∈ N. Allowing n → ∞ in (6.4), we see that r satisfies (1.8). Next we show that r is the unique reciprocal mapping satisfying (1.8) and (6.3). Suppose, let r1 : X → R be another reciprocal mapping satisfying (1.8) and (6.3), and |r1(x) −g(x)| ≤ H(x), for all x ∈ X. Then Ψ ◦r1 ◦a = r1 and hence by Theorem 1.1, r = r1, which proves that r is unique. � 142 RAVI, RASSIAS AND KUMAR Theorem 6.2. Suppose that the mapping g : X → R satisfies (6.1), for all x1,x2,x3 ∈ X, where φ is a function defined as in Theorem 6.1. Suppose there exists β ∈ (0,∞) such that β 2 < 1, (6.5) φ (x1 2k , x2 2k , x3 2k ) ≤ βφ (x1 k , x2 k , x3 k ) for all x1,x2,x3 ∈ X and k(> 1) ∈ Z+. Then there exists a unique reciprocal mapping r : X → R such that (6.6) |r(x) −g(x)| ≤ 2β 2 −β φ (x k , x k , x k ) for all x ∈ X. Proof. Replacing (x1,x2,x3) by ( x 2k , x 2k , x 2k ) in (6.1), we get∣∣∣∣g(x) − 12g (x 2 )∣∣∣∣ ≤ φ( x2k, x2k, x2k ) , for all x ∈ X. The rest of the proof is obtained by taking f = g, Ψ(z) = 1 2 z, λ = 1 2 , h(x) = φ( x 2k , x 2k , x 2k ), a(x) = 1 2 x and d(x,y) = |x−y|, for all x,y ∈ X in Theorem 1.1 and using similar arguments as in Theorem 6.1. � Corollary 6.3. Let g : X → R be a mapping and let there exist real numbers q 6= −1 and θ1 ≥ 0 such that (4.11) holds for all x1,x2,x3 ∈ X. Then there exists a unique reciprocal mapping r : X → R satisfying (1.8) and (4.12) for every x ∈ X. Proof. The proof is obtained by assuming φ(x1,x2,x3) = θ1 (∑3 i=1 |xi| q ) , for all x1,x2,x3 ∈ X and β = 2q,β = 12q in Theorems 6.1 and 6.2 respectively. � Corollary 6.4. Let g : X → R be a mapping and let there exist a real number q 6= −1. Let there exist θ2 ≥ 0 such that (4.13) holds for all x1,x2,x3 ∈ X. Then there exists a unique reciprocal mapping r : X → R satisfying (1.8) and (4.14) for every x ∈ X. Proof. It is easy to derive the required results in Corollary 6.4 by considering φ(x1,x2,x3) = θ2 (∏3 i=1 |xi| q 3 ) , for all x1,x2,x3 ∈ X and β = 2q,β = 12q in Theorems 6.1 and 6.2 respectively. � Corollary 6.5. Let θ3 ≥ 0 and q 6= −1 be real numbers, and g : X → R be a mapping satisfying the functional inequality (4.15) for all x1,x2,x3 ∈ X. Then there exists a unique reciprocal mapping r : X → R satisfying (1.8) and (4.16) for every x ∈ X. STABILITY OF A RECIPROCAL FUNCTIONAL EQUATION IN THREE VARIABLES 143 Proof. The proof is complete by choosing φ(x1,x2,x3) = θ3 (∏3 i=1 |xi| q e + (∑3 i=1 |xi| q )) , for all x1,x2,x3 ∈ X and β = 2q,β = 12q in Theorems 6.1 and 6.2 respectively. � Remark 6.6. From Section 4, Section 5 and Section 6, we observe that the results obtained in Corollaries 4.3, 5.3 and 6.3 are the same. Similarly the upper bounds in the Corollaries 4.4, 5.4 and 6.4 are identical. Also, we find that the stability results in Corollaries 4.5, 5.5 and 6.5 are similar. Therefore, from the above results, we conclude that the method of G.L. Forti is the easiest method in comparison with the direct method and fixed point method. 