A Fixed Point Approach to the Stability of a Quadratic-Additive Type Functional Equation in Non-Archimedean Normed Spaces
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1 International Journal of Mathematical Analysis Vol. 9, 015, no. 30, HIKARI Ltd, A Fied Point Approach to the Stability of a Quadratic-Additive Type Functional Equation in Non-Archimedean Normed Spaces Sun Sook Jin Department of Mathematics Education Gongju National University of Education Gongju , Republic of Korea Yang-Hi Lee Department of Mathematics Education Gongju National University of Education Gongju , Republic of Korea Copyright c 015 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 generalized Hyers Ulam stability for the functional equation f(a + y + af(y a( f( in non-archimedean normed spaces. a( f( (f(y = 0 Mathematics Subject Classification: 39B5, 39B8 Keywords: non-archimedean normed space, quadratic-additive type functional equation, stability of a functional equation
2 1478 Sun-Sook Jin and Yang-Hi Lee 1 Introduction A stability problem of functional equations was formulated by S. M. Ulam [14] in 1940, as follows: when is it true that a mapping, which approimately satisfies a functional equation, must be somehow close to an eact solution of the equation?. In following year, D. H. Hyers [5] gave a partial solution of Ulam s problem for the case of approimate additive mappings between Banach spaces. Hyers Theorem was generalized by T. Aoki [1] and D. G. Bourgin [] for additive mappings by allowing the Cauchy difference operator to be controlled by ε( p + y p. In 1978, Th.M. Rassias [1] also provided a generalization of Hyers Theorem for linear mappings by allowing the Cauchy difference operator to be controlled by ε( p + y p, which was generalized by P. Gǎvruta in [4] and S.-M. Jung in [6]. By a non-archimedean field we mean a field K equipped with a function (valuation from K into [0, such that r = 0 if and only if r = 0, rs = r s, and r + s ma{ r, s } for all r, s K. Clearly 1 = 1 and n 1 for all n N. Let X be a vector space over a scalar field K with a non-archimedean non- trivial valuation. A function : X R is a non-archimedean norm (valuation if it satisfies the following conditions: (i = 0 if and only if = 0; (ii r = r (r K, X; (iii the strong triangle inequality (ultrametric; namely, + y ma{, y } (, y X. Then (X, is called a non-archimedean space. Due to the fact that n m ma{ j+1 j : m j n 1}, (n > m, a sequence { n } is Cauchy if and only if { n+1 n } converges to zero in a non-archimedean space. By a complete non-archimedean space we mean one in which every Cauchy sequence is convergent in the space. Recently, M. S. Moslehian and Th. M. Rassias [11] discussed the Hyers Ulam stability of the Cauchy functional equation and the quadratic functional equation in non-archimedean normed spaces. Now we consider the quadratic-additive type functional equation a( a( f(a + y + af(y f( f( (f(y = 0 (1 whose solution is called a quadratic-additive mapping. For the case a = 1, Lee [10], H.-K. Kim and Y.-H. Lee [9], and G.-H. Kim and Y.-H. Lee [8] investigated the stability of the equation (1 on restricted domains, in Fuzzy spaces, and in nonarchimedean spaces, respectively. In this paper, we are going to investigate alternative stability results of the quadratic-additive functional equation (1 in non-archimedean normed spaces by using the fied point method.
3 Stability of quadratic-additive type functional equations 1479 Generalized Hyers-Ulam stability of (1 We recall the following result of the fied point theory by Diaz and Margolis. Theorem.1 ([3, 13] Suppose that a complete generalized metric space (X, d, which means that the metric d may assume infinite value, and a strictly contractive mapping J : X X with the Lipschitz constant 0 < L < 1 are given. Then, for each given element X, either d(j n, J n+1 = + for all positive integers n, or there eists a nonnegative integer k such that: (1 d(j n, J n+1 < + for all n k; ( the sequence {J n } is convergent to a fied point y of J; (3 y is the unique fied point of J in Z := {y X : d(j k, y < + }; (4 d(y, y (1/(1 Ld(y, Jy for all y Z. Let V and W be vector spaces. For a given mapping f : V W, we use the abbreviations Af(, y := f( + y f( f(y, Qf(, y := f( + y + f( y f( f(y, D a f(, y := a( f(a + y + af(y f( a( f( (f(y for all, y V, where a is a fied positive integer. We call a mapping f an additive mapping or a quadratic mapping if Af( = 0 or Qf( = 0 for all, y V, respectively. Theorem. Let V and W be vector spaces. A mapping f : V W satisfies the inequality D a f(, y = 0 for all, y V if and only if there eist a quadratic mapping g : V W and an additive mapping h : V W such that f( = g( + h( for all V.
