General solution and Ulam-Hyers Stability of Duodeviginti Functional Equations in Multi-Banach Spaces

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1 Global Jornal of Pre and Applied Mathematics. ISSN Volme 3, Nmber 7 (07), pp Research India Pblications General soltion and Ulam-Hyers Stability of Dodeviginti Fnctional Eqations in Mlti-Banach Spaces R. Mrali, M. Eshaghi-Gordji, A. Antony Raj 3,3 Department of Mathematics, Sacred Heart College, Tirpattr , TamilNad, India. Department of Mathematics, Semnan University, Iran. Abstract In this crrent wor, we carry ot the general soltion and Hyers-Ulam stability for a new form of Dodeviginti fnctional eqation in Mlti-Banach Spaces by sing fixed point techniqe.. INTRODUCTION The stability problem of fnctional eqations has a long History. When we say the fnctional eqation is stable, if for every approximate soltion, there exists an exact soltion near it. In 940, famos Ulam [] gave a wide-ranging tal before the mathematics clb of the niversity of Wisconsin in which he discssed a nmber of important nsolved problems. Among those was the qestion concerning the stability of grop homomorphisms, which was first solved by D.H.Hyers [7], in 94. Thereafter, several stability problems for different type of fnctional eqations generalized by many athors for fnctions with more general domains and codomains ([], [3], [7], [8], [0]). In this stability reslts have many applications in physics, economic theory and social and behavioral science []. In last few years, some athors have been estabilished the stability of different type of fnctional eqations in Mlti-Banach spaces [6], [],[0], [],[3], [4]. In recently M Eshaghi, S Abbaszadeh [], [3], [4] have been estabilished the stability problems in Banach Algebra. And also John M. Rassias, M. Arnmar, E. Sathya and T. Namachivayam [9] established the general soltion and also proved Stability of Nonic Fnctional Eqation in Felbin's type fzzy normed space and intitionistic fzzy normed space sing direct and fixed point method.

2 3050 R. Mrali, M. Eshaghi-Gordji and A. Antony Raj In 06, the general soltion and generalized Hyers-Ulam stability of a decic type fnctional eqation in Banach spaces, generalized -normed spaces and random normed spaces by sing direct and fixed point methods was discssed by Mohan Arnmar, Abasalt Bodaghi, John Michael Rassias and Elmalai Sathiya [5]. In K. Ravi, J.M. Rassias and B.V. Senthil Kmar [8] estabilished Ulam-Hyers stability of ndecic fnctional eqation in qasi -normed space by Fixed point method. In K. Ravi, J.M. Rassias, S. Pinelas and S.Sresh [9] estabilished the general soltion and prove the stability of this qattordecic fnctional eqation in qasi - normed spaces by Fixed point method. Very recently, R. Mrali, A. Antony Raj discssed the Hexadecic fnctional eqation and its stability in Mlti-Banach Spaces. In this crrent wor, we carry ot the general soltion and Hyers-Ulam stability for a new form of Dodeviginti fnctional eqation (, v) = ( 9 v) 8 ( 8 v) 53 ( 7 v) 86 ( 6 v) 3060 ( 5 v) 8568 ( 4 v) 8564 ( 3 v) 384 ( v) ( v) 4860 ( ) ( v) 384 ( v) 8564 ( 3 v) 8568 ( 4 v) 3060 ( 5 v) 86 ( 6 v) 53 ( 7 v) 8 ( 8 v) ( 9 v) ( v) in Mlti-Banach Spaces by sing fixed point Techniqe. (.) 8 It is easily verified that that the fnction ( ) = satisfies the above fnctional eqations. In other words, every soltion of the Dodeviginti fnctional eqation is called a Dodeviginti mapping. Now, let s recall regarding some concepts in Mlti-Banach spaces. Definition. [5] A Mlti- norm on : is a seqence. =. : sch that. is a norm on for each, x = x for each x, and the following axioms are satisfied for each with : x x x x,..., =..., for, x,..., x ; () ( ) x,..., x x... x maxi i for..., x,..., x ; x x x x,...,,0 =,...,, x,..., x, x = x,..., x for x,..., x ; for x,..., x.

