Influence of the Stepsize on Hyers Ulam Stability of First-Order Homogeneous Linear Difference Equations

Transcription

1 International Journal of Difference Equations ISSN , Volume 12, Number 2, pp (2017) ttp://campus.mst.edu/ijde Influence of te Stepsize on Hyers Ulam Stability of First-Order Homogeneous Linear Difference Equations Masakazu Onitsuka Okayama University of Science Department of Applied Matematics Okayama, , Japan Abstract Tis paper is concerned wit Hyers Ulam stability of te first-order linear difference equation x(t) ax(t) = 0 on Z, were a is a real number, x(t) = (x(t + ) x(t))/ and Z = k k Z for te stepsize > 0. It is well known tat tis equation is an approximation of te ordinary differential equation x ax = 0. Te obtained results are divided into te small stepsize case and te large stepsize case. Te main purpose of tis paper is to clarify te following: in te small stepsize case, te minimum of HUS constants of difference equation is te same as tat of ODE; in te large stepsize case, te minimum of HUS constants of difference equation is different from tat of ODE. To illustrate te obtained results, some applications to perturbed linear difference equations are included. Furtermore, some suitable examples are also attaced for a deeper understanding. AMS Subject Classifications: 39A06, 39A30, 34A30. Keywords: Hyers Ulam stability, HUS constant, linear difference equation. 1 Introduction Hyers Ulam stability originated from a stability problem in te field of functional equations wic was posed by Ulam [31,32] in Tis problem was solved by Hyers [9] in After tat, many researcers ave studied Hyers Ulam stability in te field of functional equations (see [1, 5, 10, 15, 19]). In 1998, Alsina and Ger [2] considered Hyers Ulam stability of te linear differential equation x x = 0. Tey proved tat if a differentiable function φ : I R satisfies φ (t) φ(t) for all t I, ten tere exists a solution x : I R of x x = 0 suc tat φ(t) x(t) 3 for all Received December 28, 2016; Accepted May 22, 2017 Communicated by Memet Ünal

2 282 Masakazu Onitsuka t I, were > 0 is a given arbitrary constant and I is a nonempty open interval of R. Later, tis result as been extended to various linear differential equations by many researcers (see [3,6,8,11 14,17,18,22,24,27 30,33]). Moreover, Hyers Ulam stability of te difference equations can be found in [4, 16, 23, 25, 26] and te references terein. In tis paper, we consider te first-order omogeneous linear difference equation on Z, were a is a real number and x(t) = x(t) ax(t) = 0 (1.1) x(t + ) x(t) and Z = k k Z for given > 0. We call te stepsize. Let T = Z I, were I is a nonempty open interval of R. We define T T \ max T if te maximum of T exists, = T oterwise. Note ere tat if a function x(t) exists on T, ten x(t) exists on T, and if a = 1/ olds, ten we no longer ave a first-order difference equation. Trougout tis paper, we assume tat T and T are nonempty sets of R, and a 1/. We say tat (1.1) as te Hyers Ulam stability on T if tere exists a constant K > 0 wit te following property: Let > 0 be a given arbitrary constant. If a function φ : T R satisfies φ(t) aφ(t) for all t T, ten tere exists a solution x : T R of (1.1) suc tat φ(t) x(t) K for all t T. We call suc K a HUS constant for (1.1) on T. From te following remark, we will treat only te case of a 0 and a 2/ in tis paper. Remark 1.1. If a = 0 or a = 2/, ten (1.1) does not ave Hyers Ulam stability on Z. For example, in te case a = 0, we consider te function φ(t) = t satisfying φ(t) = for all t T. Since x(t) c is te general solution of x(t) = 0, we ave φ(t) x(t) as t ; tat is, (1.1) does not ave Hyers Ulam stability on Z wen a = 0. In te case a = 2/, te function φ(t) = t( 1) t/ satisfies φ(t)+2φ(t)/ = for all t T. Since te general solution of x(t)+2x(t)/ = 0 is x(t) = c( 1) t/, we see tat φ(t) x(t) as t ; tat is, (1.1) does not ave Hyers Ulam stability on Z wen a = 2/. It is well known tat (1.1) is an approximation of te first-order omogeneous ordinary differential equation x ax = 0 (1.2) on I, were I is a nonempty open interval of R, and a is a non-zero real number. In fact, for te sake of simplicity, we consider (1.2) on [0, b], were b > 0 and [0, b] I. Let us divide te interval [0, b] into n parts, were n N. Letting = b/n, and using Euler s

