HYERS-ULAM STABILITY OF HIGHER-ORDER CAUCHY-EULER DYNAMIC EQUATIONS ON TIME SCALES

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1 Dynmic Systems nd Applictions 23 (2014) HYERS-ULAM STABILITY OF HIGHER-ORDER CAUCHY-EULER DYNAMIC EQUATIONS ON TIME SCALES DOUGLAS R. ANDERSON Deprtment of Mthemtics, Concordi College, Moorhed, MN USA Dedicted to Professor John R. Gref on the occsion of his 70th birthdy. ABSTRACT. We extend recent result on third nd fourth-order Cuchy-Euler equtions by estblishing the Hyers-Ulm stbility of higher-order liner non-homogeneous Cuchy-Euler dynmic equtions on time scles. Tht is, if n pproximte solution of higher-order Cuchy-Euler eqution exists, then there exists n exct solution to tht dynmic eqution tht is close to the pproximte one. We generlize this to ll higher-order liner non-homogeneous fctored dynmic equtions with vrible coefficients. AMS (MOS) Subject Clssifiction. 34N05, 26E70, 39A INTRODUCTION Stn Ulm [27] posed the following problem concerning the stbility of functionl equtions: give conditions in order for liner mpping ner n pproximtely liner mpping to exist. The problem for the cse of pproximtely dditive mppings ws solved by Hyers [10], who proved tht the Cuchy eqution is stble in Bnch spces, nd the result of Hyers ws generlized by Rssis [24]. Obloz [19] ppers to be the first uthor who investigted the Hyers-Ulm stbility of differentil eqution. Since then there hs been significnt mount of interest in Hyers-Ulm stbility, especilly in reltion to ordinry differentil equtions, for exmple see [7, 8, 11, 12, 13, 14, 15, 16, 17, 18, 21, 22, 25, 28]. Also of interest re mny of the rticles in specil issue guest edited by Rssis [23], deling with Ulm, Hyers-Ulm, nd Hyers- Ulm-Rssis stbility in vrious contexts. Also see Pop et l [5, 20, 21, 22]. András nd Mészáros [2] recently used n opertor pproch to show the stbility of liner dynmic equtions on time scles with constnt coefficients, s well s for certin integrl equtions. Tunç nd Biçer [26] proved the Hyers-Ulm stbility of third nd fourth-order Cuchy-Euler differentil equtions. Anderson et l [1, Corollry 2.6] proved the following concerning second-order non-homogeneous Cuchy-Euler equtions on time scles: Received September $15.00 c Dynmic Publishers, Inc.

2 654 D. R. ANDERSON Theorem 1.1 (Cuchy-Euler Eqution). Let λ 1, λ 2 R (or λ 2 = λ 1, the complex conjugte) be such tht t + λ k µ(t) 0, k = 1, 2 for ll t [, σ(b)] T, where T stisfies > 0. Then the Cuchy-Euler eqution (1.1) x (t) + 1 λ 1 λ 2 σ(t) x (t) + λ 1λ 2 tσ(t) x(t) = f(t), t [, b] T hs Hyers-Ulm stbility on [, b] T. To wit, if there exists y C 2 rd [, b] T tht stisfies y (t) + 1 λ 1 λ 2 y (t) + λ 1λ 2 y(t) f(t) σ(t) tσ(t) ε for t [, b] T, then there exists solution u C 2 rd [, b] T of (1.1) given by u(t) = e λ 1 t (t, τ 2 )y (τ 2 ) + where for ny τ 1 [, σ(b)] T the function w is given by [ w(s) = e λ 2 1(s, τ 1 ) y (τ 1 ) λ ] s 1 y(τ 1 ) + σ(s) τ 1 τ 2 e λ 1 t (t, σ(s))w(s) s, ny τ 2 [, σ 2 (b)] T, e λ 2 1 σ(s) τ 1 such tht y u Kε on [, σ 2 (b)] T for some constnt K > 0. (s, σ(ζ))f(ζ) ζ, The motivtion for this work is to extend Theorem 1.1 to the generl nth-order Cuchy-Euler dynmic eqution, nd thus extend the results in [26] s well, with n pproch different from [2]. We will show the stbility in the sense of Hyers nd Ulm of the eqution where α k M k y(t) = f(t), M 0 y(t) := y(t), M k+1 y(t) := (t) (M k y) (t), k = 0, 1,..., n 1. This is essentilly [4, (2.14)] if (t) = t nd f(t) = 0. In the lst section we will nlyze the nth-order fctored eqution with differentil opertors D nd I, where Dy = y nd Iy = y, of the form ( k D ψ k I) y(t) = f(t), t [, b] T, for right-dense continuous functions k nd ψ k, more generl dynmic eqution with vrible coefficients thn the Cuchy-Euler eqution. Throughout this work we ssume the reder hs working knowledge of time scles s cn be found in Bohner nd Peterson [3, 4], originlly introduced by Hilger [9].