3 General soltion and Ulam-Hyers Stability of Dodeviginti Fnctional Eqations 305 In this case, we say that (,. ) : is a mlti - normed space. Sppose that (,. ) : is a mlti - normed spaces, and tae. We need the following two properties of mlti - norms. They can be fond in [5]. ( a) x,..., x = x,for x, ( b) max x x,..., x x max x, x,..., x. i i i i i= i It follows from (b) that if (,. ) is a Banach space, then (,. ) is a Banach space for each ; In this case, (,. ) : is a mlti - Banach space. Definition. [5] Let (,. ) : be a mlti - normed space. A seqence ( x n) in is a mlti-nll seqence if for each > 0, there exists n0 sch that x x n n,...,. (.) sp n n 0 Let x, we say that the seqence ( x n) is mlti-convergent to x in and write limn xn = x if xn x is a mlti - nll seqence.. GENERAL SOLUTION OF (.) Theorem. If : be the vector spaces. If : be a fnction (.) for all v,, then is Viginti Do Mapping. Proof. Sbstitting =0 and v =0 in (.), we obtain that (0) = 0. Sbstitting ( v, ) with (, ) and (, ) in (.), respectively, and sbtracting two reslting eqations, we can obtain ( ) = ( ), that is to say, is an even fnction. Changing ( v, ) by (9, ) and (0, ), in (.) respectively, and sbtracting the two reslting eqations, we arrive 8 (7 ) 7 (6 ) 86 (5 ) 907 (4 ) 8568 (3 ) 9380 ( ) 384 ( ) (0 ) 4860 (9 ) 536 (8 ) 384 (7 ) (.) 8568 (5 ) (4 ) 86 (3 ) ( ) 8! ( ) = 0. Doing =8 by v= in (.), we obtain (7 ) 8 (6 ) 53 (5 ) 86 (4 ) 3060 (3 ) 8568 ( ) 8564 ( ) 384 (0 ) (9 ) 4860 (8 ) (.) (7 ) 384 (6 ) 8564 (5 ) 8568 (4 ) 3060 (3 ) 86 ( ) 8! ( ) = 0

4 305 R. Mrali, M. Eshaghi-Gordji and A. Antony Raj. Mltiplying (.) by 8, and then sbtracting (.) from the reslting eqation, we arrive 53 (6 ) 938 (5 ) 78 (4 ) 465 (3 ) ( ) 3038 ( ) 5334 (0 ) (9 ) 8834 (8 ) (.3) (7 ) 5783 (6 ) (5 ) 9340 (4 ) 5464 (3 ) ( ) 8!(9) ( ) =0. Replacing =7 by v= in (.), we obtain (6 ) 8 (5 ) 53 (4 ) 86 (3 ) 3060 ( ) 8568 ( ) 8564 (0 ) 384 (9 ) (8 ) 4860 (7 ) (6 ) (.4) 384 (5 ) 8564 (4 ) 8568 (3 ) 306 ( ) 8! ( ) = 0. Mltiplying (.4) by 53, and then sbtracting (.3) from the reslting eqation, we arrive 86 (5 ) 68 (4 ) (3 ) ( ) ( ) (0 ) (9 ) (8 ) (7 ) (.5) 64 (6 ) (5 ) 7095 (4 ) (3 ) ( ) 8!(7) ( ) =0. Doing =6 and v= in (.), we obtain (5 ) 8 (4 ) 53 (3 ) 86 ( ) 3060 ( ) 8568 (0 ) 8564 (9 ) 384 (8 ) (7 ) 4860 (6 ) (.6) (5 ) 384 (4 ) 8565 (3 ) 8586 ( ) 8! ( ) = 0. Mltiplying (.6) by 86, and then sbtracting (.5) from the reslting eqation, we get 3060 (4 ) 465 (3 ) 3350 ( ) ( ) (0 ) 0876 (9 ) (8 ) (7 ) (.7) (6 ) (5 ) (4 ) (3 ) ( ) 8!(988) ( ) =0. Doing =5 and v= in (.), we obtains (4 ) 8 (3 ) 53 ( ) 86 ( ) 3060 (0 ) 8568 (9 ) 8564 (8 ) 384 (7 ) (6 ) (.8) 4860 (5 ) (4 ) 384 (3 ) 877 ( ) 8! ( ) = 0

