Research Article On the Stability of Alternative Additive Equations in Multi-β-Normed Spaces

Transcription

1 Function Spaces Volume 206, Article ID , 7 pages Research Article On the Stability of Alternative Additive Equations in Multi-β-Normed Spaces Xiuzhong Yang, Jing Ma, and Guofen Liu College of Mathematics and Information Science, Hebei Normal University and Hebei Key Laboratory of Computational Mathematics and Applications, Shijiazhuang , China Correspondence should be addressed to Guofen Liu; liugf2003@63.com Received 7 April 206; Accepted 30 June 206 Academic Editor: Rudolf L. Stens Copyright 206 Xiuzhong Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. We introduce the notion of multi-β-normed space (0 <β ) and study the stability of the alternative additive functional equation of two forms in this type of space.. Introduction In 940, Ulam [] proposed the following stability problem: given a metric group G(, ρ), anumberε>0,andmapping f:g Gwhich satisfies the inequality ρ(f(x y), f(x) f(y)) < ε for all x, y in G, does there exist an automorphism a of G and a constant >0, depending only on G,suchthat ρ(a(x), f(x)) ε for all x in G? If the answer is affirmative, we call the equation a(x y) = a(x) a(y) of automorphism stable. One year later, Hyers [2] provided a positive partial answer to Ulam s problem. In 978, a generalized version of Hyers result was proved by Rassias in [3]. Since then, the stability problems of several functional equations have been extensively investigated by a number of authors [4 2]. In particular, we also refer the readers to the survey paper [3]forrecentdevelopmentsinUlam stypestability,[4] forrecentdevelopmentsoftheconditionalstabilityofthe homomorphism equation, and boos [5 8] for the general understanding of the stability theory. The notion of multinormed space was introduced by Dales and Polyaov [9]. This concept is somewhat similar to operator sequence space and has some connections with operator spaces. Because of its applications in and outside of mathematics, the study on the stability of various functional equations has become one of the most important research subjects in the field of functional equations and attracts much attention from many researchers worldwide. Many examples of multinormed spaces can be found in [9], and further development of the stability in multinormed spaces can be found in papers [20 24]. In order to study the stability problem in more general setting, in this paper we introduce the notion of multi-βnormed spaces which are the combination of multinormed spaces and β-normed spaces, and the definition is given as follows. In this paper we will use the following notations. Let (E, ) be a complex β-normed space with 0 < β, and let. We denote by E the linear space E E consisting of -tuples (x,...,x ),wherex,...,x E.The linear operations on E are defined coordinatewise. The zero element of either E or E is denoted by 0. Wedenotetheset N ={,2,...,}and denote by S the group of permutation on N. Definition. Amulti-β-norm on {E :}is a sequence ( )=( :)such that is β-norm on E for each, x = x foreach x E,andthefollowing axioms are satisfied for each with 2: (A) (x σ(),...,x σ() ) = (x,...,x ), (σ S, x,...,x E); (A2) (α x,...,α x ) (max i N α i β ) (x,...,x ), (α,...,α C, x,...,x E); (A3) (x,...,x,0) = (x,...,x ), (x,..., x E);

