Stability of a Functional Equation Related to Quadratic Mappings

Transcription

1 International Journal of Mathematical Analysis Vol. 11, 017, no., HIKARI Ltd, Stability of a Functional Equation Related to Quadratic Mappings Sun-Sook Jin and Yang-Hi Lee Department of Mathematics Education Gongju National University of Education Gongju 3553, Korea Copyright c 016 Sun-Sook Jin and Yang-Hi Lee. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we investigate the stability of a functional equation f(a + by) + abf( y) (a + ab)f() (b + ab)f(y) = 0, which relates to quadratic mappings. Mathematics Subject Classification: 39B8, 39B5 Keywords: stability; quadratic mapping 1 Introduction The stability problem of functional equations originated from Ulam s stability problem [10] of group homomorphisms. The functional equation which is equivalent to the following functional equation f( + y) + f( y) = f() + f(y) is called a quadratic functional equation, and every solution of a quadratic functional equation is said to be a quadratic mapping. F. Skof [9], P. W. Cholewa [1], and S. Czerwik [] proved the stability of quadratic functional equations,

2 56 Sun-Sook Jin and Yang-Hi Lee and the authors have investigated the stability problems of quadratic functional equations, see [3] and [4]. Moreover, Najati and Jung [8] have observed the Hyers-Ulam stability of a generalized quadratic functional equation f(a + by) + abf( y) = af() + bf(y) (1.1) where a, b are non-zero rational numbers with a + b = 1. In this paper we consider the stability of the following functional equation f(a + by) + abf( y) (a + ab)f() (b + ab)f(y) = 0 (1.) for real numbers a, b with ab(a + b) 0, which is a generalization of (1.1). Recall if a mapping f satisfies the functional equation f( + y) f() f(y) = 0 then we call f an additive mapping. If a mapping f is represented by sum of an additive mapping and a quadratic mapping, we call it a quadratic-additive mapping [6]. Now, in this paper, we will show that the solution of the functional equation (1.) is a quadratic-additive mapping and investigate the stability of it. In [5], H.-K. Kim and J.-R. Lee have investigated the stability of a generalized quadratic functional equation (1.) for any fied rational numbers a, b in fuzzy spaces. Main results Throughout this paper, let X be a real normed space and Y a real Banach space. Let V and W be real vector spaces and let a and b be real constants. For a given mapping f : V W, we use the following abbreviations f() f( ) f o () :=, f() + f( ) f e () :=, Af(, y) := f( + y) f() f(y), Qf(, y) := f( + y) + f( y) f() f(y), D a,b f(, y) := f(a + by) + abf( y) (a + ab)f() (b + ab)f(y) for all, y V. In the following lemma, we will show that every solution of the functional equation (1.) is a quadratic-additive mapping.

3 Stability of a functional equation 57 Lemma.1 Let a and b be fied real numbers with ab(a + b) 0 and let f : V W be a mapping satisfying the equation D a,b f(, y) = 0 (with f(0) = 0 when a + ab + b = 1). Then f is a quadratic-additive mapping such that f(a) = a f(), f((a + b)) = (a + b) f(), and f(b) = abf( ) + (b + ab)f() for all V. Proof. Assume that a mapping f : V W satisfies the functional equation (1.). Notice that f(0) = D a,bf(0,0) = 0 if a + ab + b 1. For all, y V 1 a ab b we get ( ) + y f o f o () f o (y) ( ( ) 1 b + by a ay = D a a,b f o, D a,b f o (b, 0) a D a,b f o (0, ) b(a + b) ( ) ) b + by ay a + D a,b f o, D a,b f o (by, 0) a D a,b f o (0, y) + 1 ( ab D a,bf o 0, + y ) ( ( 1 y D a,b f o, + y ) ( y + D a,b f o, + y )) a(a + b) = 0. Since f o (0) = 0, we easily show that f o ( +y ) = f o( + y) for all, y V. Therefore, we get Af o (, y) = f o ( + y) f o () f o (y) ( ) + y = f o ( + y) f o = 0 for all, y V, i.e., f o is an additive mapping. On the other hand, we have Qf e (, y) = D a,bf e ((1 + b a ), y) D a,bf e (, + y) b a Df e(y, ) b(a + b) + D a,bf e (, ) ab = 0 + D a,bf e ((1 + b ) y, 0) a a(a + b) Da,bf e ((1 + b ), 0) a ab for all, y V, i.e., f e is a quadratic mapping. Until now, we obtain that f = f e + f o is a quadratic-additive mapping. Moreover, from the equalities f(a) a f() = Df a,b f(, 0) f((a + b)) (a + b) f() = Df a,b f(, ) f(b) + abf( ) (b + ab)f() = Df a,b f(0, )

