Hyers-Ulam and Hyers-Ulam-Aoki-Rassias Stability for Linear Ordinary Differential Equations

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Avilble t http://pvu.edu/ Appl. Appl. Mth. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015, pp.

1 Avilble t Appl. Appl. Mth. ISSN: Vol. 10, Issue 1 (June 2015, pp Applictions nd Applied Mthetics: An Interntionl Journl (AAM Hyers-Ul nd Hyers-Ul-Aoki-Rssis Stbility for Liner Ordinry Differentil Equtions A. N. Mohptr Deprtent of Mthetics, Go University, Go, , Indi n@unigo.c.in Received: Jnury 23, 2013; Accepted: Mrch 7, 2015 Abstrct Here we prove the Hyers-Ul stbility nd Hyers-Ul-Aoki-Rssis stbility of the n-th order ordinry liner differentil eqution with sooth coefficients on copct nd sei-bounded intervls using successive integrtion by prts. Keywords: Ordinry differentil equtions; Hyers-Ul stbility MSC 2010 No.: 34A30; 34B30; 34C20; 39B82 1. Introduction Stnislw Mrcin Ul, in 1940, posed proble concerning the stbility of functionl eqution to give conditions in order for liner pping ner pproxite liner pping to exist. Hyers solved the proble for pir of Bnch spces, thus ce the terinology Hyers-Ul stbility (in short HU stbility. The result of Hyers ws further generlised by (Aoki, 1950 nd (Rssis, 1978, which is tered s Hyers-Ul-Aoki-Rssis stbility (in short HUAR stbility or siply Hyers-Ul-Rssis stbility or generlised Hyers-Ul stbility. Since then the stbility probles for functionl equtions hve been studied by ny theticins. The study of stbility for liner ordinry differentil equtions ws strted with the investigtion by (Obloz, 1993; Obloz, 1997 nd soon fter by (Alsin nd Ger, They studied the stbility of y (t y(t. This ws further generlised by (Miur et l.,. They studied the Hyers- 149

2 150 A. N. Mohptr Ul stbility of the differentil eqution y (t λy(t where λ is coplex nuber. After this ny hve investigted the Hyer-Ul stbility of vrious types of differentil equtions. In this note we prove the Hyers-Ul stbility nd the HUAR stbility for n-th order ordinry liner differentil eqution. For ore on Hyers-Ul type stbility of ordinry differentil equtions, we refer to (Jung, 2004; Jung, 2005; Jung, ; Miur et l., ; Qrwni, 2012; Rus, 2009; Miur et l., 2003; Miur et l., 2003b; Miur et l., b; Cipen nd Pop, Consider liner differentil eqution of n-th order L n y(x y (n + p 1 y (n p n y + q 0, (.1 on two types of intervls; copct intervl nd non-copct intervl. Assue tht the coefficient functions p 1,, p n re sufficiently sooth on the intervl under considertion. For non copct intervls, the Hyers-Ul stbility is soewht difficult to prove. For non-negtive function ɛ(t on n intervl, we sy tht n ties continuously differentible function y is n ɛ(t-pproxite solution of (.1 if y stisfies y (n (t + p 1 (ty (n 1 (t + + p n (ty(t + q(t ɛ(t, (.2 for ll t in the intervl. Siilrly we sy n ties continuously differentible function z is n exct solution of (.1 if L n z(t 0. Definition 0.1: The differentil eqution (.1 on n intervl is sid to be HU stble on n intervl if the following holds: For ny ɛ > 0 there exists constnt K > 0 (independent of ɛ such tht whenever y is n-ties differentible function stisfying L n y(x ɛ, there exists solution z of (.1 such tht y(x z(x Kɛ for ll x. Definition 0.2: The differentil eqution (.1 on n intervl is sid to be HUAR stble on if the following holds: Let ɛ(t 0 be continuous function. Then there exists n nonnegtive function ɛ 1 (t, which depends only on ɛ(t nd the coefficients of the ODE (.1, such tht whenever y is n-ties differentible function stisfying (.2, there exists solution z of (.1 such tht y(t z(t ɛ 1 (t. Consider the first order liner differentil eqution p 0 y + p 1 y + q 0, (.3 where p is nd q re ssued to be continuous functions on I (, b. In this cse ssuing (i p 0 (t 0 for ll t I, (ii p 1 (t δ for soe δ > 0, nd (iii p 1 (t p 0 dt <, the HU stbility (t ws proved in (Wng et l., 2008.