2 2 Function Spaces (A4) (x,...,x,x ) = (x,...,x ), (x,..., x E). In this case, we say that ((E, ):)is a multi-βnormed space. The following two properties of multi-β-normed spaces are easily obtained: (x,...,x) = x, (x E) ; () max i N x i (x,...,x ) x i i= max i N x i, (x,...,x E). It follows from (2) that if (E, ) is a complete β-normed space, then (E, ) is complete β-normed space for each ;inthiscase((e, ):)is a complete multi- β-normed space. In particular, if β=(e, )is Banach space, then the space ((E, ):)is multi-banach space. Now we give one example of multi-β-normed space. Example 2. Let E be an arbitrary β-normed space. The sequence (, N)on E :defined by (2) (x,...,x ) = max i N x i, (x,...,x E), (3) Obviously (6) is equivalent to the alternative Jensen equation A( x+y )= [A (x) +A(y)]. (7) 2 2 Definition 6 (see [25]). Let X, Y be linear spaces and let A be mapping from X to Y. Theequationiscalledalternative additive of the second form if A satisfies the functional equation A (x +x 2 ) +A(x x 2 ) = 2A( x 2 ). (8) Obviously (6) is equivalent to the alternative Jensen equation A( x y )= [A (x) A(y)]. (9) Stability of Alternative Additive Equation of the First Form In this section we will study the stability of the alternative additive equation of the first form in multi-β-normed space and on the restricted domain. First, we investigate the general casewherethedomainofthemappingisthewholespace.the following theorem is obtained. Theorem 7. Let X be a real normed space, and let ((Y n, ): n N)be a complete real multi-β-normed space. Suppose that δ 0;mappingf:X Ysatisfies is a multi-β-norm. Lemma 3. Let N and (x,...,x ) E.Foreach j {,...,},let(x j n ) n=,2,... be a sequence in E such that lim n x j n =x j.thenforeach(y,...,y ) E one has (f (x +y )+f(x y ) +2f( x ),...,f(x +y )+f(x y ) +2f( x )) δ; (0) lim n (x n y,...,x n y )=(x y,...,x y ). (4) Definition 4. Let ((E, ):)be a multi-β-normed space. A sequence (x n ) in E is a multinull sequence if, for each ε>0, there exists n 0 Nsuch that (x n,...,x n+ ) <ε (5) for all n n 0.Letx E;wesaythatthesequence(x n ) is multiconvergent to x in E if (x n x)is a multinull sequence. In this case, x is called the limit of the sequence (x n ) and we denote it by lim n x n =x. In this paper we will study the stability in the multi- β-normed space of alternative additive equation of the two forms, which were further studied in the normed spaces in paper [25], and their definitions are presented as follows. Definition 5 (see [25]). Let X, Y be linear spaces and let A be mapping from X to Y. Theequationiscalledalternative additive of the first form if A satisfies the functional equation A (x +x 2 ) +A(x x 2 ) = 2A( x ). (6) (f ( z )+f(z ),...,f( z )+f(z )) δ 2 β () for all x,...,x,y,...,y,z,...,z X.Thenthere exists unique alternative additive mapping of the first form A:X Y satisfying (f (z ) A(z ),...,f(z ) A(z )) for all z,...,z X. (2δ+ δ 4 β ) 2 β Proof. Letting x i =y i =0 (i=,...,)in (0) yields (2) (f (0),...,f(0)) δ. 4β (3) Setting x i =y i =z i (i=,...,)yields (f (2z )+f(0) +2f( z ),...,f(2z )+f(0) +2f( z )) δ. (4)

