vilble online t wwwisr-publitionsom/jns J Nonliner Si ppl, 0 (207, 35 322 Reserh rtile Journl Homepge: wwwtjnsom - wwwisr-publitionsom/jns Hyers-Ulm stbility of Pielou logisti differene eqution Soon-Mo Jung, Young Woo Nm Mthemtis Setion, College of Siene nd Tehnology, Hongik University, 3006 Sejong, Republi of Kore Communited by C Zhri bstrt We investigte Hyers-Ulm stbility of the first order differene eqution x i+ = x i+b x i +d, where d b =, 0 nd + d > 2 It hs Hyers-Ulm stbility if the initil point x 0 lies in some definite intervl of R The ondition + d > 2 implies tht the bove reurrene is nturl generliztion of Pielou logisti differene eqution 207 ll rights reserved Keywords: Hyers-Ulm stbility, Pielou logisti differene eqution, first order differene eqution, liner frtionl mp, Verhulst-Perl differentil eqution 200 MSC: 3930, 39B82 Introdution The differene eqution is the reursively defining sequene, eh of whih term is defined s funtion of the preeding terms The differene eqution often refers to speifi type of reurrene reltion In prtiulr, if the sequene x i } i N0 is defined s the reltion between the generl term x i nd only its first predeessor x i with the definite initil term x 0 stisfying the eqution x i+ = g(x i, then it is lled first order differene eqution In 940, Ulm [] suggested n importnt problem of the stbility of group homomorphisms: Given metri group (G, d nd funtion f : G G whih stisfies the inequlity d(f(xy, f(xf(y ε for positive number ε nd for ll x, y G, do there exist homomorphism : G G nd onstnt δ > 0 depending only on G nd ε suh tht d((x, f(x δ for ll x G? The first positive nswer to this question ws given by Hyers [3] in 94 for Cuhy dditive eqution in Bnh spes If the nswer is ffirmtive, the funtionl eqution (xy = (x(y is sid to be stble in the sense of Hyers nd Ulm (or the eqution hs the Hyers-Ulm stbility We refer the reder to [3, 4, 0, ] for the ext definition of Hyers-Ulm stbility For dedes, theory of Hyers-Ulm stbility of funtionl equtions or liner differentil equtions ws developed More reently, Hyers-Ulm stbility of differene equtions hs been given ttention For instne, see [2, 5 9] However, this stbility for differene equtions is not yet studied fr beyond the liner differene eqution s fr s we know Corresponding uthor Emil ddresses: smjung@hongikkr (Soon-Mo Jung, nmyoungwoo@hongikkr (Young Woo Nm doi:022436/jns000626 Reeived 207-02-8
S-M Jung, Y W Nm, J Nonliner Si ppl, 0 (207, 35 322 36 In this pper, we investigte Hyers-Ulm stbility of the first order liner frtionl differene eqution whih is motivted from the disretized funtion s the solution of Verhulst-Perl differentil eqution We denote by N, N 0, R, nd C the set of ll positive integers, of ll nonnegtive integers, of ll rel numbers, nd the set of ll omplex numbers, respetively We would show Hyers-Ulm stbility of the first order differene eqution of the form for ll integers i N 0, where g is the liner frtionl mp s follows x i+ = g(x i, ( g(x = x + b x + d, where, b,, d re rel numbers with d b =, 0 nd + d > 2 More preisely, we would prove tht if rel-vlued sequene i } i N0 stisfies the inequlity i+ g( i ε, for ll i N 0, then there exists solution b i } i N0 to the differene eqution ( nd positive G(ε depending only on F nd ε suh tht b i i G(ε, for ll i N 0 nd ε 0 implies tht G(ε 0 We remrk tht the differene eqution ( is disrete form of the funtionl eqution x(ξ(t = H(t, x(t, whose stbility results hve been surveyed in [] The Verhulst-Perl eqution is popultion growth model whih is given s y (t = y(t ( p qy(t, for some p, q > 0, where y is the size of popultion t the time t nd the positive onstnt p is the growth rte of popultion The nonliner term, qy(t 2 is the negtive effet on the growth due to the environment The solution is the mp s follows for some onstnt r Thus we obtin tht y(t = p q +, rq e pt y(t + = e p y(t + q p (ep y(t Disretizing the bove eqution we obtin the following reursive reltion y(n + = y(n + Cy(n, where = e p nd C = q p (ep This eqution is lled Pielou logisti differene eqution The behvior of Pielou logisti differene eqution is the sme s itertive property of some kind of liner frtionl mps Let the mp orresponding Pielou logisti eqution be s follows F(x = x Cx +, where > nd C > 0 In the sequel, we onsider the mtrix representtion of liner frtionl mp, whih lrifies the qul-
S-M Jung, Y W Nm, J Nonliner Si ppl, 0 (207, 35 322 37 ittive properties using the tre of mtrix representtion For instne, the following mp x x + b x + d, for d b 0 hs the mtrix representtion ( b M = d However, sine the mp x x+b px+pb x+d is the sme s x px+pd, the mtrix M is representtive of ny mtrix pm for ll rel numbers p 0 Thus we my ssume tht detm = only if d b > 0 If d b < 0, then we ssume tht detm = In this pper, we lwys fix the ondition d b > 0 The liner frtionl mp F for Pielou logisti differene eqution hs the mtrix representtion s follows ( 0 C The mtrix representtion of F would be lso denoted by F unless it mkes onfusion Observe the inequlity of the tre: tr(f = + > 2 In this pper, we investigte Hyers-Ulm stbility of liner frtionl mps whose tre is greter thn two These mps generte Pielou logisti differene eqution by itertion 2 Preliminries Let g be the liner frtionl mp g(x = x + b x + d, (2 for rel numbers, b, nd d, where d b = nd 0 Rell tht g ( d = Sine Hyer-Ulm stbility t is not onsidered in this rtile, suitble proper subintervl in R should be hosen The set is lled (forwrd invrint under g if g( is stisfied In this setion, we find subintervl of R invrint under g defined in (2 if the tre of the mtrix representtion of g is stritly greter thn two Lemm 2 Let g be the liner frtionl mp defined in (2 The followings re true for x R nd r > 0 (i If x + d > r, then r < g(x < 0; (ii If x + d < r, then 0 < g(x < r Proof Suppose firstly tht x + d > r Then we hve r < x + d < 0 < x + d < r r < 2 x + d < 0 d + b < r 2 x + d < 0 x d + x + b < < 0 r (x + d
S-M Jung, Y W Nm, J Nonliner Si ppl, 0 (207, 35 322 38 Moreover, we obtin tht r < + x + b x + d < 0 < x + d < r 0 < x + b x + d < r, by the similr lultions Lemm 22 Let g be the liner frtionl mp defined s (2, where, b, nd d re rel numbers, d b = nd 0 ssume tht the mtrix representtion g stisfies tr(g = 2 + τ for τ > 0 If x (+τ, then +τ < x + d or x + d < +τ Proof Sine tr(g = + d > 2, we will prove our ssertion only for the se of d Cse : ssume tht > d +d Sine > 0 nd tr(g is + d, tr(g hve ( d = + d = ( d = tr(g = tr(g = 2 + τ Using this inequlity, we n visulize this se in the following figure > ( + τ + + τ = +d > 0 Then we (+τ + (+τ ( ( x d d +τ d + +τ In view of this figure, we esily see tht x + d = x ( d > +τ Cse 2: ssume tht < d tr(g By the similr lultions of Cse, we obtin = +d < 0 Thus or ( d = + d = tr(g = tr(g = 2 + τ d > ( + τ + + τ < ( + τ + + τ On ount of the lst inequlity, we n visulize this se in the following figure, d +τ d + +τ ( ( x d (+τ + (+τ By onsidering this figure, we show tht d x > +τ or x + d < +τ
S-M Jung, Y W Nm, J Nonliner Si ppl, 0 (207, 35 322 39 Proposition 23 Let g be the liner frtionl mp defined s (2, where, b, nd d re rel numbers, d b = nd 0 ssume tht the mtrix representtion g stisfies tr(g = 2 + τ, for τ > 0 Then g mps } the intervl x R : x + d > +τ into itself Proof Denote the following two intervls x R : x + d > + τ }, x R : x }, (22 ( + τ by S τ nd by T τ, respetively Then Lemm 22 implies tht T τ S τ nd Lemm 2 implies tht g(s τ T τ Hene, the intervl S τ is invrint under g s follows: whih ompletes the proof g(s τ T τ S τ, (23 3 Hyers-Ulm stbility Suppose tht rel-vlued sequene n } n N0 stisfies the inequlity i+ F(i, i ε, for positive number ε nd for ll i N 0, where is the bsolute vlue of rel number If there exists the sequene b i } i N0 whih stisfies tht b i+ = F(i, b i, (3 for eh i N 0, nd i b i G(ε for ll i N 0, where the positive number G(ε onverges to zero s ε 0, then we sy tht the differene eqution (3 hs Hyers-Ulm stbility Theorem 3 Let g be liner frtionl mp defined s (2 of whih mtrix representtion stisfies tht tr(g = 2 + τ for τ > 0 For ny given 0 < ε < τ (+τ, let the rel-vlued sequene i} i N0 stisfy the inequlity for ll i N 0 If 0 is in the intervl S τ = stisfies nd for eh i N 0 b i i i+ g( i ε, x + d > +τ x R : b i+ = g(b i, ( + τ 2i b i ε 0 0 + ( + τ 2j, j=0 }, then there exists sequene b i } i N0 whih Proof First, we lim tht n S τ for ll n N 0 Rell tht R \ S τ nd T τ re bounded disjoint intervls beuse g(s τ T τ S τ by Lemm 22 nd Proposition 23 Let x R \ S τ nd x T τ From the definitions of S τ nd T τ, we hve nd d + τ x d + + τ, (32 ( + τ x + ( + τ (33 There re only two ses for the lotion of the bounded disjoint intervls R \ S τ nd T τ s we see in the following figure
