A Fixed Point Approach of Quadratic Functional Equations

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1 Int. Journl of Mth. Anlysis, Vol. 7, 03, no. 30, HIKARI Ltd, A Fixed Point Approch of Qudrtic Functionl Equtions Mudh Almhlebi Deprtment of Mthemtics, Fculty of Sciences University of Ibn Tofil, Kenitr, Morocco mudh979@hotmil.fr Copyright c 03 Mudh Almhlebi. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. Abstrct. In this pper, by using the fixed point method in Bnch spces, we prove the generlized Hyers-Ulm-Rssis stbility for the qudrtic functionl eqution f(x+y +z)+f(x+y z)+f(x y +z)+f(x y z) = f(x)+4f(y)+4f(z). Keywords: Hyers-Ulm-Rssis Stbility, Fixed point method, qudrtic functionl eqution. Introduction nd preliminries Questions concerning the stbility of functionl eqution seem to hve been first rised by Ulm [7]. Hyers [6] showed tht, if we hve δ > 0 nd the mpping f : E F, where E nd F re Bnch spces, such tht f(x + y) f(x) f(y) δ for ll x, y E, then there exists unique mpping S : E F such tht S(x + y) = S(x) + S(y) nd f(x) S(x) δ. The pper of Rssis [5] hs provided lot of influence in the development of wht we cll generlized Hyers-Ulm stbility or s Hyers-Ulm-Rssis stbility of functionl equtions. A generliztion of the Rssis theorem ws obtined by Găvrut [5] by replcing the unbounded Cuchy difference by generl control function in the spirit of Rssis pproch. The functionl eqution f(x + y) + f(x y) = f(x) + f(y) (.) is clled qudrtic functionl eqution. In prticulr, every solution of the qudrtic functionl eqution is sid to be qudrtic function. A generlized

2 47 Mudh Almhlebi Hyers-Ulm stbility problem for the qudrtic functionl eqution hs been studied in [9] nd [3]. We recll fundmentl result in fixed point theory. Let X be set. A function d : X X [0, ] is clled generlized metric on X if d stisfies () d(x, y) = 0 if nd only if x = y; () d(x, y) = d(y, x) for ll x, y X; (3) d(x, z) d(x, y) + d(y, z) for ll x, y, z X. Theorem.. [4] Suppose we re given complete generlized metric spce (X, d) nd strictly contrctive mpping J : X X, white the Lipshitz constnt L <. If there exists nonnegtive integer k such tht d(j k x, J k+ x) < for some x X, then the following re true: () the sequence J n x converges to fixed point x of J; () x is the unique fixed point of J in the set Y = {y X : d(j k x, y) < }; (3) d(y, x ) d(y, Jy) for ll y Y. L In this pper, ssume tht X is normed vector spce with norm. nd tht Y is Bnch spce with norm.. For f : X Y nd φ : X 3 [0, ), we consider the functionl eqution f(x+y+z)+f(x+y z)+f(x y+z)+f(x y z) = f(x)+4f(y)+4f(z) (.) nd the difference opertor Df : X 3 Y s Df(x, y, z) = f(x + y + z) + f(x + y z) + f(x y + z) + f(x y z) f(x) 4f(y) 4f(z). The im of this pper is to prove the generlized Hyers-Ulm-Rssis stbility for functionl eqution (.) by using the fixed point method in Bnch spces.. Hyers-Ulm stbility of (.) In this section, we prove the generlized Hyers-Ulm-Rssis stbility for the functionl eqution(.). The functionl eqution (.) is connected with the eqution (.) s follows Lemm.. A function f : X Y stisfies the functionl eqution f(x+y+z)+f(x+y z)+f(x y+z)+f(x y z) = f(x)+4f(y)+4f(z) (.) for x, y, z in X, If nd only if f stisfies the functionl eqution for x, y X. f(x + y) + f(x y) = f(x) + f(y) (.) Proof. Suppose tht f stisfies (.), then by putting x = y = z = 0 in (.), we obtin f(0) = 0. By putting y = z = 0 in (.), we get f(x) = 4f(x). Putting z = 0 in (.), we obtin tht f stisfies (.).