3 Function Spaces 3 It follows from (), (3), and (4) that (f (2z ) 2f(z ),...,f(2z ) 2f(z )) (f (2z )+f(0) +2f( z ),...,f(2z ) +f(0) +2f( z )) + ( 2 (f ( z )+f(z )),..., 2 (f ( z )+f(z ))) + ( f (0),..., f(0)) 2δ+ δ 4 β. Therefore, we have ( 2 n f(2n z ) f(z ),..., 2 n f(2n z ) f(z )) (2δ+ δ n 4 ) β 2, β ( 2 n+m f(2n+m z ) 2 n f(2n z ),..., 2 n f(2n z )) = 2 n+m f(2n+m z ) (2δ+ δ 4 β ) n+m =n+2 β for all m, n N, m. It follows from (A2)and(7)that ( 2 n+m f(2n+m x) 2 n f(2n x),..., 2 n+m+ f(2 n+m+ x) 2 n+ f(2n+ x)) = ( 2 n+m f(2n+m x) 2 n f(2n x),..., 2 ( 2 n+m f(2n+m (2 x)) 2 n f(2n (2 x)))) 2 n f(2n x),..., f(2 n (2 x))) ( 2 n+m f(2n+m x) 2 n+m f(2n+m (2 x)) 2 n (2δ+ δ 4 β ) n+m =n+ 2. β (5) (6) (7) (8) Hence {(/2 n )f(2 n x)} is Cauchy sequence, which must be convergent in complete real multi-β-normed space; that is, there exists mapping A : X Y such that A(x) fl lim n (/2 n )f(2 n x). Hence, for arbitrary ε>0, there exists n 0 N;ifn n 0,thenwehave ( 2 n f(2n x) A (x),..., 2 n+ f(2n+ x) A(x)) <ε. Considering (2), we obtain (9) lim n 2 x) A(x) =0, x X. (20) If we let n=0in (7), then we have ( 2 m f(2m z ) f(z ),..., 2 m f(2m z ) f(z )) (2δ+ δ m 4 ) β 2. β = (2) Letting m and maing use of Lemma 3 and (20), we now that mapping A satisfies (2). Let x, y X.Settingx = =x =2 n x, y = =y = 2 n y in (0) and dividing both sides by 2 nβ yield ( 2 n f(2n (x + y)) + 2 n f(2n (x y)) n f( 2n x),..., 2 n f(2n (x+y)) + 2 n f(2n (x y)) + 2 which together with () implies 2 n f( 2n x)) δ 2, nβ 2 n f(2n (x+y))+ 2 n f(2n (x y))+2 2 n f( 2n x) δ 2. nβ Taing limit as n,wehave (22) (23) A(x+y)+A(x y)+2a( x) = 0, (x, y X). (24) So A is the alternative additive mapping of the first form. It remainstoshowthata is uniquely determined. Let A :X Y be another alternative additive mapping of the first form that satisfies (2). It follows from (24) that some properties of mapping A are obtained: () If we let y=0,wegeta(x) = A(x), soa is odd mapping. (2) If we let x=y=0,wehavea(0) = 0. (3) Putting y = xyields A(2x) = 2A(x); thatis, A(x) = (/2)A(2x). (4) Replacing x, y with 2x, respectively, yields A(2 2 x) = 2 2 A(x);henceA(x) = (/2 2 )A(2 2 x).

4 4 Function Spaces (5) Replacing x, y with 2 2 x, respectively, yields A(2 3 x) = 2 3 A(x);thatis,A(x) = (/2 3 )A(2 3 x). Proceeding in an obvious fashion yields A(x) = (/2 n )A(2 n x). Similarly,wehaveA (x) = (/2 n )A (2 n x). Letting z = =z =xin (2) and in view of () we obtain f (x) A(x) (2δ+ δ 4 ) β 2 β. (25) Similarly we have f (x) A (x) (2δ+ δ 4 ) β 2 β. (26) Therefore, A (x) A (x) = 2 n A(2n x) 2 n A (2 n x) x) A (2 n x) 2 nβ A(2n 2 nβ A(2n x) f (2 n x) + x) A (2 n x) 2 nβ f(2n 2 (2δ + δ nβ 4 ) 2 β 2 β. Taing limit as n,wehavea =A. (27) It is a time to study the stability of this type mapping on the local domain. We only prove the stability result when the target spaces are real multi-banach spaces, that is, the special case of real multi-β-normed space when β=.for 0<β<, it is an interesting open problem. The following are our results. Theorem 8. Let X be a real normed space, let ((Y n, ):n N) bearealmulti-banachspace,andletd>0, δ 0.Suppose that mapping f:x Ysatisfies Proof. Fix.LetX =(x,...,x ) and Y =(y,...,y ) satisfy (x,...,x ) + (y,...,y ) <d.ifx = Y = 0, then let T = (t,...,t ) X and T =d.ifx = 0 or Y = 0,let ( + d ) X, X { X Y ; T = {( + d ) Y, X { Y < Y. (30) If X Y,weget T = X +d>d.if X < Y, we