S-M Jung, Y W Nm, J Nonliner Si ppl, 0 (207, 35 322 320 R\S τ [ x ] T τ [ ] x or T τ [ ] x R\S τ [ ] x ording to (32 nd the first figure, we get x d + + τ or by (33 nd the seond figure, we hve Hene, we obtin ( + τ x, x + ( + τ d + τ x ( x x + d + τ + ( + τ = 2 + τ ( + τ + + τ = ( + τ τ = ( + τ > ε Sine 0 S τ, g( 0 T τ by Lemm 2 or (23 Moreover, g( 0 ε Thus, by (34, / R \ S τ, tht is, S τ Then, by indution, we n show tht n S τ for ll n N 0 Sine g (x =, g (x+d 2 hs uniform upper bound in S τ s follows g (x = x + d 2 = 2 x + d 2 < ( + τ 2 < Thus, g is Lipshitz mp on S τ with the Lipshitz onstnt (+τ 2 Finlly, we n esily pply indution to prove b i i = g(b i g( i + g( i i g(b i g( i + g( i i ( + τ 2 b i i + ε (34 for eh i N 0 ( + τ 2i b i ε 0 0 + ( + τ 2j, j=0 4 pplition The Pielou logisti differene eqution n be treted s the itertion of the liner frtionl mp F(x = x x Cx + = C x +, for > nd C > 0 (The lst expression of F is given in the form of d b = Then tr(f = +
S-M Jung, Y W Nm, J Nonliner Si ppl, 0 (207, 35 322 32 The invrint set S τ under F is s follows: S τ = x R : x + C > + }, (4 C where τ = + 2 > 0 by (22 stbility Exmple 4 Let F be the liner frtionl mp s follows Then the Pielou logisti differene eqution hs Hyers-Ulm F(x = x Cx +, for > nd C > 0 For every given 0 < ε < 2+ ( +C, let sequene i} i N0 stisfy the inequlity i+ F( i ε, for ll i N 0 If 0 is in S τ defined in (4, then there exists sequene b i } i N0 whih stisfies nd i b i for eh i N 0, where τ = + 2 > 0 b i+ = F(b i, ( + τ 2i b i ε 0 0 + ( + τ 2j, j=0 Proof By the diret lultion, we obtin τ ( + τ C = 2 + ( + C Then Theorem 3 implies tht if the inequlity 0 < ε < 2+ ( +C sequene b i } i N0 suh tht b i+ = F(b i, nd i b i for eh i N 0, whih ompletes the proof knowledgment ( + τ 2i b i ε 0 0 + ( + τ 2j, j=0 holds nd 0 S τ, then there exists This reserh ws supported by Bsi Siene Reserh Progrm through the Ntionl Reserh Foundtion of Kore (NRF funded by the Ministry of Edution (No 206RDB039306 This work ws supported by 207 Hongik University Reserh Fund Referenes [] J Brzdȩk, K Ciepliński, Z Leśnik, On Ulm s type stbility of the liner eqution nd relted issues, Disrete Dyn Nt So, 204 (204, 4 pges [2] J Brzdȩk, D Pop, B Xu, The Hyers-Ulm stbility of nonliner reurrenes, J Mth nl ppl, 335 (2007, 443 449 [3] D H Hyers, On the stbility of the liner funtionl eqution, Pro Nt d Si U S, 27 (94, 222 224
S-M Jung, Y W Nm, J Nonliner Si ppl, 0 (207, 35 322 322 [4] D H Hyers, G Is, T M Rssis, Stbility of funtionl equtions in severl vribles, Progress in Nonliner Differentil Equtions nd their pplitions, Birkhäuser Boston, In, Boston, M, (998 [5] S-M Jung, Hyers-Ulm stbility of the first-order mtrix differene equtions, dv Differene Equ, 205 (205, 3 pges [6] S-M Jung, Y W Nm, On the Hyers-Ulm stbility of the first-order differene eqution, J Funt Spes, 206 (206, 6 pges [7] S-M Jung, D Pop, M T Rssis, On the stbility of the liner funtionl eqution in single vrible on omplete metri groups, J Globl Optim, 59 (204, 65 7 [8] S-M Jung, M T Rssis, liner funtionl eqution of third order ssoited with the Fiboni numbers, bstr ppl nl, 204 (204, 7 pges [9] D Pop, Hyers-Ulm-Rssis stbility of liner reurrene, J Mth nl ppl, 309 (2005, 59 597 [0] T M Rssis, On the stbility of the liner mpping in Bnh spes, Pro mer Mth So, 72 (978, 297 300 [] S M Ulm, olletion of mthemtil problems, Intersiene Trts in Pure nd pplied Mthemtis, Intersiene Publishers, New York-London, (960