Malaya J. Mat. 2(1)(2014) 49 60

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1 Malaya J. Mat. 2(1)(2014) Functonal equaton orgnatng from sum of hgher owers of arthmetc rogresson usng dfference oerator s stable n Banach sace: drect and fxed ont methods M. Arunumar a, and G. Brtto Antony Xaver b a Deartment of Mathematcs, Government Arts College, Truvannamala , Taml Nadu, Inda. b Deartment of Mathematcs, Sacred Heart College, Truattur , Taml Nadu, Inda. Abstract In ths aer, the authors has roved the soluton of a new tye of functonal equaton f ( j x j ) ( j f (x j ) ),, 1 whch s orgnatng from sum of hgher owers of an arthmetc rogresson. Its generalzed Ulam - Hyers stablty n Banach sace usng drect and fxed ont methods are nvestgated. An alcaton of ths functonal equaton s also studed. Keywords: Addtve functonal equatons, strlng numbers, olynomal factoral, dfference oerator, generalzed Ulam - Hyers stablty, fxed ont MSC: 39B52, 39B72, 39B82. c 2012 MJM. All rghts reserved. 1 Introducton Durng the last seven decades, the erturbaton roblems of several functonal equatons have been extensvely nvestgated by a number of authors [1, 2, 12, 13, 20, 21, 26, 28]. The termnology generalzed Ulam - Hyers stablty orgnates from these hstorcal bacgrounds. These termnologes are also aled to the case of other functonal equatons. For more detaled defntons of such termnologes, one can refer to [5, 8, 9, 10, 14, 15, 16, 22, 23, 24, 27]. One of the most famous functonal equatons s the addtve functonal equaton f (x + y) f (x) + f (y). (1.1) In 1821, t was frst solved by A.L. Cauchy n the class of contnuous real-valued functons. It s often called an addtve Cauchy functonal equaton n honor of Cauchy. The theory of addtve functonal equatons s frequently aled to the develoment of theores of other functonal equatons. Moreover, the roertes of addtve functonal equatons are owerful tools n almost every feld of natural and socal scences. Every soluton of the addtve functonal equaton (1.1) s called an addtve functon. Corresondng author. E-mal addresses: annarun2002@yahoo.co.n (M. Arunumar), shcbrtto@yahoo.co.n (G. Brtto Antony Xaver).

2 50 M. Arunumar et al. / Functonal equaton orgnatng... The soluton and stablty of the followng varous addtve functonal equatons f (2x y) + f (x 2y) 3 f (x) 3 f (y), (1.2) f (x + y 2z) + f (2x + 2y z) 3 f (x) + 3 f (y) 3 f (z), (1.3) f (m(x + y) 2mz) + f (2m(x + y) mz) 3m[ f (x) + f (y) f (z)] m 1, (1.4) ( ) ( ) ( ) n 1 n 1 n 1 f a x 2ax n + f 2a x ax n 3a f (x ) f (x n ) n 3, (1.5) f (2x ± y ± z) f (x ± y) + f (x ± z) (1.6) f (qx ± y ± z) f (x ± y) + f (x ± z) + (q 2) f (x), q 2 (1.7) were dscussed by D.O. Lee [11], K. Rav, M. Arunumar [25], M. Arunumar [3, 4]. Also M. Arunumar et. al., [7] nvestgated the generalzed Ulam-Hyers stablty of a functonal equaton f (y) f (y + z) + f (y z) 2 whch s orgnatng from arthmetc mean of consecutve terms of an arthmetc rogresson usng drect and fxed ont methods. Infact M. Arunumar et. al.,[6] has roved the soluton and generalzed Ulam - Hyers - Rassas stablty of a n dmensonal addtve functonal equaton f (x) n l1 ( ) f (x + lyl ) + f (x ly l ) where n s a ostve nteger, whch s orgnatng from arthmetc mean of n consecutve terms of an arthmetc rogresson. In ths aer, the authors establshed the soluton and the generalzed Ulam - Hyers stablty of a new tye of addtve functonal equaton f ( j x j ) 2 l ( j f (x j ) ),, 1 (1.8) whch s orgnatng from sum of hgher owers of an arthmetc rogresson. An alcaton of ths functonal equaton s also studed. In Secton 2, some basc relmnares about dfference oerator s dscussed. In Secton 3, the general soluton of the functonal equaton (1.8) s gven. In Secton 4 and 5, the generalzed Ulam - Hyers stablty of the addtve functonal equaton (1.8) usng drect and fxed ont methods are resectvely roved. An alcaton of the addtve functonal equaton (1.8) s dscussed n Secton 6. 2 Basc relmnares on dfference oerator Defnton 2.1. [18] If {y } s a sequence of numbers, then we defne the dfference oerator as Lemma 2.1. [18] From (2.1) and the shft relaton, E(y ) y +1, we obtan (y ) y +1 y. (2.1) E + 1. (2.2) Defnton 2.2. [18] If n s ostve nteger, then the ostve olynomal factoral s defned as Lemma 2.2. [18] If S n r s are the Strlng numbers of second nd, then (n) ( 1)( 2)...( (n 1)). (2.3) n n Sr n (r). (2.4)

3 M. Arunumar et al. / Functonal equaton orgnatng Defnton 2.3. [18] For the ostve nteger n, the nverse oerators are defned as f Lemma 2.3. [18] If m, are ostve ntegers and > m, then Theorem 2.1. [18] If s ostve nteger, then n (z ) y, then z n (y ). (2.5) 1 (m) (m+1) + c, where c s constant. (2.6) (m + 1) 1 (y ) Theorem 2.2. If and are ostve ntegers then Proof. The roof follows by Lemmas 2.2, 2.3 and Theorem 2.1. y ( r) + c, where c s constant. (2.7) ( + 1 r) n S n () [ + 1](r+1) r. (2.8) [r + 1] 3 General soluton of the functonal equaton(1.8) In ths secton, the general soluton of the functonal equaton (1.8) s gven. Theorem 3.3. Let X and Y be real vector saces. The mang f : X Y satsfes the functonal equaton (1.1) for all x, y X f and only f f : X Y satsfes the functonal equaton (1.8) for all x 1, x 2,, x X. Proof. The roof follows by the addtve roerty. Hereafter though out ths aer, let us consder X and Y to be a normed sace and a Banach sace, resectvely. 4 Stablty results: Drect method In ths secton, the generalzed Hyers - Ulam - Rassas stablty of the addtve functonal equaton (1.8) s rovded. Theorem 4.4. Let { 1, 1} and α : X [0, ) be a functon such that α [ t x 1, t x 2,, t ] x t0 t converges n R (4.1) for all x 1, x 2, x 3, x X. Let f : X Y be a functon satsfyng the nequalty [ ] f j [ x j j f [x j ] ] α [x 1, x 2, x 3, x ] (4.2) for all x 1, x 2, x 3, x X. Then there exsts a unque addtve mang A : X Y satsfyng the functonal equaton (1.8) and 1 α[ s x, s x,, s x] s (4.3) s 1 2 where for all x X. The mang A[x] s defned by Sr [ + 1] (r+1) [r + 1] (4.4) for all x X. A[x] lm t f [ t x] t (4.5)

4 52 M. Arunumar et al. / Functonal equaton orgnatng... Proof. Assume 1. Relacng [x 1, x 2,, x ] by [x, x,, x] n (4.2), we get f [[ ] x] [ ] f [x] α [x, x,, x] (4.6) for all x X. The above equaton can be rewrtten as [{ } ] [ ] f [ + 1 r] x [ + 1 r] f [x] α [x, x,, x] (4.7) for all x X. Usng Theorem 2.2, we have [{ } f Sr [ + 1] (r+1) [r + 1] x ] [ Sr ] [ + 1] (r+1) [r + 1] f [x] α [x, x,, x] (4.8) for all x X. Defne Sr n the above equaton and re modfyng, we arrve f [ x] f [x] [ + 1] (r+1) [r + 1] α [x, x,, x] (4.9) for all x X. Now relacng x by x and dvdng by n (4.9), we get f [x] f [2 x] 2 α [x, x,, x] 2 (4.10) for all x X. From (4.8) and (4.10), we obtan f [x] f [2 y] 2 f [x] f [x] + f [x] f [2 x] 2 1 { } α [x, x,, x] α [x, x,, x] + (4.11) for all x X. In general for any ostve nteger t, we get f [x] f [t x] t 1 1 t 1 s0 for all x X. In order to rove the convergence of the sequence α [ s x, s x,, s x] s (4.12) α [ s x, s x,, s x] s0 s { f [ t x] relace x by l x and dvdng by l n (4.12), for any t, l > 0, we deduce f [ l x] l f [l+t x] [l+t] 1 l f [l x] f [t l x] t t t 1 α s0 α s0 }, [ s+l x, s+l x,, s+l x s+l [ s+l x, s+l x,, s+l x as l s+l ] ]

5 M. Arunumar et al. / Functonal equaton orgnatng { f [ t } x] for all x X. Hence the sequence, s a Cauchy sequence. Snce Y s comlete, there exsts a mang A : X Y such that t A[x] lm t f [ t x] t x X. Lettng t n (4.12) we see that (4.3) holds for all x X. To show that A satsfes (1.8), relacng [x 1, x 2, x 3, x ] by [ t x 1, t x 2,, t x ] and dvdng by t n (4.2) and usng the defnton of A(x), and then lettng t, we see that A satsfes (1.8) for all x 1, x 2,, x X. To rove that A s unque, let B[x] be another addtve mang satsfyng (1.8) and (4.3), then A[x] B[x] 1 t A[ t x] B[ t x] 1 t { A[ t x] f [ t x] + f [ t x] B[ t x] } 2 α[ t+s x, t+s x,, t+s x] s0 t+s 0 as s for all x X. Hence A s unque. For 1, we can rove a smlar stablty result. Ths comletes the roof of the theorem. The followng corollary s an mmedate consequence of Theorem 4.4 concernng the Ulam-Hyers [13], Ulam-Hyers-Rassas [21], Ulam-Gavruta-Rassas [20] and Ulam-JRassas [26] stabltes of (1.8). Corollary 4.1. Let and q be nonnegatve real numbers. Let a functon f : X Y satsfes the nequalty [ ] f j [ x j j f [x j ] ], { } x j q, q < 1 or q > 1; x j q, q < 1 or q > 1 ; { { }} x j q + x j q, q < 1 or q > 1 ; for all x 1, x 2,, x X. Then there exsts a unque addtve functon A : X Y such that 1, x q q, x q q ( + 1) x q q (4.13) (4.14) for all x X. 5 Stablty results: Fxed ont method In ths secton, we aly a fxed ont method for achevng stablty of the addtve functonal equaton (1.8). Now, we resent the followng theorem due to B. Margols and J.B. Daz [17] for fxed ont theory.

6 54 M. Arunumar et al. / Functonal equaton orgnatng... Theorem 5.5. [17] Suose that for a comlete generalzed metrc sace (Ω, δ) and a strctly contractve mang T : Ω Ω wth Lschtz constant L. Then, for each gven x Ω, ether d(t n x, T n+1 x) n 0, or there exsts a natural number n 0 such that (FP1) d(t n x, T n+1 x) < for all n n 0 ; (FP2) The sequence (T n x) s convergent to a fxed to a fxed ont y of T (FP3) y s the unque fxed ont of T n the set {y Ω : d(t n 0 x, y) < }; (FP4) d(y, y) 1 L 1 d(y, Ty) for all y. Usng the above theorem, we now obtan the generalzed Ulam - Hyers stablty of (1.8). Theorem 5.6. Let f : X Y be a mang for whch there exst a functon α : X [0, ) wth the condton α [ µ tx 1, µ tx 2,, µ tx ] t0 µ t converges n R (5.1) where µ f 0 and µ 1 1 f 1 such that the functonal nequalty f [ j x j ] for all x 1, x 2, x 3, x X. If there exsts L L() < 1 such that the functon x γ[x] 1 [ x α, x,, x ], [ j f [x j ] ] α [x 1, x 2, x 3, x ] (5.2) has the roerty γ[x] L µ γ [µ x]. (5.3) Then there exsts a unque addtve mang A : X Y satsfyng the functonal equaton (1.8) and for all x X. L1 γ[x] (5.4) 1 L Proof. Consder the set Ω {/ : X Y, [0] 0} and ntroduce the generalzed metrc on Ω, d(, q) nf{k (0, ) : [x] q[x] Kγ[x], x X}. It s easy to see that (Ω, d) s comlete. Defne T : Ω Ω by T[x] 1 µ [µ x], for all x E. Now, q Ω, d(, q) K [x] q[x] Kγ[x], x X. 1 [µ µ x] 1 q[µ µ x] 1 Kγ[µ µ x], x X, 1 [µ µ x] 1 q[µ µ x] LKγ[x], x X, T[x] Tq[x] LKγ[x], x X, d(, q) LK.