have T = Y +d>d. Therefore, X T + Y + T 2 T ( X + Y ) d; X T + Y T 2 T ( X + Y ) d; X 2T + Y 2 T ( X + Y ) d; X ± T T X =( X +d) X =d, X ± T T X T X + T d. It follows from (28) that for X Y ; =( Y +d) X =d, for Y X ; (f (x +y )+f(x y ) +2f( x ),...,f(x +y )+f(x y ) +2f( x )) (f (x +y ) +f(x y 2t ) +2f( (x t )),...,f(x +y ) (3) (f (x +y )+f(x y ) +2f( x ),...,f(x +y )+f(x y ) +2f( x )) δ, (28) +f(x y 2t ) + 2f ( (x t ))) + (f (x +y 2t )+f(x y ) +2f( (x t )),...,f(x +y 2t ) (f (z )+f( z ),...,f(z )+f( z )) δ 2 for all x,...,x,y,...,y,z,...,z X that satisfy (x,...,x ) + (y,...,y ) dand (z,...,z ) d. Then there exists unique alternative additive mapping of the first form A:X Ysuch that (f (z ) A(z ),...,f(z ) A(z )) 95 4 δ (29) for all z,...,z X. f(x y ) + 2f ( (x t ))) + (f (x +y 2t )+f(x y 2t ) +2f( (x 2t )),...,f(x +y 2t ) +f(x y 2t ) + 2f ( (x 2t ))) + (2f ((t x )+t )+2f( x ) +4f(x t ),...,2f((t x )+t )+2f( x ) +4f(x t )) + (4f (x t )

5 Function Spaces 5 +4f( (x t )),...,4f(x t ) +4f( (x t ))) 7δ, (2f (z )+2f( z ),...,2f(z )+2f( z )) (2f (z )+f( z +t ) +f( z t ),...,2f(z )+f( z +t ) +f( z t )) + ( f ( z t ) f(z +t ),..., f( z t ) f(z +t )) + ( f (z t ) f( z +t ),..., f(z t ) f( z +t )) + (2f ( z )+f(z t ) +f(z +t ),...,2f( z )+f(z t ) +f(z +t )) 5δ. (32) It follows from Theorem 7 that there exists unique alternative additive mapping of the first form A:X Ysatisfying (29) for all z,...,z X. Corollary 9. Let ((X n, ) : n N)be a real multinormed space, and let ((Y n, ) : n N)be a multi-banach space. Mapping f:x Ysatisfies alternative additive equation of the first form if and only if, for each,if (x,...,x ) + (y,...,y ) and (z,...,z ),onehas (f (x +y )+f(x y ) +2f( x ),...,f(x +y )+f(x y ) +2f( x )) 0; (f (z )+f( z ),...,f(z )+f( z )) 0. (33) 3. Stability of Alternative Additive Equation of the Second Form In this section we will study the stability of the alternative additive equation of the second form in multi-β-normed space and on the restricted domain. First, we investigate the general case where the domain of the mapping is the whole space. The following theorem is obtained. Theorem 0. Let X be a real normed space, and let ((Y n, ): n N)be a complete real multi-β-normed space. Suppose that δ 0;mappingf:X Ysatisfies (f (x +y ) f(x y ) +2f( y ),...,f(x +y ) f(x y ) +2f( y )) δ (34) for all x,...,x,y,...,y X. Then there exists unique alternativeadditivemappingofthesecondforma:x Y such that (f (x ) A(x ),...,f(x ) A(x )) 22β +2 β (35) + 2 β (2 β ) δ for all x,...,x X. Proof. Let x i =y i =0 (i=,...,)in (34); we get (f (0),...,f(0)) δ. 2β (36) Replacing y i (i=,...,)with x i,weobtain (f (2x ) f(0) +2f( x ),...,f(2x ) (37) f(0) +2f( x )) δ. Let x = =x =0and replace y i (i=,...,)with x i ;we obtain (f (x )+f( x ),...,f(x )+f( x )) δ. (38) Hence for all x,...,x Xwe have (f (2x ) 2f(x ),...,f(2x ) 2f(x )) (f (2x ) f(0) +2f( x ),...,f(2x ) f(0) +2f( x )) + (2 (f (x )+f( x )),..., 2(f(x )+f( x ))) + (f (0),...,f(0)) 22β +2 β + δ. 2 β Therefore, for all m, n N, m,wehave ( 2 n f(2n x ) f(x ),..., 2 n f(2n x ) f(x )) ( 22β +2 β + δ) 2 β 2, β ( 2 n+m f(2n+m x ) 2 n f(2n x ),..., n = 2 n+m f(2n+m x ) 2 n f(2n x )) ( 22β +2 β + δ) 2 β n+m =n+ 2. β (39) (40) We omit the following arguments because they are similar to that of Theorem 7.

