SUPERSTABILITY OF DIFFERENTIAL EQUATIONS WITH BOUNDARY CONDITIONS

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1 Electronic Journl of Differentil Equtions, Vol. 01 (01), No. 15, pp. 1. ISSN: URL: or ftp ejde.mth.txstte.edu SUPERSTABILITY OF DIFFERENTIAL EQUATIONS WITH BOUNDARY CONDITIONS JINGHAO HUANG, QUSUAY H. ALQIFIARY, YONGJIN LI Abstrct. In this rticle, we estblish the superstbility of differentil equtions of second order with boundry conditions or with initil conditions s well s the superstbility of differentil equtions of higher order with initil conditions. 1. Introduction In 190, Ulm [] posed problem concerning the stbility of functionl equtions: Give conditions in order for liner function ner n pproximtely liner function to exist. A yer lter, Hyers [7] gve n nswer to the problem of Ulm for dditive functions defined on Bnch spces: Let X 1 nd X be rel Bnch spces nd ε > 0. Then for every function f : X 1 X stisfying f(x + y) f(x) f(y) ε (x, y X 1 ), there exists unique dditive function A : X 1 X with the property f(x) A(x) ε (x X 1 ). After Hyers s result, mny mthemticins hve extended Ulm s problem to other functionl equtions nd generlized Hyers s result in vrious directions (see [,, 1, ]). A generliztion of Ulm s problem ws recently proposed by replcing functionl equtions with differentil equtions: The differentil eqution ϕ ( f, y, y,..., y (n)) = 0 hs the Hyers-Ulm stbility if for given ε > 0 nd function y such tht ϕ ( f, y, y,..., y (n)) ε, there exists solution y 0 of the differentil eqution such tht y(t) y 0 (t) K(ε) nd lim ε 0 K(ε) = 0. Ob loz seems to be the first uthor who hs investigted the Hyers-Ulm stbility of liner differentil equtions (see [1, 19]). Therefter, Alsin nd Ger published their pper [1], which hndles the Hyers-Ulm stbility of the liner differentil eqution y (t) = y(t): If differentible function y(t) is solution of the inequlity 000 Mthemtics Subject Clssifiction. A10, 39B, 3A0, 6D10. Key words nd phrses. Hyers-Ulm stbility; superstbility; liner differentil equtions; boundry conditions; initil conditions. c 01 Texs Stte University - Sn Mrcos. Submitted Jnury, 01. Published October 1, 01. 1

2 J. HUANG, Q. H. ALQIFIARY, Y. LI EJDE-01/15 y (t) y(t) ε for ny t (, ), then there exists constnt c such tht y(t) ce t 3ε for ll t (, ). Those previous results were extended to the Hyers-Ulm stbility of liner differentil equtions of first order nd higher order with constnt coefficients in [17, 6, 7] nd in [16], respectively. Furthermore, Jung hs lso proved the Hyers- Ulm stbility of liner differentil equtions (see [9, 10, 11]). Rus investigted the Hyers-Ulm stbility of differentil nd integrl equtions using the Gronwll lemm nd the technique of wekly Picrd opertors (see [, 5]). Recently, the Hyers-Ulm stbility problems of liner differentil equtions of first order nd second order with constnt coefficients were studied by using the method of integrl fctors (see [15, 9]). The results given in [10, 15, 17] hve been generlized by Cimpen nd Pop [3] nd by Pop nd Rş [0, 1] for the liner differentil equtions of nth order with constnt coefficients. Furthermore, the Lplce trnsform method ws recently pplied to the proof of the Hyers-Ulm stbility of liner differentil equtions (see [3]). In 1979, Bker, Lwrence nd Zorzitto [] proved new type of stbility of the exponentil eqution f(x + y) = f(x)f(y). More precisely, they proved tht if complex-vlued mpping f defined on normed vector spce stisfies the inequlity f(x + y) f(x)f(y) δ for some given δ > 0 nd for ll x, y, then either f is bounded or f is exponentil. Such phenomenon is clled the superstbility of the exponentil eqution, which is specil kind of Hyers-Ulm stbility. It seems tht the results of Gǎvruţ, Jung nd Li [5] re the erliest one concerning the superstbility of differentil equtions. In this pper, we prove the superstbility of the liner differentil equtions of second order with initil nd boundry conditions s well s liner differentil equtions of higher order in the form of (3.1) with initil conditions. First of ll, we give the definition of superstbility with initil nd boundry conditions. Definition 1.1. Assume tht for ny function y C n [, b], if y stisfies the differentil inequlity ϕ ( f, y, y,..., y (n)) ɛ for ll x [, b] nd for some ɛ 0 with initil(or boundry) conditions, then either y is solution of the differentil eqution ϕ ( f, y, y,..., y (n)) = 0 (1.1) or y(x) Kɛ for ny x [, b], where K is constnt not depending on y explicitly. Then, we sy tht (1.1) hs superstbility with initil (or boundry) conditions.. Preliminries Lemm.1. Let y C [, b] nd y() = 0 = y(b), then mx y(x) mx y (x). Proof. Let M = mx{ y(x) : x [, b]}. Since y() = 0 = y(b), there exists x 0 (, b) such tht y(x 0 ) = M. By Tylor s formul, we hve y() = y(x 0 ) + y (x 0 )(x 0 ) + y (ξ) (x 0 ),