7 M. Arunumar et al. / Functonal equaton orgnatng Ths mles d(t, Tq) Ld(, q), for all, q Ω..e., T s a strctly contractve mang on Ω wth Lschtz constant L. From (4.9), we arrve f [ x] f [x] γ[x] (5.5) for all x X. Usng (5.3) for the case 0 t reduces to f [x] f [x] Lγ[x] for all x X,.e., d( f, T f ) L d( f, T f ) L L 1 <. Agan relacng x x n (5.5), we get, [ ] [ ] x x f [x] f γ (5.6) for all x X. Usng (5.3) for the case 1 t reduces to ( ) x f [x] f γ[x] for all x X, In above cases, we arrve.e., d( f, T f ) 1 d( f, T f ) 1 L 0 <. d( f, T f ) L 1. Therefore (FP1) holds. By (FP2), t follows that there exsts a fxed ont A of T n Ω such that A[x] lm t f [µ t x] µ t x X. (5.7) To order to rove A : X Y s addtve, relacng [x 1,, x ] by [ µ t x 1,, µ t x ] and dvdng by µ t n (5.2) and usng the defnton of A(x), and then lettng t, we see that A satsfes (1.8) for all x 1,, x X. By (FP3), A s the unque fxed ont of T n the set {A Ω : d( f, A) < }, A s the unque functon such that for all x X and K > 0. Fnally by (FP4), we obtan Kγ[x] d( f, A) 1 1 L d( f, T f ) ths mles whch yelds ths comletes the roof of the theorem. d( f, A) L1 1 L L1 1 L γ[x] The followng corollary s an mmedate consequence of Theorem 5.6 concernng the Ulam-Hyers [13], Ulam-Hyers-Rassas [21], Ulam-Gavruta-Rassas [20] and Ulam-JRassas [26] stabltes of (1.8).

8 56 M. Arunumar et al. / Functonal equaton orgnatng... Corollary 5.2. Let f : X Y be a mang and there exts real numbers and q such that [ ] f j [ x j j f [x j ] ] (), { } () x j q, q < 1 or q > 1; () x j q, q < 1 or q > 1 ; { { }} (v) x j q + x j q, q < 1 or q > 1 ; for all x 1, x 2,, x X. Then there exsts a unque addtve functon A : X Y such that () 1, for all x X. () () (v) x q q, x q q ( + 1) x q q (5.8) (5.9) Proof. Settng for allx 1, x 2,, x X.. Now, 1 α[µ t x 1, µ t x 2,, µ t x ] µ t, { } x j q, α[x 1, x 2,, x ] x j q, µ t, { µ t µ t { { }} x j q + x j q, µ t x j q } µ t x j q,, { { }} µ t µ t x j q + µ t x j q, Thus, (5.1) s holds. But we have γ[x] 1 γ [x] has the roerty γ[x] L µ γ [µ x] for all x X. Hence γ[x] 1 α[x, x,, x] x q, x q, ( + 1) x q. 0 as t, 0 as t, 0 as t, 0 as t.