6 6 Function Spaces Theorem. Let X be a real normed space, let ((Y n, ):n N) be a complete real multi-banach space, and let d>0, δ 0. Suppose that f:x Ysatisfies (f (x +y ) f(x y ) +2f( y ),...,f(x +y ) f(x y ) +2f( y )) δ, (f (z )+f( z ),...,f(z )+f( z )) δ 2 (4) for x,...,x,y,...,y,z,...,z X that satisfy (x,...,x ) + (y,...,y ) dand (z,...,z ) d.then there exists unique alternative additive mapping of the second form A:X Ysuch that (f (x ) A(x ),...,f(x ) A(x )) 49 2 δ (42) for all x,...,x X. Proof. Fix N;chooseX = (x,...,x ) and Y = (y,...,y ) with (x,...,x ) + (y,...,y ) <d.ifx = Y = 0, thenlett =(t,...,t ) X and T =d.ifx = 0 or Y = 0,thenlet ( + d ) X, X { X Y ; T = {( + d ) Y, X { Y < Y. (43) If X Y,then T = X +d>d.if X < Y, then T = Y +d>d. Therefore, X T + Y + T 2 T ( X + Y ) d; X T + Y T 2 T ( X + Y ) d; X 2T + Y 2 T ( X + Y ) d; T + Y d; T Y d; T + Y d. It follows from (4) that (f (x +y ) f(x y ) +2f( y ),...,f(x +y ) f(x y ) +2f( y )) (f (x +y ) f(x y 2t ) +2f( (y +t )),...,f(x +y ) f(x y 2t ) + 2f ( (y +t ))) + (f (x +y 2t ) f(x y ) (44) +2f( (y t )),...,f(x +y 2t ) f(x y )+2f( (y t ))) + (f (x +y 2t ) f(x y 2t ) +2f( y ),...,f(x +y 2t ) f(x y 2t )+2f( y )) + (2f (t +y ) 2f(t y ) +4f( y ),...,2f(t +y ) 2f(t y ) +4f( y )) + (2f (t +y ) 2f( (t +y )),...,2f(t +y ) +2f( (t y ))) 7δ, (2f (z )+2f( z ),...,2f(z )+2f( z )) (2f (z )+f( z +t ) f(z +t ),...,2f(z )+f( z +t ) f(z +t )) + (2f ( z )+f(z +t ) f( z +t ),...,2f( z )+f(z +t ) f( z +t )) 4δ. (45) According to Theorem 0, there exists unique alternative additive mapping of the second form A:X Ysuch that (42) holds true. Corollary 2. Let ((X n, ) : n N) be a real multinormedspaceandlet((y n, ) : n N) be a multi- Banach space. Mapping f : X Y satisfies alternative additive equation of the second form if and only if, for each N if (x,...,x ) + (y,...,y ) and (z,...,z ),onehas (f (x +y ) f(x y ) +2f( y ),...,f(x +y ) f(x y ) +2f( y )) 0; (f (z )+f( z ),...,f(z )+f( z )) 0. Competing Interests The authors declare that they have no competing interests. Authors Contributions (46) All authors conceived of the study, participated its design and coordination, drafted the paper, participated in the sequence alignment,andreadandapprovedthefinalpaper.

