Alireza Kamel Mirmostafaee

Transcription

1 Bull. Korean Math. Soc. 47 (2010), No. 4, pp DOI /BKMS STABILITY OF THE MONOMIAL FUNCTIONAL EQUATION IN QUASI NORMED SPACES Alireza Kael Mirostafaee Abstract. Let X be a linear space and Y be a coplete quasi p-nor space. We will show that for each function f : X Y, which satisfies the inequality n xf(y) n!f(x) ϕ(x, y) for suitable control function ϕ, there is a unique onoial function M of degree n which is a good approxiation for f in such a way that the continuity of t f(tx) and t ϕ(tx, ty) iply the continuity of t M(tx). 1. Introduction The concept of stability of a functional equation arises when one replaces a functional equation by an inequality which acts as a perturbation of the equation. In 1940, Ula [18] posed the first stability proble. In 1941, D. H. Hyers [10] gave the first significant partial solution to his question. Hyers theore was generalized for additive appings by T. Aoki [3] in 1950 and D. G. Bourgin [5] in In 1978, Th. M. Rassias [17] solved the proble for linear appings by considering an unbounded Cauchy difference. The phenoenon that was introduced and proved by Th. M. Rassias in the year 1978, is called the Hyers-Ula-Rassias stability. Let X and Y be linear spaces and Y X be the vector space of all functions fro X to Y. Following [11], for each x X, define x : Y X Y X by Inductively, we define x f(y) = f(x + y) f(y), f Y X, y X. x1,...,x n f(y) = x1,...,x n 1 ( xn f(y)) for each x 1,..., x n, y X and f Y X, we write n xf(y) = x1,...,x n f(y) Received February 17, Matheatics Subject Classification. Priary 39B52, 39B82; Secondary 47H10. Key words and phrases. quasi p-nor, onoial functional equation, fixed point alternative, Hyers Ula Rassias stability. This research is supported by Ferdowsi University of Mashhad, No: MP88094MIM. 777 c 2010 The Korean Matheatical Society

2 778 ALIREZA KAMEL MIRMOSTAFAEE if x 1 = = x n = x. By induction on n, it can be easily verified that n ( ) n (1.1) n xf(y) = ( 1) n k f(kx + y), n N, x, y X. k k=0 It can be easily verified that every polynoial of degree at ost n satisfies the functional equation n+1 x f(y) = 0. Hence the functional equation n xf(y) = 0 is called the polynoial functional equation of degree n 1. The functional equation (1.2) n xf(y) = n!f(x) is called the onoial functional equation of degree n, since the function f(x) = cx n is a solution of the functional equation. Every solution of the onoial functional equation of degree n is said to be a onoial apping of degree n. In particular additive, quadratic, cubic and quartic functions are onoials of degree one, two, three and four respectively. The research on the stability of polynoial or onoial equations was initiated by D. H. Hyers in [11]. The proble has been recently considered by soe authors see e.g. M. H. Albert and J. A. Baker [1], L. Cãdariu and V. Radu [6], A. Gilányi [8, 9], Z. Kaiser [13], Y.-H. Lee [14] and D. Wolna [19]. In Section 3, we use the fixed point alternative theore to prove the Hyers- Ula-Rassias stability of onoial functional equation of an arbitrary degree in coplete quasi-nored spaces. More precisely, we will show that if a function f fro a linear space X to a coplete p-nored space Y for soe n N satisfies the inequality n xf(y) n!f(x) ϕ(x, y), x, y X for suitable control function ϕ, then f can be suitably approxiated by a unique onoial M : X Y of degree n. In Section 4, we will show that, for each x X, the continuity of s f(sx) and s ϕ(sx, sy) guarantee the continuity of s M(sx). It follows that in this case, M(tx) = t n M(x) for each t R and x X. 2. Preliinaries In this section, we give soe preliinaries, which will be used in this paper. We start by the following definition. Definition 2.1. The pair (X, d) is called a generalized coplete etric space if X is a nonepty set and d : X 2 [0, ] satisfies the following conditions: (a) d(x, y) 0 and the equality holds if and only if x = y, (b) d(x, y) = d(y, x), (c) d(x, z) d(x, y) + d(y, z), (d) every d-cauchy sequence in X is d-convergent. Note that the distance between two points in a generalized etric space is peritted to be infinity.