3 Stepsize and Hyers Ulam Stability 283 metod, we get (1.1) on [0, b] since x (t) can be approximated by x(t) (see [7, pp ]). Hyers Ulam stability of (1.2) and its generalized equations on I as been studied by [11 13, 22, 24, 27, 29, 30, 33]. We say tat (1.2) as te Hyers Ulam stability on I if tere exists a constant K > 0 wit te following property: Let > 0 be a given arbitrary constant. If a differentiable function φ : I R satisfies φ (t) aφ(t) for all t I, ten tere exists a solution x : I R of (1.2) suc tat φ(t) x(t) K for all t I. We call suc K a HUS constant for (1.2) on I. Te following result is obtained by using one of te results presented by Jung [13] et al. Teorem 1.2 (See [13, 22, 24, 29]). Suppose tat a 0. Ten (1.2) as Hyers Ulam stability wit a HUS constant 1/ a on R. Furtermore, te solution x(t) of (1.2) satisfying φ(t) x(t) / a for all t R is te only one (unique), were φ(t) is a differentiable function satisfying φ (t) aφ(t) for all t R. A natural question now arises. Can we find an explicit solution corresponding to te solution x(t) of (1.2) in Teorem 1.2? Onitsuka and Soji [24] gave te answer to tis question. Te obtained result is as follows. Teorem 1.3 (See [24, Teorem 1]). Let > 0 be a given arbitrary constant. Suppose tat a differentiable function φ : I R satisfies φ (t) aφ(t) for all t I, were a 0. Ten one of te following olds: (i) if a > 0 and sup I exists, ten lim φ(t) exists were τ = sup I, and any solution t τ 0 x(t) of (1.2) wit lim φ(t) x(τ) t τ 0 < /a satisfies tat φ(t) x(t) < /a for all t I; (ii) if a > 0 and sup I does not exist, ten lim φ(t)e at exists, and tere exists exactly ( ) one solution x(t) = lim φ(t)e at e at of (1.2) suc tat φ(t) x(t) /a for all t I; (iii) if a < 0 and inf I exists, ten lim φ(t) exists were σ = inf I, and any solution t σ+0 x(t) of (1.2) wit lim φ(t) x(σ) t σ+0 < / a satisfies tat φ(t) x(t) < / a for all t I; (iv) if a < 0 and inf Idoes not exist, ten lim t φ(t)e at exists, and tere exists exactly ( ) one solution x(t) = lim t φ(t)e at e at of (1.2) suc tat φ(t) x(t) / a for all t I.

4 284 Masakazu Onitsuka Using (ii) or (iv) in Teorem 1.3, we can find an explicit solution corresponding to te solution x(t) of (1.2) in Teorem 1.2. We will recall tat (1.1) is an approximation of (1.2). In te case tat = 1, te following teorem is obtained by using a result presented by Brzdęk, Popa and Xu [4]. Teorem 1.4 (See [4, Teorem 3]). Suppose tat a 0, 1, 2 and = 1. Ten (1.1) as Hyers Ulam stability wit a HUS constant 1/ a on Z. Furtermore, te solution x(t) of (1.1) satisfying φ(t) x(t) / a for all t Z is te only one, were φ(t) is a function satisfying 1 φ(t) aφ(t) for all t Z. Comparing Teorem 1.2 wit Teorem 1.4, we see tat if a 0 and a > 1, ten te minimums of HUS constants in Teorems 1.2 and 1.4 are te same. However, if a 2 and a < 1, ten tey are different. Now, important questions arise: (Q1) Wat is te minimum of HUS constants for (1.1) on Z? (Q2) Can we find an explicit solution corresponding to te solution x(t) of (1.1) in Teorem 1.4? (Q3) How does te stepsize influence te minimum of HUS constants for (1.1) on Z? Te purpose of tis paper is to answer te above questions. Te answers to tese questions are derived in Section 4 from te following two teorems. Teorem 1.5. Let > 0 be a given arbitrary constant. Suppose tat a function φ : T R satisfies φ(t) aφ(t) for all t T, were a 0 and a > 1/. Ten one of te following olds: (i) if a > 0 and max T exists, ten any solution x(t) of (1.1) wit φ(τ) x(τ) < /a satisfies tat φ(t) x(t) < /a for all t T, were τ = max T ; (ii) if a > 0 and max T does not exist, ten lim φ(t)(a + 1) t/ exists, and tere exists exactly one solution x(t) = lim φ(t)(a + t 1) (a + 1) t of (1.1) suc tat φ(t) x(t) /a for all t T ; (iii) if 1/ < a < 0 and min T exists, ten any solution x(t) of (1.1) wit φ(σ) x(σ) < / a satisfies tat φ(t) x(t) < / a for all t T, were σ = min T ; (iv) if 1/ < a < 0 and min T does not exist, ten lim φ(t)(a + t 1) t/ exists, and tere exists exactly one solution x(t) = lim t φ(t)(a + 1) t of (1.1) suc tat φ(t) x(t) / a for all t T. (a + 1) t