4 58 Sun-Sook Jin and Yang-Hi Lee for all V, we get the equalities f(a) = a f(), f((a + b)) = (a + b) f(), and f(b) = abf( ) + (b + ab)f() for all V. For the special rational number case, we can show that f is itself a quadratic mapping. Lemma. Let a be a fied rational number with ab(a + b)(a 1) 0, or let a + b be a fied rational number with ab(a + b)(a + b 1) 0, or let b be a fied rational number with ab(a + b) 0. If a mapping f : V W satisfies the functional equation (1.), then f is a quadratic mapping such that f(a) = a f() and f(b) = b f() for all V. Proof. By Lemma.1, f o is an additive mapping such that f o (a) = a f o (), f o ((a + b)) = (a + b) f o (), and f o (b) = abf o ( ) + (b + ab)f o () for all V. Observe that, since Af o 0, f o (r) = rf o () for all rational numbers r and V. In the first, if a is a nonzero rational number with a 1, then the equality f o (a) = af o () = a f o () implies that f o 0. Secondary, if a + b is a nonzero rational number with a + b 1, then the equality f o ((a + b)) = (a + b)f o () = (a + b) f o () implies that f o 0. Finally, suppose that b is a nonzero rational number. Then the equality f o (b) = bf o () = abf o ( ) + (b + ab)f o () implies the equality (b + a 1)bf o () = 0. In this case, for a nonzero rational number b, we only need to check the case of a Q or a + b Q or a = 1 or a + b = 1. So (b + a 1)b 0, which means that f o 0. Therefore, we have shown that f f e which proved this lemma. H.-M. Kim and J.-R. Lee [5] showed that for fied rational numbers a and b if f : V W is a solution of the functional equation (1.), then f is a quadratic mapping. In the following lemma, we get the converse of it for some conditions. Lemma.3 Let a and b be fied rational numbers with ab(a + b) 0. A mapping f : V W satisfies the functional equation (1.) (with f(0) = 0 when a + ab + b = 1) if and only if f is a quadratic mapping. Proof. Assume that a mapping f : V W satisfies the functional equation (1.). By Lemmas.1 and., we conclude that f is a quadratic mapping. Conversely, assume that f is a quadratic mapping. Notice that f() = f( ) and f(r) = r f() for all V and all rational numbers r. Now, we will prove that D a,b f(, y) = 0 for all, y V and all positive integers a and b. We apply an induction on i {1,,..., a} and j {1,,..., b} to prove D a,b f(, y) = 0 for all, y V. For i = 1 and j = 1, we have the following equality D 1,1 f(, y) = Qf(, y) = 0

5 Stability of a functional equation 59 for all, y V. Assume that D 1,j f(, y) = 0 for all, y V and for all j k. Then D 1,k+1 f(, y) = D 1,k f( + y, y) + (k + 1)Qf(, y) = 0 for all, y V. Hence we have D 1,b f(, y) = 0 for all, y V and all positive integers b. Observe that D,b f(, y) = D 1,b f(, y) + Qf( + by, ) f(by) + b f(y) = 0 for all, y V. Now assume that D i,b f(, y) = 0 for all, y V, b Z +, and all i k with k. Then D k+1,b f(, y) = D k,b f(, y) D k 1,b f(, y) + Qf(k + by, ) = 0 for all, y V and all positive integers b. So we have the equality D a,b f(, y) = 0 for all, y V and all positive integers a, b. Let a = p and b = r be positive q s rational numbers with p, q, r, s Z +. Since ps, qr are nonzero positive integers and f( ) = f(), we have qs q s for all, y V. From the equality for all, y V, we conclude that ( D a,b f(, y) = D ps,qr f qs, y ) = 0 qs D a, b f(, y) = D a,b f(, y), D a,b f(, y) = 0 for all, y V and all rational numbers a, b with ab > 0. We now consider the case for ab < 0 with a + b 0. The equality D a,b f(, y) = 0 follows from the equalities D a,b f(, y) = D a, a b f( y, y) when a(a + b) > 0 and D a,b f(, y) = D a b,b f(, y ) when b(a + b) > 0. In the following theorem, using Lemma.1, we can prove the stability of the functional equation (1.).