9 M. Arunumar et al. / Functonal equaton orgnatng Now, 1 µ γ[µ x] µ µ µ x q, µ µ x qs, ( + 1) µ µ x q. µ 1 µ q 1 µ q 1 µ q 1, x q, x q, ( + 1) x q. µ 1 γ[x], γ[x], µ q 1 µ q 1 µ q 1 γ[x], γ[x]. Hence the nequalty (5.3) holds ether, L 1 for q 0 f 0 and L 1 1 for q 0 f 1. Now from (5.4), we rove the followng cases for condton (). Case:1 L 1 for q 0 f 0 Case:2 L 1 1 for q 0 f 1 ( ( 1)(0 1)) ( 1)(0 1) ( ) ( 1)(0 1) 1 1 ( 1)(0 1) Also the nequalty (5.3) holds ether, L q 1 for q < 1 f 0 and L 1 q 1 for q > 1 f 1. Now from (5.4), we rove the followng cases for condton (). Case:1 L q 1 for q < 1 f 0 ( (q 1)) (q 1) x q (q 1) s x q (q 1) x q q Case:2 L 1 2 s 1 for s > 1 f 1 ( ) (q 1) 1 1 x q (q 1) s x q (q 1) x q s (q 1) Agan, the nequalty (5.3) holds ether, L q 1 for q < 1 1 f 0 and L for q > 1 q 1 f 1. Now from (5.4), we rove the followng cases for condton (). Case:1 L q 1 for q < 1 f 0 ( (q 1)) (q 1) x q (q 1) q x q (q 1) x q q

10 58 M. Arunumar et al. / Functonal equaton orgnatng... Case:2 L 1 q 1 for q > 1 f 1 ( ) (q 1) 1 1 x q (q 1) q x q (q 1) x q q (q 1) Fnally the nequalty (5.3) holds ether, L q 1 for q < 1 1 f 0 and L for q > 1 q 1 roof of condton (v) s smlar lnes to that of condton (). Hence the roof s comlete. f 1. The 6 Alcaton of the functonal equaton (1.8) We now that the followng sums of owers arthmetc rogresson ( + 1) ( + 1)(2 1) 2 In general, usng Strlng numbers of second nd, one can arrve S r [ + 1] (r+1). [r + 1] Wth the hel of the above dscusson, the authors transform the sum of th ower of frst natural numbers as a functonal equaton havng addtve soluton. f ( j x j ) ( j f (x j ) ),, 1 References [1] J. Aczel and J. Dhombres, Functonal Equatons n Several Varables, Cambrdge Unv, Press, [2] T. Ao, On the stablty of the lnear transformaton n Banach saces, J. Math. Soc. Jaan, 2 (1950), [3] M. Arunumar, Soluton and stablty of Arun-addtve functonal equatons, Internatonal Journal Mathematcal Scences and Engneerng Alcatons, Vol 4, No. 3, August 2010, [4] M. Arunumar, G. Ganaathy, S. Murthy, S. Kartheyan, Stablty of the generalzed Arun-addtve functonal equaton n Instutonstc fuzzy normed saces, Internatonal Journal Mathematcal Scences and Engneerng Alcatons Vol.4, No. V, December 2010, [5] M. Arunumar, C. Leela Sabar, Soluton and stablty of a functonal equaton orgnatng from a chemcal equaton, Internatonal Journal Mathematcal Scences and Engneerng Alcatons Vol. 5 No. II (March, 2011), 1-8. [6] M. Arunumar, S. Hema latha, C. Dev Shaymala Mary, Functonal equaton orgnatng from arthmetc Mean of consecutve terms of an arthmetc Progresson are stable n banach sace: Drect and fxed ont method, JP Journal of Mathematcal Scences, Volume 3, Issue 1, 2012, Pages [7] M. Arunumar, G. Vjayanandhraj, S. Kartheyan, Soluton and Stablty of a Functonal Equaton Orgnatng From n Consecutve Terms of an Arthmetc Progresson, Unversal Journal of Mathematcs and Mathematcal Scences, Volume 2, No. 2, (2012),

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12 60 M. Arunumar et al. / Functonal equaton orgnatng... Receved: Arl 4, 2013; Acceted: November 25, 2013 UNIVERSITY PRESS Webste: htt://