3 A fixed point pproch of qudrtic functionl equtions 473 Assume tht f stisfies (.), then by putting x = y = 0, we get f(0) = 0 nd by putting y = x in (.), we get f(x) = 4f(x). Then, f(x+y +z)+f(x+y z)+f(x y +z)+f(x y z) = 4f(x)+4f(y)+4f(z) = f(x) + 4f(y) + 4f(z). Using the fixed point method, we prove the generlized Hyers -Ulm-Rssis stbility of the qudrtic functionl eqution Df(x, y, z) = 0. Theorem.. Let f : X Y be mpping with f(0) = 0 for which there exists function φ : X 3 [0, ) such tht there exists n L < such tht φ(x, x, x) 4Lφ(x, x, x ) for ll x X, nd lim n + 4 n φ( n x, n y, n z) = 0 (.3) Df(x, y, z) φ(x, y, z) (.4) for ll x, y, z X. Then, there exists unique qudrtic mpping Q : X Y stisfying (.) nd f(x) Q(x) φ(x, x, x) (.5) 8 8L for ll x X. Proof. Let us consider the set S = {g : X Y } nd we introduce the generlized metric on S s follows : d(g, h) = inf {K [0, ) : g(x) h(x) Kφ(x, x, x), x X}. It is esy to show tht (S, d) is complete (see for exmple the proof of Theorem.5 in []). Now, let us consider the liner mpping J : S S such tht Jg(x) := g(x) (.6) 4 for ll x X. First we ssert tht J is strictly contrctive on X. Given g, h S, let K [0, ) be n rbitrry constnt with d(g, h) K, tht is g(x) h(x) Kφ(x, x, x). So we hve Jg(x) Jf(x) = 4 g(x) f(x) K φ(4x, x, x) KLφ(x, x, x) 4 tht is, d(jg, Jh) Ld(g, h) (.7) for ll g, h S. Letting x = x, y = x nd z = x in (.4), we get for ll x X. So, we obtin tht f(x) 8f(x) φ(x, x, x) (.8) d(f, Jf) 8. By Theorem., there exists mpping Q : X Y stisfying the following

4 474 Mudh Almhlebi () Q is fixed point of J, tht is, Q(x) = 4Q(x) for ll x X. The Q is unique fixed point of J in the set M = {g S : d(f, g) }. This implies tht Q is unique mpping such tht there exists K (0, ) stisfying f(x) Q(x) Kφ(x, x, x), for ll x X. () lim J n f( n x) f(x) = lim = Q(x) (.9) n + n + 4 n for ll x X. (3) d(f, Q) d(f, Jf), (.0) L which implies the inequlity d(f, Q) 8 8L. (.) This implies tht the inequlity (.5) holds. From (.3),(.4) nd (.9), we get DQ(x, y, z) = lim n + 4 n Df(n x, n y, n z) lim n + 4 n φ(n x, n y, n z) = 0 for ll x, y, z X. So, Df(x, y, z) = 0 for ll x, y, z X. By lemm (.), the mpping Q : X Y is qudrtic. Corollry.3. Let p < nd θ 0 be rel numbers, nd let f : X Y be mpping such tht ( x Df(x, y, z) θ p + y p + z p) (.) for ll x, y, z X. Then, there exists unique qudrtic mpping Q : X Y stisfying (.) nd f(x) Q(x) 3θ 8 p+ x p, x X. (.3) Proof. We get the result from Theorem. by tking φ(x, y, z) := θ( x p + y p + z p ) (.4) for ll x, y, z X. Then, we cn choose L = p nd we get the desired result. Corollry.4. Let p, q, r such tht p + q + r < nd θ 0 be rel numbers, nd let f : X Y be mpping such tht ( x Df(x, y, z) θ p. y q. z r) (.5) for ll x, y, z X. Then, there exists unique qudrtic mpping Q : X Y stisfying (.) nd f(x) Q(x) θ 8 p+q+r+ x p+q+r, x X. (.6)

5 A fixed point pproch of qudrtic functionl equtions 475 Proof. We get the result from Theorem. by tking φ(x, y, z) := θ( x p. y q. z r ) (.7) for ll x, y, z X. Then, we cn choose L = p+q+r nd we get the desired result. For convenience, we use the following bbrevition for given N nd mpping f : X Y D f(x, y, z) := f(x + y + z) + f(x + y z) + f(x y + z) + f(x y z) for ll x, y, z X. 4 f(x) 4f(y) 4f(z) (.8) Theorem.5. Let f : X Y be mpping with f(0) = 0 for which there exists function φ : X 3 [0, ) stisfying D f(x, y, z) φ(x, y, z) (.9) lim n + 4 n φ( n x, n y, n z) = 0 (.0) for ll x, y, z X. Let 0 < L < be constnt such tht the mpping x ψ(x, y, z) := φ( x, y, z) + φ(x, x, 0) + 3 φ(x, 0, 0) stisfying ψ(x, x, x) 4Lψ(x, x, x ) for ll x X, then there exists unique qudrtic mpping Q : X Y stisfying (.) nd f(x) Q(x) ψ(x, x, x), x X. (.) 8 8L Proof. From (.9), we hve D f(x, y, z) D f(x, 0, 0) φ(x, y, z) + φ(x, 0, 0) (.) for ll x, y, z X. Therefore, f(x + y + z) + f(x + y z) + f(x y + z) + f(x y z) 4f(x) 4f(y) 4f(z) φ(x, y, z) + φ(x, 0, 0) (.3) for x, y, z X. Replcing x by x in (.3), we get f(x+y+z)+f(x+y z)+f(x y+z)+f(x y z) 4f(x) 4f(y) 4f(z) Letting y = x nd z = 0 in (.4), we get From (.4) nd (.5), we get φ( x, y, z) + φ(x, 0, 0) (.4) f(x) 4f(x) φ(x, x, 0) + φ(x, 0, 0) (.5) f(x+y+z)+f(x+y z)+f(x y+z)+f(x y z) f(x) 4f(y) 4f(z) ψ(x, y, z) (.6)