3 HYERS-ULAM STABILITY HYERS-ULAM STABILITY FOR HIGHER-ORDER CAUCHY-EULER DYNAMIC EQUATIONS In this section we estblish the Hyers-Ulm stbility of the higher-order nonhomogeneous Cuchy-Euler dynmic eqution on time scles of the form (2.1) where α k M k y(t) = f(t), M 0 y(t) := y(t), M k+1 y(t) := (t) (M k y) (t), k = 0, 1,..., n 1 for given constnts α k R with α n 1, nd for functions, f C rd [, b] T, using the following definition. Definition 2.1 (Hyers-Ulm stbility). Let, f C rd [, b] T nd n N. If whenever M k x C rd [, b] T stisfies α k M k x(t) f(t) ε, t [, b] T there exists solution u of (2.1) with M k u C rd [, b] T for k = 0, 1,..., n 1 such tht x u Kε on [, σ n (b)] T for some constnt K > 0, then (2.1) hs Hyers-Ulm stbility [, b] T. Remrk 2.2. Before proving the Hyers-Ulm stbility of (2.1) we will need the following lemm, which llows us to fctor (2.1) using the elementry symmetric polynomils [6] in the n symbols ρ 1,..., ρ n given by s n 1 = s 1 (ρ 1,...,ρ n ) = i ρ i s n 2 = s 2(ρ 1,...,ρ n ) = i<j ρ i ρ j s n 3 = s 3 (ρ 1,...,ρ n ) = s n 4 = s 4(ρ 1,...,ρ n ) =. s n t = s t(ρ 1,...,ρ n ) = i<j<k i<j<k<l ρ i ρ j ρ k ρ i ρ j ρ k ρ l i 1 <i 2 < <i t ρ i1 ρ i2... ρ it. s n n = s n (ρ 1,...,ρ n ) = ρ 1 ρ 2 ρ 3...ρ n.

4 656 D. R. ANDERSON In generl, we let s j i represent the ith elementry symmetric polynomil on j symbols. Then, given the α k in (2.1), introduce the chrcteristic vlues λ k C vi the elementry symmetric polynomil s n t on the n symbols λ 1,..., λ n, where α n = s 0 1 nd (2.2) α k = s n n k = s n k ( λ 1,..., λ n ) = i 1 <i 2 < <i n k ( 1) n k λ i1 λ i2 λ in k. Lemm 2.3 (Fctoriztion). Given y, C rd [, b] T nd α k R with α n 1, let M k y C rd [, b] T, where M 0 y(t) := y(t) nd M k+1 y(t) := (t) (M k y) (t) for k = 0, 1,..., n 1. Then we hve the fctoriztion (2.3) α k M k y(t) = (D λ k I) y(t), n N, where the differentil opertor D is defined vi Dx = x for x C rd [, b] T, nd I is the identity opertor. Proof. We proceed by mthemticl induction on n N, utilizing the substitution defined in (2.2). For n = 1, α k M k y(t) = α 0 M 0 y(t) + α 1 M 1 y(t) = s 1 ( λ 1 )y(t) + 1 (t)y (t) = (D λ 1 I)y(t) nd the result holds. Assume (2.3) holds for n 1. Then we hve α n+1 1 nd n+1 α k M k y(t) = α 0 y(t) + α k M k y(t) + M n+1 y(t) = s n+1 n+1y(t) + s n+1 n+1 k M ky(t) + (t) (M n y) (t) = λ n+1 s n n y(t) + ( ) s n n+1 k λ n+1 s n n k Mk y(t) + (t)d (M n y)(t) = λ n+1 [s n ny(t) + ] s n n km k y(t) + s n n+1 km k y(t) + (t)d (M n y)(t) ( ) = λ n+1 s n n km k y(t) + (t)d s n n+1 km k 1 y(t) + M n y (t) = λ n+1 s n n k M ky(t) + (t)d = λ n+1 s n n k M ky(t) + (t)d ( n 1 ) s n n k M ky(t) + M n y (t) s n n k M ky(t)