5 Stepsize and Hyers Ulam Stability 285 Remark 1.6. If te stepsize > 0 is sufficiently small, ten we can coose a so tat 0 < < 1/ a ; tat is, 1/ < a olds in te case tat a is negative. From tis fact, we can conclude tat te assertions (iii) and (iv) in Teorem 1.5 are te results wen te stepsize is sufficiently small. Teorem 1.7. Let > 0 be a given arbitrary constant. Suppose tat a function φ : T R satisfies φ(t) aφ(t) for all t T, were a 2/ and a < 1/. Ten one of te following olds: (i) if 2/ < a < 1/ and min T exists, ten any solution x(t) of (1.1) wit φ(σ) x(σ) < /() satisfies tat φ(t) x(t) < /() for all t T, were σ = min T ; (ii) if 2/ < a < 1/ and min T does not exist, ten exists, and tere exists exactly one solution x(t) = lim t φ(t)(a + 1) t (a + 1) t of (1.1) suc tat φ(t) x(t) /() for all t T ; lim φ(t)(a + 1) t/ t (iii) if a < 2/ and max T exists, ten any solution x(t) of (1.1) wit φ(τ) x(τ) < / satisfies tat φ(t) x(t) < / for all t T, were τ = max T ; (iv) if a < 2/ and max T does not exist, ten lim φ(t)(a + 1) t/ exists, and tere exists exactly one solution x(t) = lim φ(t)(a + t 1) (a + 1) t of (1.1) suc tat φ(t) x(t) / for all t T. Remark 1.8. If te stepsize > 0 is suitably large, ten we can coose a so tat eiter a < 2/ or 2/ < a < 1/. From tis fact, we can conclude tat te assertions in Teorem 1.7 are te results wen te stepsize is suitably large. In te next section, we will consider small stepsize linear difference equations, and present te proof of Teorem 1.5. In Section 3, we will consider large stepsize linear difference equations, and present te proof of Teorem 1.7. Using Teorems 1.5 and 1.7, we will give te answers to questions (Q1) (Q3) in Section 4. In te final section, we will discuss te applications to perturbed linear difference equations corresponding to linear difference equation (1.1). For illustration of te obtained results, we will take some concrete examples.

6 286 Masakazu Onitsuka 2 Small Stepsize Linear Difference Equations In tis section, we consider te case of a 0 and a > 1/. First we give some preparations for te proof of Teorem 1.5. Lemma 2.1. Suppose tat a 0 and a > 1/. Let > 0 be a given arbitrary constant and let φ(t) be a real-valued function on T. Ten te inequality φ(t) aφ(t) olds for all t T if and only if te inequality olds for all t T. ( 0 φ(t) ) (a + 1) t 2(a + 1) t+ a Proof. Te statement of Lemma 2.1 is clearly true since te equality olds for all t T. ( φ(t) ) (a + 1) t a = 1 ( φ(t + ) ) ( (a + 1) φ(t) ) (a + 1) t+ a a = ( φ(t) aφ(t) + )(a + 1) t+ Proposition 2.2. Let > 0 be a given arbitrary constant. Suppose tat a function φ : T R satisfies φ(t) aφ(t) for all t T, were a 0 and a > 1/. Ten tere exist a nondecreasing function u : T R and a nonincreasing function v : T R suc tat and one of te following old: φ(t) = u(t)(a + 1) t + a = v(t)(a + 1) t a (i) if a > 0 and max T exists, ten te inequality olds for all t T, were τ = max T ; (2.1) u(t) u(τ) < v(τ) v(t) (2.2) (ii) if a > 0 and max T does not exist, ten lim u(t) and lim v(t) exist, and olds for all t T ; u(t) lim u(t) = lim v(t) v(t) (2.3)