3 EJDE-01/15 SUPERSTABILITY OF DIFFERENTIAL EQUATIONS 3 thus y(b) = y(x 0 ) + y (x 0 )(b x 0 ) + y (η) (b x 0 ) ; y (ξ) = M (x 0 ), M y (η) = (b x 0 ). In the cse x 0 (, +b ], we hve M (x 0 ) M (b ) / = M (b ) ; In the cse x 0 [ +b, b), we hve M (x 0 b) M (b ) / = M (b ). So Therefore, mx y (x) mx y(x) M (b ) = mx y(x). (b ) mx y (x). Lemm.. Let y C [, b] nd y() = 0 = y (), then mx y(x) Proof. By Tylor formul, we hve mx y (x). y(x) = y() + y ()(x ) + y (ξ) (x ). We hve (x ) (b ). Therefore, Thus y(x) y (ξ) (b ). mx y(x) mx y (x). Theorem.3 ([5]). Consider the differentil eqution with boundry conditions y (x) + β(x)y(x) = 0 (.1) y() = 0 = y(b), (.) where y C [, b], β(x) C[, b], < < b < +. If mx β(x) < /(b ). Then (.1) hs the superstbility with boundry conditions (.). Theorem. ([5]). Consider the differentil eqution (.1) with initil conditions y() = 0 = y (), (.3) where y C [, b], β(x) C[, b], < < b < +. If mx β(x) < /(b ). Then (.1) hs the superstbility with initil conditions (.3).

4 J. HUANG, Q. H. ALQIFIARY, Y. LI EJDE-01/15 3. Min results In the following theorems, we investigte the superstbility of the differentil eqution y (x) + p(x)y (x) + q(x)y(x) = 0 (3.1) with boundry conditions y() = 0 = y(b) (3.) or initil conditions y() = 0 = y (), (3.3) where y C [, b], p C 1 [, b], q C 0 [, b], < < b < +. Theorem 3.1. If mx{ q(x) 1 p (x) p (x) } < /(b ). (3.) Then (3.1) hs the superstbility with boundry conditions (3.). Proof. Suppose tht y C [, b] stisfies the inequlity for some ɛ > 0. Let for ll x [, b], nd define z(x) by y (x) + p(x)y (x) + q(x)y(x) ɛ (3.5) u(x) = y (x) + p(x)y (x) + q(x)y(x), (3.6) y(x) = z(x) exp By substitution (3.7) in (3.6), we obtin ( 1 z (x) + ( q(x) 1 p (x) p (x)) ( 1 z(x) = u(x) exp ) p(τ)dτ. (3.7) ) p(τ)dτ. Then it follows from inequlity (3.5) tht z (x) + ( q(x) 1 p (x) p (x)) z(x) = u(x)exp( 1 p(τ)dτ) ( 1 x ) exp p(τ)dτ ɛ. From (3.) nd (3.7) we hve z() = 0 = z(b). (3.) Define β(x) = q(x) 1 p (x) p (x), then it follows from (3.) nd by Lemm.1, mx z(x) mx z (x) [mx z (x) + β(x)z(x) + mx β(x) mx z(x) ] (b { ( ) 1 x mx exp p(τ)dτ )}ɛ + mx β(x) mx z(x). Obviously, mx{exp( 1 p(τ)dτ)} < on [, b]. Hence, there exists constnt K > 0 such tht z(x) Kɛ for ll x [, b].