3 STABILITY OF THE MONOMIAL EQUATIONS 779 Definition 2.2. Let (X, d) be a generalized coplete etric space. A apping Λ : X X satisfies a Lipschitz condition with Lipschitz constant L 0 if d(λ(x), Λ(y)) Ld(x, y), x, y X. If L < 1, then Λ is called a strictly contractive operator. In 2003, Radu [15] eployed the following result, due to Diaz and Margolis [7], to prove the stability of Cauchy additive functional equation. Using such an elegant idea, several authors applied the ethod to investigate the stability of soe functional equations, see [6, 12, 16]. Proposition 2.3 (The fixed point alternative principle). Suppose that a coplete generalized etric space (E, d) and a strictly contractive apping J : E E with the Lipschitz constant 0 < L < 1 are given. Then, for a given eleent x E, exactly one of the following assertions is true: either (a) d(j n x, J n+1 x) = for all n 0 or (b) there exists soe integer k such that d(j n x, J n+1 x) < for all n k. Actually, if (b) holds, then the sequence {J n x} is convergent to a fixed point x of J and (b1) x is the unique fixed point of J in F := {y E, d(j k x, y) < }; (b2) d(y, x ) d(y,jy) 1 L for all y F. Reark 2.4. The fixed point x, if it exists, is not necessarily unique in the whole space E; it ay depend on x. Actually, if (b) holds, then (F, d) is a coplete etric space and J(F) F. Therefore the properties (b1) and (b2) follow fro The Banach fixed point theore. Definition 2.5. A quasi-nor on a real vector space X is a function x x fro X to [0, ) which satisfies (i) x > 0 for every x 0 in X, (ii) tx = t. x for every t R and x X, (iii) there is k 1 such that x+y k( x + y ) for every x, y X. Aoki [2] (see also [4]) has shown that every quasi-nored space (X, ) adits an equivalent quasi-nor such that for soe 0 < p 1, (2.1) x + y p x p + y p, x, y X. In this case, (X, ) is called a quasi p-nored space. In special case, when p = 1, (X, ) turns into a nored linear space. 3. Stability of onoial functional equations Throughout the reainder of this paper, unless otherwise stated, we will assue that 0 < p 1 and q = 1 p, X is a real vector space and Y is a coplete quasi p-nor space. In 2006, D. Wolna in [19] proved the following result:

4 780 ALIREZA KAMEL MIRMOSTAFAEE Lea 3.1. For every apping f : X Y and, k N, the following identity holds: (3.1) f(2x) 2 f(x) = 1! where and [ ( ) F 1 (x) ( ) ] F +1 (x) H(x), x X, F i (x) = x f(ix)!f(x), x X; i = 1,..., + 1 Proof. [19, Lea 2, page 102]. H(x) = 2xf(x)!f(2x), x X. Corollary 3.2. If f : X Y is a onoial of degree, then (3.2) f(2x) = 2 f(x), x X. Proof. If f is a onoial of degree, then F 1 (x) = = F +1 (x) = H(x) = 0 for each x X. Hence the result follows fro Lea 3.1. Lea 3.3. Let ψ : X [0, ) be a function and E = Y X. Define d(g, h) = inf{a > 0 : g(x) h(x) a q ψ(x) x X}, g, h E. Then d is a generalized coplete etric on E. Proof. Clearly d(f, g) 0 for each f, g E and the equality holds only when f = g. By the definition, d is syetric. Let g, g, g E, d(g, g ) < a 1 and d(g, g ) < a 2. Then g(x) g (x) a q 1 ψ(x) and g (x) g (x) a q 2 ψ(x) for each x X. It follows that g(x) g (x) p g(x) g (x) p + g (x) g (x) p ( p, (a 1 + a 2 ) ψ(x)) x X. Therefore d(g, g ) a 1 + a 2. Since this holds for each a 1, a 2 with d(g, g ) < a 1 and d(g, g ) < a 2, d(g, g ) d(g, g ) + d(g, g ). Let {f n } be a Cauchy sequence in E. Then for each ε > 0, there is soe n 0 N such that d(f n, f ) < ε, n, n 0. By the definition, for each n, n 0, (3.3) f n (x) f (x) ε q ψ(x), x X. Hence, for each x X, {f n (x)} is a Cauchy sequence in coplete space Y. It follows that f(x) = li n f n (x) exists for each x X. By (3.3) for each n n 0, f(x) f n (x) = li f (x) f n (x) ε q ψ(x), x X.