7 Stepsize and Hyers Ulam Stability 287 (iii) if 1/ < a < 0 and min T exists, ten te inequality olds for all t T, were σ = min T ; v(t) v(σ) < u(σ) u(t) (2.4) (iv) if 1/ < a < 0 and min T does not exist, ten lim u(t) and lim v(t) exist, t t and v(t) lim v(t) = lim u(t) u(t) (2.5) t t olds for all t T. Proof. Suppose tat a 0 and a > 1/. Define te functions u(t) and v(t) as follows: ( u(t) = φ(t) ) ( (a + 1) t and v(t) = φ(t) + ) (a + 1) t a a for t T. Ten (2.1) olds, and terefore, we obtain and u(t) = v(t) 2 a t (a + 1) (2.6) < v(t) if a > 0, u(t) > v(t) if 1 < a < 0 (2.7) for t T. Using te assertion in Lemma 2.1, we ave te inequality 0 u(t) 2(a + 1) t+ for t T. Noticing tat (a + 1) t/ = a(a + 1) (t+)/ and using (2.6) and te above inequality, we get 2(a + 1) t+ v(t) 0 for t T. Terefore, we see tat u(t) is nondecreasing and v(t) is nonincreasing. First we prove assertion (i). From above-mentioned facts and te assumptions in (i), u(τ) and v(τ) become te maximum of u(t) and te minimum of v(t) on T, respectively. Tus, using (2.7), we ave (2.2), and terefore, assertion (i) is true. We next prove assertion (ii). Let s I be a fixed number. From (2.7) wit a > 0, we ave u(t) < v(s) for all t T ; tat is, u(t) is bounded above. Hence, we see tat lim u(t) exists. Using (2.6) and a > 0, we get lim u(t) = lim v(t), and terefore, (2.3) is satisfied for t T since u(t) is nondecreasing and v(t) is nonincreasing. Tus, assertion (ii) is true. Using te same arguments in te proofs of (i) and (ii), we can easily see tat assertions (iii) and (iv) are also true. Te proof is now complete.

8 288 Masakazu Onitsuka Next we will prove Teorem 1.5 by using te idea of te proof of Teorem 1.3 (see [24]). Proof of Teorem 1.5. First we prove case (i). From assertion (i) in Proposition 2.2, we can find two functions u : T R and v : T R suc tat (2.1) and (2.2) old for t T. We now consider te solution x(t) of (1.1) wit φ(τ) x(τ) < /a and a > 0, were τ = max T. Ten, tis solution is expressed as x(t) = x(τ)(a + 1) t τ for t T. Since φ(τ) x(τ) < /a and (2.1) old, we ave u(τ) < x(τ)(a + 1) τ < v(τ). Using (2.1), (2.2) and tis inequality, we get φ(t) x(t) u(τ) x(τ)(a + 1) τ t (a + 1) + a < a and φ(t) x(t) v(τ) x(τ)(a + 1) τ t (a + 1) a > a for t T. Terefore, we obtain te inequality φ(t) x(t) < /a for t T. Next we prove case (ii). From assertion (ii) in Proposition 2.2, we can find two functions u : T R and v : T R suc tat (2.1) and (2.3) old for t T. Since lim u(t) exists and (2.1) olds for t T, te function φ(t)(a + 1) t/ also as te same limiting value lim φ(t)(a + 1) t/ = lim u(t). We consider te function x(t) = lim φ(t)(a + t 1) (a + 1) t (2.8) for t T. Ten, tis function is a solution of (1.1). From (2.1) and (2.3), we ave φ(t) x(t) = u(t) lim φ(t)(a + 1) t (a + 1) t + a a and φ(t) x(t) = v(t) lim φ(t)(a + 1) t (a + 1) t a a (2.9) (2.10) for t T. Hence, we get te inequality φ(t) x(t) /a for t T. Noticing tat if we coose a constant c so tat c lim φ(t)(a + 1) t, ten te function x(t) = c(a + 1) t/ becomes a solution of (1.1), owever, it does not satisfy (2.9) or (2.10) for t sufficiently large. Tus, (2.8) is exactly one solution of (1.1) satisfying φ(t) x(t) /a for t T.