6 476 Mudh Almhlebi for x, y, z X, where ψ(x, y, z) := φ( x, y, z) + φ(x, x, 0) + 3 φ(x, 0, 0). (.7) Let S := {g : X Y }. We introduce generlized metric on S s d(g, h) = inf {K [0, ) : g(x) h(x) Kψ(x, x, x), x X}. (.8) We consider the mpping J : S S defined by Jg(x) := g(x), g S, x X. (.9) 4 Similr to the proof of Theorem., we get the desired result. References [] L. Cădriu nd V. Rdu, Fixed points nd the stbility of Jensen s functionl eqution, Journl of Inequlities in Pure nd Applied Mthemtics, vol. 4, no., rticle 4, 003. [] L. Cădriu nd V. Rdu, On the stbility of the Cuchy functionl eqution: fixed point pproch, Grzer Mth. Ber., 346(004), [3] St. Czerwik, On the stbility of the qudrtic mpping in normed spces, Abhndlungen us dem Mthemtischen Seminr der Universitt Hmburg, vol. 6, pp , 99. [4] J. B. Diz nd B. Mrgolis, A fixed point theorem of the lterntive, for contrctions on generlized complete metric spce, Bulletin of the Americn Mthemticl Society, vol. 74, pp , 968. [5] P. Găvrut, A generliztion of the Hyers-Ulm-Rssis stbility of pproximtely dditive mppings, Journl of Mthemticl Anlysis nd Applictions, vol. 84, no. 3, pp , 994. [6] D. H.Hyers, On the stbility of the liner functionl eqution, Proceedings of the Ntionl Acdemy of Sciences of the United Sttes of Americ, vol. 7, no. 4, pp. 4, 94. [7] D.H. Hyers, G. Isc,nd Th.M. Rssis Stbility of Functionl Equtions in Severl Vribles, Birkhuser, Bsel 998. [8] G. Isc nd Th. M. Rssis, Stbility of ψ-dditive mppings: pplictions to nonliner nlysis, Interntionl Journl of Mthemtics nd Mthemticl Sciences, vol. 9, no., pp. 9 8, 996. [9] K.-W. Jun nd Y.-H. Lee, On the Hyers-Ulm-Rssis stbility of pexiderized qudrtic inequlity, Mthemticl Inequlities Applictions, vol. 4, no., pp. 93 8, 00. [0] S.-M. Jung, On the Hyers-Ulm stbility of the functionl equtions tht hve the qudrtic property, Journl of Mthemticl Anlysis nd Applictions, vol., no., pp. 6 37, 998. [] M. Mirzvziri nd M. S. Moslehin, A fixed point pproch to stbility of qudrtic eqution, Bulletin of the Brzilin Mthemticl Society, vol. 37, no. 3, pp , 006. [] C.-G. Prk, On the stbility of the qudrtic mpping in Bnch modules, Journl of Mthemticl Anlysis nd Applictions, vol. 76, no., pp , 00 [3] C.-G. Prk nd Th. M. Rssis, Hyers-Ulm stbility of generlized Apollonius type qudrtic mpping, Journl of Mthemticl Anlysis nd Applictions, vol. 3, no., pp , 006.

7 A fixed point pproch of qudrtic functionl equtions 477 [4] C. Prk, Fixed points nd Hyers-Ulm-Rssis stbility of Cuchy-Jensen functionl equtions in Bnch lgebrs, Fixed Point Theory nd Applictions, vol. 007, Article ID 5075, 5 pges, 007. [5] Th. M. Rssis, On the stbility of functionl equtions in Bnch spces, Journl of Mthemticl Anlysis nd Applictions, vol. 5, no., pp , 000. [6] F. Skof, Locl properties nd pproximtion of opertors, Rendiconti del Seminrio- Mtemtico e Fisico di Milno, vol. 53, pp. 39, 983. [7] S. M. Ulm, Problems in Modern Mthemtics, John Wiley Sons, New York, NY, USA, 964. Received: August 5, 0