3 AAM: Intern. J., Vol. 10, Issue 1 (June However, (Jung, considered the eqution (.3 in coplex Bnch spce X with coplex vlued continuous coefficients. He proved the HUAR stbility of (.3: Theore 0.3: ((Jung, Let X be coplex Bnch spce, nd let q : I X strongly continuous function. Let p 1 be coplex vlued continuous function nd ɛ(t be non-negtive function on I. Denote G by G (t e R t ( p1(udu. Assue tht (i p 1 (t, exp t p 1(udu q(t re integrble on (, c for ech c I, (ii ɛ(texp t p 1(udu is integrble on I. Let y : I X be n ɛ(t-pproxite solution of (.3 with p 0 (t 1, where the derivtive is understood to exist in the strong sense. Then there exists unique x 0 X given by ( 1 t x 0 s li t b G (t y(t + q(u G (u du, ( such tht the function y 1 (t G (t x 0 t q(u du G (u is n unique exct solution of (.3 (with p 0 (t 1 nd stisfies y(t y 1 (t G (t t ɛ(u G (u du. This result gve n ipetus to study the stbility (in ters of unique solution of higher order liner differentil equtions. In the generl cse of n-th order, for constnt nd non constnt coefficients, the HUAR stbility ws proved by (?, nd (Pop nd Ros, 2012 respectively. In this cse the rguent for the n-th order liner eqution ws bsiclly successive ppliction of the Theore 0.3, ssuing tht the liner prt of the eqution is fctorised into product of first order ters (lthough not entioned explicitly ( ( ( d d dy dx + 1(x dx + 2(x dx + n(xy + q(x, nd on certin conditions on i (x. For n 2, the HU stbility ws proved in (Li nd Shen, 2010 using the bove fctoristion. The conditions on the i (x s cn be replced by soe integrbility conditions to prove the stbility for these equtions. Also there re other ethods, such s reducing second order liner non hoogeneous eqution to first order eqution using known solution of the corresponding second order hoogeneous eqution (Jvdin et l., 2011, or reducing the second order non hoogeneous eqution to first order liner non hoogeneous eqution if the second order eqution is exct (Ghei et l., For third order, the stbility ws studied explicitly using the bove fctoristion ethod in (Jung, 2012 nd (Abdollhpour et l., As it hs been noted, the constrints on the coefficient functions for stbility for higher order eqution is firly strong. However, if the underlying intervl is copct, the conditions on the coefficients cn be relxed nd hence the bove techniques work under less nuber of conditions.

4 152 A. N. Mohptr Here we ssue tht the intervl under considertion is either copct or seibounded nd we prove the HU nd HUAR stbility of n-th order liner differentil eqution with sooth vrible coefficients by successively integrting it nd converting it to n integrl eqution, where certin initil or terinl conditions re stisfied. This ethod is siple nd sees to hve been either reined unnoticed so fr or is considered too eleentry to be discussed in reserch rticle. We note tht on copct intervls the HU stbility ws studied for liner differentil equtions in (Li nd Shen, 2009; Gvrut et l., 2011; Qrwni, 2012; Abdollhpour nd Njti, 2011; Abdollhpour et l., 2012; Li nd Shen, 2010, using different ethods. The intervls on which we prove the stbility re either copct or sei bounded. 2. Hyers-Ul stbility of liner ODE on copct nd seibounded intervls For the reining prt of our discussion we will denote I 1 [, b], < < b <, I 2 [, b, < < b, I 3 (, b], < b <. Here we need few les which re required for the in result: Le 0.4: ( Let f be continuous function on n intervl I, where I I 1 or I I 2. Then n-successive integrtions ner the end point yield where t n I. (i (ii 1 dt n 1 dt n 2 1 dt n 1 dt n 2 f(t dt f(t (t n t n 1 (n 1! dt. (.4 dt (t n n, (.5 (n! (b Let f be continuous function on I, where I I 1 or I I 3. Then n-successive integrtions ner the end point b give where t n I. (iii (iv dt n 1 t n t n dt n 1 t n 1 dt n 2 t n 1 dt n 2 t 1 t 1 f(t dt f(t (t t n n 1 t n (n 1! dt. (.6 dt (b t n n, (.7 (n! Proof: we will prove (i by induction. For n 1 the identity is trivilly true. Assue it to hold for n 1, i.e. 1 1 dt n 2 f(t dt f(t (t n 1 t n 2 dt. (n 2!

5 AAM: Intern. J., Vol. 10, Issue 1 (June Then for n, dt n dt n 2 f(t dt dt n 1 f(t (t n 1 t n 2 (n 2! dt. With chnge of region using t t n 1 t n, one hs the t-integrl fro to t n nd t n 1 integrl fro t to t n. So the right hnd side of the integrl in the bove becoes (t n 1 t n 2 dt f(t dt n 1 t (n 2! f(t (t n t n 1 (n 1! dt. This proves prt (i. Now prt (ii follows fro prt (i by setting f(t 1. The proofs of prts (iii nd (iv re siilr to tht of prts (i nd (ii respectively. So we oit the proof. Le 0.5: ( Let ξ be n k ties continuously differentible on I, where I I 1 or I I 2, such tht ξ( ξ ( ξ (k 1 ( 0. Then for ny k ties continuously differentible function f on I nd for ny t, t k I, t k 1 t (i f(uξ (k (udu ( 1 j f (j (tξ (k j (t + ( 1 k j0 f (k (tξ(tdt, (.8 (ii tk dt k 1 tk 1 dt k 2 k ( k tk tk 1 ( 1 dt k 1 0 f(uξ (k (udu tk +1 dt k 2 f ( (uξ(udu, (.9 where the ter for 0 is understood to be f(t k ξ(t k. (b Let ξ be n n ties continuously differentible on I where I I 1 or I I 3 such tht ξ(b ξ (b ξ (k 1 (b 0. Then for ny k ties continuously differentible function f on I nd for ny t k I, dt k 1 dt k 2 t k t k 1 k ( k b ( 1 k 0 t k dt k 1 t 1 f(uξ (k (udu t k 1 dt k 2 where the ter for 0 is understood to be f(t k ξ(t k. t k +1 f ( (uξ(udu, (.10 Proof: We will prove prt ((i by induction. Note tht for k 1, the conclusion holds trivilly by integrtion by prts. Assuing tht it is true for k for ny 1 k n 1, we will prove it

6 154 A. N. Mohptr for k + 1. Now, the hypothesis tht it is true for k nd n integrtion by prts yield t f(uξ (k+1 (udu f(tξ (k (t f(tξ (k (t t f (uξ (k (udu } t ( 1 j f (j+1 (tξ k j (t + ( 1 k f (k+1 (uξ(udu { k 1 j0 k 1 t f(tξ (k (t + ( 1 j+1 f (j+1 (tξ (k j (t + ( 1 k+1 f (k+1 (uξ(udu j0 k t ( 1 l f (l (tξ (k+1 j (t + ( 1 k+1 f (k+1 (uξ(udu. l0 Prt ((ii cn lso be proved using induction. This identity is stisfied for k 1. Assue tht prt (ii holds for k. We will prove it for k + 1. For k + 1, n integrtion by prt, the conditions on ξ t nd the hypothesis tht (.9 holds for k yield t2 tk dt k dt k 1 f(uξ (k+1 (udu tk t2 ( dt k dt k 1 dt 1 f(uξ (k+1 (udu t2 dt k dt 1 (f(t 1 ξ (k (t 1 f (uξ (k (udu dt k dt k t2 dt 1 f(t 1 ξ (k (t 1 f(tξ (k (tdt k ( k tk+1 tk ( 1 j dt k dt k 1 j j0 dt k [ k i0 dt k f (uξ (k (udu ( tk dt k dt k 1 f (uξ (k (udu tk j+2 ( k tk tk 1 ( 1 i dt k 1 i f (j (uξ(udu ] tk i+1 dt k 2 f (i+1 (uξ(udu. Now tht the l-th ter of the first su dds up to the (l 1-th ter of the second su in the bove to give ( n + 1 tk+1 tk tk l+2 ( 1 l dt k dt k 1 f (l (uξ(udu. l After suing up there re k +1 ters which re the ters of the expnsion for the cse k +1. The proof of prt (b is se s tht of prt ((ii. So we oit its proof.

7 AAM: Intern. J., Vol. 10, Issue 1 (June Rerk 0.6: We y cll the identities (.9, (.10 s Leibnitz forule for successive integrtion. Le 0.7: ( Let I I 1 or I 2, nd ssue tht p i C n i (I for 1 i n. Suppose tht ξ is solution of the differentil eqution y (n (x + p 1 (xy (n 1 (x + + p n (xy(x g(x, (.11 on I with ξ (k ( 0 for 0 k n 1, where g is given continuous function on I. Then for ny t n I g(t (t n t n 1 (n 1! dt ξ(t n + n j n ( n j ( 1 j1 0 [ p ( j (t (t ] n t j+ 1 ξ(t dt. (j + 1! (.12 (b Let I I 1 or I I 3. Let ζ be solution of the differentil eqution (.11 on I, with ζ (k (b 0 for 0 k n 1, where g nd p i be s in the prt( of this le. Then for ny t n I g(t (t t n n 1 dt t n (n 1! n ( 1 n ζ(t n + j1 ( 1 n j n j ( n j b [ 0 t n p ( j (t (t n t j+ 1 (j + 1! ] ζ(t dt. (.13 Proof: We will oit the proof of prt (b since it is siilr to tht of prt (. To prove prt(, using Les 0.4(i nd 0.5(ii we hve

8 156 A. N. Mohptr g(t (t n t n 1 dt (n 1! ξ(t n ξ(t n + ξ(t n + j+1 ξ(t n + ξ(t n + dt n 1 1 dt n 2 1 dt n 1 dt n 2 dt n 1 1 dt n 1 1 n g(udu ( 1 2 dt n 1 dt n 2 j1 n j1 dt [ ξ (n (t + p 1 (tξ (n 1 (t + + p n (tξ(t ] dt n 3 p 1 (tξ (n 1 (tdt + ( j dt n 2 dt n j 1 p j (tξ (n j (t dt + dt n 2 dt n 1 dt n 1 j+1 p n (uξ(udu dt n j ( j ( n j ( n j tn j dt n j ( 1 0 n j n ( n j tn ( 1 dt n 1 0 n j n ( n j tn [ ( 1 j1 j1 0 dt n j 1 p j (tξ (n j (t dt j +1 dt n j 1 j +1 p ( j (t (t n t j+ 1 (j + 1! p ( j (tξ(t dt p ( j (tξ(t dt ] ξ(t dt. (.14 Our in result is s follows: Theore 0.8: Consider the differentil eqution (.1 on n intervl I. Assue tht the coefficients p k re n k ties continuously differentible on I for 1 k n. Assue tht q is coplex vlued continuous function on I. Let ɛ(t be n rbitrry nonnegtive continuous function on I. ( Assue tht the bove hypotheses hold on I I 1 or I I 2. Then there exists nonnegtive function ɛ 1 (x (depending on ɛ(x nd the coefficient functions p i only such tht if n n-ties continuously differentible function y stisfies the inequlity (.2, then there exists nonzero solution z 1 of (.1 such tht y(x z 1 (x ɛ 1 (x, (.15

9 AAM: Intern. J., Vol. 10, Issue 1 (June where ɛ 1 (x is given by (.18. (b Assue tht the bove hypotheses hold on I, where I I 1 or I I 3. Then there exists nonnegtive function ɛ 2 (x (depending on ɛ(x nd the coefficient functions p i only such tht if n n-ties continuously differentible function y stisfies the inequlity (.2, then there exists nonzero solution z 2 of (.1 stisfying where [ ] ( (t b xn 1 n ɛ 2 (x ɛ(t dt exp (n 1! x y(x z 2 (x ɛ 2 (x, (.16 x j1 ( 1 n j n j ( n j 0 p ( j (t xj+ 1 (t (j + 1! dt. (.17 Proof: For prt (, for siplicity, we will denote L n y to be the left hnd side of (.1. Suppose tht L n y(t ɛ(t for ll t I. Let z 1 stisfies L n z(t 0 nd tht z (k 1 ( y (k ( for 0 k n 1. Then L n y(t L n z 1 (t ɛ(t. Setting g(t L n y(t L n z 1 (t, nd ξ(t y(t z 1 (t, note tht g(t ɛ(t, nd tht g nd ξ stisfy the hypotheses of Le 0.7. So ξ (n (t + p 1 (tξ (n 1 (t + + p n (tξ(t g(t. Upon integrting successively n ties ner we obtin (.12 using Le 0.7(. Using tringle inequlity of the bsolute vlue, nd tht g(t ɛ(t, we hve n j n ( n j tn [ ξ(t n ( 1 j1 0 n j n ξ(t ( n j tn [ n + ( 1 j1 0 g(t (t n t n 1 dt (n 1! Setting t n x in the bove inequlity, we hve p ( j (t (t n t j+ 1 (j + 1! p ( j (t (t n t j+ 1 (j + 1! ɛ(t (t n t n 1 (n 1! dt. ] ξ(t dt ] ξ(t dt ξ(x x (x tn 1 ɛ(t (n 1! dt + x [ n n j ( n j ( 1 j1 0 p ( j ] (x tj+ 1 (t (j + 1! ξ(t dt. So by Gronwll s inequlity (see Theore of (Pchptte, 1998 nd reclling tht ξ(x

