Zygfryd Kominek REMARKS ON THE STABILITY OF SOME QUADRATIC FUNCTIONAL EQUATIONS

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1 Opuscula Mathematica Vol. 8 No To the memory of Professor Andrzej Lasota Zygfryd Kominek REMARKS ON THE STABILITY OF SOME QUADRATIC FUNCTIONAL EQUATIONS Abstract. Stability problems concerning the functional equations of the form f(x + y) = 4f(x) + f(y) + f(x + y) f(x y), f(x + y) + f(x y) = 8f(x) + f(y) are investigated. We prove that if the norm of the difference between the LHS the RHS of one of equations () or (), calculated for a function g is say, dominated by a function ϕ in two variables having some stard properties then there exists a unique solution f of this equation the norm of the difference between g f is controlled by a function depending on ϕ. Keywords: quadratic functional equations, stability. Mathematics Subject Classification: 39B, 39B5, 39B7.. INTRODUCTION In paper [8], C. Park J. Su An considered the following functional equations f(x + y) = 4f(x) + f(y) + f(x + y) f(x y) () f(x + y) + f(x y) = 8f(x) + f(y) () in the class of functions transforming a real linear space X into another real linear space Y. They have proved that any of equations () () the quadratic functional equation Q(x + y) + Q(x y) = Q(x) + Q(y), x, y X (3) are equivalent. Stability of equation (3) was widely considered (cf., e.g., [, 3, 5, 6]). In the case of Y a Banach space, the authors of [8] also considered the problem of 57

2 58 Zygfryd Kominek Hyers-Ulam-Rassias stability (see [4]) of equations () (). In particular, they have proved that if α (0, ) is a constant g(x + y) 4g(x) g(y) g(x + y) + g(x y) θ( x α + y α ), (4) for all x, y X, then there exists a unique quadratic function Q : X Y such that g(x) Q(x) θ α 4 x α, x X. In this note we will show that equations (), () (3) are equivalent in a more general case. We will also show that in the stability results one may replace the right-h-side of (4) by a function ϕ in two variables having some natural properties (cf. [,6]). In particular, we cover the case of inequality (4) with α. However, we obtain somewhat larger estimation constant.. EQUIVALENCE OF EQUATIONS (), () AND (3) Theorem. Let X Y be commutative groups, the latter without elements of order two. Then in the class of functions transforming X into Y, equations (), () (3) are equivalent. Proof. Assume that f : X Y is a solution of equation (). Putting x = y = 0 in () we obtain 4f(0) = 0, whence f(0) = 0. Setting y = 0 in (), we get f(x) = 4f(x) for x = 0 it follows from () that f(y) = f( y). For arbitrary x, y X, there is 4f(x + y) + 4f(x y) = f(x + y) + f(x y) = = 4f(x) + f(y) + f(x + y) f(x y)+ + 4f(x) + f(y) + f(x y) f(x + y) = = 8f(x) + 8f(y), which means that f satisfies equation (3). Assume that Q is a solution of equation (3). Then Q is even Q(x + y) + Q( y) = Q(x + (x + y)) + Q(x (x + y)) = = Q(x) + Q(x + y) = = Q(x) + Q(x + y) + [Q(x) + Q(y) Q(x y)] = = 4Q(x) + Q(x + y) + Q(y) Q(x y). Therefore, Q fulfils equation (). Assume that f satisfies (). Setting x = y = 0, we get 8f(0) = 0 hence f(0) = 0. If y = 0, then f(x) = 4f(x) if x = 0, then f( y) = f(y). Therefore, 4f(x + y) + 4f(x y) = f(x + y) + f(x y) = 8f(x) + f(y) = which means that f satisfies (3). = 8f(x) + 8f(y),

