ON THE GENERALIZED SUPERSTABILITY OF nth ORDER LINEAR DIFFERENTIAL EQUATIONS WITH INITIAL CONDITIONS

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1 PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle série, tome , DOI: 10.98/PIM H ON THE GENERALIZED SUPERSTABILITY OF nth ORDER LINEAR DIFFERENTIAL EQUATIONS WITH INITIAL CONDITIONS Jingho Hung, Qusuy H. Alqifiry, nd Yongjin Li Abstrct. We estblish the generlized superstbility of differentil equtions of nth-order with initil conditions nd investigte the generlized superstbility of differentil equtions of second order in the form of y x + pxy x + qxyx = 0 nd the superstbility of liner differentil equtions with constnt coefficients with initil conditions. 1. Introduction In 1940, Ulm [8] 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 fx + y fx fy ε x, y X 1, there exists unique dditive function A : X 1 X with the property fx Ax ε x X 1. A generlized solution to Ulm s problem for pproximtely liner mppings ws proved by Rssis in 1978 []. He considered mpping f : E 1 E such tht t ftx is continuous in t for ech fixed x. Assume tht there exists θ 0 nd 0 p < 1 such tht fx + y fx fy θ x p + y p for ny x, y E 1. After Hyers s result, mny mthemticins hve extended Ulm s problem to other functionl equtions nd generlized Hyers s result in vrious directions [4, 8, 1]. A generliztion of Ulm s problem ws recently proposed by 010 Mthemtics Subject Clssifiction: Primry 44A10, 39B8; Secondry 34A40, 6D10. Key words nd phrses: Hyers Ulm stbility; superstbility; generlized superstbility; liner differentil equtions; initil conditions. Communicted by Grdimir Milovnović. 43

2 44 HUANG, ALQIFIARY, AND LI 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 ϕf, y, y,..., y n ε, there exists solution y 0 of the differentil eqution such tht yt y 0 t Kε nd lim ε 0 Kε = 0. Obłoz seems to be the first uthor who hs investigted the Hyers Ulm stbility of liner differentil equtions [18,19]. Therefter, Alsin nd Ger published their pper [1], which hndles the Hyers Ulm stbility of the liner differentil eqution y t = yt: If differentible function yt is solution of the inequlity y t yt ε for ny t,, then there exists constnt c such tht yt ce t 3ε for ll t,. Those previous results were extended to the Hyers Ulm stbility of liner differentil equtions of the first nd higher orders with constnt coefficients in [16,6,7] nd in [17], respectively. Furthermore, Jung hs lso proved the Hyers Ulm stbility of liner differentil equtions [9 11]. Rus investigted the Hyers Ulm stbility of differentil nd integrl equtions using the Gronwll lemm nd the technique of wekly Picrd opertors [4, 5]. Recently, the Hyers Ulm stbility problems of liner differentil equtions of the first nd second orders with constnt coefficients were studied by using the method of integrl fctors [15, 9]. The results given in [10,15,16] 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 [3]. In 1979, Bker, Lwrence nd Zorzitto [] proved new type of stbility of the exponentil eqution fx + y = fxfy. More precisely, they proved tht if complex-vlued mpping f defined on normed vector spce stisfies the inequlity fx + y fxfy 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. Here we investigte the generlized superstbility of liner differentil eqution of the nth order in the form 1.1 y n x + βxyx = 0, with initil conditions 1. y = y = = y n 1 = 0, where n N +, y C n [, b], β C 0 [, b], < < b < +. In ddition to tht we investigte the generlized superstbility of differentil equtions of the second order in the form of y x + pxy x + qxyx = 0 nd the superstbility of liner differentil equtions with constnt coefficients. First of ll, we give the definition of superstbility nd generlized superstbility with initil nd boundry conditions.

3 GENERALIZED SUPERSTABILITY 45 Definition 1.1. Assume tht for ny function y C n [, b], if y stisfies the differentil inequlity ϕf, y, y,..., y n ǫ for ll x [, b] nd for some ǫ 0 with initil or boundry conditions, then either y is solution of the differentil eqution 1.3 ϕf, y, y,..., y n = 0 or yx Kǫ for ny x [, b], where K is constnt not depending on y explicitly. Then, we sy tht 1.3 hs superstbility with initil or boundry conditions. Definition 1.. Assume tht for ny function y C n [, b], if y stisfies the differentil inequlity ϕf, y, y,..., y n ϕx for ll x [, b] nd for some function ϕ : [, b] [0, with initilor boundry conditions, then either y is solution of the differentil eqution 1.3 or yx Φx for ny x [, b], where Φ : I [0, is function not depending on y explicitly. Then, we sy tht 1.3 hs generlized superstbility with initil or boundry conditions.. Min results In this section, given the closed intervl I = [, b], we ssume tht ϕ : I [0, nd let Mpx denote mx τ [,x] pτ for every p CI,R. Theorem.1. If βx < /b n for every x I, then 1.1 hs generlized superstbility with initil conditions 1.. Proof. Suppose tht function y C n I,R stisfies the inequlity y n x + βxyx ϕx, for ll x I, By the Tylor formul, we hve yx = y + y x + + yn 1 n 1! x n 1 + yn ξ x n. Therefore, yx = y n ξ x n My n x n x for every x [, b]. Then, Myx M My n x n x M My n x x n M = My n x n x.