5 STABILITY OF THE MONOMIAL EQUATIONS 781 Therefore d(f, f n ) ε for each n n 0. Hence li n d(f, f n ) = 0. This copletes the proof of the lea. Theore 3.4. Let ϕ : X X [0, ) and f : X Y satisfy the inequality (3.4) x f(y)!f(x) ϕ(x, y), x, y X for soe N. Define (3.5) ψ(x) = 1 ( [ ( ) ] p [ p ) q, 2 ϕ(x, (i + 1)x) + ϕ(2x, x)] x X.! i If for soe L < 1, (3.6) i=0 2 ψ(2x) Lψ(x), x X and li n 2 n ϕ(2 n x, 2 n y) = 0 for all x, y X, then there exists a unique onoial apping M : X Y of degree such that (3.7) Proof. By Lea 3.1 M(x) f(x) ψ(x) (1 L p ) q, x X. (3.8) 2 f(2x) f(x) ψ(x), x X. Let E = Y X and define d(g, h) = inf{a > 0 : g(x) h(x) a q ψ(x), x X}, g, h E. By Lea 3.3, d is a coplete generalized etric on E. Define J : E E by J(g)(x) = 2 g(2x) for each g E and x X. Let ε > 0 and a = d(g, h) + ε. Then by the definition, Thanks to (3.6), for each x X, g(x) h(x) a q ψ(x), x X. J(g)(x) J(h)(x) = 2 g(2x) 2 h(2x) 2 a q ψ(2x) La q ψ(x). By the definition, d(j(g), J(h)) L p a = L p (d(g, h) + ε). Since ε > 0 was arbitrary, d(j(g), J(h)) L p d(g, h), g, h E. This eans that J is a contractive apping with Lipschitz constant L p < 1. By (3.8), d(f, J(f)) 1, therefore, by Proposition 2.3, J has a unique fixed point M : X Y in the set F = {g E : d(f, g) < }, where M is defined by (3.9) M(x) := li n J n (f)(x) = li n 2 n f(2 n x), x X. Moreover, d(f, M) d(f, J(f)) 1 1 L p 1 L p.

6 782 ALIREZA KAMEL MIRMOSTAFAEE This eans that (3.7) holds. Thanks to (1.1) and (3.9), for each x, y X, we have x M(y)!M(x) ( ) = ( 1) j M(jx + y)!m(x) j j=0 = li n 2 n ( ( 1) j j j=0 = li n 2 n 2 n xf(2 n y)!f(2 n x) li n 2 n ϕ(2 n x, 2 n y) = 0. ) f(2 n jx + 2 n y)!f(2 n x) Hence M is a onoial of degree. To prove the uniqueness assertion, let us assue that there exists a onoial M : X Y of degree, which satisfies (3.7). Thanks to Corollary 3.2, M is a fixed point of J in F. However, by Proposition 2.3, J has only one fixed point in F, hence M M. The following result can be obtained by iitating the proof of Theore 3.4. Theore 3.5. With the notations of Theore 3.4, let (3.4) hold for soe N. If for soe positive L < 1, and li n 2 n ϕ(2 n x, 2 n y) = 0. M : X Y of degree such that 2 ψ(2 1 x) Lψ(x), x X M(x) f(x) Then there exists a unique onoial Lψ(x) (1 L p ) q, x X. 4. Continuity behavior of onoial appings In this section, we investigate continuity of onoial appings in quasi p- nored spaces. In fact, we will show that under soe conditions on f and ψ, the onoial apping s M(sx) is continuous. Theore 4.1. Let the conditions of theore 3.4 (or Theore 3.5) hold. If for each x X, the function s f(sx) fro R to Y is continuous at soe point s 0 and for soe x 0 X, the function s ψ(sx 0 ) fro R to [0, ) is bounded in a neighborhood of s 0, then s M(sx 0 ) fro R to Y is continuous at s 0. Proof. We prove the theore under conditions of Theore 3.4. The proof for the other case is siilar. Let α > 0 and δ 1 > 0 be such that s s 0 < δ 1 ψ(sx 0 ) < α.

7 STABILITY OF THE MONOMIAL EQUATIONS 783 Given ε > 0, choose soe n 0 so that L n 0 α (1 L p ) q < ε 3 q. Since 0 < L < 1, such a choice is possible. By the continuity of s f(2 n0 sx 0 ), we can find positive δ δ 1 such that (4.1) s s 0 < δ f(2 n 0 sx 0 ) f(2 n 0 ε s 0 x 0 ) < 3 q 2 n. 0 Let s s 0 < δ. Then M(sx 0 ) M(s 0 x 0 ) p = 2 n0 M(2 n0 sx 0 ) 2 n0 M(2 n0 s 0 x 0 ) p 2 n0p( M(2 n0 sx 0 ) f(2 n0 sx 0 ) p + f(2 n0 sx 0 ) f(2 n0 s 0 x 0 ) p + f(2 n 0 s 0 x 0 ) M(2 n 0 s 0 x 0 ) p) < εp 3 + εp 3 + εp 3 = εp. That is s s 0 < δ M(sx 0 ) M(s 0 x 0 ) < ε. The above inequality proves continuity of s M(sx 0 ) at s 0. Corollary 4.2. If s f(sx) and s ψ(sx) fro R to Y and R + respectively are continuous, then s M(sx) is continuous. In this case, M(tx) = t M(x) for each t R and x X. Proof. Follows fro Theore 4.1 and [13, Lea 5]. Corollary 4.3. Let ( X, ) be a nored space. Let for soe ε > 0 and positive real nuber β, f : X Y satisfy the inequality ( x f(y)!f(x) ε x β + y β), x, y X. (i) If β <, there is a unique onoial M : X Y of degree such that f(x) M(x) ε x β ( [ ( ) p ) q (1 + (i + 1) )] 2! (1 β + (2 β + 1) p) q, x X. 2 (βp p) i i=0 (ii) If β >, there is a unique onoial M : X Y of degree such that f(x) M(x) ε 2 ( β) x β ( [ ( ) p ) q (1 + (i + 1) )] 2! (1 β + (2 β + 1) p) q, x X. 2 (p βp) i i=0