5 General soltion and Ulam-Hyers Stability of Dodeviginti Fnctional Eqations Mltiplying (.8) by 3060, and then sbtracting (.7) from the reslting eqation,we arrive 8568 (3 ) ( ) ( ) (0 ) (9 ) (8 ) (7 ) (6 ) (.9) (5 ) (4 ) (3 ) ( ) 8!(4048) ( ) = 0. Doing x=4x and y= x in (.), we get (3 ) 8 ( ) 53 ( ) 86 (0 ) 3060 (9 ) 8568 (8 ) 8564 (7 ) 384 (6 ) (5 ) (.0) (4 ) 439 (3 ) 3640 ( ) 8! ( ) = 0. Mltiplying (.0) by 8568, and then sbtracting (.9) from the reslting eqation, we arrive 8564 ( ) 3038 ( ) 38 (0 ) 0876 (9 ) (8 ) (7 ) (6 ) (5 ) (4 ) (.) (3 ) ( ) 8!(66) ( ) = 0. Changing =3 and v= in (.), we obtain ( ) 8 ( ) 53 (0 ) 86 (9 ) 3060 (8 ) 8568 (7 ) 8565 (6 ) (.) 384 (5 ) 439 (4 ) (3 ) 4688 ( ) 8! ( ) = 0. Mltiplying (.) by 8564, and then sbtracting (.) from the reslting eqation, we arrive 384 ( ) (0 ) (9 ) 0048 (8 ) (7 ) (6 ) (5 ) (4 ) (.3) (3 ) ( ) 8!(380) ( ) = 0. Changing = and v= in (.), we obtain ( ) 8 (0 ) 53 (9 ) 86 (8 ) 306 (7 ) 8586 (6 ) (.4) 877 (5 ) 3640(4 ) 4688 (3 ) 5788 ( ) 8! ( ) = 0. Mltiplying (.4) by 384, and then sbtracting (.3) from the reslting eqation, we arrive (0 ) (9 ) (8 ) (7 ) (6 ) (5 ) (4 ) (3 ) (.5) ( ) 8!(63004) ( ) = 0. Changing = and v= in (.), we obtain

6 3054 R. Mrali, M. Eshaghi-Gordji and A. Antony Raj (0 ) 8 (9 ) 54 (8 ) 834 (7 ) 33 (6 ) (.6) 9384 (5 ) 64 (4 ) 4039 (3 ) 63 ( ) 8! ( ) = 0. Mltiplying (.6) by 43758, and then sbtracting (.5) from the reslting eqation, we arrive 4860 (9 ) (8 ) (7 ) (6 ) (5 ) (4 ) (3 ) (.7) ( ) 8!(0676) ( ) = 0. Changing =0 and v= in (.), we obtain (9 ) 8 (8 ) 53 (7 ) 86 (6 ) 3060 (5 ) (.8) 8568 (4 ) 8564 (3 ) 384 ( ) ( ) = 0. Mltiplying (.8) by 4860, and then sbtracting (.7) from the reslting eqation, we arrive ( ) ( ) = 0 (.9). It follows from (.9), we reach. Hence is a Dodeviginti Mapping. This completes the proof. 8 ( ) = ( ) (.0) 3. STABILITY OF THE FUNCTIONAL EQUATION (.) IN MULTI- BANACH SPACES Theorem 3. 3 Let be an linear space and let (,. ) : K be a mlti- Banach space. Sppose that is a non-negative real nmber and : be a fnction flfills sp (, v ),..., (, v ) (3.),...,, v,..., v. Then there exists a niqe Dodeviginti fnction : sch that sp 4033 ( ) ( ),..., ( ) ( ). (3.) forall i, where i=,,...,. Proof. Taing =0 i and Changing i v by i in (3.), and dividing by in the reslting eqation, we arrive

7 General soltion and Ulam-Hyers Stability of Dodeviginti Fnctional Eqations 3055 sp (8 ) 8 (6 ) 53 (4 ) 86 ( ) 3060 (0 ) 8568 (8 ) 8564 (6 ) 384 (4 ) ( ),..., (8 ) 8 (6 ) 53 (4 ) 86 ( ) 3060 (0 ) 8568 (8 ) 8564 (6 ) 384 (4 ) ( )) (3.3) Plgging,..., into 9,...,9 and Changing v, v,..., v by,..., in (3.), we have sp ( (8 ) 8 (7 ) 53 (6 ) 86 (5 ) 3060 (4 ) 8568 (3 ) 8564 ( ) 384 ( ) (0 ) 4860 (9 ) (8 ) 384 (7 ) 8564 (6 ) 8568 (5 ) 3060 (4 ) 86 (3 ) 53 ( ) 8! ( ),..., (8 ) 8 (7 ) 53 (6 ) 86 (5 ) 3060 (4 ) 8568 (3 ) 8564 ( ) 384 ( ) (0 ) 4860 (9 ) (8 ) 384 (7 ) 8564 (6 ) 8568 (5 ) 3060 (4 ) 86 (3 ) 53 ( ) 8! ( ) (3.4) Combining (3.3) and (3.4), we have sp 8 (7 ) 7 (6 ) 86 (5 ) 907 (4 ) 8568 (3 ) 9380 ( ) 384 ( ) (0 ) 4860 (9 ) 536 (8 ) 384 (7 ) 8568 (5 ) (4 ) 86 (3 ) ( ) 8! ( ),...,8 (7 ) 7 (6 ) 86 (5 ) 907 (4 ) 8568 (3 ) 9380 ( ) 384 ( ) (0 ) 4860 (9 ) 536 (8 ) 384 (7 ) 8568 (5 ) (4 ) 3 86 (3 ) ( ) 8! ( ) (3.5) Taing,..., by 8,...,8 and Changing y, y,..., y by,..., in (3.) and sing evenness of f, we arrive sp (7 ) 8 (6 ) 53 (5 ) 86 (4 ) 3060 (3 ) 8568 ( ) 8564 ( ) 384 (0 ) (9 ) 4860 (8 ) (7 ) 384 (6 ) 8564 (5 ) 8568 (4 ) 3060 (3 ) 86 ( ) 8! ( ),..., (7 ) 8 (6 ) 53 (5 ) 86 (4 ) 3060 (3 ) 8568 ( ) 8564 ( ) 384 (0 ) (9 ) 4860 (8 ) (7 )

8 3056 R. Mrali, M. Eshaghi-Gordji and A. Antony Raj 384 (6 ) 8564 (5 ) 8568 (4 ) 3060 (3 ) 86 ( ) 8! ( ) (3.6) Mltiplying by 8 on both sides of (3.6),then it follows from (3.5) and the reslting eqation we get sp 53 (6 ) 938 (5 ) 78 (4 ) 465 (3 ) ( ) 3038 ( ) 5334 (0 ) (9 ) 8834 (8 ) (7 ) 5783 (6 ) (5 ) 9340 (4 ) 5464 (3 ) ( ) 8!(9) ( ),...,53 (6 ) 938 (5 ) 78 (4 ) 465 (3 ) ( ) 3038 ( ) 5334 (0 ) (9 ) 8834 (8 ) (7 ) 5783 (6 ) (5 ) 9340 (4 ) (3 ) ( ) 8!(9) ( ) (3.7),...,. Taing,..., by 7,...,7 and Changing y, y,..., y by,..., in (3.) and sing evenness of f, we arrive sp (6 ) 8 (5 ) 53 (4 ) 86 (3 ) 3060 ( ) 8568 ( ) 8564 (0 ) 384 (9 ) (8 ) 4860 (7 ) (6 ) 384 (5 ) 8564 (4 ) 8568 (3 ) 306 ( ) 8! ( ),..., (6 ) 8 (5 ) 53 (4 ) 86 (3 ) 3060 ( ) 8568 ( ) 8564 (0 ) 384 (9 ) (8 ) 4860 (7 ) (6 ) 384 (5 ) 8564 (4 ) 8568 (3 ) 306 ( ) 8! ( ) (3.8) Mltiplying (3.8) by 53, then it follows from (3.7) and the reslting eqation we arrive sp 86 (5 ) 68 (4 ) (3 ) ( ) ( ) (0 ) (9 ) (8 ) (7 ) 64 (6 ) (5 ) 7095 (4 ) (3 ) ( ) 8!(7) ( ),..., 86 (5 ) 68 (4 ) (3 ) ( ) ( ) (0 ) (9 ) (8 ) (7 ) 64 (6 ) (5 ) (4 ) (3 ) ( ) 8!(7) ( ) (3.9)