5 EJDE-01/15 SUPERSTABILITY OF DIFFERENTIAL EQUATIONS 5 Moreover, mx{exp( 1 constnt K > 0 such tht p(τ)dτ)} < on [, b] which implies tht there exists y(x) = ( z(x) exp 1 { ( mx exp 1 Kɛ. ) p(τ)dτ )} p(τ)dτ Kɛ Thus (3.1) hs superstbility stbility with boundry conditions (3.). As in Theorem., we cn prove the following theorem. Theorem 3.. If mx{q(x) 1 p (x) p (x) } < /(b ). Then (3.1) hs superstbility stbility with initil conditions (3.3). Now, s exmples, we investigte the superstbility of the differentil eqution with boundry conditions nd initil conditions α(x)y (x) + β(x)y (x) + γ(x)y(x) = 0 (3.9) y() = 0 = y(b) (3.10) y() = 0 = y (), (3.11) where y C [, b], α, β, γ C 1 [, b], < < b < + nd α(x) 0 on [, b]. Theorem 3.3. (1) If mx{ γ(x) α(x) 1 ( β(x) α(x) ) 1 ( β(x) α(x) ) } < /(b ), then (3.9) hs superstbility with boundry conditions (3.10). () If mx{ γ(x) α(x) 1 ( β(x) α(x) ) 1 ( β(x) α(x) ) } < /(b ), then (3.9) hs superstbility with initil conditions (3.11). Corollry 3.. (1) If mx{ l(x) k(x) 1 d k (x) dx k(x) (k (x)/k(x)) } < /(b ), then d dx [k(x)y (x)] + l(x)y(x) = 0 (3.1) hs superstbility with boundry conditions (3.10), where k C 1 [, b], k(x) 0 on [, b] nd l C 0 [, b]. () If mx{ l(x) k(x) 1 d k (x) dx k(x) (k (x)/k(x)) } < /(b ), then (3.1) hs superstbility with initil conditions (3.11).

6 6 J. HUANG, Q. H. ALQIFIARY, Y. LI EJDE-01/15 Exmple 3.5. The differentil eqution y (x) + y (x) + y(x) = 0 (3.13) hs the superstbility with boundry conditions (3.10) on ny closed bounded intervl [, b] nd the superstbility with initil conditions (3.11) on ny closed bounded intervl [, b]. In the following theorem, we investigte the stbility of differentil eqution of higher order of the form y (n) (x) + β(x)y(x) = 0 (3.1) with initil conditions y() = y () = = y (n 1) () = 0, (3.15) where n N +, y C n [, b], β C 0 [, b], < < b < +. Theorem 3.6. If mx β(x) < initil conditions (3.15). (b ) n. Then (3.1) hs the superstbility with Proof. For every ɛ > 0, y C [, b], if y (n) (x) + β(x)y(x) ɛ nd y() = y () = = y (n 1) () = 0. Similrly to the proof of Lemm., y(x) = y() + y ()(x ) + + y(n 1) () (n 1)! (x )n 1 + y(n) (ξ) (x ) n. Thus y(x) = y(n) (ξ) (x ) n mx y (n) (b )n (x) for every x [, b]; so, we obtin (b )n mx y(x) [mx y (n) (x) + β(x)y(x) ] + (b )n (b )n ɛ + (b )n mx β(x) mx y(x). Let η = (b )n mx β(x), K = (b )n (1 η). It is esy to see tht y(x) Kɛ. Hence (3.1) hs superstbility with initil conditions (3.15). mx β(x)y(x) Acknowledgements. The uthors would like to thnk the nonymous referee for his or her corrections nd suggestions. Yongjin Li ws supported by the Ntionl Nturl Science Foundtion of Chin (107113). References [1] C. Alsin, R. Ger; On some inequlities nd stbility results relted to the exponentil function, J. Inequl. Appl. (199) [] J. Bker, J. Lwrence, F. Zorzitto; The stbility of the eqution f(x + y) = f(x)f(y), Proc. Amer. Mth. Soc. 7 (1979), -6. [3] D. S. Cimpen, D. Pop; On the stbility of the liner differentil eqution of higher order with constnt coefficients, Appl. Mth. Comput. 17 (010), [] S. Czerwik; Functionl Equtions nd Inequlities in Severl Vribles, World Scientific, Singpore, 00. [5] P. Gǎvruţ, S. Jung, Y. Li; Hyers-Ulm stbility for second- order liner differentil equtions with boundry conditions, Electronic J. Diff. Equ. 011 (011), 1-5.