3 Remarks on the stability of some quadratic functional equations 59 Now assume that Q satisfies (3). Then Q(0) = 0, Q(y) = Q( y) Q(x) = 4Q(x). Thus which ends the proof. Q(x + y) + Q(x y) = Q(x) + Q(y) = 8Q(x) + Q(y), Remark. The assumption that Y has no elements of order two is essential. To see this, consider the group 0,,, 3, 4, 5, 6, 7} with the usual addition mod 8. Then f 6 is a solution of () but it does not satisfy equation (3). Moreover, f is a solution of () but it satisfies neither equation () nor (3). 3. GENERAL LEMMA ON STABILITY In the rest of the paper we assume that: X is a commutative group, Y is a real Banach space. Moreover, we use the convention: X = X \ 0}; If not stated otherwise, any formula containing variables x /or y is valid for all x, y X. We start with some lemmas. Lemma. Let g : X Y be a function satisfying the inequality r α i g(γ i x + δ i y) ϕ(x, y), (5) i= where we are given: a positive integer r, real constants α i, integer constants γ i, δ i, i,..., r} such that γ i δ i 0 for some i,..., r} δ i γ j δ j γ i for every j i, j,..., r}, real constants λ n, n N, λ 0 =, integer constants β n, n N, β 0 =, while ϕ : X X [0, ) is a function satisfying the conditions limn λ n ϕ(β n x, β n y) = 0; λ (6) nϕ(β n x, β n x) <. If there exists a constant K > 0 such that λ n+ g(β n+ x) λ n g(β n x) Kλ n ϕ(β n x, β n x), n N 0}, (7) then for every x X the sequence (λ n g(β n x)) n N converges to a function f : X Y fulfilling the equation r α i f(γ i x + δ i y) = 0 (8) i=

4 50 Zygfryd Kominek the estimate g(x) f(x) K λ n ϕ(β n x, β n x). (9) Proof. It follows from (7) that for given positive integers n, k there is k λ n+k g(β n+k x) λ n g(β n x) λ n+j+ g(β n+j+ x) λ n+j g(β n+j x) j=0 n+k K λ j ϕ(β j x, β j x). j=n Therefore (λ n g(β n x)) n N is a Cauchy sequence, whence it is convergent. Define a function f : X Y by the equality f (x) = lim n λ ng(β n x). In (5) let us put β n x instead of x, β n y instead of y multiply both sides of (5) by λ n. On account of (6), taking the limit as n tends to infinity, we obtain where Moreover, f(x) = r α i f(γ i x + δ i y) = 0, i= f (x), x X, lim n λ n g(0), x = 0. n g(x) λ n g(β n x) λ k+ g(β k+ x) λ k g(β k x) k=0 K λ k ϕ(β k x, β k x). k=0 As n tends to infinity, we get g(x) f(x) K λ n ϕ(β n x, β n x). 4. LEMMAS ON EQUATIONS () AND () Lemma. If f : X Y satisfies () for all x, y X, then it satisfies () for all x, y X.

5 Remarks on the stability of some quadratic functional equations 5 Proof. Setting, successively, y = x, y = x, y = x, x 0 in (), we get Adding (0) (), we obtain, thanks to (), f(3x) = 5f(x) + f(x) f(0), (0) f(4x) = 4f(x) + f(x) + f(3x) f( x), () f(x) = 3f(x) + f( x) + f(0). () f(4x) = 9f(x) + f(x) f( x) f(0), 3f(x) + f( x) + f(0) = 9f(x) + f(x) f( x) f(0). Applying () once more, we observe that 3f(x) + f( x) + f(0) + 3f( x) + f(x) + f(0) + f(0) = 9f(x) f( x) f(0), which implies that 4f(0) = 5[f(x) f( x)]. Consequently, f(0) = 0, f( x) = f(x) f(x) = f(x). Now it is easy to verify that () is fulfilled for all x, y X. Lemma 3. If f : X Y satisfies () for all x, y X, then it satisfies () for all x, y X. Proof. Setting y = x, x 0 in (), we get f(3x) = 9f(x), (3) the substitution y = x,x 0 in () yields f(3x) = 7f(x) + f( x), whence f(x) = f( x). If we put, successively, y = x, y = 4x, x 0 in (), then f(4x) + f(0) = 8f(x) + f(x), f(6x) + f(x) = 8f(x) + f(4x). (4) On account of (4) (3) we obtain 9f(x) + f(x) = 8f(x) + 6f(x) + 4f(x) f(0) whence f(x) = 4f(x) f(0). (5) 3 According to (5) (3), f(6x) = 4f(3x) 3 f(0) = 36f(x) 3 f(0).