4 1480 Sun-Sook Jin and Yang-Hi Lee Proof (Necessity We decompose f into the even part and the odd part by putting f( + f( f( f( g( =, h( = for all V. Notice that f(0 = Daf(0,0 a(a+1 Qg(, y = D ag(y, ( D a g( + y, a( = 0, = 0. From the equalities D ag(, y Ah(, y = D ah(y, a + (D a h(, 0 D a h( + y, 0 a( = 0 + D ag(, a + D ah(, y for all, y V, we conclude that g is a quadratic mapping and h is an additive mapping. (Sufficiency If there eist a quadratic mapping g : V W and an additive mapping h : V W such that f( = g( + h( for all V, then g(0 = h(0 = 0, g( = g(, and D 1 g(, y = Qg(, y = 0 for all, y V. Assume that D k g(, y = 0 for all, y V and k 1. Then D k+1 g(, y = D k g(, + y + (k + 1Qg(, y = 0 for all, y V. By induction, it follows that D a g(, y = 0 for all, y V. Since h is an additive mapping, we have h( = h(, h(a = ah(, and D a h(, y = Ah(a, y + aah(y, = 0 for all, y V. Now we prove the generalized Hyers-Ulam stability of the functional equation Df 0, using Theorem.1. Theorem.3 Let V be a vector space over a scalar field K, and let (Y, be a complete non-archimedean space over K. Let ϕ : V [0, be a function such that there eists an L < 1 with ( ϕ(, y ( Lϕ, y (
5 Stability of quadratic-additive type functional equations 1481 for all, y V. Suppose that f : V Y is a mapping satisfying D a f(, y ϕ(, y (3 for all, y V. Then there eists a unique quadratic-additive mapping T : V Y such that f( T ( for all V, where ϕ is given by ϕ( ( (1 L ϕ( := ma{ϕ(,, ϕ(, } for all V. In particular the mapping T is represented by T ( = lim [ f (( n + f ( ( n + f ((n f ( ( n ] n ( n ( n for all V. Proof. Let S be the set of all functions g : V Y with g(0 = 0. We introduce a generalized metric on S by d(g, h := inf {K [0, ] g( h( K ϕ( for all V }. It is not difficult to show that (S, d is a complete generalized metric space (see [7]. Now we consider the operator J : S S defined by (4 Jg( := g(( g( ( ( + g(( + g( ( ( for all V. Let g, h S be given such that d(g, h < β; by the definition, g( h( β ϕ( for all V. Hence, we have { (a + (g(( h(( Jg( Jh( ma (, a(g( ( h( ( } ( { } a + ϕ(( a ϕ( ( β ma, ( ( { } ϕ(( ϕ( ( β ma, ( ( βl ma {( ϕ(, ϕ( }
6 148 Sun-Sook Jin and Yang-Hi Lee for all V, which implies that d(jg, Jh Ld(g, h for any g, h S. That is, J is a strict contraction with the Lipschitz constant L. Moreover, by (, we see that a f( Jf( = ( D af(, a + ( D af(, { } a + ϕ(, a ϕ(, ma, ( ( 1 ma {ϕ(,, ϕ(, } ( 1 for all V, i.e., d(f, Jf < (see the definition of d. Therefore, (a+1 by Theorem.1 (, the sequence {J n } is convergent to a fied point T of J, i.e., T ( := lim J n f( n = lim n [ f(( n f( ( n + f((n + f( ( n ( n ( n for all V. Moreover, we have d(f, T 1 1 L d(f, Jf 1 ( (1 L which implies the validity of (4. Replacing, y by ( n, ( n y, respectively, in ( and (3, we have D a T (, y D a f(( n, ( n y D a f( ( n, ( n y = n lim ( n + D af(( n, ( n y + D a f( ( n, ( n y ( n { ( n + 1 lim ma n ( n ϕ((n, ( n y, ( n } + 1 ( n ϕ( (n, ( n y { 1 n lim ma ( n ϕ((n, ( n y, } 1 ( n ϕ( (n, ( n y lim L n ma {ϕ(, y, ϕ(, y} n = 0 ]
7 Stability of quadratic-additive type functional equations 1483 for all, y V. This follows that T is a quadratic-additive mapping. To prove the uniqueness assertion, let us assume that there eists another quadraticadditive mapping T : V Y which satisfies (4. Then, by Theorem., we get QT e 0 and AT o 0 which follows that T ( = T e( + T o( = T e((a T o((a + 1 ( ( = JT ( for all V, i.e., T is a fied point of J in S. However, by Theorem.1, J has only one fied point in S, and hence T = T. This completes the proof. Theorem.4 Let V be a vector space over a scalar field K, and let (Y, be a complete non-archimedean space over K. Let ϕ : V [0, be a function such that there eists an L < 1 with L ϕ(, y ϕ ((, (y (5 for all, y V. Suppose that f : V Y is a mapping satisfying D a f(, y ϕ(, y (6 for all, y V. Then there eists a unique quadratic-additive mapping T : V Y such that f( T ( L ϕ( (1 L (7 for all V, where ϕ is given by ϕ( := ma{ϕ(,, ϕ(, } for all V. In particular the mapping T is represented by for all V. [ ( ( ( ( n T ( = n lim f + f ( n ( n ( ( ( ] (n + f f ( n ( n Proof. Let S be the set of all functions g : V Y with g(0 = 0. We introduce a generalized metric on S by d(g, h := inf {K [0, ] g( h( K ϕ( for all V }.