7 Function Spaces 7 Acnowledgments This wor is ported by the National Natural Science Foundation of China (Grant 379), the Key Foundation of Education Department of Hebei Province (Grant ZD206023), and Natural Science Foundation of Education Department of Hebei Province (Grant Z20403). References [] S. M. Ulam, Problems in Modern Mathematics,Wiley,NewYor, NY, USA, 960. [2] D. H. Hyers, On the stability of the linear functional equation, Proceedings of the National Academy of Sciences of the United States of America,vol.27,no.4,pp ,94. [3] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proceedings of the American Mathematical Society, vol. 72,no.2,pp ,978. [4]R.P.Agarwal,B.Xu,andW.Zhang, Stabilityoffunctional equations in single variable, Mathematical Analysis and Applications,vol.288,no.2,pp ,2003. [5] Z. Gajda, On stability of additive mappings, International Mathematics and Mathematical Sciences, vol.4,no. 3, pp , 99. [6] P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, Mathematical Analysis and Applications,vol.84,no.3,pp ,994. [7] S.-M. Jung and J. M. Rassias, A fixed point approach to the stability of a functional equation of the spiral of Theodorus, Fixed Point Theory and Applications, vol. 2008, Article ID 94500, 7 pages, [8] C.ParandJ.M.Rassias, StabilityoftheJensen-typefunctional equation in C * -algebras: a fixed point approach, Abstract and Applied Analysis, vol. 2009, Article ID , 7 pages, [9] B. Xu and J. Brzde, Hyers-Ulam stability of a system of first order linear recurrences with constant coefficients, Discrete Dynamics in Nature and Society, vol.205,articleid269356, 5pages,205. [0]B.Xu,J.Brzdę,andW.Zhang, Fixed-pointresultsandthe Hyers-Ulam stability of linear equations of higher orders, Pacific Mathematics,vol.273,no.2,pp ,205. [] X. Yang, L. Chang, G. Liu, and G. Shen, Stability of functional equations in (n;β)-normed spaces, Inequalities and Applications,vol.205,no.2,8pages,205. [2] X. Zhao, X. Yang, and C.-T. Pang, Solution and stability of a general mixed type cubic and quartic functional equation, Function Spaces and Applications, vol.203,article ID 67380, 8 pages, 203. [3] N. Brillouët-Belluot, J. Brzdę, and K. Cieplińsi, On some recent developments in Ulam s type stability, Abstract and Applied Analysis, vol. 202, Article ID 76936, 4 pages, 202. [4] J. Brzdę, W. Fechner, M. S. Moslehian, and J. Siorsa, Recent developments of the conditional stability of the homomorphism equation, Banach Mathematical Analysis, vol.9,no. 3, pp , 205. [5] D.H.Hyers,G.Isac,andTh.M.Rassias,Stability of Functional Equations in Several Variables, Birhäuser, Basel, Switzerland, 998. [6] S. Czerwi, Functional Equations and Inequalities in Several Variables, World Scientific, Hacensac, NJ, USA, [7] P. Kannappan, Functional Equations and Inequalities with Applications, Springer Monographs in Mathematics, Springer, New Yor, NY, USA, [8] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis,vol.48ofSpringer Optimization and Its Applications, Springer, New Yor, NY, USA, 20. [9] H. G. Dales and M. E. Polyaov, Multi-normed spaces and multi-banach algebras. [20] H. G. Dales and M. S. Moslehian, Stability of mappings on multi-normed spaces, Glasgow Mathematical Journal, vol. 49, no. 2, pp , [2] M. S. Moslehian, Superstability of higher derivations in multi- Banach algebras, Tamsui Oxford Mathematical Sciences,vol.24,no.4,pp ,2008. [22] M. S. Moslehian, K. Niodem, and D. Popa, Asymptotic aspect of the quadratic functional equation in multi-normed spaces, Mathematical Analysis and Applications,vol.355,no. 2,pp ,2009. [23] L. Wang, B. Liu, and R. Bai, Stability of a mixed type functional equation on multi-banach spaces: a fixed point approach, Fixed Point Theory and Applications, vol.200,articleid283827,9 pages, 200. [24] X. Yang, L. Chang, and G. Liu, Orthogonal stability of mixed additive-quadratic Jensen type functional equation in multi- Banach spaces, Advances in Pure Mathematics,vol.5,no.6,pp , 205. [25] J. M. Rassias and M. J. Rassias, Asymptotic behavior of alternative JENsen and JENsen type functional equations, Bulletin des Sciences Mathématiques,vol.29,no.7,pp ,2005.

8 Advances in Operations Research Advances in Decision Sciences Applied Mathematics Algebra Probability and Statistics The Scientific World Journal International Differential Equations Submit your manuscripts at International Advances in Combinatorics Mathematical Physics Complex Analysis International Mathematics and Mathematical Sciences Mathematical Problems in Engineering Mathematics Discrete Mathematics Discrete Dynamics in Nature and Society Function Spaces Abstract and Applied Analysis International Stochastic Analysis Optimization