8 784 ALIREZA KAMEL MIRMOSTAFAEE Furtherore, if f is continuous, then M is continuous and M(tx) = t M(x) for each t R and x X. ) Proof. Let ϕ(x, y) = ε ( x β + y β and ψ be defined by (3.5). Then (4.2) ψ(x) = ε x β 2! ( i=0 [ ( ) p (1 + (i + 1) )] β + (2 β + 1) p) q, x X. i If β <, then 2 ψ(2x) = 2 +β ψ(x), x X. Hence for L = 2 +β < 1, the conditions of Theore 3.4 hold. Therefore, we can find a unique onoial function of degree such that f(x) M(x) ψ(x) (1 2 (βp p) ) q, x X. This proves (i). In case (ii), when β >, by (4.2), we can write 2 ψ(2 1 x) = 2 β ψ(x), x X. By Theore 3.5 for L = 2 β, there is a unique onoial function M : X Y of degree such that f(x) M(x) The last assertion follows fro Theore ( β) ψ(x) (1 2 (p βp) ) q, x X. Corollary 4.4. Let for soe ε > 0, f : X Y satisfy the inequality x f(y)!f(x) ε, x, y X. Then there is a unique continuous onoial apping M : X Y of degree such that ( ( ε ) ) p q i=0 i + 1 f(x) M(x) (2!)(1 2 p ) q, x X. Furtherore if f is continuous, then M is continuous and M(tx) = t M(x) for each t R and x X. Proof. Apply Theore 3.4 for L = 2 and ϕ(x, y) = ε, x, y X. References [1] M. Albert and J. A. Baker, Functions with bounded nth differences, Ann. Polon. Math. 43 (1983), no. 1, [2] T. Aoki, Locally bounded linear topological spaces, Proc. Ip. Acad. Tokyo 18 (1942), [3], On the stability of the linear transforation in Banach spaces, J. Math. Soc. Japan 2 (1950), [4] Y. Benyaini and J. Lindenstrauss, Geoetric Nonlinear Functional Analysis. Vol. 1, Aerican Matheatical Society Colloquiu Publications, 48. Aerican Matheatical Society, Providence, RI, 2000.

9 STABILITY OF THE MONOMIAL EQUATIONS 785 [5] D. G. Bourgin, Classes of transforations and bordering transforations, Bull. Aer. Math. Soc. 57 (1951), [6] L. Cǎdariu and V. Radu, Rearks on the stability of onoial functional equations, Fixed Point Theory 8 (2007), no. 2, [7] J. B. Diaz and B. Margolis, A fixed point theore of the alternative, for contractions on a generalized coplete etric space, Bull. Aer. Math. Soc. 74 (1968), [8] A. Gilányi, Hyers-Ula stability of onoial functional equations on a general doain, Proc. Natl. Acad. Sci. USA 96 (1999), no. 19, [9], On the stability of onoial functional equations, Publ. Math. Debrecen 56 (2000), no. 1-2, [10] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), [11], Transforations with bounded th differences, Pacific J. Math. 11 (1961), [12] S.-M. Jung, T.-S. Ki, and K.-S. Lee, A fixed point approach to the stability of quadratic functional equation, Bull. Korean Math. Soc. 43 (2006), no. 3, [13] Z. Kaiser, On stability of the onoial functional equation in nored spaces over fields with valuation, J. Math. Anal. Appl. 322 (2006), no. 2, [14] Y.-H. Lee, On the stability of the onoial functional equation, Bull. Korean Math. Soc. 45 (2008), no. 2, [15] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), no. 1, [16] J. M. Rassias, Alternative contraction principle and Ula stability proble, Math. Sci. Res. J. 9 (2005), no. 7, [17] Th. M. Rassias, On the stability of the linear apping in Banach spaces, Proc. Aer. Math. Soc. 72 (1978), no. 2, [18] S. M. Ula, Probles in Modern Matheatics, Science Editions John Wiley & Sons, Inc., New York [19] D. Wolna, The stability of onoial functions on a restricted doain, Aequationes Math. 72 (2006), no. 1-2, Departent of Pure Matheatics Center of Excellence in Analysis on Algebraic Structures Ferdowsi University of Mashhad P.O. 1159, Mashhad 91775, Iran E-ail address: irostafaei@ferdowsi.u.ac.ir