6 60 Sun-Sook Jin and Yang-Hi Lee Theorem.4 Let a and b be real constants with ab(a + b) 0 and let ϕ : V [0, ) be a function satisfying either or for all, y V. If a mapping f : V Y satisfies ϕ(a i, a i y) a i < (.1) ( a i ϕ a, y ) < (.) i a i D a,b f(, y) ϕ(, y) (.3) for all, y V with f(0) = 0, then there eists a unique solution F : V Y of the functional equation (1.) such that the inequality { 1 ϕ(a i, 0) if ϕ satisfites (.1), a f() F () i+ ai ϕ (, 0 ) (.4) if ϕ satisfites (.) a i+1 holds for all V. Proof. We will prove the theorem in two cases, either ϕ satisfies (.1) or (.). Case 1. Let ϕ satisfy (.1). It follows from (.3) that for all V 1 a n f(an ) 1 a n+m f(an+m ) 1 D a,bf(a i+1, 0) a i 1 a i ϕ(ai+1, 0) (.5) So, it is easy to show that the sequence { f(an ) } is a Cauchy sequence for all a n V. Since Y is complete and f(0) = 0, the sequence { f(an ) } converges for a n all V. Hence, we can define a mapping F : V Y by F () := lim n f(a n ) a n for all V. Moreover, if we put n = 0 and let m in (.5), we obtain the first inequality in (.4). From the definition of F, we get D a,b F (, y) = lim D a,b f ( a n, a n y ) n a n lim ϕ ( a n, a n y) = 0 n a n

7 Stability of a functional equation 61 i.e., D a,b F (, y) = 0 for all, y V. To prove the uniqueness, we assume now that there is another solution F : V Y of the functional equation (1.) which satisfies the first inequality in (.4). Since the equality F () = F (a n ) a n holds for all n N by Lemma.1, we obtain lim f(a n ) F () a n = lim f(a n ) n F (a n ) a n a n ϕ(a i+n, a i+n ) a n+i+ n = 0 ϕ(a i, a i ) a i+ for all V, i.e, F () = lim n f(a n ) a n = F () for all V. Case. Let ϕ satisfy (.). It follows from (.3) that ( ) an f a n ( ) a n+m f a n+m a i D a,b f ( ) a i ϕ a, 0 i+1 ( a i+1, 0) (.6) for all V. So, it is easy to show that the sequence {a n f( )} is a Cauchy a n sequence for all V. Since Y is complete and f(0) = 0, the sequence {a n f( )} converges for all V. Hence, we can define a mapping F : V a n Y by ( ) F () := lim a n f n a n for all V. Moreover, if we put n = 0 and let m in (.6), we obtain the second inequality in (.4). From the definition of F, we get (.4) for all V and D a,b F (, y) = lim n ( an D a,b f a, y ) ( lim n a n n an ϕ a, y ) = 0 n a n for all, y V i.e., D a,b F (, y) = 0 for all, y V. To prove the uniqueness, we assume now that there is another solution F : V W of the functional equation (1.) which satisfies the second inequality in (.4). Since the equality