9 General soltion and Ulam-Hyers Stability of Dodeviginti Fnctional Eqations 3057 Taing,..., by 6,...,6 and Changing v, v,..., v by,..., in (3.) and sing evenness of we arrive sp (5 ) 8 (4 ) 53 (3 ) 86 ( ) 3060 ( ) 8568 (0 ) 8564 (9 ) 384 (8 ) (7 ) 4860 (6 ) (5 ) 384 (4 ) 8565 (3 ) 8586 ( ) 8! ( ),..., (5 ) 8 (4 ) 53 (3 ) 86 ( ) 3060 ( ) 8568 (0 ) 8564 (9 ) 384 (8 ) (7 ) 4860 (6 ) (5 ) 384 (4 ) 8565 (3 ) 8586 ( ) 8! ( ) (3.0) Mltiplying (3.0) by 86, then it follows from (3.9) and the reslting eqation we arrive sp 3060 (4 ) 465 (3 ) 3350 ( ) ( ) (0 ) 0876 (9 ) (8 ) (7 ) (6 ) (5 ) (4 ) (3 ) ( ) 8!(988) ( ),...,3060 (4 ) 465 (3 ) 3350 ( ) ( ) (0 ) 0876 (9 ) (8 ) (7 ) (6 ) (5 ) (4 ) (3 ) ( ) 8!(988) ( ) (3.0) Plgging,..., into 5,...,5 and Changing v, v,..., v by,..., in (3.) and sing evenness of, we arrive sp (4 ) 8 (3 ) 53 ( ) 86 ( ) 3060 (0 ) 8568 (9 ) 8564 (8 ) 384 (7 ) (6 ) 4860 (5 ) (4 ) 384 (3 ) 877 ( ) 8! ( ),..., (4 ) 8 (3 ) 53 ( ) 86 ( ) 3060 (0 ) 8568 (9 ) 8564 (8 ) 384 (7 ) (6 ) 4860 (5 ) (4 ) 384 (3 ) 877 ( ) 8! ( ) (3.) Mltiplying (3.) by 3060, then it follows from (3.) and the reslting eqation we arrive sp 8568 (3 ) ( ) ( ) (0 ) (9 ) (8 ) (7 ) (6 ) (5 ) (4 ) (3 ) ( ) 8!(4048) ( ),..., 8568 (3 ) ( ) ( ) (0 ) (9 )

10 3058 R. Mrali, M. Eshaghi-Gordji and A. Antony Raj (8 ) (7 ) (6 ) (5 ) (4 ) (3 ) ( ) 8!(4048) ( ) (3.3) Plgging,..., into 4,...,4 and Changing y, y,..., y by,..., in (3.) and sing evenness of f, we arrive sp (3 ) 8 ( ) 53 ( ) 86 (0 ) 3060 (9 ) 8568 (8 ) 8564 (7 ) 384 (6 ) (5 ) (4 ) 439 (3 ) 3640 ( ) 8! ( ),..., (3 ) 8 ( ) 53 ( ) 86 (0 ) 3060 (9 ) 8568 (8 ) 8564 (7 ) 384 (6 ) (5 ) (4 ) 439 (3 ) 3640 ( ) 8! ( ) (3.4) Mltiplying (3.4) by 8568, then it follows from (3.3) and the reslting eqation we arrive sp 8564 ( ) 3038 ( ) 38 (0 ) 0876 (9 ) (8 ) (7 ) (6 ) (5 ) (4 ) (3 ) ( ) 8!(66) ( ),..., 8564 ( ) 3038 ( ) 38 (0 ) 0876 (9 ) (8 ) (7 ) (6 ) (5 ) (4 ) (3 ) ( ) 8!(66) ( ) (3.5) Taing,..., by 3,..., and Changing y, y,..., y by,..., in (3.) and sing evenness of f, we arrive sp ( ) 8 ( ) 53 (0 ) 86 (9 ) 3060 (8 ) 8568 (7 ) 8565 (6 ) 384 (5 ) 439 (4 ) (3 ) 4688 ( ) 8! ( ),..., ( ) 8 ( ) 53 (0 ) 86 (9 ) 3060 (8 ) 8568 (7 ) 8565 (6 ) 384 (5 ) 439 (4 ) (3 ) 4688 ( ) 8! ( ) (3.6) Mltiplying (3.6) by 8564, then it follows from (3.5) and the reslting eqation we arrive sp 384 ( ) (0 ) (9 ) 0048 (8 ) (7 ) (6 ) (5 ) (4 ) (3 ) ( ) 8!(380) ( ),...,