7. COUNTER-EXAMPLES The following example illustrates the fact that the functional equation (1.8) is not stable for q = −1 in Corollary 4.3. Example 7.1. Let ϕ : X → R be a mapping defined by ϕ(x) = { c1 x for x ∈ (1,∞) c1 otherwise where c1 > 0 is a constant, and define a mapping g : X → X by g(x) = ∞∑ n=0 ϕ(2−nx) 2n , for all x ∈ X. Then the mapping g satisfies the inequality (7.1) |D3g(x1,x2,x3)| ≤ 3c1 ( 3∑ i=1 |xi|−1 ) for all xi ∈ X,i = 1, 2, 3. Therefore there do not exist a reciprocal mapping r : X → R and a constant δ > 0 such that (7.2) |g(x) −r(x)| ≤ δ|x|−1 for all x ∈ X. Proof. |g(x)| ≤ ∑∞ n=0 |ϕ(2−nx)| |2n| ≤ ∑∞ n=0 c1 2n = 2c1. Hence g is bounded by 2c1. If(∑3 i=1 |xi| −1 ) ≥ 1, then the left hand side of (4.1) is less than 3c1. Now, suppose that 0 < (∑3 i=1 |xi| −1 ) < 1. Then there exists a positive integer m such that (7.3) 1 2m+1 < ( 3∑ i=1 |xi|−1 ) < 1 2m . Hence (∑3 i=1 |xi| −1 ) < 1 2m implies 2m|xi|−1 < 1, for i = 1, 2, 3 or xi 2m > 1, for i = 1, 2, 3 or xi 2m−1 > 2 > 1, for i = 1, 2, 3 144 RAVI, RASSIAS AND KUMAR and consequently 1 2m−1 (2(k − 1)x1 + x2 + x3), 1 2m−1 ((k − 1)x1 + x2), 1 2m−1 ((k − 1)x1 + x3) > 1. Therefore, for each value of n = 0, 1, 2, . . . ,m− 1, we obtain 1 2n (2(k − 1)x1 + x2 + x3), 1 2n ((k − 1)x1 + x2), 1 2n ((k − 1)x1 + x3) > 1 and D3ϕ ( 1 2n x1, 1 2n x2, 1 2n x3 ) = 0, for n = 0, 1, 2, . . . ,m − 1. Using (7.3) and the definition of g, we obtain |D3g(x1,x2,x3)| (|x1|−1 + |x2|−1 + |x3|−1) ≤ ∞∑ n=m ∣∣∣∣ϕ (2−n(2(k − 1)x1 + x2 + x3)) − ϕ(2−n((k−1)x1+x2))ϕ(2−n((k−1)x1+x3))ϕ(2−n((k−1)x1+x2))+ϕ(2−n((k−1)x1+x3)) ∣∣∣∣ 2n (|x1|−1 + |x2|−1 + |x3|−1) ≤ ∞∑ k=0 3 2 c1 2k2m (|x1|−1 + |x2|−1 + |x3|−1) ≤ ∞∑ k=0 3 2 c1 2k = 3 2 c1 ( 1 − 1 2 )−1 = 3c1, for all x,y ∈ X. That is, the inequality (7.1) holds true. Now, assume that there exists a reciprocal mapping r : X → R satisfying (7.2). Therefore, we have (7.4) |g(x)| ≤ (δ + 1)|x|−1. However, we can choose a positive integer p with pc1 > δ + 1. If x ∈ (1, 2p−1), then 2−nx ∈ (1,∞) for all n = 0, 1, 2, . . . ,p− 1 and therefore |g(x)| = ∞∑ n=0 ϕ(2−nx) 2n ≥ m−1∑ n=0 c1 2−nx 2n = pc1 x > (δ + 1)x−1 which contradicts (7.4). Therefore, the reciprocal type of functional equation (1.8) is not stable for q = −1 in Corollary 4.3. � The following example illustrates the fact that the functional equation (1.8) is not stable for q = −1 in Corollary 4.5. 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Ulam, Problems in Modern Mathematics, Chapter VI, Wiley-Interscience, New York, 1964. 1PG & Research Department of Mathematics, Sacred Heart College, Tirupattur - 635 601, TamilNadu, India 2Pedagogical Department E.E., Section of Mathematics and Informatics, National and Capodistrian University of Athens, 4, Agamemnonos Str., Aghia Paraskevi, Athens, Attikis 15342, GREECE 3Department of Mathematics, C. Abdul Hakeem College of Engg. and Tech., Melvisharam - 632 509, TamilNadu, India ∗Corresponding author