Research Article Generalized Hyers-Ulam Stability of the Second-Order Linear Differential Equations

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1 Hindwi Publihing Corportion Journl of Applied Mthemtic Volume 011, Article ID , 10 pge doi: /011/ Reerch Article Generlized Hyer-Ulm Stbility of the Second-Order Liner Differentil Eqution A. Jdin, 1 E. Sorouri, G. H. Kim, 3 nd M. Ehghi Gordji 1 Deprtment of Phyic, Semnn Unierity, P. O. Box , Semnn, Irn Deprtment of Mthemtic, Semnn Unierity, P. O. Box , Semnn, Irn 3 Deprtment of Mthemtic, Kngnm Unierity, Yongin, Gyeonggi , Republic of Kore Correpondence hould be ddreed to G. H. Kim, ghkim@kngnm.c.kr Receied 6 September 011; Accepted 3 October 011 Acdemic Editor: Kupplplle Vjrelu Copyright q 011 A. Jdin et l. Thi i n open cce rticle ditributed under the Cretie Common Attribution Licene, which permit unretricted ue, ditribution, nd reproduction in ny medium, proided the originl work i properly cited. We proe the generlized Hyer-Ulm tbility of the nd-order liner differentil eqution of the form y pxy qxy fx, with condition tht there exit nonzero y 1 : I X in C I uch tht y 1 pxy 1 qxy 1 0ndI i n open interl. A conequence of our min theorem, we proe the generlized Hyer-Ulm tbility of eerl importnt well-known differentil eqution. 1. Introduction The tbility problem of functionl eqution trted with the quetion concerning tbility of group homomorphim propoed by Ulm 1 during tlk before Mthemticl Colloquium t the Unierity of Wiconin, Mdion. In 1941, Hyer ge prtil olution of Ulm problem for the ce of pproximte dditie mpping in the context of Bnch pce. In 1978, Ri 3 generlized the theorem of Hyer by conidering the tbility problem with unbounded Cuchy difference fx y fx fy ɛ x p y p, ɛ >0, p 0, 1. Thi phenomenon of tbility tht w introduced by Ri 3 i clled the Hyer-Ulm-Ri tbility or the generlized Hyer-Ulm tbility. Let X be normed pce oer clr field K, ndleti be n open interl. Aume tht for ny function f : I X tifying the differentil inequlity n ty n t n 1 ty n 1 t 1 ty t 0 tyt ht ɛ 1.1

2 Journl of Applied Mthemtic for ll t I nd for ome ɛ 0, there exit function f 0 : I X tifying n ty n t n 1 ty n 1 t 1 ty t 0 tyt ht 0, ft f0 t Kɛ 1. for ll t I; here Kt i n expreion for ɛ with lim ɛ 0 Kɛ 0. Then, we y tht the boe differentil eqution h the Hyer-Ulm tbility. If the boe ttement i lo true when we replce ɛ nd Kɛ by ϕt nd φt, where ϕ, φ : I 0, re function not depending on f nd f 0 explicitly, then we y tht the correponding differentil eqution h the Hyer-Ulm-Ri tbility or the generlized Hyer-Ulm tbility. The Hyer-Ulm tbility of differentil eqution y y w firt inetigted by Alin nd Ger 4. Thi reult h been generlized by Tkhi et l. 5 for the Bnch pce-lued differentil eqution y λy. In6, Miur et l. lo proed the Hyer-Ulm- Ri tbility of liner differentil of firt order, y gtyt 0, where gt i continuou function, while the uthor 7 proed the Hyer-Ulm-Ri tbility of liner differentil of the form cty t yt. Jung8 proed the Hyer-Ulm-Ri tbility of liner differentil of firt order of the form cty tgtytht 0. In thi pper, we inetigte the generlized Hyer-Ulm tbility of differentil eqution of the form y pxy qxy fx. 1.3 We ume tht X i complex Bnch pce, I, b i n rbitrry interl, nd y 1 : I X i nonzero olution of correponding homogeneou eqution of 1.3, where y 1 pxy 1 qxy Min Reult Tking ome ide from 8, we re going to inetigte the tbility of the nd-order liner differentil eqution. For the ke of conenience, ll the integrl nd derition will be iewed exiting nd Rω denote the rel prt of complex number ω. Moreoer, let I, b be n open interl, where, b R {± } re rbitrrily gien with <b. Theorem.1. Let X be complex Bnch pce. Aume tht p, q : I C nd f : I X re continuou function nd y 1 : I X i nonzero twice continuouly differentible function which tifie the differentil eqution 1.4. If twice continuouly differentible function y : I X tifie y pxy qxy fx ϕx.1

3 Journl of Applied Mthemtic 3 for ll x I, wherek y/y 1 X nd ϕ : I 0, i continuou function, then there exit unique x 0 X uch tht yx y 1x exp y 1 u [ { f ] x 0 y 1 exp d d k y 1 u y1 x x exp R y 1 u b ( } ϕt pudu y 1 u y1 t dt d.. Proof. We ume tht x yx y 1 x.3 for ll x I. It follow from 1.4,.1, nd.3 tht ( xy 1 x ( px xy1 x ( qx xy1 x fx ( x y 1 xxy 1 x px ( x y 1 x xy 1 x qxxy 1 x fx x y 1 x x ( y 1 x pxy 1 x x ( y 1 x pxy 1 x qxy 1 x fx x y 1 x x ( y 1 x pxy 1 x fx y1 x ( x x y1 x px fx y 1 x y 1 x.4 ϕx, o, we he ( x x y1 x px fx y 1 x y 1 x ϕx y1 x..5

4 4 Journl of Applied Mthemtic For implicity, we ue the following nottion: { ( y z : exp y 1 u y 1 ( { f y 1 exp d y 1 u.6 for ll I. By mking ue of thi nottion nd by.5,weget { z zl exp ( y y 1 u y 1 ( { f y 1 exp d y 1 u { l ( yl exp y 1 u y 1 l l ( f ( y 1 exp d y 1 u ( t ( ( yt dt exp l y 1 u y 1 t t ( { f y 1 exp d y 1 u (( { y1 t t pt exp l y 1 t y 1 u ( yt t y1 u exp{ ( yt y 1 t y 1 u y 1 t ( ft { t y 1 t exp dt y 1 u { t exp l y 1 u (( y1 t ( yt ( yt pt ft dt y 1 t y 1 t y 1 t y 1 t ( } ϕt pudu y 1 u y1 t dt l.7

5 Journl of Applied Mthemtic 5 for ll l, x I. Since exp{r t y 1u /y 1 u pudu ϕt/ y 1 t i umed to be integrble on I, we my elect l 0 I, for ny gien ɛ>0, uch tht l, x l 0 implie zx zl <ɛ.thti,{zl} l I i Cuchy net. By completene of X, there exit n x 0 X uch tht zl conerge to x 0 l b. It follow from.7 nd the preiou rgument tht, for ny x I, yx y 1x exp y 1 u [ { f ] x 0 y 1 exp d d k y 1 u y 1x exp z x 0 d y 1 u y1 x x exp R z zl d y 1 u y 1 x x exp R zl x 0 d y 1 u y 1 x x exp R y 1 u ( ϕt pudu l y 1 u y 1 t dt d y1 x x exp R zl x 0 d y 1 u y1 x x exp R y 1 u b ( ϕt pudu y 1 u y1 t dt d.8 l b. Moreoer, ( { y 0 x y 1 x exp y 1 u [ { f ].9 x 0 y 1 exp d d k y 1 u i olution of 1.3.

6 6 Journl of Applied Mthemtic Now, we proe the uniquene property of x 0. Aume tht x 1,x X tify inequlity. in plce of x 0. Then, we he y 1x x ( { exp y1 x x x x 1 d y 1 u exp R y 1 u b ( ϕt pudu y 1 u y1 t dt d,.10 thu, x x 1 x ( { exp R } b Adu Adu (ϕt/ y1 t dt d x }, exp R Adu d.11 where A denote y 1 u /y 1 u pu. It follow from the integrbility hypothei tht b ( ϕt pudu y 1 u y1 t dt 0.1 b. Thi implie tht x 1 x nd the proof i complete. Remrk.. It follow from Theorem.1 tht ( { yx y 1 x exp y 1 u [ { f.13 c 1 y 1 exp d ]d c y 1 u i the generl olution of the differentil eqution 1.3, where c 1,c re rbitrry element of X nd y 1 x i nonzero olution of the correponding homogeneou eqution 1.3. Remrk.3. If we replce C by R in the proof of Theorem.1 nd we ume tht p, q re rel-lued continuou function, then we cn ee tht Theorem.1 i true for rel Bnch pce X.

7 Journl of Applied Mthemtic 7 Hence, eery nd-order liner differentil eqution h the generlized Hyer-Ulm tbility with the condition tht there exit olution of correponding homogeneou eqution or there exit generl olution in the ordinry differentil eqution. Exmple.4. Conider the econd-order liner differentil eqution with contnt coefficient y by cy fx..14 Let b 4c 0, m b ± b 4c/, nd let f : I R, ϕ : I 0, be continuou function. Aume tht y : I R i twice continuouly differentil function tifying the differentil inequlity y by cy fx ϕx.15 for ll x I. On the other hnd, by ordinry differentil eqution, we know tht y 1 x expmx i olution of correponding homogeneou eqution of.14. It follow from Theorem.1, Remrk.3, nd.14 tht there exit olution y 0 : I R of.14 uch tht ( y 0 x expmx exp m b [ ].16 x 0 f expm b m bd d k for ll x I nd tht yx y0 x expmx x ( exp m b b ϕt expm bt dt expmx d..17 Exmple.5. Conider.14. Letb 4c <0, m b ± b 4c/ α ± iβ, ndletf : I R,ϕ : I 0, be continuou function. Let y : I R be twice continuouly differentil function tifying the differentil inequlity of.15 for ll x I. It follow