11 General soltion and Ulam-Hyers Stability of Dodeviginti Fnctional Eqations ( ) (0 ) (9 ) 0048 (8 ) (7 ) (6 ) (5 ) (4 ) (3 ) ( ) 8!(380) ( ) (3.7) Taing,..., by,..., and Changing y, y,..., y by,..., in (3.) and sing evenness of f, we arrive sp ( ) 8 (0 ) 53 (9 ) 86 (8 ) 306 (7 ) 8586 (6 ) 877 (5 ) 3640(4 ) 4688 (3 ) 5788 ( ) 8! ( ),..., ( ) 8 (0 ) 53 (9 ) 86 (8 ) 306 (7 ) 8586 (6 ) 877 (5 ) 3640(4 ) 4688 (3 ) 5788 ( ) 8! ( ) (3.8) Mltiplying (3.8) by 384, then it follows from (3.7) and the reslting eqation we arrive sp (0 ) (9 ) (8 ) (7 ) (6 ) (5 ) (4 ) (3 ) ( ) 8!(63004) ( ),..., (0 ) (9 ) (8 ) (7 ) (6 ) (5 ) (4 ) (3 ) ( ) 8!(63004) ( ) (3.9) Changing y, y,..., y by,..., in (3.) and sing evenness of f, we arrive sp (0 ) 8 (9 ) 54 (8 ) 834 (7 ) 33 (6 ) 9384 (5 ) 64 (4 ) 4039 (3 ) 63 ( ) 8! ( ),..., (0 ) 8 (9 ) 54 (8 ) 834 (7 ) 33 (6 ) 9384 (5 ) 64 (4 ) 4039 (3 ) 63 ( ) 8! ( ) (3.0) Mltiplying (3.0) by 43758, then it follows from (3.9) and the reslting eqation we arrive sp 4860 (9 ) (8 ) (7 ) (6 ) (5 ) (4 ) (3 ) ( ) 8!(0676) ( ),...,4860 (9 ) (8 ) (7 ) (6 ) (5 ) (4 )

12 3060 R. Mrali, M. Eshaghi-Gordji and A. Antony Raj (3 ) ( ) 8!(0676) ( ) (3.) Taing = =,...,= = 0 and Changing y, y,..., y by,..., (3.), we obtain sp (9 ) 8 (8 ) 53 (7 ) 86 (6 ) 3060 (5 ) 8568 (4 ) 8564 (3 ) 384 ( ) ( ),..., (9 ) 8 (8 ) 53 (7 ) 86 (6 ) 3060 (5 ) 8568 (4 ) 8564 (3 ) 384 ( ) ( ) (3.) Mltiplying (3.) by 4860, then it follows from (3.) and the reslting eqation we arrive sp ( ) ( ),..., ( ) ( ) (3.3) Dividing on both sides by in (3.3), we arrive sp 645 ( ) 644 ( ),..., ( ) 644 ( ) (3.4) 8! Again, dividing on both sides by 644 in (3.4), we arrive 645 ( ) ( ),.., ( ) ( ) sp !(644) (3.5) forall,...,. = l: l(0) = 0 and introdce the generalized metric d defined on by s Let d ( l, m ) = inf [0, ] l ( ) m ( ),..., l ( ) m ( ),..., sp Then it is easy to show that,d is a generalized complete metric space, See [6]. We define an operator : by l( ) = l( ) 8 We assert that is a strictly contractive operator. Given lm,, let [0, ] be an arbitary constant with d( l, m). From the definition if follows that l( ) m( ),..., l( ) m( ),...,. sp