5 = ((t)d λ n+1 I) = ((t)d λ n+1 I) = ((t)d λ n+1 I) nd the proof is complete. HYERS-ULAM STABILITY 657 s n n km k y(t) α k M k y(t) (D λ k I)y(t) Theorem 2.4 (Hyers-Ulm Stbility). Given y,, f C rd [, b] T with A > 0 for some constnt A, nd α k R with α n 1, consider (2.1) with M k y C rd [, b] T for k = 0,...,n 1. Using the λ k from the fctoriztion in Lemm 2.3, ssume (2.4) (t) + λ k µ(t) 0, k = 1, 2,..., n for ll t [, σ n 1 (b)] T. Then (2.1) hs Hyers-Ulm stbility on [, b] T. Proof. Let ε > 0 be given, nd suppose there is function x, with M k x C rd [, b] T, tht stisfies α k M k x(t) f(t) ε, t [, b] T. We will show there exists solution u of (2.1) with M k u C rd [, b] T for k = 0, 1,..., n 1 such tht x u Kε on [, σ n (b)] T for some constnt K > 0. To this end, set This implies by Lemm 2.3 tht so tht g 1 = x λ 1 x = (D λ 1 I) x g 2 = g 1 λ 2 g 1 = (D λ 2 I)g 1. g k = g k 1 λ k g k 1 = (D λ k I)g k 1. g n = g n 1 λ n g n 1 = (D λ n I)g n 1. g n (t) f(t) = α k M k x(t) f(t), g n (t) f(t) ε, t [, b] T. By the construction of g n we hve g n 1 λ n g n 1 f ε, tht is g n 1 λ n g n 1 f ε ε A.

6 658 D. R. ANDERSON By [1, Lemm 2.3] nd (2.4) there exists solution w 1 C rd [, b] T of (2.5) w (t) λ n f(t) w(t) (t) (t) = 0, or (t)w (t) λ n w(t) f(t) = 0, t [, b] T, where w 1 is given by w 1 (t) = e λn (t, τ 1 )g n 1 (τ 1 ) + nd there exists n L 1 > 0 such tht τ 1 e λn (t, σ(s)) f(s) (s) s, ny τ 1 [, σ(b)] T, g n 1 (t) w 1 (t) L 1 ε/a, t [, σ(b)] T. Since g n 1 = g n 2 λ n 1 g n 2, we hve tht g n 2 λ n 1g n 2 (t) w 1 (t) L 1 ε/a, t [, σ(b)] T. Agin we pply [1, Lemm 2.3] to see tht there exists solution w 2 C rd [, σ(b)] T of w (t) λ n 1 (t) w(t) w 1(t) (t) = 0, or (t)w (t) λ n 1 w(t) w 1 (t) = 0, t [, σ(b)] T, where w 2 is given by w 2 (t) = e λ n 1 (t, τ 2 )g n 2 (τ 2 ) + nd there exists n L 2 > 0 such tht e λ n 1 τ 2 (t, σ(s)) w 1(s) (s) s, ny τ 2 [, σ 2 (b)] T, g n 2 (t) w 2 (t) L 2 L 1 ε/a 2, t [, σ 2 (b)] T. Continuing in this mnner, we see tht for k = 1, 2,..., n 1 there exists solution w k C rd [, σk 1 (b)] T of w (t) λ n k+1 (t) w(t) w k 1(t) (t) t [, σ k 1 (b)] T, where w k is given by (2.6) w k (t) = e λ n k+1 (t, τ k )g n k (τ k )+ nd there exists n L k > 0 such tht In prticulr, for k = n 1, = 0, or (t)w (t) λ n k+1 w(t) w k 1 (t) = 0, e λ n k+1 τ k g n k (t) w k (t) (t, σ(s)) w k 1(s) s, ny τ k [, σ k (b)] T, (s) k L j ε/a k, t [, σ k (b)] T. j=1 n 1 g 1 (t) w n 1 (t) L j ε/a n 1, j=1 t [, σ n 1 (b)] T