7 EJDE-01/15 SUPERSTABILITY OF DIFFERENTIAL EQUATIONS 7 [6] R. Ger, P. Šemrl; The stbility of the exponentil eqution, Proc. Amer. Mth. Soc. 1 (1996), [7] D. H. Hyers; On the stbility of the liner functionl eqution, Proc. Ntl. Soc. USA 7 (191),. [] D. H. Hyers, G. Isc, Th. M. Rssis; Stbility of Functionl Equtions in Severl Vribles, Birkhäuser, Boston, 199. [9] S.-M. Jung; Hyers-Ulm stbility of liner differentil equtions of first order, Appl. Mth. Lett. 17 (00), [10] S.-M. Jung; Hyers-Ulm stbility of liner differentil equtions of first order, III, J. Mth. Anl. Appl. 311 (005), [11] S.-M. Jung; Hyers-Ulm stbility of liner differentil equtions of first order, II, Appl. Mth. Lett. 19 (006), 5 5. [1] S.-M. Jung; Hyers-Ulm-Rssis Stbility of Functionl Equtions in Nonliner Anlysis, Springer, New York, 011. [13] S.-M. Jung; On the superstbility of the Functionl Eqution f(x y ) = yf(x), Abh. Mth. Sem. Univ. Hmburg 67 (1997), [1] S.-M. Jung; On the superstbility of the functionl eqution f(x x m) = f(x 1 )... f(x m), Comm. Koren Mth. Soc. 1 (1999), No.1, pp [15] Y. Li, Y. Shen; Hyers-Ulm stbility of liner differentil equtions of second order, Appl. Mth. Lett. 3 (010) [16] T. Miur, S. Miyjim, S. E. Tkhsi; Hyers-Ulm stbility of liner differentil opertor with constnt coefficients, Mth. Nchr. 5 (003), [17] T. Miur, S. Miyjim, S. E. Tkhsi, A chrcteriztion of Hyers-Ulm stbility of first order liner differentil opertors, J. Mth. Anl. Appl. 6 (003), [1] M. Ob loz; Hyers stbility of the liner differentil eqution, Rocznik Nuk.-Dydkt. Prce Mt. 13 (1993), [19] M. Ob loz; Connections between Hyers nd Lypunov stbility of the ordinry differentil equtions, Rocznik Nuk.-Dydkt. Prce Mt. 1 (1997), [0] D. Pop, I. Rş; On the Hyers-Ulm stbility of the liner differentil eqution, J. Mth. Anl. Appl. 31 (011), [1] D. Pop, I. Rş; Hyers-Ulm stbility of the liner differentil opertor with non-constnt coefficients, Appl. Mth. Comput. 19 (01), [] Th. M. Rssis; On the stbility of the liner mpping in Bnch spces, Proc. Amer. Mth. Soc. 7 (197), [3] H. Rezei, S.-M. Jung, Th. M. Rssis; Lplce trnsform nd Hyers-Ulm stbility of liner differentil equtions, J. Mth. Anl. Appl. 03 (013), 51. [] I. A. Rus; Remrks on Ulm stbility of the opertoril equtions, Fixed Point Theory 10 (009), [5] I. A. Rus; Ulm stbility of ordinry differentil equtions, Stud. Univ. Bbes-Bolyi Mth. 5 (009), [6] S. E. Tkhsi, T. Miur, S. Miyjim; On the Hyers-Ulm stbility of the Bnch spcevlued differentil eqution y = λy, Bull. Koren Mth. Soc. 39 (00), [7] S. E. Tkhsi, H. Tkgi, T. Miur, S. Miyjim; The Hyers-Ulm stbility constnts of first order liner differentil opertors, J. Mth. Anl. Appl. 96 (00), [] S. M. Ulm; Problems in Modern Mthemtics, Chpter VI, Scince Editors, Wiley, New York, [9] G. Wng, M. Zhou, L. Sun; Hyers-Ulm stbility of liner differentil equtions of first order, Appl. Mth. Lett. 1 (00), Jingho Hung Deprtment of Mthemtics, Sun Yt-Sen University, Gungzhou, Chin E-mil ddress: hjingho@mil.sysu.edu.cn Qusuy H. Alqifiry Deprtment of Mthemtics, University of Belgrde, Belgrde, Serbi. University of Al-Qdisiyh, Al-Diwniy, Irq E-mil ddress: qhq010@gmil.com

8 J. HUANG, Q. H. ALQIFIARY, Y. LI EJDE-01/15 Yongjin Li (corresponding uthor) Deprtment of Mthemtics, Sun Yt-Sen University, Gungzhou, Chin E-mil ddress: