UNIVERZITET U BEOGRADU MATEMATIČKI FAKULTET. Qusuay Hatim Eghaar Alqifiary

Transcription

1 UNIVERZITET U BEOGRADU MATEMATIČKI FAKULTET Qusuy Htim Eghr Alqifiry STABILNOST REŠENJA DIFERENCIJALNIH JEDNAČINA U SMISLU HAJERSA I ULAMA DOKTORSKA DISERTACIJA BEOGRAD, 2015

2 UNIVERSITY OF BELGRADE FACULTY OF MATHEMATICS Qusuy Htim Eghr Alqifiry HYERS-ULAM STABILITY OF THE SOLUTIONS OF DIFFERENTIAL EQUATIONS DOCTORAL DISSERTATION BELGRADE, 2015

3 Abstrct This thesis hs been written under the supervision of my mentor Prof. dr. Julk Knezević-Miljnović t the University of Belgrde in the cdemic yer The im of this study is to investigte Hyers-Ulm stbility of some types of differentil equtions, nd to study generlized Hyers-Ulm stbility nd s well s specil cse of the Hyers-Ulm stbility problem, which is clled the superstbility. Therefore, when there is differentil eqution, we nswer the three min questions: 1- Does this eqution hve Hyers -Ulm stbility? 2- Wht re the conditions under which the differentil eqution hs stbility? 3- Wht is Hyers-Ulm constnt of the differentil eqution? The thesis is divided into three chpters. Chpter 1 is divided into 3 sections. In this chpter, we introduce some sufficient conditions under which ech solution of the liner differentil eqution u (t + ( 1 + ψ(t u(t = 0 is bounded. Aprt from this we prove the Hyers-Ulm stbility of it nd the nonliner differentil equtions of the form u (t + F (t, u(t = 0, by using the Gronwll lemm nd we prove the Hyers-Ulm stbility of the second-order liner differentil equtions with boundry conditions. In ddition to tht we estblish the superstbility of liner differentil equtions of second-order nd higher order with continuous coefficients nd with constnt coefficients, respectively. Chpter 2 is divided into 2 sections. In this chpter, by using the Lplce trnsform method, we prove tht the liner differentil eqution of the nth-order y (n (t + n 1 α k y (k (t = f(t hs the generlized Hyers-Ulm stbility. And we prove lso the Hyers-Ulm- Rssis stbility of the second-order liner differentil equtions with initil nd boundry conditions, s well s liner differentil equtions of higher order in the form of y (n (x + β(xy(x = 0, with initil conditions. Furthermore, 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+p(xy (x+q(xy(x = 0. Chpter 3 is divided into 2 sections. In this chpter, by pplying the fixed point lterntive method, we give necessry nd sufficient condition in order tht the first order liner k=0

4 Alqifiry Abstrct ii system of differentil equtions ż(t + A(tz(t + B(t = 0 hs the Hyers-Ulm- Rssis stbility nd find Hyers-Ulm stbility constnt under those conditions. In ddition to tht, we pply this result to second-order differentil eqution ÿ(t + f(tẏ(t + g(ty(t + h(t = 0. Also, we pply it to differentil equtions with constnt coefficient in the sme sense of proofs. And we give sufficient condition in order tht the first order nonliner system of differentil equtions hs Hyers-Ulm stbility nd Hyers-Ulm-Rssis stbility. In ddition, we present the reltion between prcticl stbility nd Hyers-Ulm stbility nd lso Hyers- Ulm-Rssis stbility. Scientific field (nučn oblst: Mthemtics (mtemtik(34a40, 34A12, 34D10 Nrrow scientific field (už nučn oblst: Differentil Equtions (Diferencijlne jednčine UDC: (043.3

5 Podci o mentoru i člnovim komisije: MENTOR: redovni profesor dr Julk Knezević-Miljnović Mtemtički fkultet, Univerzitet u Beogrdu ČLANOVI KOMISIJE : redovni profesor dr Boško Jovnović Mtemtički fkultet, Univerzitet u Beogrdu redovni profesor dr Miodrg Splević Mtemtički fkultet, Univerzitet u Beogrdu Dtum odbrne:

6 Acknowledgements I would like to express my specil pprecition nd thnks to my supervisor Professor Julk Knezević-Miljnović, for ll the help she hs given over the lst three yers. Her dvice on my reserch hs been priceless. I would like to cknowledge those who hve shped my mthemticl eduction. The stff of professors t the deprtment of mthemtics. A specil thnks to my fmily nd friends, for their full support, ptience nd understnding. To ech of the bove, I express my deepest grtitude. Qusuy Alqifiry

7 Introduction This subject dtes bck to the tlk given by the Polish-Americn mthemticin Ulm t the University of Wisconsin in 1940 (see [40]. In tht tlk, Ulm sked whether n pproximte solution of functionl eqution must be ner n exct solution of tht eqution. This sking of Ulm is stted s follows: Theorem Let G 1 be group nd let G 2 be metric group with metric d(.,.. Given ε > 0, does there exist δ > 0 such tht if function h : G 1 G 2 stisfies the inequlity d(h(xy.h(xh(y < δ for ll x, y G 1, then there is homomorphism H : G 1 G 2 with d(h(x, H(x < ε for ll x G 1? One yer lter, prtil nswer to this question ws given by D. H. Hyers [5] for dditive functions defined on Bnch spces: Theorem Let f : X 1 X 2 be function between Bnch spces such tht f(x + y f(x f(y δ, for some δ > 0 nd for ll x, y X 1. Then the limit A(x = lim n 2 n f(2 n x exists for ech x X 1, nd A : X 1 X 2 is the unique dditive function such tht f(x A(x δ for every x X 1. Moreover, if f(tx is continuous in t for ech fixed x X 1, then the function A is liner. This result is clled the Hyers-Ulm Stbility of dditive Cuchy eqution g(x + y = g(x+g(y. After Hyers s result, mny mthemticins hve extended Ulm s

8 Alqifiry Introduction vi problem to other functionl equtions nd generlized Hyers s result in vrious directions (see [6, 32, 38, 44]. Ten yers fter the publiction of Hyerss theorem, D. G. Bourgin extended the theorem of Hyers nd stted it in his pper [4] without proof. Unfortuntely, it seems tht this result of Bourgin filed to receive ttention from mthemticins t tht time. No one hs mde use of this result for long time. In 1978, Rssis [44] introduced new functionl inequlity tht we cll Cuchy- Rssis inequlity nd succeeded in extending the result of Hyers, by wekening the condition for the Cuchy differences to unbounded mp s follows: Theorem Let f : X 1 X 2 be function between Bnch spces. If f stisfies the functionl inequlity f(x + y f(x f(y θ( x p + y p for some θ 0, p with 0 p < 1 nd for ny x, y X 1, then there exists unique dditive function A : X 1 X 2 such tht f(x A(x 2θ 2 2 p x p for ech x X 1. If, in ddition, f(tx is continuous in t for ech fixed x X 1, then the function A is liner. The stbility phenomenon of this kind is clled the Generlized Hyers-Ulm Stbility (or Hyers-Ulm-Rssis stbility. 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 Hyers-Ulm stbility if for given ε > 0 nd function y such tht φ(f, y, y,..., y (n ε, there exists solution y of the differentil eqution such tht y(t y (t K(ε nd lim ε 0 K(ε = 0. If the preceding sttement is lso true when we replce ε nd K(ε by φ(t nd Φ(t, where φ, Φ re pproprite functions not depending on y nd y explicitly, then we sy tht the corresponding differentil eqution hs the generlized Hyers-Ulm stbility (or Hyers-Ulm-Rssis stbility. Ob loz seems to be the first uthor who hs investigted the Hyers-Ulm stbility of liner differentil equtions (see [21, 22]. Therefter, Alsin nd Ger published their pper [3], 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 y (t y(t ε for ny t (,, then there exists constnt c such tht y(t ce t 3ε for ll t (,. Since then, this problem now known s the problem of Hyers-Ulm stbility - hs been extensively investigted for the lgebric, functionl, differentil, integrl, nd opertor equtions.

9 Alqifiry Introduction vii Those previous results were extended to the Hyers-Ulm stbility of liner differentil equtions of first order [33 37, 42, 43]. Rus investigted the Hyers-Ulm stbility of differentil nd integrl equtions using the Gronwll lemm nd the technique of wekly Picrd opertors (see [13, 14]. Recently, The results given in [36, 42, 45] hve been generlized by Cimpen nd Pop [10] nd by Pop nd Rş [8, 9] for the liner differentil equtions. In 1979, J.Bker, J. Lwrence nd F. Zorzitto[16] proved new type of stbility of the exponentil eqution f(x + y = f(xf(y. More precisely, they proved tht if complex-vlued mpping f defined on normed vector spce stisfies the inequlity f(x + y f(xf(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 P. Gǎvruţ, S. Jung nd Y. Li [23] re the erliest one concerning the superstbility of differentil equtions. This thesis is bout stbility of some types of differentil equtions, where we introduce this thesis in three chpters. Chpter one is titled by Hyers-Ulm stbility of Differentil Equtions. This chpter consists of three sections. In section 1.1, we introduce some sufficient conditions under which ech solution of the liner differentil eqution (1.1.2 is bounded. As well s we prove the Hyers-Ulm stbility of the liner differentil equtions of the form ( In section 1.2, we prove the Hyers-Ulm stbility of the nonliner differentil equtions of the form (1.2.1 by using the Gronwll lemm. In section 1.3, we prove the Hyers-Ulm stbility of the second-order liner differentil equtions with boundry conditions. Furthermore, the superstbility of liner differentil equtions with constnt coefficients. Chpter two is titled by Generlized Hyers-Ulm stbility of Differentil equtions. This chpter consists of two sections. In section 2.1, by using the Lplce trnsform method, we prove tht the liner differentil eqution of the nth-order n 1 y (n (t + α k y (k (t = f(t k=0 hs the generlized Hyers-Ulm stbility, where α k is sclr, y nd f re n times continuously differentible nd of exponentil order, respectively. In section 2.2, 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 + p(xy (x + q(xy(x = 0. In dditionl, we prove the Hyers-Ulm-Rssis stbility of the second-order liner differentil equtions with initil nd boundry conditions s well s liner differentil equtions of higher order in the form of y (n (x + β(xy(x = 0,

10 Alqifiry Introduction viii with initil conditions y( = y ( = = y (n 1 ( = 0, where n N +, y C n [, b], β C 0 [, b], < < b < +. Chpter three is titled by Hyers-Ulm stbility of system of differentil equtions. This chpter consists of two sections. In section 3.1, by pplying the fixed point lterntive method, we give necessry nd sufficient condition in order tht the first order liner system of differentil equtions ż(t + A(tz(t + B(t = 0 hs the Hyers-Ulm-Rssis stbility nd find Hyers-Ulm stbility constnt under those conditions. In ddition to tht, we pply this result to second order differentil eqution ÿ(t + f(tẏ(t + g(ty(t + h(t = 0. Also, we pply it to differentil equtions with constnt coefficient in the sme sense of proofs. In section 3.2, we give sufficient condition in order tht the first order nonliner system of differentil equtions hs Hyers-Ulm stbility nd Hyers-Ulm-Rssis stbility. In ddition, we present the reltion between prcticl stbility nd Hyers-Ulm stbility nd lso Hyers-Ulm-Rssis stbility.

11 Contents Abstrct Acknowledgements Introduction i iv v 1 Hyers-Ulm Stbility of Differentil Equtions Hyers-Ulm Stbility of Liner Differentil Equtions Boundedness of Solutions of Second Order Differentil Eqution Hyers-Ulm Stbility of Liner Differentil Equtions of Second Order Hyers-Ulm Stbility of Nonliner Differentil Equtions of Second Order Hyers-Ulm Stbility of Differentil Equtions with Boundry Conditions Generlized Hyers-Ulm Stbility of Differentil Equtions Generlized Hyers-Ulm Stbility of Liner Differentil Equtions Generlized Hyers-Ulm Stbility of Differentil Equtions with boundry Conditions Generlized Superstbility of Differentil Equtions with Initil Conditions Hyers-Ulm-Rssis Stbility of Liner Differentil Equtions with Boundry Conditions Hyers-Ulm Stbility of System of Differentil Equtions 47

12 Alqifiry Contents x 3.1 Hyers-Ulm Stbility of Liner System of Differentil Equtions Preliminries nd Auxiliry Results Hyers-Ulm Stbility of First Order System of Differentil Equtions Hyers-Ulm Stbility of Second Order Differentil Eqution Hyers-Ulm Stbility of Nonliner System of Differentil Equtions Preliminries nd Auxiliry Results Hyers-Ulm Stbility of System of Differentil Equtions 58 List of ppers 62 Bibliogrphy 63

13 Chpter 1 Hyers-Ulm Stbility of Differentil Equtions 1.1 Hyers-Ulm Stbility of Liner Differentil Equtions Boundedness of Solutions of Second Order Differentil Eqution In this subsection, we first introduce nd prove lemm which is kind of the Gronwll inequlity. Lemm [28] Let u, v : [0, [0, be integrble functions, c > 0 be constnt, nd let 0 be given. If u stisfies the inequlity for ll t, then for ll t. u(t c + u(τv(τdτ (1.1.1 ( u(t c exp v(τdτ Proof. It follows from (1.1.1 tht u(tv(t c + u(τv(τdτ v(t

14 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 2 for ll t. Integrting both sides of the lst inequlity from to t, we obtin ( ln c + u(τv(τdτ ln c v(τdτ or c + ( u(τv(τdτ c exp v(τdτ for ech t, which together with (1.1.1 implies tht ( u(t c exp v(τdτ for ll t. In the following theorem, using Lemm 1.1.1, we investigte sufficient conditions under which every solution of the differentil eqution is bounded. u (t + ( 1 + ψ(t u(t = 0 (1.1.2 Theorem [28] Let ψ : [0, R be differentible function. Every solution u : [0, R of the liner differentil eqution (1.1.2 is bounded provided tht 0 ψ (t dt < nd ψ(t 0 s t. Proof. First, we choose lrge enough so tht 1 + ψ(t 1/2 for ll t. Multiplying (1.1.2 by u (t nd integrting it from to t, we obtin 1 2 u (t u(t2 + ψ(τu(τu (τdτ = c 1 for ll t. Integrting by prts, this yields 1 2 u (t u(t ψ(tu(t2 1 2 for ny t. Then it follows from (1.1.3 tht ψ (τu(τ 2 dτ = c 2 ( u(t2 1 2 u (t u(t2 1 2 u (t ( 1 + ψ(t u(t 2 2 = c ψ (τu(τ 2 dτ

15 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 3 for ll t. Thus, it holds tht u(t 2 4c ψ (τu(τ 2 dτ 4 c ψ (τ u(τ 2 dτ (1.1.4 for ny t. In view of Lemm 1.1.1, (1.1.4 nd our hypothesis, there exists constnt M 1 > 0 such tht ( u(t 2 4 c 2 exp 2 ψ (τ dτ < M1 2 for ll t. On the other hnd, since u is continuous, there exists constnt M 2 > 0 such tht u(t M 2 for ll 0 t, which completes the proof. Corollry [28] Let ϕ : [0, R be differentible function stisfying ϕ(t 1 s t. eqution is bounded provided 0 ϕ (t dt <. Every solution u : [0, R of the liner differentil u (t + ϕ(tu(t = 0 ( Hyers-Ulm Stbility of Liner Differentil Equtions of Second Order Given constnts L > 0 nd 0, let U(L; denote the set of ll functions u : [, R with the following properties: (i u is twice continuously differentible; (ii u( = u ( = 0; (iii u (τ dτ L. We now prove the Hyers-Ulm stbility of the liner differentil eqution (1.1.2 by using the Gronwll inequlity. Theorem [28] Given constnts L > 0 nd 0, ssume tht ψ : [, R is differentible function with C := ψ (τ dτ < nd λ := inf t t0 ψ(t > 1. If function u U(L; stisfies the inequlity u (t + ( 1 + ψ(t u(t ε (1.1.6

16 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 4 for ll t nd for some ε 0, then there exist solution u 0 U(L; of the differentil eqution (1.1.2 nd constnt K > 0 such tht u(t u 0 (t K ε (1.1.7 for ny t, where K := ( 2L 1 + λ exp C 2(1 + λ. Proof. We multiply (1.1.6 with u (t to get ε u (t u (tu (t + u(tu (t + ψ(tu(tu (t ε u (t for ll t. If we integrte ech term of the lst inequlities from to t, then it follows from (ii tht ε u (τ dτ 1 2 u (t u(t2 + for ny t. Integrting by prts nd using (iii, we hve εl 1 2 u (t u(t ψ(tu(t2 1 2 ψ(τu(τu (τdτ ε for ll t. Since 1 + λ > 0 holds for ll t, it follows from (1.1.8 tht or 1 + λ 2 u (τ dτ ψ (τu(τ 2 dτ εl (1.1.8 u(t u (t λ u(t u (t ( 1 + ψ(t u(t 2 2 εl εl ψ (τu(τ 2 dτ for ny t. Applying Lemm 1.1.1, we obtin u(t 2 2Lε ( λ exp 1 + λ ψ (τ u(τ 2 dτ u(t 2 2Lε 1 + λ + 1 ψ (τ u(τ 2 dτ 1 + λ ψ (τ dτ 2Lε ( C 1 + λ exp 1 + λ

17 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 5 for ll t. Hence, it holds tht ( u(t exp C 2(1 + λ 2Lε 1 + λ for ny t. Obviously, u 0 (t 0 stisfies the eqution (1.1.2 nd the conditions (i, (ii, nd (iii such tht u(t u 0 (t K ε ( 2L for ll t, where K = exp C. 1+λ 2(1+λ If we set ϕ(t := 1+ψ(t, then the following corollry is n immedite consequence of Theorem Corollry [28] Given constnts L > 0 nd 0, ssume tht ϕ : [, R is differentible function with C := ϕ (τ dτ < nd λ := inf t t0 ϕ(t > 0. If function u U(L; stisfies the inequlity u (t + ϕ(tu(t ε for ll t nd for some ε 0, then there exist solution u 0 U(L; of the differentil eqution (1.1.5 nd constnt K > 0 such tht u(t u 0 (t K ε for ny t, where K := exp ( C 2L. 2λ λ Exmple [28] Let ϕ : [0, R be constnt function defined by ϕ(t := for ll t 0 nd for constnt > 0. Then, we hve C = 0 ϕ (τ dτ = 0 nd λ = inf t 0 ϕ(t =. Assume tht twice continuously differentible function u : [0, R stisfies u(0 = u (0 = 0, 0 u (τ dτ L, nd u (t + ϕ(tu(t = u (t + u(t ε for ll t 0 nd for some ε 0 nd L > 0. According to Corollry 1.1.5, there exists solution u 0 : [0, R of the differentil eqution, y (t + y(t = 0, such tht u(t u 0 (t 2L ε

18 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 6 for ny t 0. Indeed, if we define function u : [0, R by u(t := α (t + 1 cos 2α t + sin t α, 2 (t where we set α = + L, then u stisfies the conditions stted in the first prt +2 of this exmple, s we see in the following. It follows from the definition of u tht u (t = ( 2α (t α (t cos t ( α (t α (t nd hence, we get u(0 = u (0 = 0. Moreover, we obtin u (t 2 + ( 4 α (t + 1 α (t sin t nd 0 u (τ 2 + ( 4 dτ = 0 (τ + 1 αdτ + α (τ + 1 dτ 3 = ( 2 + ( 2 α + 1 α = L. For ny given ε > 0, if we choose the constnt α such th < α then we cn esily see tht u (t + u(t ( 8 (t α cos t 3 (t + 1 ( (t α sin t α 3 (t ( 8 (t α + 3 (t = α ε for ny t 0. ( 4 (t (t ε , α + α

19 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 7 Theorem [28] Given constnts L > 0 nd 0, ssume tht ψ : [, (0, is monotone incresing nd differentible function. If function u U(L; stisfies the inequlity (1.1.6 for ll t nd for some ε > 0, then there exists solution u 0 U(L; of the differentil eqution (1.1.2 such tht for ny t. 2Lε u(t u 0 (t ψ( (1.1.9 Proof. We multiply (1.1.6 with u (t to get ε u (t u (tu (t + u(tu (t + ψ(tu(tu (t ε u (t for ll t. If we integrte ech term of the lst inequlities from to t, then it follows from (ii tht ε u (τ dτ 1 2 u (t u(t2 + ψ(τu(τu (τdτ ε for ny t. Integrting by prts, the lst inequlities together with (iii yield εl 1 2 u (t u(t ψ(tu(t2 1 2 for ll t. Then we hve 1 2 ψ(tu(t2 1 2 for ny t. Applying Lemm 1.1.1, we obtin ψ (τu(τ 2 dτ + εl εl ψ(tu(t2 εl exp ( u (τ dτ ψ (τu(τ 2 dτ εl ψ (τ ψ(τ dτ = εl ψ(t ψ( ψ (τ ψ(τ u(τ2 ψ(τ 2 dτ for ll t, since ψ : [, (0, is monotone incresing function. Hence, it holds tht 2Lε u(t ψ( for ny t. Obviously, u 0 (t 0 stisfies the eqution (1.1.2, u 0 U(L;, s well s the inequlity (1.1.9 for ll t.

20 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 8 Corollry [28] Given constnts L > 0 nd 0, ssume tht ϕ : [, (1, is monotone incresing nd differentible function with ϕ( = 2. If function u U(L; stisfies the inequlity u (t + ϕ(tu(t ε for ll t nd for some ε > 0, then there exists solution u 0 U(L; of the differentil eqution (1.1.5 such tht for ny t. u(t u 0 (t 2Lε If we set ϕ(t := ψ(t, then the following corollry is n immedite consequence of Theorem Corollry [28] Given constnts L > 0 nd 0, ssume tht ϕ : [, (, 0 is monotone decresing nd differentible function with ϕ( = 1. If function u U(L; stisfies the inequlity u (t + ( 1 ϕ(t u(t ε for ll t nd for some ε > 0, then there exists solution u 0 U(L; of the differentil eqution such tht for ny t. u (t + ( 1 ϕ(t u(t = 0 u(t u 0 (t 2Lε Exmple [28] Let ϕ : [0, (, 0 be monotone decresing function defined by ϕ(t := e t 2 for ll t 0. Then, we hve ϕ(0 = 1. Assume tht twice continuously differentible function u : [0, R stisfies u(0 = u (0 = 0, 0 u (τ dτ L, nd u (t + ( 1 ϕ(t u(t = u (t + ( 3 e t u(t ε

21 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 9 for ll t 0 nd for some ε > 0 nd L > 0. According to Corollry 1.1.8, there exists solution u 0 : [0, R of the differentil eqution, y (t + ( 3 e t y(t = 0, such tht for ny t 0. u(t u 0 (t 2Lε Indeed, if we define function u : [0, R by u(t := α (t + 1 sin t + 1 α 3 2 (t + 1 cos t α 2 2, where α is rel number with α 2 ε, then u stisfies the conditions stted 43 in the first prt of this exmple, s we see in the following. It follows from the definition of u tht u (t = 3α (t + 1 sin t 1 α 4 2 (t + 1 sin t 2 nd hence, we get u(0 = u (0 = 0. Moreover, we obtin nd 0 u (t 3 α (t α 4 2 (t u (τ 3 α dτ 0 (τ + 1 dτ + 1 α dτ =: L < (τ We cn see tht u (t + ( 3 e t u(t 12α 3α sin t (t (t + 1 cos t + ( 4 e t α 4 (t + 1 sin t e t α 3 e t cos t α 2 (t α (t α 5 (t α 4 (t α 3 (t α for ny t α ε

22 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions Hyers-Ulm Stbility of Nonliner Differentil Equtions of Second Order In the following theorems, we investigte the Hyers-Ulm stbility of the nonliner differentil eqution u (t + F (t, u(t = 0. (1.2.1 Theorem [28] Given constnts L > 0 nd 0, ssume tht F : [, R (0, is function stisfying F (t, u(t/f (t, u(t > 0 nd F (t, 0 = 1 for ll t nd u U(L;. If function u : [, [0, stisfies u U(L; nd the inequlity u (t + F (t, u(t ε (1.2.2 for ll t nd for some ε > 0, then there exists solution u 0 : [, [0, of the differentil eqution (1.2.2 such tht for ny t. u(t u 0 (t Lε Proof. We multiply (1.2.2 with u (t to get ε u (t u (tu (t + F (t, u(tu (t ε u (t for ll t. If we integrte ech term of the lst inequlities from to t, then it follows from (ii tht ε u (τ dτ 1 2 u (t 2 + F (τ, u(τu (τdτ ε for ny t. Integrting by prts nd using (iii, the lst inequlities yield εl 1 2 u (t 2 + F (t, u(tu(t for ll t. Then we hve F (t, u(tu(t εl + εl + u (τ dτ F (τ, u(τu(τdτ εl F (τ, u(τu(τdτ F (τ, u(τ F (τ, u(τu(τdτ F (τ, u(τ

23 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 11 for ny t. Applying Lemm 1.1.1, we obtin ( F (t, u(tu(t εl exp F (τ, u(τ F (τ, u(τ dτ = εlf (t, u(t for ll t. Hence, it holds tht u(t Lε for ny t. Obviously, u 0 (t 0 stisfies the eqution (1.2.1 nd u 0 U(L; such tht u(t u 0 (t Lε for ll t. In the following theorem, we investigte the Hyers-Ulm stbility of the Emden- Fowler nonliner differentil eqution of second order for the cse where α is positive odd integer. u (t + h(tu(t α = 0 (1.2.3 Theorem [28] Given constnts L > 0 nd 0, ssume tht h : [, (0, is differentible function. Let α be n odd integer lrger thn 0. If function u : [, [0, stisfies u U(L; nd the inequlity u (t + h(tu(t α ε (1.2.4 for ll t nd for some ε > 0, then there exists solution u 0 : [, [0, of the differentil eqution (1.2.3 such tht for ny t, where β := α + 1. u(t u 0 (t Proof. We multiply (1.2.4 with u (t to get ( 1/β βlε h( ε u (t u (tu (t + h(tu(t α u (t ε u (t for ll t. If we integrte ech term of the lst inequlities from to t, then it follows from (ii tht ε u (τ dτ 1 2 u (t 2 + h(τu(τ α u (τdτ ε u (τ dτ

24 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 12 for ny t. Integrting by prts nd using (iii, the lst inequlities yield εl 1 2 u (t 2 + h(t u(tα+1 α + 1 h (τ u(τα+1 dτ εl α + 1 for ll t. for ll t. Then we hve h(t u(tα+1 α + 1 εl + for ny t. Applying Lemm 1.1.1, we obtin h(t u(tα+1 α + 1 for ll t, from which we hve εl + h (τ u(τα+1 α + 1 dτ h (τ h(τu(τα+1 h(τ α + 1 dτ ( εl exp h (τ h(τ dτ εl h(t h( u(t α+1 (α + 1Lε h( for ll t. Hence, it holds tht u(t ( 1/β βlε h( for ny t, where we set β = α + 1. Obviously, u 0 (t 0 stisfies the eqution (1.2.3 nd u 0 U(L;. Moreover, we get u(t u 0 (t ( 1/β βlε h( for ll t. Given constnts L 0, M > 0, nd 0, let U(L; M; denote the set of ll functions u : [, R with the following properties: (i u is twice continuously differentible; (ii u( = u ( = 0;

25 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 13 (iii u(t L for ll t ; (iv u (τ dτ M for ll t. We now investigte the Hyers-Ulm stbility of the differentil eqution of the form where β is positive odd integer. u (t + u(t + h(tu(t β = 0, (1.2.5 Theorem [28] Given constnts L 0, M > 0 nd 0, ssume tht h : [, [0, is function stisfying C := h (τ dτ <. Let β be n odd integer lrger thn 0. If function u U(L; M; stisfies the inequlity u (t + u(t + h(tu(t β ε (1.2.6 for ll t nd for some ε > 0, then there exists solution u 0 : [, R of the differentil eqution (1.2.5 such tht u(t u 0 (t ( CL β 1 2Mε exp β + 1 for ny t. Proof. We multiply (1.2.6 with u (t to get ε u (t u (tu (t + u(tu (t + h(tu(t β u (t ε u (t for ll t. If we integrte ech term of the lst inequlities from to t, then it follows from (ii tht ε u (τ dτ 1 2 u (t u(t2 + h(τu(τ β u (τdτ ε u (τ dτ for ny t. Integrting by prts nd using (ii nd (iv, the lst inequlities yield εm 1 2 u (t u(t2 + h(t β + 1 u(tβ+1 1 β + 1 h (τu(τ β+1 dτ εm

26 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 14 for ll t. Then it follows from (iii tht 1 2 u(t2 εm + 1 h (τu(τ β+1 dτ β + 1 εm + 2 β u(τ2 h (τu(τ β 1 dτ εm + 2 β + 1 εm + 2Lβ 1 β t 0 t 1 t 0 t 2 u(τ2 h (τ u(τ β 1 dτ 1 2 u(τ2 h (τ dτ for ny t. Applying Lemm 1.1.1, we obtin ( 1 t 2L β 1 2 u(t2 εm exp β + 1 h (τ dτ for ll t. Hence, it holds tht u(t ( CL β 1 2Mε exp β + 1 ( 2CL β 1 εm exp β + 1 for ny t. Obviously, u 0 (t 0 stisfies the eqution (1.2.5 nd u 0 U(L; M;. Furthermore, we get u(t u 0 (t ( CL β 1 2Mε exp β + 1 for ll t. 1.3 Hyers-Ulm Stbility of Differentil Equtions with Boundry Conditions Lemm [29] Let I = [, b] be closed intervl with < < b <. If y C 2 (I, R nd y( = 0 = y(b, then (b 2 mx y(x x I 8 mx x I y (x.

27 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 15 Proof. Let M := mx y(x. Since y( = 0 = y(b, there exists x 0 (, b such x I tht y(x 0 = M. By the Tylor s theorem, we hve y( = y(x 0 + y (x 0 ( x 0 + y (ξ 2 ( x 0 2, y(b = y(x 0 + y (x 0 (b x 0 + y (η 2 (b x 0 2 for some ξ, η [, b]. Since y( = y(b = 0 nd y (x 0 = 0, we get y (ξ = If x 0 (, ( + b/2 ], then we hve If x 0 [ ( + b/2, b, then we hve 2M ( x 0, 2 y (η = 2M (b x M ( x 0 2M 2 ( b 2 = 8M (b. 2 2 Hence, we obtin 2M (b x 0 2M 2 ( b 2 = 8M (b. 2 2 mx x I y (x 8M (b = 8 2 (b mx y(x. 2 x I Therefore, which ends the proof. (b 2 mx y(x x I 8 mx x I y (x, Lemm [29] Let I = [, b] be closed intervl with < < b <. If y C 2 (I, R nd y( = 0 = y (, then (b 2 mx y(x x I 2 Proof. By the Tylor s theorem, we hve mx x I y (x. y(x = y( + y ((x + y (ξ (x 2 2

28 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 16 for some ξ [, b]. Since y( = y ( = 0 nd (x 2 (b 2, we get for ny x I. Thus, we obtin y(x y (ξ (b 2 2 (b 2 mx y(x x I 2 mx x I y (x, which completes the proof. In the following theorems, we prove the Hyers-Ulm stbility of the following liner differentil eqution with boundry conditions or with initil conditions y (x + β(xy(x = 0 (1.3.1 y( = 0 = y(b (1.3.2 y( = 0 = y ( (1.3.3 where I = [, b], y C 2 (I, R, β C(I, R, nd < < b <. Theorem [29] Given closed intervl I = [, b], let β C(I, R be function stisfying mx β(x < 8/(b x I 2. If function y C 2 (I, R stisfies the inequlity y (x + β(xy(x ε, (1.3.4 for ll x I nd for some ε 0, s well s the boundry conditions in (1.3.2, then there exist constnt K > 0 nd solution y 0 C 2 (I, R of the differentil eqution (1.3.1 with the boundry conditions in (1.3.2 such tht for ny x I. y(x y 0 (x Kε

29 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 17 Proof. By Lemm 1.3.1, we hve Thus, it follows from (1.3.4 tht (b 2 mx y(x x I 8 (b 2 ε + 8 (b 2 mx y(x x I 8 mx x I y (x + β(xy(x + (b 2 mx β(x mx 8 x I x I mx x I y (x. (b 2 8 y(x. mx x I β(x mx y(x Let C := (b 2 8 nd K := C 1 C mx β(x. Obviously, y 0 0 is solution of (1.3.1 with the boundry conditions in (1.3.2 nd y(x y 0 (x Kε x I for ny x I. Theorem [29] Given closed intervl I = [, b], let β : I R be function stisfying mx β(x < 2/(b x I 2. If function y C 2 (I, R stisfies the inequlity (1.3.4 for ll x I nd for some ε 0 s well s the initil conditions in (1.3.3, then there exist solution y 0 C 2 (I, R of the differentil eqution (1.3.1 with the initil conditions in (1.3.3 nd constnt K > 0 such tht for ny x I. y(x y 0 (x Kε Proof. On ccount of Lemm 1.3.2, we hve Thus, it follows from (1.3.4 tht (b 2 mx y(x x I 2 (b 2 ε + 2 (b 2 mx y(x x I 2 mx x I y (x + β(xy(x + (b 2 mx β(x mx 2 x I x I mx x I y (x. (b 2 2 y(x. mx x I β(x mx y(x x I

30 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 18 Let C := (b 2 2 nd K := C 1 C mx β(x. Obviously, y 0 0 is solution of (1.3.1 with the initil conditions in (1.3.3 nd y(x y 0 (x Kε for ll x I. In the following theorems, we investigte the Hyers-Ulm stbility of the differentil eqution with boundry conditions or with initil conditions y (x + p(xy (x + q(xy(x = 0 (1.3.5 y( = 0 = y(b (1.3.6 y( = 0 = y ( (1.3.7 where y C 2 (I, R, p C 1 (I, R, q C(I, R, nd I = [, b] with < < b <. Let us define function β : I R by for ll x I. β(x := q(x 1 2 p (x 1 4 p(x2 Theorem [29] Assume tht there exists constnt L 0 with L x p(τdτ L (1.3.8 for ny x I nd mx β(x < 8/(b x I 2. If function y C 2 (I, R stisfies the inequlity y (x + p(xy (x + q(xy(x ε (1.3.9 for ll x I nd for some ε 0 s well s the boundry conditions in (1.3.6, then there exist constnt K > 0 nd solution y 0 C 2 (I, R of the differentil eqution (1.3.5 with the boundry conditions in (1.3.6 such tht for ny x I. y(x y 0 (x Ke L ε

31 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 19 Proof. Suppose y C 2 (I, R stisfies the inequlity (1.3.9 for ll x I. Let us define u(x := y (x + p(xy (x + q(xy(x, ( ( 1 x z(x := y(x exp p(τdτ ( for ll x I. By ( nd (1.3.11, we obtin z (x + (q(x 12 p (x 14 ( 1 p(x2 z(x = u(x exp 2 for ll x I. Now, it follows from (1.3.8 nd (1.3.9 tht z (x + (q(x 12 p (x 14 ( p(x2 z(x 1 = u(x exp 2 tht is, z (x + β(xz(x εe L/2 for ny x I. Moreover, it follows from ( tht z( = 0 = z(b. x x p(τdτ p(τdτ εe L/2, In view of Theorem 1.3.3, there exists constnt K > 0 nd function z 0 C 2 (I, R such tht z 0(x + (q(x 12 p (x 14 p(x2 z 0 (x = 0, ( nd z 0 ( = 0 = z 0 (b z(x z 0 (x Kεe L/2 ( for ll x I. We now set ( y 0 (x := z 0 (x exp 1 2 x p(τdτ. (1.3.14

32 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 20 Then, since ( y 0(x = z 0(x exp 1 2 ( y 0(x = z 0(x exp 1 2 x x p(τdτ 1 ( 2 p(xz 0(x exp 1 2 p(τdτ 1 ( 2 p (xz 0 (x exp ( 4 p(x2 z 0 (x exp 1 2 x x ( p(xz 0(x exp p(τdτ p(τdτ, 1 2 it follows from (1.3.12, (1.3.14, (1.3.15, nd ( tht y 0(x + p(xy 0(x + q(xy 0 (x ( = z 0(x + (q(x 12 p (x 14 p(x2 z 0 (x = 0 x x ( exp 1 2 p(τdτ, p(τdτ x p(τdτ ( ( for ll x I. Hence, y 0 stisfies (1.3.5 nd the boundry conditions in ( Finlly, it follows from (1.3.8 nd ( tht ( y(x y 0 (x = z(x exp 1 x ( p(τdτ z 0 (x exp 1 x p(τdτ 2 2 ( = z(x z 0 (x exp 1 x p(τdτ 2 ( Kεe L/2 exp 1 x p(τdτ 2 for ll x I. Ke L ε Theorem [29] Assume tht there exists constnt L 0 such tht (1.3.8 holds for ll x I. Assume moreover tht mx β(x < 2/(b x I 2. If function y C 2 (I, R stisfies the inequlity (1.3.9 for ll x I nd for some ε 0 s well s the initil conditions in (1.3.7, then there exist constnt K > 0 nd solution y 0 C 2 (I, R of the differentil eqution (1.3.5 with the initil conditions in (1.3.7 such tht y(x y 0 (x Ke L ε

33 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 21 for ny x I. Proof. Suppose y C 2 (I, R stisfies the inequlity (1.3.9 for ny x I. Let us define u(x nd z(x s in ( nd (1.3.11, respectively. By ( nd (1.3.11, we obtin z (x + (q(x 12 p (x 14 ( 1 p(x2 z(x = u(x exp 2 for ll x I. Now, it follows from (1.3.8 nd (1.3.9 tht z (x + (q(x 12 p (x 14 ( p(x2 z(x 1 = u(x exp 2 tht is, z (x + β(xz(x εe L/2 for ll x I. Furthermore, in view of (1.3.11, we hve z( = 0 = z (. x x p(τdτ p(τdτ εe L/2, By Theorem 1.3.4, there exists constnt K > 0 nd function z 0 C 2 (I, R such tht z 0(x + (q(x 12 p (x 14 p(x2 z 0 (x = 0, nd z 0 ( = 0 = z 0( z(x z 0 (x Kεe L/2 for ny x I. We now set ( y 0 (x := z 0 (x exp 1 2 x p(τdτ. Moreover, since ( y 0(x = z 0(x exp 1 2 x p(τdτ 1 ( 2 p(xz 0(x exp 1 2 x p(τdτ

34 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 22 nd ( y 0(x = z 0(x exp 1 2 x p(τdτ 1 ( 2 p (xz 0 (x exp ( 4 p(x2 z 0 (x exp 1 2 x x ( p(xz 0(x exp p(τdτ p(τdτ, 1 2 x p(τdτ we hve y 0(x + p(xy 0(x + q(xy 0 (x ( = z 0(x + (q(x 12 p (x 14 p(x2 z 0 (x = 0 ( exp 1 2 x p(τdτ for ny x I. Hence, y 0 stisfies (1.3.5 long with the initil conditions in ( Finlly, it follows tht ( y(x y 0 (x = z(x exp 1 x ( p(τdτ z 0 (x exp 1 x p(τdτ 2 2 ( = z(x z 0 (x exp 1 x p(τdτ 2 ( Kεe L/2 exp 1 x p(τdτ 2 Ke L ε for ll x I. In similr wy, we investigte the Hyers-Ulm stbility of the differentil eqution with boundry conditions or with initil conditions y (x + k (x k(x y (x + l(x y(x = 0 ( k(x y( = 0 = y(b ( y( = 0 = y ( (1.3.19

35 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 23 where y C 2 (I, R, k C 1 (I, R\{0}, l C(I, R, nd < < b <. Given closed intervl I = [, b], we set β(x := l(x k(x 1 d k (x 2 dx k(x 1 ( k 2 (x 4 k(x for ll x I. Theorem [29] Assume tht there exists constnt L 0 with L x k (τ dτ L ( k(τ for ny x I nd mx β(x < 8/(b x I 2. If function y C 2 (I, R stisfies the inequlity y (x + k (x k(x y (x + l(x k(x y(x ε, ( for ll x I nd some ε 0, s well s the boundry conditions in (1.3.18, then there exist constnt K > 0 nd solution y 0 C 2 (I, R of the differentil eqution ( with the boundry conditions in ( such tht for ny x I. y(x y 0 (x Ke L ε Proof. Suppose y C 2 (I, R stisfies ( for ll x I. Let us define u(x := y (x + k (x k(x y (x + l(x y(x, ( k(x ( 1 x k (τ z(x := y(x exp 2 k(τ dτ ( for ll x I. By ( nd (1.3.23, we obtin ( l(x z (x + k(x 1 d k (x 2 dx k(x 1 ( k 2 ( (x 1 z(x = u(x exp 4 k(x 2 Further, it follows from ( nd ( tht ( l(x z (x + k(x 1 d k (x 2 dx k(x 1 ( k 2 ( (x z(x 1 4 k(x = u(x exp 2 ( 1 x ε exp 2 εe L/2, x x k (τ k(τ dτ. k (τ dτ k(τ k (τ k(τ dτ

36 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 24 tht is, z (x + β(xz(x εe L/2 for ll x I. Moreover, it follows from ( nd ( tht z( = 0 = z(b. By Theorem 1.3.3, there exists constnt K > 0 nd function z 0 C 2 (I, R such tht ( l(x z 0(x + k(x 1 d k (x 2 dx k(x 1 ( k 2 (x z 0 (x = 0, 4 k(x nd for ny x I. We now set Then, since ( y 0(x = z 0(x exp 1 2 z 0 ( = 0 = z 0 (b z(x z 0 (x Kεe L/2 ( y 0 (x := z 0 (x exp 1 x k (τ 2 k(τ dτ. x nd ( y 0(x = z 0(x exp 1 x k (τ 2 k(τ dτ k (x 1 ( k (x z 0 (x exp( 1 x 2 k(x ( k 2 (x z 0 (x exp( 1 x 4 k(x 2 we hve y 0(x + k (x k(x y 0(x + l(x k(x y 0(x ( ( l(x = z 0(x + k(x 1 ( k (x 2 k(x = 0. k (τ k(τ dτ 1 ( k (x 2 k(x z 0(x exp 1 x k (τ 2 k(τ dτ 1 4 ( k(x z 0(x exp k (τ k(τ dτ k (τ k(τ dτ, 1 2 x k (τ k(τ dτ ( k 2 ( (x z 0 (x exp 1 x k (τ k(x 2 k(τ dτ

37 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 25 Hence, y 0 stisfies ( long with the boundry conditions in ( Finlly, it follows tht ( y(x y 0 (x = z(x exp 1 x ( k (τ 2 k(τ dτ z 0 (x exp 1 x k (τ dτ 2 k(τ ( = z(x z 0 (x exp 1 x k (τ 2 k(τ dτ ( Kεe L/2 exp 1 x k (τ 2 k(τ dτ Ke L ε for ll x I. By similr method s we pplied to the proof of Theorem 1.3.6, we cn prove the following theorem. Hence, we omit the proof. Theorem [29] Assume tht mx β(x < 2/(b x I 2 nd there exists constnt L 0 for which the inequlity ( holds for ll x I. If function y C 2 (I, R stisfies the inequlity ( for ll x I nd for some ε 0 s well s the boundry conditions in (1.3.19, then there exist constnt K > 0 nd solution y 0 C 2 (I, R of the differentil eqution ( with the boundry conditions in ( such tht for ny x I. y(x y 0 (x Ke L ε Now, we give the definition of superstbility with initil nd boundry conditions. Definition [18] Assume tht for ny function y C n [, b], if y stisfies the differentil inequlity φ ( f, y, y,..., y (n ϵ for ll x [, b] nd for some ϵ 0 with initil(or boundry conditions, then either y is solution of the differentil eqution φ ( f, y, y,..., y (n = 0 (1.3.24

38 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 26 or y(x Kϵ for ny x [, b], where K is constnt not depending on y explicitly. Then, we sy tht Eq.( hs superstbility with initil(or boundry conditions. In the following theorem, we investigte the stbility of differentil eqution of higher order in the form of with initil conditions y (n (x + β(xy(x = 0 ( y( = y ( = = y (n 1 ( = 0, ( where n N +, y C n [, b], β C 0 [, b], < < b < +. Theorem [18] If mx β(x < n! (b n. Then ( hs the superstbility with initil conditions ( Proof. For every ϵ > 0, y C 2 [, b], if y (n (x + β(xy(x ϵ nd y( = y ( = = y (n 1 ( = 0. Similrly to the proof of Lemm 1.3.2, Thus y(x = y( + y ((x + + y(n 1 ( (n 1! (x n 1 + y(n (ξ (x n. n! y(x = y (n (ξ (x n mx y (n (x (b n n! n! for every x [, b]; so, we obtin mx y(x (b n [mx y (n (b n (x + β(xy(x ] + mx β(xy(x n! n! (b n (b n ϵ + mx β(x mx y(x. n! n! Let η = (b n mx β(x, K = (b n. It is esy to see tht n! n!(1 η y(x Kϵ. Hence ( hs superstbility with initil condtions ( In the following theorems, we investigte the superstbility of the differentil eqution y (n (x + n 1 y (n 1 (x y (x + 0 y(x = 0 ( with initil conditions y( = y ( = = y (n 1 ( = 0, ( where y C n (I, C, i R(i = 0, 1,, n 1, I = [, b], < < b < +.

39 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 27 Lemm [19] Assume tht y C 1 (I, C nd C {z C z < 1 }. If b y (x Cy(x ε with y( = 0, then there exists constnt K > 0 such tht y(x Kε. Proof. Let y(x = A(x+i B(x, where i denotes imginry unit nd A(x, B(x C 1 (I, R. Since y( = 0, we hve By Tylor formul, we obtin nd A( = 0 nd B( = 0; mx A(x (b mx A (x CA(x + C (b mx A(x (b mx y (x Cy(x + C (b mx A(x (b ε + C (b mx A(x mx B(x (b ε + C (b mx B(x. Since C {z C z < 1 }, there exists constnt K such tht b mx y(x mx A(x 2 + mx B(x 2 Kε. Theorem [19] If ll the roots of the chrcteristic eqution re in the disc {z C z < 1 }, then ( hs superstbility with initil conditions ( b Proof. Assume tht λ 1, λ 2,, 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 y(x nd g i (x = g i 1(x λ i g i 1 (x(i = 2, 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 y(x ε, nd g i ( = 0 for every i = 1, 2,, n 1. Since the bsolute vlue of λ n < 1 nd g b n 1( = 0, it follows from Lemm tht there exists K 1 > 0 such tht g n 1 (x K 1 ε.

40 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 28 Recll g n 1 (x = g n 2(x λ n 1 g n 2 (x, we hve g n 2(x λ n 1 g n 2 (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 y(x Kε. This completes the proof of our theorem.

41 Chpter 2 Generlized Hyers-Ulm Stbility of Differentil Equtions 2.1 Generlized Hyers-Ulm Stbility of Liner Differentil Equtions Throughout this section, F will denote either the rel field R or the complex field C. A function f : (0, F is sid to be of exponentil order if there re constnts A, B R such tht f(t Ae tb for ll t > 0. For ech function f : (0, F of exponentil order, we define the Lplce trnsform of f by F (s = 0 f(te st dt. There exists unique number σ < such tht this integrl converges if R(s > σ nd diverges if R(s < σ, where R(s denotes the rel prt of the (complex number s. The number σ is clled the bsciss of convergence nd denoted by σ f. It is well known tht F (s 0 s R(s. Furthermore, f is nlytic on the open right hlf plne {s C : R(s > σ} nd we hve d ds F (s = 0 te st f(tdt (R(s > σ. The Lplce trnsform of f is sometimes denoted by L(f. It is well known tht L is liner nd one-to-one.

42 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 30 Conversely, let f(t be continuous function whose Lplce trnsform F (s hs the bsciss of convergence σ f, then the formul for the inverse Lplce trnsforms yields f(t = 1 α+it 2πi lim F (se st ds = 1 e (α+iyt F (α + iydy T α it 2π for ny rel constnt α > σ f, where the first integrl is tken long the verticl line R(s = α nd converges s n improper Riemnn integrl nd the second integrl is used s n lterntive nottion for the first integrl (see [1]. Hence, we hve L(f(s = 0 L 1 (F (t = 1 2π f(te st dt (R(s > σ f e (α+iyt F (α + iydy (α > σ f. The convolution of two integrble functions f, g : (0, F is defined by Then L(f g = L(fL(g. (f g(t := Lemm [12] Let P (s = n k=0 0 f(t xg(xdx. α k s k nd Q(s = m β k s k, where m, n re nonnegtive integers with m < n nd α k, β k re sclrs. Then there exists n infinitely differentible function g : (0, F such tht nd g (i (0 = L(g = Q(s P (s where σ P = mx{r(s : P (s = 0}. (R(s > σ P k=0 { 0 (for i {0, 1,..., n m 2}, β m /α n (for i = n m 1 Lemm [12] Given n integer n > 1, let f : (0, F be continuous function nd let P (s be complex polynomil of degree n. Then there exists n n times continuously differentible function h : (0, F such tht L(h = L(f P (s (R(s > mx{σ P, σ f }, where σ P = mx{r(s : P (s = 0} nd σ f is the bsciss of convergence for f. In prticulr, it holds tht h (i (0 = 0 for every i {0, 1,..., n 1}.

43 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 31 Let F denote either R or C. In the following theorem, using the Lplce trnsform method, we investigte the generlized Hyers-Ulm stbility of the liner differentil eqution of first order y (t + αy(t = f(t. (2.1.1 Theorem [30] Let α be constnt in F nd let φ : (0, (0, be n integrble function. If continuously differentible function y : (0, F stisfies the inequlity y (t + αy(t f(t φ(t (2.1.2 for ll t > 0, then there exists solution y α eqution (2.1.1 such tht : (0, F of the differentil for ny t > 0. y(t y α (t e R(αt e R(αx φ(xdx 0 Proof. If we define function z : (0, F by z(t := y (t + αy(t f(t for ech t > 0, then y(0 + L(f L(y = L(z s + α s + α. (2.1.3 If we set y α (t := y(0e αt + (E α f(t, where E α (t = e αt, then y α (0 = y(0 nd y(0 + L(f L(y α = = y α(0 + L(f. (2.1.4 s + α s + α Hence, we get L ( y α(t + αy α (t = sl(y α y α (0 + αl(y α = L(f. Since L is one-to-one opertor, it holds tht y α(t + αy α (t = f(t. Thus, y α is solution of ( Moreover, by (2.1.3 nd (2.1.4, we obtin L(y L(y α = L(E α z. Therefore, we hve y(t y α (t = (E α z(t. (2.1.5 In view of (2.1.2, it holds tht z(t φ(t (2.1.6

44 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 32 for ll t > 0, nd it follows from the definition of convolution, (2.1.5, nd (2.1.6 tht y(t y α (t = (E α z(t = E α (t xz(xdx 0 0 e α(t x φ(xdx e R(αt e R(αx φ(xdx for ll t > 0. (We remrk tht 0 er(αx φ(xdx exists for ech t > 0 provided φ is n integrble function. 0 Corollry [30] Let α be constnt in F nd let φ : (0, (0, be n integrble function such tht 0 e R(α(x t φ(xdx Kφ(t (2.1.7 for ll t > 0 nd for some positive rel constnt K. If continuously differentible function y : (0, F stisfies the inequlity (2.1.2 for ll t > 0, then there exists solution y α : (0, F of the differentil eqution (2.1.1 such tht y(t y α (t Kφ(t for ny t > 0. In the following remrk, we show tht there exists n integrble function φ : (0, (0, stisfying the condition ( Remrk [30] Let α be constnt in F with R(α > 1. φ(t = Ae t for ll t > 0 nd for some A > 0, then we hve for ech t > 0. 0 e R(α(x t φ(xdx = 0 e R(α(x t Ae x dx 1 ( = Ae t Ae R(αt 1 + R(α R(α φ(t If we define

45 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 33 Now, we pply the Lplce trnsform method to the proof of the generlized Hyers-Ulm stbility of the liner differentil eqution of second order y (t + βy (t + αy(t = f(t. (2.1.8 Theorem [30] Let α nd β be constnts in F such tht there exist, b F with + b = β, b = α, nd b. Assume tht φ : (0, (0, is n integrble function. If twice continuously differentible function y : (0, F stisfies the inequlity y (t + βy (t + αy(t f(t φ(t (2.1.9 for ll t > 0, then there exists solution y c : (0, F of the differentil eqution (2.1.8 such tht for ll t > 0. y(t y c (t er(t b 0 e R(x φ(xdx + er(bt b 0 e R(bx φ(xdx Proof. If we define function z : (0, F by z(t := y (t+βy (t+αy(t f(t for ech t > 0, then we hve L(z = ( s 2 + βs + α L(y [sy(0 + βy(0 + y (0] L(f. ( In view of (2.1.10, function y 0 : (0, F is solution of (2.1.8 if nd only if ( s 2 + βs + α L(y 0 sy 0 (0 [βy 0 (0 + y 0(0] = L(f. ( Now, since s 2 + βs + α = (s (s b, ( implies tht If we set L(y sy(0 + [βy(0 + y (0] + L(f (s (s b y c (t := y(0 et be bt b = L(z (s (s b. ( [βy(0 + y (0]E,b (t + (E,b f(t, ( where E,b (t := et e bt, then y b c (0 = y(0. Moreover, since y c(t = y(0 2 e t b 2 e bt + [βy(0 + y (0] et be bt + d b b dt (E,b f(t, (E,b f(t = et b 0 e x f(xdx ebt b 0 e bx f(xdx,

46 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 34 we hve y c(0 = y(0 2 b 2 b + [βy(0 + y (0] b b = ( + by(0 + βy(0 + y (0 = y (0. It follows from ( tht L(y c = sy c(0 + [βy c (0 + y c(0] + L(f. ( (s (s b Now, ( nd ( imply tht y c is solution of ( Applying ( nd ( nd considering the fcts tht y c (0 = y(0, y c(0 = y (0, nd L(E,b z = L(z, we obtin (s (s b L(y L(y c = L(E,b z or equivlently, y(t y c (t = (E,b z(t. In view of (2.1.9, it holds tht z(t φ(t, nd it follows from the definition of the convolution tht y(t y c (t = (E,b z(t er(t b 0 e R(x φ(xdx + er(bt b 0 e R(bx φ(xdx for ny t > 0. We remrk tht 0 e R(x φ(xdx nd 0 e R(bx φ(xdx exist for ny t > 0 provided φ is n integrble function. Corollry [30] Let α nd β be constnts in F such tht there exist, b F with + b = β, b = α, nd b. Assume tht φ : (0, (0, is n integrble function for which there exists positive rel constnt K with 0 ( e R((t x + e R(b(t x φ(xdx Kφ(t ( for ll t > 0. If twice continuously differentible function y : (0, F stisfies the inequlity (2.1.9 for ll t > 0, then there exists solution y c : (0, F of the differentil eqution (2.1.8 such tht for ll t > 0. y(t y c (t K b φ(t

47 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 35 We now show tht there exists n integrble function φ : (0, (0, which stisfies the condition ( Remrk [30] Let α nd β be constnts in F such tht there exist, b F with + b = β, b = α, R( < 1, R(b < 1, nd b. If we define φ(t = Ae t for ll t > 0 nd for some A > 0, then we get ( e R((t x + e R(b(t x φ(xdx 0 for ll t > 0. = = 0 ( e R((t x + e R(b(t x Ae x dx A ( e t e R(t A ( + e t e R(bt 1 R( 1 R(b ( 1 1 R( + 1 φ(t 1 R(b Similrly, we pply the Lplce trnsform method to investigte the generlized Hyers-Ulm stbility of the liner differentil eqution of nth order n 1 y (n (t + α k y (k (t = f(t ( k=0 Theorem [30] Let α 0, α 1,..., α n be sclrs in F with α n = 1, where n is n integer lrger thn 1. Assume tht φ : (0, (0, is n integrble function of exponentil order. If n n times continuously differentible function y : (0, F stisfies the inequlity n 1 y(n (t + α k y (k (t f(t φ(t ( k=0 for ll t > 0, then there exist rel constnts M > 0 nd σ g nd solution y c : (0, F of the differentil eqution ( such tht for ll t > 0 nd α > σ g. y(t y c (t M 0 e α(t x φ(xdx

48 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 36 Proof. Applying the integrtion by prts repetedly, we derive L ( y (k = s k L(y k s k j y (j 1 (0 for ny integer k > 0. Using this formul, we cn prove tht function y 0 : (0, F is solution of ( if nd only if L(f = = n α k s k L(y 0 k=0 n α k s k L(y 0 k=0 = P n,0 (sl(y 0 j=1 n k=1 n α k n j=1 k=j n j=1 where P n,j (s := n α k s k j for j {0, 1,..., n}. k=j Let us define function z : (0, F by k j=1 s k j y (j 1 0 (0 α k s k j y (j 1 0 (0 P n,j (sy (j 1 0 (0, ( n 1 z(t := y (n (t + α k y (k (t f(t ( k=0 for ll t > 0. Then, similrly s in (2.1.18, we obtin Hence, we get L(z = P n,0 (sl(y L(y 1 P n,0 (s n P n,j (sy (j 1 (0 L(f. j=1 ( n P n,j (sy (j 1 (0 + L(f j=1 = L(z P n,0 (s. ( Let σ f be the bsciss of convergence for f, let s 1, s 2,..., s n be the roots of the polynomil P n,0 (s, nd let σ P = mx{r(s k : k {1, 2,..., n}}. For ny s with R(s > mx{σ f, σ P }, we set ( n 1 G(s := P n,j (sy (j 1 (0 + L(f. ( P n,0 (s j=1

49 Alqifiry Hyers-Ulm Sbility of The Solutions of Differentil Equtions 37 By Lemm 2.1.2, there exists n n times continuously differentible function f 0 such tht L(f 0 = L(f ( P n,0 (s for ll s with R(s > mx{σ f, σ P } nd for ny i {0, 1,..., n 1}. For j {1, 2,..., n}, we note tht f (i 0 (0 = 0 ( P n,j (s P n,0 (s = 1 s j j 1 k=0 α k s k s j P n,0 (s ( for every s with R(s > mx{0, σ P }. Applying Lemm for the cse of Q(s = j 1 α k s k nd P (s = s j P n,0 (s, we cn find n infinitely differentible k=0 function g j such tht L(g j = j 1 k=0 α k s k s j P n,0 (s ( nd g (k j (0 = 0 for k {0, 1,..., n 1}. Let f j (t := tj 1 (j 1! g j(t ( for j {1, 2,..., n}. Then we hve { 0 (for i {0, 1,..., j 2, j, j + 1,..., n 1}, f (i j (0 = 1 (for i = j 1. ( If we define y c (t := n y (j 1 (0f j (t + f 0 (t, j=1 then the conditions ( nd ( imply tht y (i c (0 = y (i (0 (2.1.28