8 6 Sun-Sook Jin and Yang-Hi Lee F () = a n F ( ) holds for all n N by Lemma.1, we get a n ( ) ( ) ( ) lim n an f F () a n = lim n an f a n F a n a n ( ) a n+i ϕ a, n+i a n+i ( a i ϕ a, ) i a i = 0 for all V, i.e, F () = lim n a n f ( ) a = F () for all V. n From the Lemma. and the previous theorem, we obtained the following corollary clearly. Corollary.5 Let a be a fied rational number with ab(a + b)(a 1) 0, or let a + b be a fied rational number with ab(a + b)(a + b 1) 0, or let b be a fied rational number with ab(a+b) 0. And let ϕ : V [0, ) be a function satisfying either the condition (.1) or the condition (.) for all, y V. If a mapping f : V Y satisfies f(0) = 0 and (.3) for all, y V, then there eists a unique quadratic mapping F : V Y satisfying the equation (1.) and the equality (.4) for all V. Now, we obtain another stability result of the functional equation (1.) by following. Theorem.6 Let a and b be real constants with ab(a + b) > 0 and and let ϕ : V [0, ) be a function satisfying either ϕ(b i, b i y) < (.7) b i or ( (b + ab) i ϕ b, y ) < (.8) i b i for all, y V. If a mapping f : V Y satisfies (.3) for all, y V with f(0) = 0, then there eists a unique solution F : V Y of the functional equation (1.) such that ( ϕ(0,b i )+ϕ(0, b i ) + ϕ(0,bi )+ϕ(0, b i ) ) b i+ (b +ab) i+1 if ϕ satisfites (.7), f() F () (.9) b i +(b +ab) ( ( ) ( )) i ϕ 0, b + ϕ 0, i+1 b i+1 if ϕ satisfites (.8)

9 Stability of a functional equation 63 for all V. Proof. We will prove the theorem in two cases, either ϕ satisfies (.7) or (.8). Case 1. Let ϕ satisfy (.7). It follows from (.3) that f(b n ) + f( b n ) f(bn+m ) + f( b n+m ) b n b n+m 1 b i+ Da,b f(0, b i ) D a,b f(0, b i ) f(b n ) f( b n ) f(bn+m ) f( b n+m ) (b + ab) n (b + ab) n+m ϕ(0, b i ) + ϕ(0, b i ) b i+, (.10) 1 (b + ab) i+1 Da,b f(0, b i ) + D a,b f(0, b i ) (.11) ϕ(0, b i ) + ϕ(0, b i ) (b + ab) i+1 (.1) for all V. So, it is easy to show that the sequences { f(bn )+f( b n ) } and b n } are Cauchy sequences for all V. Since Y is complete and { f(bn ) f( b n ) (b +ab) n f(0) = 0, the sequences { f(bn )+f( b n ) } and { f(bn ) f( b n ) } converge for all b n (b +ab) n V. Hence, we can define mappings F, F : V Y by F () := lim n f(b n ) + f( b n ) b n, F () := lim n f(b n ) f( b n ) (b + ab) n for all V. From the definitions of F and F, we obtain that F (b) = b F (), F (b) = (b + ab)f (), DF (, y) = 0, and DF (, y) = 0 for all, y V. Therefore, F and F are quadratic-additive mapping by Lemma.1. Moreover, if we put n = 0 and let m in (.10) and (.1), we obtain the inequalities f() + f( ) F () ϕ(0, b i ) + ϕ(0, b i ) b i+,

10 64 Sun-Sook Jin and Yang-Hi Lee f() f( ) F () ϕ(0, b i ) + ϕ(0, b i ) (b + ab) i+1 hold for all V. Put F () := F ()+F () for all V, then F () = F e () and F () = F o () for all V. Since f() F () = f() + f( ) F () + f() f( ) F () for all V, we obtain the first inequality in (.9). To prove the uniqueness, we assume now that there is another solution F : V W of the functional equation (1.) which satisfies the first inequality in (.9). Since the equalities F e (b) = b F e () and F o (b) = (b +ab)f o () hold by Lemma.1, we obtain for all n N f e (b n ) + f o(b n ) b n (b + ab) F () n = f e (b n ) F b n e () + f o(b n ) (b + ab) F o() n = f e (b n ) F e(b n ) b n b n + f o (b n ) (b + ab) F o(b n ) n (b + ab) n ( ϕ(0, b i+n ) + ϕ(0, b i+n ) + ϕ(0, bi+n ) + ϕ(0, b i+n ) b i++n (b + ab) i+1 b n + ϕ(0, bi+n ) + ϕ(0, b i+n ) + ϕ(0, ) bi+n ) + ϕ(0, b i+n ) b i+ (b + ab) n (b + ab) n+i+1 ( ) ϕ(0, b i+n ) + ϕ(0, b i+n ) = b i++n ( ) ϕ(0, b i ) + ϕ(0, b i ) b i+ i.e, F () = lim n ( f(b n )+f( b n ) b n + f(bn ) f( b n ) (b +ab) n ) = F () for all V. Case. Let ϕ satisfy (.8). Together with (.3), it follows that ( ( ) ( b n f + f b )) ( ( ) ( + bn+m f + f )) b n n b n+m b n+m ( ) ( b i D a,bf 0, + D b i+1 a,b f 0, ) b i+1 ( ( ) ( )) b i ϕ 0, + ϕ 0,, b i+1 b i+1

11 Stability of a functional equation 65 ( ( ) ( (b + ab) n f + f b n b n (b + ab) i D a,bf ( ( (b + ab) i ϕ 0, )) + (b + ab) n+m ( 0, b i+1 ( ( f ) D a,b f b i+1 ) b n+m ) ( 0, ( )) + ϕ 0, b i+1 ( )) + f b n+m b i+1 ) for all V. From the above inequalities, we obtain that there eists a unique solution F : V Y of the functioal equation (.4) satisfying the second inequality in (.9) by using the similar method used in the case 1. From the Lemma. and the previous theorem, we obtained the following corollary clearly. Corollary.7 Let a be a fied rational number with ab(a+b)(a 1) 0, or let a + b be a fied rational number with ab(a + b)(a + b 1) 0, or let b be a fied rational number with ab(a + b) 0. Suppose that ab > 0 and ϕ : V [0, ) is a function satisfying either the condition (.7) or the condition (.8) for all, y V. If a mapping f : V Y satisfies f(0) = 0 and (.3) for all, y V, then there eists a unique quadratic mapping F : V Y satisfying the inequality (1.) and the inequality (.9) for all V. We can easily following stability theorem by using the similar method used in the previous theorems. Theorem.8 Let a or b be a real constants with ab(a + b) (a + b) < 0 and let ϕ : V [0, ) be a function satisfying either or ϕ(b i, b i y) < (.13) (b + ab) i ( b i ϕ b, y ) < (.14) i b i for all, y V. If a mapping f : V Y satisfies (.3) for all, y V with f(0) = 0, then there eists a unique solution F : V Y of the functional equation (1.) satisfying the inequality (.9) for all V. Corollary.9 Let a be a fied rational number with ab(a+b)(a 1)(a+b) 0, or let a + b be a fied rational number with ab(a + b)(a + b 1)(a + b) 0, or let b be a fied rational number with ab(a + b)(a + b) 0. Suppose that

12 66 Sun-Sook Jin and Yang-Hi Lee ab < 0 and ϕ : V [0, ) is a function satisfying either the condition (.13) or the condition (.14) for all, y V. If a mapping f : V Y satisfies f(0) = 0 and (.3) for all, y V, then there eists a unique quadratic mapping F : V Y satisfying the functional equation (1.) and the inequality (.9) for all V. Recall the following theorem shows the stability of the functional equation (1.) on the restricted domain V \{0}. Theorem.10 ( Theorem 4.3 and Theorem 4.4 in [7]) Let a and b be real constants with ab(a + b) 0 and let ϕ : (V \{0}) [0, ) be a function satisfying either or ϕ((a + b) i, (a + b) i y) (a + b) i <, (.15) ( ) (a + b) i ϕ (a + b), y < (.16) i (a + b) i for all, y V \{0}. If a mapping f : V Y satisfies f(0) = 0 and (.3) for all, y V \{0}, then there eists a unique mapping F : V Y satisfying the equation (1.) for all V \{0} and the inequality { ϕ((a+b) i,(a+b) i ) if ϕ satisfites (.15), (a+b) f() F () i+ (a + b)i ϕ ( ), if ϕ satisfites (.16) (a+b) i+1 (a+b) i+1 (.17) for all V \{0}. Using the previous theorem and Lemma., we get the following corollary. Corollary.11 Let a be a fied rational number with ab(a + b)(a 1) 0, or let a + b be a fied rational number with ab(a + b)(a + b 1) 0, or let b be a fied rational number with ab(a + b) 0. Let ϕ : V [0, ) be a function satisfying either the condition (.15) or the condition (.16) for all, y V. If a mapping f : V Y satisfies f(0) = 0 and (.3) for all, y V \{0}, then there eists a unique quadratic mapping F : V Y satisfying the inequality (1.) and the inequality (.17) for all V \{0}. The following corollary follows from Theorem.4 and Theorem.10 by taking ϕ(, y) = θ( p + y p ) defined on a real normed space X.