8 1484 Sun-Sook Jin and Yang-Hi Lee It is not difficult to show that (S, d is a complete generalized metric space (see [7]. Now we consider the operator J : S S defined by Jg( := ( ( g ( + g ( + ( g ( ( g for all V. Let g, h S be given such that d(g, h < β; by the definition, for all V. Hence, we have g( h( β ϕ( Jg( Jh( { (( ( ( ( + ( ma g h, (( ( ( ( a 1 } g h { ( ( + β ma ϕ, (a + ( 1 a 1 } ϕ { ( ( } β ma ϕ, ϕ βl ma {( ϕ(, ϕ( } for all V, which implies that d(jg, Jh Ld(g, h for any g, h S. That is, J is a strict contraction with the Lipschitz constant L. Moreover, by (5, we see that ( f( Jf( = D af, ( ϕ, L ma {ϕ(,, ϕ(, } for all V, i.e., d(f, Jf L < (see the definition of d. Therefore, a+1 by Theorem.1 (, the sequence {J n } is convergent to a fied point T of J, i.e., ( ( ( T ( := n lim J n f( = n lim ( ( (n + f [ ( n ( n f f + f ( n ]. ( ( n ( n
9 Stability of quadratic-additive type functional equations 1485 Moreover, we have d(f, T 1 1 L d(f, Jf L (1 L y which implies the validity of (7. Replacing, y by,, respectively, (a+1 n (a+1 n in (5 and (6, we have ( D a T (, y = lim n ( ( n D a f (, y n ( n ( D a f (, y n ( n ( ( (n + D a f (, y n ( n ( +D a f (, y n ( n { ( n + ( n ( lim ma ϕ n (, y, n ( n ( n ( n ( } ϕ (, y n ( n { ( n ( n lim ma ϕ (, y, n ( n ( n ( } ϕ (, y n ( n n lim L n ma {ϕ(, y, ϕ(, y} = 0 for all, y V. To prove the uniqueness assertion, let us assume that there eists a quadratic-additive mapping T : X Y which satisfies (3.4. Then, by Theorem., we get QT e 0 and AT o 0 which follows that ( ( T ( = T e( + T o( = ( T e + (T o = JT ( for all V, i.e., T is a fied point of J in S. However, by Theorem.1, J has only one fied point in S, and hence T = T. This completes the proof. References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, (1950,
10 1486 Sun-Sook Jin and Yang-Hi Lee [] D.G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951, [3] J. B. Diaz and B. Margolis, A fied point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968, [4] P. Gǎvruta, A generalization of the Hyers Ulam Rassias stability of approimately additive mappings, J. Math. Anal. Appl. 184 (1994, [5] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 7 (1941, 4. [6] S.-M. Jung, On the Hyers Ulam Rassias stability of approimately additive mappings, J. Math. Anal. Appl. 04 (1996, [7] S.-M. Jung, T.-S. Kim and K.-S. Lee, A fied point approach to the stability of quadratic functional equation, Bull. Korean Math. Soc. 43 (006, no. 3, [8] G.-H. Kim and Y.-H. Lee, Stability of the Cauchy additive and quadratic type functional equation in non-archimedean normed spaces, Far East J. Math. Sci. 76 (013, [9] H.-M. Kim and Y.-H. Lee, Stability of functional equation and inequality in fuzzy normed spaces, J. Chungcheong Math. Soc. 6 (013, [10] Y.-H. Lee, Hyers-Ulam-Rassias stability of a quadratic-additive type functional equation on a restricted domain, Int. J. Math. Analysis 7 (013, [11] M. S. Moslehian and Th. M. Rassias, Stability of functional equations in non-archimedean spaces, Appl. Anal. Discrete Math. 10 (007, [1] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 7 (1978,
11 Stability of quadratic-additive type functional equations 1487 [13] V. Radu, The fied point alternative and the stability of functional equations, Fied Point Theory 4 (003, no. 1, [14] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, Received: March 7, 015; Published: May 1, 015
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