4 46 HUANG, ALQIFIARY, AND LI Thus Myx My n x n x x n M y n x + βxyx + x n Mϕx + b n Let C 1 = 1 b n mx βx. It is esy to see tht Myx x n M βx Myx mx βx Myx. x n Mϕx. C 1 Moreover, yx Myx, which completes the proof of our theorem. In the following theorem, we investigte the generlized superstbility of the differentil eqution.1 y x + pxy x + qxyx = 0 with initil conditions. y = 0 = y, where y C [, b], p C[, b], q C 0 [, b], < < b < +. Theorem.. If mx { qx 1 p x p x} 4 < /b, then.1 hs generlized superstbility with initil conditions.. Proof. Suppose tht y C [, b] stisfies the inequlity.3 y x + pxy x + qxyx ϕx. Let.4 ux = y x + pxy x + qxyx, for ll x [, b], nd define zx by.5 yx = zx exp 1 By substitution.5 in.4, we obtin z x + qx 1 p x p x 4 pτdτ. zx = ux exp 1 Then it follows from inequlity.3 tht z x + qx 1 p x p x zx = ux 1 exp 4 1 ϕx exp pτdτ. pτdτ pτdτ.

5 GENERALIZED SUPERSTABILITY 47 From. nd.5 we hve z = 0 = zb. It follows from Theorem.1 tht there exists constnt C 1 > 0 such tht x n 1 x zx M ϕx exp pτdτ. C 1 From.5 we hve x n yx M C 1 ϕx exp 1 pτdτ exp 1 Thus.1 hs generlized superstbility with initil conditions.. pτdτ. In the following theorems, we investigte the superstbility of the differentil eqution.6 y n x + n 1 y n 1 x y x + 0 yx = 0 with initil conditions.7 y = y = = y n 1 = 0, where y C n I,C, i R i = 0, 1,..., n 1, I = [, b], < < b < +. Lemm.1. Assume tht y C 1 I,C nd C { z C z < 1 b }. If y x Cyx ε with y = 0, then there exists constnt K > 0 such tht yx Kε. Proof. Let yx = Ax+iBx, where i is the imginry unit nd Ax, Bx C 1 I,R. Since y = 0, we hve A = 0 nd B = 0; so, similr to Theorem.1, we obtin mx Ax b mx A x CAx + C b mx Ax b mx y x Cyx + C b mx Ax b ε + C b mx Ax mx Bx b ε + C b mx Bx. Since C { z C z < 1 b }, there exists constnt K such tht mx yx mx Ax + mx Bx Kε. Theorem.3. If ll the roots of the chrcteristic eqution re in the disc { z C z < 1 b }, then.6 hs superstbility with initil conditions.7. Proof. Assume tht λ 1, λ,..., λ n re the roots of the chrcteristic eqution λ n + n 1 λ n λ + 0 = 0. Define g 1 x = y x λ 1 yx nd g i x = g i 1 x λ ig i 1 x i =, 3,, n 1, thus g n 1 x λ n g n 1 x = y n x + n 1 y n 1 x y x + 0 yx ε, nd g i = 0 for every i = 1,,..., n 1.

6 48 HUANG, ALQIFIARY, AND LI Since the bsolute vlue of λ n < 1 b nd g n 1 = 0, it follows from Lemm.1 tht there exists K 1 > 0 such tht g n 1 x K 1 ε. Recll g n 1 x = g n x λ n 1g n x, we hve g n x λ n 1g n x K 1 ε. By n rgument similr to the bove nd by induction, we cn show tht there exists constnt K > 0 such tht yx Kε. Remrk.1. In the present pper, we hve discussed the cse tht the solution f of differentil inequlity is bounded. In fct, the cse tht f is the exct solution of the corresponding differentil eqution fx = 0 0ε is lso included in it. Acknowledgements. The uthors thnk the nonymous referee for his or her suggestions nd corrections. This work ws supported by the Ntionl Nturl Science Foundtion of Chin References 1. C. Alsin, R. Ger, On some inequlities nd stbility results relted to the exponentil function, J. Inequl. Appl. 1998, J. Bker, J. Lwrence, F. Zorzitto, The stbility of the eqution fx + y = fxfy, Proc. Am. Mth. Soc , D. S. Cimpen, D. Pop, On the stbility of the liner differentil eqution of higher order with constnt coefficients, Appl. Mth. Comput , S. Czerwik, Functionl Equtions nd Inequlities in Severl Vribles, World Scientific, Singpore, P. Gǎvruţ, S. Jung, Y. Li, Hyers Ulm stbility for second-order liner differentil equtions with boundry conditions, Electronic J. Diff. Equ , R. Ger, P. Šemrl, The stbility of the exponentil eqution, Proc. Am. Mth. Soc , D. H. Hyers, On the stbility of the liner functionl eqution, Proc. Ntl. Soc. USA , D. H. Hyers, G. Isc, Th. M. Rssis, Stbility of Functionl Equtions in Severl Vribles, Birkhäuser, Boston, S.-M. Jung, Hyers Ulm stbility of liner differentil equtions of first order, Appl. Mth. Lett , , Hyers Ulm stbility of liner differentil equtions of first order, III, J. Mth. Anl. Appl , , Hyers Ulm stbility of liner differentil equtions of first order, II, Appl. Mth. Lett , , Hyers Ulm-Rssis Stbility of Functionl Equtions in Nonliner Anlysis, Springer, New York, , On the superstbility of the Functionl Eqution fx y = yfx, Abh. Mth. Sem. Univ. Hmburg , , On the superstbility of the functionl eqution fx x m = fx 1 fx m, Comm. Koren Mth. Soc , Y. Li, Y. Shen, Hyers Ulm stbility of liner differentil equtions of second order, Appl. Mth. Lett , T. Miur, S. Miyjim, S. E. Tkhsi, A chrcteriztion of Hyers Ulm stbility of first order liner differentil opertors, J. Mth. Anl. Appl , , Hyers Ulm stbility of liner differentil opertor with constnt coefficients, Mth. Nchr , M. Obłoz, Hyers stbility of the liner differentil eqution, Rocznik Nuk.-Dydkt. Prce Mt ,

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