13 Stability of a functional equation 67 Corollary.1 Let a, b, p, and θ be real constants such that ab(a + b) 0, a 1, (a + b) 1, p and p, θ > 0. If a mapping f : X Y satisfies the inequality D a,b f(, y) θ( p + y p ) (.18) for all, y X, then there eists a unique mapping F : X Y satisfying the equation (1.) and the inequality { } θ p f() F () min (a + b) a + b p, θ p (.19) a a p for all X\{0}. The following corollary follows from Lemma., Theorem.4, and Theorem.10. Corollary.13 Let a be a fied rational number with ab(a + b)(a 1) 0, or let a + b be a fied rational number with ab(a + b)(a + b 1) 0, or let b be a fied rational number with ab(a + b) 0. Let p and θ be positive real numbers such that p. If a mapping f : X Y satisfies the inequality (.18) for all, y X, then there eists a unique quadratic mapping F : X Y satisfying the inequality (.19) for all X\{0}. The following corollary follows from Theorem.10 with the negative p. Corollary.14 Let a, b, p, and θ be real constants such that ab(a + b) 0, p < 0 and θ > 0. If a mapping f : X Y satisfies the inequality (.18) for all, y X\{0} with f(0) = 0, then the mapping f : X Y satisfies the equation (1.) for all, y X\{0}. Proof. According to Theorem.10, there eists a unique mapping F : V Y satisfying the equation (1.) for all, y X\{0} and the inequality f() F () θ p (a + b) a + b p (.0) for all X\{0}. Since the equality D a,b F (, y) = 0 holds for all, y X\{0} and the inequality (.0) holds for all X\{0}, we obtain the inequality ab f() F () (D a,b f D a,b F )((k + 1), k) + (F f)(((a + b)k + 1)) + a(a + b) (f F )((k + 1)) + b(a + b) (f F )(k) ( ) (a + b)k + 1 p + a(a + b) (k + 1) p + b(a + b) k p ) + (k + 1) p + k p θ p (a + b) a + b p 0, as k,

14 68 Sun-Sook Jin and Yang-Hi Lee for all X\{0}. Acknowledgements. This work was supported by Gongju National University of Education Grant. References [1] P. W. Cholewa, Remarks on the stability of functional equations, Aeq. Math., 7 (1984), [] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Hamburg, 6 (199), [3] S.-S. Jin and Y.-H. Lee, Generalized Hyers-Ulam stability of a 3- dimensional quadratic functional equation, Int. J. Math. Anal., 10 (016), [4] S.-S. Jin and Y.-H. Lee, Generalized Hyers-Ulam stability of a 3- dimensional quadratic functional equation in modular spaces, Int. J. Math. Anal., 10 (016), [5] H.-M. Kim and J.-R. Lee, Approimate Euler-Lagrange Quadratic Mappings in Fuzzy Banach Spaces, Abstr. Appl. Anal., 013 (013), Article ID 86974, [6] Y.-H. Lee, On the quadratic additive type functional equations, Int. J. Math. Anal., 7 (013), [7] Y.-H. Lee and S.-M. Jung, A general theorem on the stability of a class of functional equations including monomial equations, Journal of Inequalities and Applications, 015 (015), [8] A. Najati and S.-Jung, Approimately quadratic mappings on restricted domains, Journal of Inequalities and Applications, 010 (010), Article ID [9] F. Skof, Proprieta locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano, 53 (1983), [10] S.M. Ulam, Problems in Modern Mathematics, Wiley, New York, Received: November, 016; Published: January 17, 017