8 8 Journl of Applied Mthemtic from the ordinry differentil eqution tht y 1 x expαx coβx. Then it follow from Theorem.1, Remrk.3,nd.15 tht there exit olution y 0 : I R of.14 uch tht y 0 x expαx co ( βx ( x 0 co ( β x expαx co ( βx exp α b co ( β expα b co ( β d ( f exp α b co ( β d exp α b d k.18 for ll x I, where k y/expα coβ nd x 0 R i unique nd yx y0 x expαx co ( βx x ( exp αb co ( β b co ( βt expαbtϕtdt d..19 Exmple.6. Conider the eqution y x ( 1 x y 1 x y 6 1 x..0 Let I, b be n open interl, where, b 1, re rbitrrily gien with < b, f : I R nd ϕ : I 0, re continuou function. Aume tht y : I R i twice continuouly differentil function tifying the differentil inequlity y x 1 x y ( 1 x y 6 1 x ϕx.1 for ll x I. Bythetrilofy 0 x x, we ee tht it i olution of correponding homogeneou eqution of.0. Then it follow from Theorem.1, Remrk.3, nd.1 tht there exit olution y 0 : I R of.0 uch tht ( ( 1 y 0 x x x 0 k ( ( x 1 x x 4 3x. for ll x I, where k y/ nd x 0 R i unique nd yx y 0 x x x (( 1 b t 1 t ϕtdt d..3

9 Journl of Applied Mthemtic 9 Remrk.7. We know tht Eulr differentil eqution of econd order h the generl olution in ordinry differentil eqution, then we cn ue Theorem.1 nd Remrk.3 for the Hyer-Ulm-Ri tbility in thi ce. Let p be rel contnt, nd I 1, 1. We know tht Legender differentil eqution ( 1 x y xy p ( p 1 y 0.4 h the generl olution y 0 y 1 x 1 y x,.5 where y 1 x 1 p( p 1 ( ( ( p p p 1 p 3 x x 4, 4! y x x P 1( p ( ( ( ( p 3 p 1 p p 4 x 3 x 5 3! 5!.6 nd 0, 1 re rbitrry contnt. By Theorem.1 nd Remrk.3, Legender differentil eqution h Hyer-Ulm-Ri tbility. Hermite differentil eqution y xy py 0,.7 where p i rel contnt, h the generl olution y 0 y 1 x 1 y x.8 tht y 1 x 1 1 n n p ( p (p n x n, n! n1 y x x 1n n( p 1 ( p 3 (p n 1 n 1! x n1.9 for ll x R, nd 0, 1 re rbitrry contnt. Thu Hermite differentil eqution h generlized Hyer-Ulm tbility. It i well known from the ordinry differentil eqution tht y 1 x J p x 1 n ( x np, n!γ ( n p 1.30 n0

10 10 Journl of Applied Mthemtic for ll x R, i olution of Beel differentil eqution ( x y xy x p y 0.31 tht p 0. Then Beel differentil eqution h Hyer-Ulm-Ri tbility. We know from the ordinry differentil eqution tht Lguerre, Chebyhe, nd Gu hypergeometric differentil eqution he the generl olution. Then we cn how tht thoe he generlized Hyer-Ulm tbility. Acknowledgment The third uthor of thi work w prtilly upported by Bic Science Reerch Progrm through the Ntionl Reerch Foundtion of Kore NRF funded by the Minitry of Eduction, Science nd Technology Grnt no Reference 1 S. M. Ulm, Problem in Modern Mthemtic, chpter 6, John Wiley & Son, New York, NY, USA, D. H. Hyer, On the tbility of the liner functionl eqution, Proceeding of the Ntionl Acdemy of Science of the United Stte of Americ, ol. 7, pp. 4, T. M. Ri, On the tbility of the liner mpping in Bnch pce, Proceeding of the Americn Mthemticl Society, ol. 7, no., pp , C. Alin nd R. Ger, On ome inequlitie nd tbility reult relted to the exponentil function, Journl of Inequlitie nd Appliction, ol., no. 4, pp , S.-E. Tkhi, T. Miur, nd S. Miyjim, On the Hyer-Ulm tbility of the Bnch pce-lued differentil eqution y λy, Bulletin of the Koren Mthemticl Society, ol. 39, no., pp , T. Miur, S. Miyjim, nd S.-E. Tkhi, A chrcteriztion of Hyer-Ulm tbility of firt order liner differentil opertor, Journl of Mthemticl Anlyi nd Appliction, ol. 86, no. 1, pp , S.-M. Jung, Hyer-Ulm tbility of liner differentil eqution of firt order, Applied Mthemtic Letter, ol. 17, no. 10, pp , S.-M. Jung, Hyer-Ulm tbility of liner differentil eqution of firt order. II, Applied Mthemtic Letter, ol. 19, no. 9, pp , 006.

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