7 HYERS-ULAM STABILITY 659 implies by the definition of g 1 tht x (t) λ 1 (t) x(t) w n 1 n 1(t) (t) L j ε/a n, t [, σ n 1 (b)] T. j=1 Thus there exists solution w n C rd [, σn 1 (b)] T of w (t) λ 1 (t) w(t) w n 1(t) (t) t [, σ n 1 (b)] T, where w n is given by (2.7) w n (t) = e λ 1 (t, τ n )x(τ n ) + = 0, or (t)w (t) λ 1 w(t) w n 1 (t) = 0, τ n e λ 1 nd there exists n L n > 0 such tht (2.8) x(t) w n (t) Kε := By construction, (t, σ(s)) w n 1(s) s, ny τ n [, σ n (b)] T, (s) L j ε/a n, t [, σ n (b)] T. j=1 (D λ 1 I)w n (t) = w n 1 (t) 2 (D λ k I)w n (t) = (D λ 2 I)w n 1 (t) = w n 2 (t). (D λ k I)w n (t) = (D λ n I) w 1 (t) (2.5) = f(t) on [, σ n 1 (b)] T, so tht u = w n is solution of (2.1), with u C rd [, σn 1 (b)] T nd x(t) w n (t) Kε for t [, σ n (b)] T by (2.8). Moreover, using (2.7) nd (2.6), we hve n itertive formul for this solution u = w n in terms of the function x given t the beginning of the proof. 3. EXAMPLE Letting Dy = y nd I be the identity opertor, consider the non-homogeneous fifth-order Cuchy-Euler dynmic eqution (3.1) [ (td) (tD) (tD) (tD) tD + 120I ] y(t) = f(t) for some right-dense continuous function f, on [, b] T ; in fctored form it is (td + 5I)(tD + 4I)(tD + 3I)(tD + 2I)(tD + I)y(t) = f(t). If T = R, this is equivlent to the non-homogeneous fifth-order Cuchy-Euler differentil eqution t 5 y (5) + 25t 4 y (4) + 200t 3 y + 600t 2 y + 600ty + 120y = f(t), By Theorem 2.4 we hve tht (3.1) hs Hyers-Ulm stbility.

8 660 D. R. ANDERSON 4. HIGHER-ORDER LINEAR NON-HOMOGENEOUS FACTORED DYNAMIC EQUATIONS WITH VARIABLE COEFFICIENTS Generlizing wy from higher-order Cuchy-Euler equtions, we consider the following higher-order liner non-homogeneous fctored dynmic equtions with vrible coefficients given by (4.1) ( k D ψ k I)y(t) = f(t), t [, b) T, where k, ψ k, f C rd [, b) T for k = 1, 2,..., n, Dy(t) = y (t), I is the identity opertor, nd k (t) A > 0 for ll t [, b) T, for some constnt A > 0. Here we llow for b = for those time scles tht re unbounded bove. Before our min result in this section we need the following lemm. Lemm 4.1. Let, ψ, f C rd [, b) T with (t) A > 0 for some constnt A, nd ssume (4.2) (t) + µ(t)ψ(t) 0 nd e 1 ψ σ(τ)) (t, (τ) τ < L for ll t [, b) T, for some constnt 0 < L <. Then the first-order dynmic eqution hs Hyers-Ulm stbility on [, b) T. (D ψi)y f = 0 Proof. Suppose there exists function x such tht for some ε > 0, for ll t [, b) T. Set (D ψi) x(t) f(t) ε q(t) = (D ψi) x(t) f(t), t [, b) T. Clerly q(t) ε for ll t [, b) T, nd we cn solve for x to obtin x(t) = e ψ(t, )x() + q(τ) + f(τ) e ψ(t, σ(τ)) τ. (τ) Let y be the unique solution of the initil-vlue problem (D ψi) y(t) f(t) = 0, y() = x(). Then y is given by nd y(t) x(t) = by condition (4.2). y(t) = e ψ(t, )x() + e ψ(t, σ(τ)) f(τ) (τ) τ, e ψ(t, σ(τ)) q(τ) (τ) τ t ε e 1 σ(τ)) (t, ψ (τ) τ Lε