10 158 A. N. Mohptr y(x z 1 (x, one hs y(x z 1 (x [ x ] (x tn 1 ɛ(t dt (n 1! ( x n j n ( n j exp ( 1 ɛ 1 (x, j1 0 p ( j (x tj+ 1 (t (j + 1! dt (.18 for ll x. The proof of prt (b is siilr to tht of prt (, where z 2 is solution of (.1, z (k 2 (b y(k (b for 0 k n 1, nd uses Les 0.4(b, 0.5(b nd 0.7(b. Rerk 0.9: If the intervl under considertion is I 1 nd ɛ(x ɛ, then it follows tht the function ɛ 1 (x in (.18 nd ɛ 2 (x in (.17 re bounded by Kɛ for soe K > 0. Hence in this cse the liner ODE is HU stble. Rerk 0.10: It is interesting to copre the error estites in the bove Theore 0.3 with tht of Theore 0.8. Note tht for n 1, z 2 (x y 1 (x, where z 2 nd y 1 re obtined in Theores 0.8(b nd 0.3 respectively. So z 2 in Theore 0.8 is unique. For n > 1, the function z 1 stisfying (.18 is not unique, s cn be seen fro the next exple. Exple 0.11: Consider the differentil eqution u x on the intervl I [0, 1]. Let ɛ 1/4 nd y(x x Here K sup x [0,1] x Then y x x2 16 (2 x ɛ. Since y(0 1/16 nd y (0 0, ccording to Theore 0.8, z 1 (x x nd y(x z 1 (x ɛ x2 2 ɛ 2 1/8. Let z 2 (x x Then z 2 stisfies z 2 x nd, since x2 16 x4 192 for ll x [0, 1],

11 AAM: Intern. J., Vol. 10, Issue 1 (June Hence, z is not unique. y(x z 2 (x x 2 16 x x2 16 x x2 (12 x ɛ 2. However, if we insist on z j (j 1, 2 nd its derivtives upto (n 1-th order to hve the se initil or terinl vlue s tht of the derivtives of y, then z j is unique (which follows fro the uniqueness of solutions of the initil vlue proble nd we hve y(t z j (t ɛ j (t for j 1, 2, ppering in (.15 nd (.16 respectively. Rerk 0.12: All the results in this section cn esily be generlised to liner differentil eqution in coplex Bnch spce X, where the differentibility is considered in the strong sense. More precisely, Theore 0.13: Let J J j, j 1, 2, where J 1 I 1 or J 1 I 2, nd J 2 I 1 or J 2 I 3 respectively. Let X be coplex Bnch spce. Let ɛ : J [0, be continuous function. If y : J X is strongly n-ties continuously differentible function stisfying (.11, where p i re in C (n i (J, C functions such tht, whenever g(x ɛ(x, there exist non-negtive functions ɛ j (x, j 1, 2 (independent of y, nd Bnch spce vlued n-ties strongly differentil functions z j, j 1, 2, stisfying L n z j (x 0, z (k j (s j y (k (s j (for j 1, 2, 0 k n 1, with s 1, s 2 b, such tht y(x z j (x ɛ j (x for j 1, 2. The proof of it goes lost in verbti with tht of the bove theore using results siilr to Les 0.5, 0.7 for Bnch spce vlued functions. 3. Conclusion Here we prove the Hyers-Ul stbility nd Hyers-Ul-Aoki-Rssis stbility of n-th order liner ordinry differentil eqution with sooth coefficients on copct nd sei-bounded intervls using successive integrtion by prts. The ide here is s follows: if y stisfies (.2 on I, where I is one of the for I 1 or I 2 or I 3, then choose solution z of (.1 which longwith ll upto its n 1 derivtives gree with those of y t the finite end point of the intervl. This solution z is used to prove tht the differentil eqution (.1 is HUAR stble. This is chieved by pplying the corresponding differentil opertor on y z nd integrting successively n ties ner this end point (t which y nd z longwith their first n 1 derivtives gree nd king use of Gronwll s inequlity.