13 General soltion and Ulam-Hyers Stability of Dodeviginti Fnctional Eqations 306 l m l m 8 Therefore, sp ( ) ( ),..., ( ) ( ),...,. Hence,it holds that lm,. This Means that L =. 8 d( l, m) d( l, m) d( l, m) 8 8 is strictly contractive operator on with the Lipschitz constant 645 By (3.5), we have d(, ). Applying the Theorem. in [7], we 8!(644) dedce the existence of a fied point of that is the existence of mapping : sch that n Moreover, we have 8 ( ) = ( ). d, 0, which implies for all. Also, n n ( ) ( ) = lim ( ) = lim n d(, ) d(, ) implies the ineqality n 8n d(, ) d(, ) n n Taing =,...,= =, v =,...,= v = v in (3.) and divide both sides by Now, applying the property (a) of mlti-norms, we obtain n n (, v) = lim f 8, v n n 8n. lim =0 n 8n for all v,. The niqeness of follows from the fact that is the niqe fixed point of with the property that there exists (0, ) sch that

14 306 R. Mrali, M. Eshaghi-Gordji and A. Antony Raj for all,...,. Hence is Dodeviginti fnction. ( ) ( ),..., ( ) ( ) sp Example We consider the fnction where :. Let : be defined by 8 ( ) =, <, 8n n ( ) = ( ) (4.) for all. If the fnction defined in (4.) satisfies the fnctional ineqality n= (644) ( v, ) (4.) 644 for all v,, then there do not exist an Dodeviginti fnction : and a constant >0 sch that 8 ( ) ( ),. (4.3) proof: 8n n ( ) = ( ) = 8n n=0 n= Therefore, we see that bonded by on. Now, sppose that 0 < <. 8 Then there exists a positive integer sch that < (4.4) 8 8 and n ( 9 v), n ( 8 v), n ( 7 v), n ( 6 v), n ( 5 v), n ( 4 v), n ( 3 v), n ( v), n ( v), n ( ), n ( v), n ( v), n ( 3 v), n ( 4 v), n ( 5 v),

15 General soltion and Ulam-Hyers Stability of Dodeviginti Fnctional Eqations 3063 n ( 6 v), n ( 7 v), n ( 8 v), n ( 9 v), n ( v) (,) forall n= 0,,,...,. Hence for n= 0,,,...,, From the definition of and the ineqality, we obtain that (, ) 8n v n= (644) (644) Therefore f satisfies (4.) for all v,. we prove that the fnctional eqation (.) is not stable. Sppose on the contrary that there exists an Dodeviginti fnction : and a constant >0 sch that 8 f ( ) ( ),. for all. Then there exists a constant c sch that nmbers. So we arrive that 8 = c for all rational 8 ( ) c. (4.5) m for all. Tae m with m> c. If is a rational nmber in (0, ), then n (0,) for all n= 0,,... m, and for this, we get ( ) = ( ) = ( m ) > c. n= 8 n which contradicts (4.5). Hence the fnctional eqation (.) is not stable. REFERENCES [] Aoi.T, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Jpn. (950), [] Aczel.J, A short corse on fnctional eqations, D. Reidel Pbl. Co. Dordrecht., (987). [3] Czerwi.S, Fnctional Eqations and Ineqalities in Several Variables, World Scientific Pblishing Co., Singapore, New Jersey, London, (00). [4] Diaz.J.B and Margolis.B, A fixed point theorem of the alternative, for contraction on a generalized complete metric space, Blletin of the American Mathematical Society, vol. 74 (968), [5] Dales, H.G and Moslehian, Stability of mappings on mlti-normed spaces, Glasgow Mathematical Jornal, 49 (007), 3-33.