6 5 Zygfryd Kominek On the other h, by virtue of (4) (5), we get whence 36f(x) f(0) + f(x) = 8f(x) + [8f(x) + f(x) f(0)] 3 f(0) = 0 f(x) = 4f(x), by (5). Using these equalities together with f(x) = f( x), one can easily verify that () is fulfilled for all x, y X. 5. STABILITY OF EQUATION () We use these Lemmas in the proofs of Theorems 3, in which we put D := (x, x), ( x, x), (x, x), ( x, x); x X }. Theorem. Let g : X Y be a function satisfying the inequality g(x + y) 4g(x) g(y) g(x + y) + g(x y) ω(x, y), (6) where ω : X X [0, ) is a function fulfilling the following conditions: limn 9 n ω(3 n x, 3 n y) = 0; 9 n ω(3 n u, 3 n v) < for all (u, v) D. Then there exists a unique quadratic function Q : X Y satisfying the estimate where Q(x) g(x) 9 n+ ϕ(3n x, 3 n x) + ψ(x), (7) ϕ(x, y) = [ω(x, y) + ω( x, y) + ω(x, y) + ω( x, y)] (8) ψ(x) = 6 [ [ω(x, x) + ω( x, x)] + ω(x, x) + ω( x, x) + [ω(x, x) + ω( x, x)]]. (9) Proof. First observe that, because of (8) the limit properties of the function ω stated in (6), the function ϕ satisfies (6). Let p h be the even the odd part, respectively, of the function g, i.e., p(x) = It is not hard to check that g(x) + g( x), h(x) = g(x) g( x), x X. p(x + y) 4p(x) p(x + y) p(y) + p(x y) [ω(x, y) + ω( x, y)] (0)

7 Remarks on the stability of some quadratic functional equations 53 h(x + y) 4h(x) h(x + y) h(y) + h(x y) [ω(x, y) + ω( x, y)]. () Setting y = x then y = x in inequality (0), we get p(3x) 5p(x) p(x) + p(0) [ω(x, x) + ω( x, x)] 4p(x) p(0) + p(x) [ω(x, x) + ω( x, x)]. Consequently, p(3x) 9p(x) [ω(x, x) + ω( x, x) + ω(x, x) + ω( x, x)]. Thus, because of (8), 9 p(3x) p(x) ϕ(x, x). () 9 It follows from inequality () that 9 n+ p(3n+ x) 9 n p(3n x) 9 9 n ϕ(3n x, 3 n x). Taking λ n = 9 n, β n = 3 n, n N, from Lemmas Theorem, we infer that there exists a quadratic function Q : X Y fulfilling the estimate Q(x) p(x) (Note that (0) is of form (5) with r = 5.) Now we are going to check the inequality 9 n+ ϕ(3n x, 3 n x). (3) h(x) ψ(x). (4) Since h is odd, then h(0) = 0. Setting y = x, y = x finally y = x in (), we obtain h(3x) 5h(x) h(x) [ω(x, x) + ω( x, x)], h(x) 4h(x) ω(x, x) + ω( x, x), h(3x) 3h(x) + h(x) [ω(x, x) + ω( x, x)]. Consequently, by the triangle inequality h(x) x) + ω( x, x) [ω(x, + ω(x, x) + ω( x, x)+ 6 ω(x, x) + ω( x, x) + ]. (5)

8 54 Zygfryd Kominek By virtue of (9) (5), we obtain estimate (4). Because of (3), this is (7). To prove the uniqueness of Q assume that Q : X Y is a quadratic function satisfying estimate (7). On account of a theorem proved in [7], Thus Q(3x) = 9Q(x) as well as Q (3x) = 9Q (x), x X. Q(x) Q (x) = 9 k Q(3k x) Q (3 k x) 9 k Q(3k x) f(3 k x) + Q (3 k x) f(3 k x) } 9 k = j=k 9 n+ ϕ(3n+k x, 3 n+k x)) + ψ(3 k x)} = 9 j+ ϕ(3j x, 3 j x) + 9 k ψ(3k x). By our assumption, the last expression tends to zero, as k. This completes the proof of Theorem. Theorem 3. Assume that X is a commutative group uniquely divisible by 3. Let g : X Y be a function satisfying the inequality g(x + y) 4g(x) g(y) g(x + y) + g(x y) ω(x, y), where ω : X X [0, ) is a function fulfilling the following conditions limn 9 n ω(3 n x, 3 n y) = 0; 9n ω(3 n u, 3 n v) < for all (u, v) D. Then there exists a unique quadratic function Q : X Y satisfying the estimate Q(x) g(x) 9 n ϕ(3 n x, 3 n x) + ψ(x), where ϕ ψ are defined as in Theorem. Proof. The proof runs similarly to the proof of Theorem. Let ϕ ψ be defined as in Theorem. We consider inequalities (0) (). In the proof of Theorem, we obtained the following inequalities h(x) ψ(x) p(3x) 9p(x) ϕ(x, x). (6)

9 Remarks on the stability of some quadratic functional equations 55 Replacing x by x 3 in (6), we get 9p( x 3 ) p(x) ϕ(x 3, x 3 ) whence 9 n+ x p( 3 n+ ) 9n p( x 3 n ) x 9n ϕ( 3 n+, x 3 It follows from the assumptions of Theorem 3 that limn 9 n ϕ(3 n x, 3 n y) = 0, 9n ϕ(3 n x, 3 n x) <. n+ ). On account of Lemmas 3, as well as Theorem, there exists a quadratic function Q : X Y satisfying the following estimate Therefore, Q(x) p(x) Q(x) g(x) 9 n x ϕ( 3 n+, x 3 n+ ). 9 n x ϕ( 3 n+, x ) + ψ(x). 3n+ The proof of the uniqueness of Q is quite similar to that of Theorem. 6. STABILITY OF EQUATION () Theorem 4. Let g : X Y be a function satisfying the inequality g(x + y) g(x y) 8g(x) g(y) ϕ(x, y), (7) where ϕ : X X [0, ) is a function fulfilling the conditions: limn 9 n ϕ(3 n x, 3 n y) = 0; 9 n ϕ(3 x x, 3 n x) is convergent. Then there exists a unique quadratic function Q : X Y satisfying the following estimate Q(x) g(x) 9 n+ ϕ(3n x, 3 n x)). (8) Proof. Putting y = x in (7), we get g(3x) 9g(x) ϕ(x, x). (9) Now we argue quite similarly as in the proof of Theorem (the even case), obtaining the existence of a unique quadratic function Q : X Y fulfilling estimate (8).

10 56 Zygfryd Kominek Theorem 5. Assume that X is a commutative group uniquely divisible by 3. Let g : X Y be a function satisfying the inequality g(x + y) + g(x y) 8g(x) g(y) ϕ(x, y), where ϕ : X X [0, ) is a function fulfilling the following conditions: limn 9 n ϕ(3 n x, 3 n y) = 0; 9n ϕ(3 n x, 3 n x) <. Then there exists a unique quadratic function Q : X Y satisfying the estimate Q(x) g(x) 9 n ϕ(3 n x, 3 n x). Proof. Setting x 3 instead of x in (9), we obtain g(x) 9g( x 3 ) ϕ(x 3, x 3 ). Now we argue as in the proof of Theorem 3, obtaining a unique quadratic function Q fulfilling the following estimate Q(x) g(x) 9 n x ϕ( 3 n+, x 3 n+ ). Concluding remark. Let θ 0 ω(x, y) = θ( x α + y α ), or ω(x, y) = θ x β y β. Theorems 4 can be applied to these functions ω with α < β <, whereas Theorems 3 5 with α > β >. Thus our theorems cover the cases considered by several other authors, in particular, by C. Park J. Su An. Acknowledgements The author is indebted to the referee for his valuable remarks suggestions concerning this paper. REFERENCES [] D.G. Bourgin, Classes of transformations bordering transformations, Bull. Amer. Math. Soc. 57 (95), [] P.W. Cholewa, Remark on the stability of functional equations, Aequationes Math. 7 (984), [3] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 6 (99),

11 Remarks on the stability of some quadratic functional equations 57 [4] S. Czerwik, Stability of Functional Equations of Hyers-Ulam-Rassias Type, Hadronic Press, Florida, USA, 003. [5] S.-M. Jung, Stability of the quadratic equation of Pexider type, Abh. Math. Sem. Univ. Hamburg 70 (000), [6] W. Fechner, On Hyers-Ulam stability of functional equations connected with additive quadratic mappings, J. Math. Anal. Appl. 3 (006), [7] Z. Kominek, K. Troczka, Some remarks on subquadratic functions, Demonstratio Math. 39 (006) 4, [8] C. Park, J. Su An, Hyers-Ulam stability of quadratic functional equations, (submitted). Zygfryd Kominek zkominek@ux.math.us.edu.pl Silesian University Institute of Mathematics ul. Bankowa 4, Katowice, Pol Received: January 4, 008. Revised: October 4, 008. Accepted: July 4, 008.