12 160 A. N. Mohptr Acknowledgents We re grteful to the nonyous reviewers nd the editor Professor Alikbr Montzer Hghighi for their helpful coents nd suggestions which indeed iproved the qulity of this nuscript. REFERENCES Abdollhpour, M. nd Njti, A. (2011. Stbility of liner differentil eqution of third order. Applied Mthetics Letters, 24: Abdollhpour, M., Njti, A., nd Ki, H. (2012. Hyers-ul stbility of differentil eqution of third order. Interntionl Journl of Mtheticl Anlysis, 6(59: Alsin, C. nd Ger, R. (1998. On soe inequlities nd stbility results relted to the exponentil function. Journl of Inequlities nd Applictions, 2: Aoki, T. (1950. On the stbility of the liner trnsfortion in bnch spces. Journl of the Mtheticl Society of Jpn, 2: Cipen, D. nd Pop, D. (2010. On the stbility of liner differentil eqution with constnt coefficients. Applied Mthetics nd Coputtion, 217: Gvrut, P., Jung, S., nd Li, Y. (2011. Hyers-ul stbility for second order liner differentil equtions with boundry conditions. Electronics Journl of Differentil Equtions, 2011(80:1 5. Ghei, M., Gordji, M., Alizdeh, B., nd Prk, C. (2012. Hyers-ul stbility of exct second order liner differentil equtions. Advnces in Difference Equtions, 2012(36:1 7. Jvdin, A., Soruri, E., Ki, G., nd Gordgi, M. (2011. Generlized hyers-ul stbility of the second-order liner differentil equtions. Journl of Applied Mthetics, 2011(doi: /2011/813137:1 10. Jung, S. Hyers-ul stbility of liner differentil equtions of first order (ii. Jung, S. (2004. Hyers-ul stbility of liner differentil equtions of first order. Applied Mthetics Letters, 17: Jung, S. (2005. Hyers-ul stbility of liner differentil equtions of first order (iii. Journl of Mtheticl Anlysis nd Applictions, 311: Jung, S. (2012. Approxite solutions of liner differentil eqution of third order. Bulletin of the Mlysin Mtheticl Sciences Society, 35(4: Li, Y. nd Shen, Y. (2009. Hyers-ul stbility of nonhoogeneous liner differentil equtions of second order. Interntionl Journl of Mthetics nd Mtheticl Anlysis, 2009:1 7. Li, Y. nd Shen, Y. (2010. Hyers-ul stbility of liner differentil equtions of second order. Applied Mthetics Letters, 23: Miur, T., Jung, S., nd Tkhsi, S. Hyers-ul-rssis stbility of bnch spce vlued liner differentil equtions y λy. Miur, T., Jung, S., nd Tkhsi, S. Hyers-ul stbility of bnch spce vlued liner differentil equtions y λy. Miur, T., Miyji, S., nd Tkhsi, S. (2003. A chrcteristion of hyers-ul stbility of first order liner differentil opertors. Journl of Mtheticl Anlysis nd Applictions, 286:

13 AAM: Intern. J., Vol. 10, Issue 1 (June Miur, T., Miyji, S., nd Tkhsi, S. (2003b. Hyers-ul stbility of first order liner differentil opertor with constnt coefficients. Mthetische Nchrichten, 286: Obloz, M. (1993. Hyers stbility of the liner differentil equtions. Rocznik Nukowo- Dydktyczny. Prce Mtetyczne, 13: Obloz, M. (1997. Connection between hyers nd lypunov stbility of the ordinry differentil equtions. Rocznik Nukowo-Dydktyczny. Prce Mtetyczne, 14: Pchptte, B. (1998. Inequlities for Differentil nd Integrl equtions. Acdeic Press, Sn Deigo, London. Pop, D. nd Ros, I. (2012. Hyers-ul stbility of the liner differentil opertor with nonconstnt coefficients. Applied Mthetics nd Coputtion, 219: Qrwni, M. (2012. Hyers-ul stbility of liner nd nonliner differentil equtions of second order. Interntionl Journl of Applied Mthetics Reserch, 1(4: Rssis, T. (1978. On the stbility liner pping in bnch spces. Proceedings of the Aericn Mtheticl Society, 72: Rus, L. (2009. Ul stbility of ordinry differentil equtions. Studi Mthetic, Bbes- Bolyi Mth, 54: Wng, G., Zhou, M., nd Sun, L. (2008. Hyers-ul stbility of liner differentil eqution of first order. Applied Mthetics Letters, 21:

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

PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle série, tome 9811 015, 43 49 DOI: 10.98/PIM15019019H ON THE GENERALIZED SUPERSTABILITY OF nth ORDER LINEAR DIFFERENTIAL EQUATIONS WITH INITIAL CONDITIONS