16 3064 R. Mrali, M. Eshaghi-Gordji and A. Antony Raj [6] Fridon Moradlo, Approximate Eler-Lagrange-Jensen type Additive mapping in Mlti-Banach Spaces: A Fixed point Approach, Commn. Korean Math. Soc. 8 (03), [7] Hyers.D.H, On the stability of the linear fnctional eqation, Proc. Natl. Acad. Sci. USA 7 (94), -4. [8] Hyers.D.H, Isac.G, Rassias.T.M, Stability of Fnctional Eqations in Several Variables, Birhäser, Basel, (998). [9] John M. Rassias, Arnmar.M, Sathya.E and Namachivayam.T, Varios generalized Ulam-Hyers Stabilities of a nonic fnctional eqations, Tbilisi Mathematical Jornal 9() (06), [0] Jn.K, Kim.H, The Generalized Hyers-Ulam-Rassias stability of a cbic fnctional eqation, J. Math. Anal. Appl. 74 (00), [] Ligang Wang, Bo Li and Ran Bai, Stability of a Mixed Type Fnctional Eqation on Mlti-Banach Spaces: A Fixed Point Approach, Fixed Point Theory and Applications (00), 9 pages. [] M Eshaghi, S Abbaszadeh, On The Orthogonal Pexider Derivations In Orthogonality Banach Algebras, Fixed Point Theory 7() (06), [3] M Eshaghi, S Abbaszadeh, M de la Sen, Z Farhad, A Fixed Point reslt and the stability problem in Lie speralgebras, Fixed Point Theory and Applications () (05), -. [4] ME Gordji, S Abbaszadeh, Theory of Approximate Fnctional Eqations: In Banach Algebras, Inner Prodct Spaces and Amenable Grops, Academic Press. [5] Mohan Arnmar, Abasalt Bodaghi, John Michael Rassias and Elmalai Sathiya, The general soltion and approximations of a decic type fnctional eqation in varios normed spaces, Jornal of the Chngcheong Mathematical Society, 9() (06). [6] Mihet.D and Rad.V, On the stability of the additive Cachy fnctional eqation in random normed spaces, Jornal of mathematical Analysis and Applications, 343 (008), [7] Rad.V, The fixed point alternative and the stability of fnctional eqations, Fixed Point Theory 4 (003), [8] K. Ravi, J.M. Rassias and B.V. Senthil Kmar, Ulam-Hyers stability of ndecic fnctional eqation in qasi-beta normed spaces fixed point method, Tbilisi Mathematical Science 9 () (06), [9] K. Ravi, J.M. Rassias, S. Pinelas and S.Sresh, General soltion and stability of Qattordecic fnctional eqation in qasi beta normed spaces, Advances in pre mathematics 6 (06), 9-94.

17 General soltion and Ulam-Hyers Stability of Dodeviginti Fnctional Eqations 3065 [0] Sattar Alizadeh, Fridon Moradlo, Approximate a qadratic mapping in mlti-banach Spaces, A Fixed Point Approach. Int. J. Nonlinear Anal. Appl., 7. No. (06), [] Tian Zho X, John Michael Rassias and Wan Xin X, Generalized Ulam - Hyers Stability of a General Mixed AQCQ fnctional eqation in Mlti- Banach Spaces: A Fixed point Approach, Eropean Jornal of Pre and Applied Mathematics 3 (00), [] Ulam.S.M, A Collection of the Mathematical Problems, Interscience, New Yor, (960). [3] Xizhong Wang, Lidan Chang, Gofen Li, Orthogonal Stability of Mixed Additive-Qadratic Jenson Type Fnctional Eqation in Mlti-Banach Spaces, Advances in Pre Mathematics 5 (05), [4] Zhiha Wang, Xiaopei Li and Themistocles M. Rassias, Stability of an Additive-Cbic-Qartic Fnctional Eqation in Mlti-Banach Spaces, Abstract and Applied Analysis (0), pages.

18 3066 R. Mrali, M. Eshaghi-Gordji and A. Antony Raj

Stability and nonstability of octadecic functional equation in multi-normed spaces

Arab. J. Math. 208 7:29 228 https://doi.org/0.007/s40065-07-086-0 Arabian Journal of Mathematics M. Nazarianpoor J. M. Rassias Gh. Sadeghi Stability and nonstability of octadecic functional equation in