9 HYERS-ULAM STABILITY 661 Remrk 4.2. The convergence condition on the integrl in (4.2) is essentilly the sme s S1 nd S2 in [2], nd cn be met for vrious functions. For exmple, if T = R, (w) = w 2, ψ(w) = sin w, nd = 1, then e 1 t σ(τ)) (t, ψ (τ) τ = 1 [ ] esin w(t, τ) w2 τ 2dτ 1 e 1+1 t, 1 + e 1 1 t 1 for ll t 1, nd so clerly converges on [1, ) R. If T = N, (w) = w, ψ(w) = 5/2, nd = 3, then t 1 e 1 ψ σ(τ)) (t, (τ) τ = τ=3 Γ(1 + τ)γ(t 5/2) τγ(t)γ(τ 3/2) = Γ(t 5/2) 5 π Γ(t) [0, 2/5) for ll integers t 3, nd so clerly converges on [3, ) N, where Γ is the gmm function. Theorem 4.3 (Hyers-Ulm Stbility). Given k, ψ k, f C rd [, b) T with k (t) A > 0 for some constnt A, ssume (4.3) k (t) + µ(t)ψ k (t) 0 nd e 1 ψ k (t, σ(τ)) k k (τ) τ < L k 1 for k = 1, 2,..., n nd for ll t [, b) T, where 0 < L k 1 < is some constnt. Then (4.1) hs Hyers-Ulm stbility on [, b) T. Proof. Suppose there exists function x such tht ( k D ψ k I)x(t) f(t) ε for some ε > 0, for ll t [, b) T. Define the new functions x 0 := x, y n := f, nd (4.4) x k := ( k D ψ k I)x k 1, k = 1,...,n. Then tht cn be solved to yield for k = 1,...,n. Note tht so x k (t) = k (t)x k 1(t) ψ k (t)x k 1 (t), x k 1 (t) = e ψ k k (t, )x k 1 () + x n y n (t) = x n f (t) ε, e ψ k k (t, σ(τ)) x k(τ) k (τ) τ n x n 1 ψ n x n 1 y n = n x n 1 ψ n x n 1 f ε. Hyers-Ulm stbility of this first-order eqution by Lemm 4.1 implies there exists function y n 1 such tht x n 1 y n 1 L n 1 ε

10 662 D. R. ANDERSON nd n (t)y n 1 (t) ψ n(t)y n 1 (t) = y n (t) = f(t), for some constnt L n 1 > 0, where y n 1 is given by Then y n 1 (t) = e ψn (t, )y n 1 () + n e ψn (t, σ(τ)) y n(τ) n n (τ) τ. n 1 x n 2 ψ n 1x n 2 y n 1 L n 1 ε, so gin Hyers-Ulm stbility of the first-order eqution implies there exists function y n 2 such tht x n 2 y n 2 L n 2 L n 1 ε nd n 1 (t)y n 2(t) ψ n 1 (t)y n 2 (t) = y n 1 (t), for some constnt L n 2 > 0, where y n 2 is given by y n 2 (t) = e ψ n 1 n 1 (t, )y n 2 () + Continuing in this wy, we obtin function y 0 such tht n 1 (4.5) x 0 y 0 = x y 0 ε e ψ n 1 n 1 (t, σ(τ)) y n 1(τ) n 1 (τ) τ. j=0 L j nd 1 (t)y 0 (t) ψ 1(t)y 0 (t) = y 1 (t), for some constnt L 0 > 0, where y 0 is given by y 0 (t) = e ψ 1 1 (t, )y 0 () + e ψ 1 1 (t, σ(τ)) y 1(τ) 1 (τ) τ. Note tht by construction of y 0 nd generlly y k, we hve ( k D ψ k I)y 0 (t) = =. ( k D ψ k I) y 1 (t) k=2 ( k D ψ k I) y 2 (t) k=3 = ( n D ψ n I)y n 1 (t) = y n (t) = f(t), mking y 0 solution of (4.1). This fct, together with inequlity (4.5), shows tht (4.1) hs Hyers-Ulm stbility.

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