9 Stepsize and Hyers Ulam Stability 289 We will prove case (iii). From assertion (iii) in Proposition 2.2, we can find two functions u : T R and v : T R suc tat (2.1) and (2.4) old for t T. We consider te solution x(t) = x(σ)(a + 1) t σ of (1.1) wit φ(σ) x(σ) < / a and 1/ < a < 0, were σ = min T. Since φ(σ) x(σ) < / a and (2.1) old, we get v(σ) < x(σ)(a + 1) σ < u(σ). Using (2.1), (2.4) and tis inequality, we obtain φ(t) x(t) < / a for t T. Finally we prove case (iv). By means of assertion (iv) in Proposition 2.2, tere exist two functions u(t) and v(t) satisfying (2.1) and (2.5) for t T. Since lim u(t) exists t and (2.1) olds for t T, we ave lim φ(t)(a + t 1) t/ = lim u(t). Consider te t function x(t) = lim φ(t)(a + 1) t (a + 1) t, (2.11) wic becomes a solution of (1.1). From (2.1) and (2.5), we get te inequality φ(t) x(t) / a for t T. Using te same argument as in te proof of case (ii), we can conclude tat (2.11) is exactly one solution of (1.1) satisfying φ(t) x(t) / a for t T. Tis completes te proof of Teorem Large Stepsize Linear Difference Equations In tis section, we consider te case of a 2/ and a < 1/. First we give some preparations for te proof of Teorem 1.7. Lemma 3.1. Suppose tat a 2/ and a < 1/. Let > 0 be a given arbitrary constant and let φ(t) be a real-valued function on T. Ten te inequality φ(t) aφ(t) olds for all t T if and only if te inequality ) 0 ( olds for all t T. φ(t) + ( 1) t (a + 1) t Proof. Since te equality ( ) φ(t) + ( 1) t (a + 1) t ( = φ(t 1 t+ ( 1) + ) + (a + 1) = φ(t) aφ(t) + ( 1) t+ (a + 1) t+ φ(t) + ( 1) t 2 a + 1 t+ ) (a + 1) t+

10 290 Masakazu Onitsuka olds for all t T, te statement of Lemma 3.1 is clearly true. Proposition 3.2. Let > 0 be a given arbitrary constant. Suppose tat a function φ : T R satisfies φ(t) aφ(t) for all t T, were a 2/ and a < 1/. Ten tere exist a nondecreasing function u : T R and a nonincreasing function v : T R suc tat φ(t) = u(t)(a + 1) t ( 1) t t = v(t)(a + 1) t ( 1) + (3.1) and one of te following old: (i) if 2/ < a < 1/ and min T exists, ten te inequality olds for all t T, were σ = min T ; v(t) v(σ) < u(σ) u(t) (3.2) (ii) if 2/ < a < 1/ and min T does not exist, ten lim u(t) and lim v(t) t t exist, and v(t) lim v(t) = lim u(t) u(t) (3.3) t t olds for all t T ; (iii) if a < 2/ and max T exists, ten te inequality olds for all t T, were σ = min T ; u(t) u(τ) < v(τ) v(t) (3.4) (iv) if a < 2/ and max T does not exist, ten lim u(t) and lim v(t) exist, and olds for all t T. Proof. Define te functions u(t) = u(t) lim u(t) = lim v(t) v(t) (3.5) ( φ(t) + ( 1) t ) (a + 1) t and v(t) = ( φ(t) ( 1) t ) (a + 1) t

11 Stepsize and Hyers Ulam Stability 291 for t T, were a 2/ and a < 1/. Ten (3.1) and u(t) = v(t) + 2 t a + 1 (3.6) old for t T. Terefore, we ave > v(t) if 2 u(t) < a < 1, < v(t) if a < 2 (3.7) for t T. From te assertion in Lemma 3.1, we see tat 0 u(t) 2 a + 1 t+ for t T. Moreover, using equalities (3.6) and a + 1 t 1 a + 1 = a + 1 t+ = and te above inequality, we get 2 a + 1 t+ v(t) 0 ( ) 2 + a a + 1 t+, for t T. Terefore, we see tat u(t) is nondecreasing and v(t) is nonincreasing. First we prove assertion (i). Since u(t) is nondecreasing and v(t) is nonincreasing, and using (3.7) wit 2/ < a < 1/, we obtain (3.2) for t T ; tat is, assertion (i) is true. We next prove assertion (ii). Let s I be a fixed number. From (3.7) wit 2/ < a < 1/, we ave u(t) > v(s) for all t T ; tat is, u(t) is bounded below. Since u(t) is a nondecreasing function, we can conclude tat lim t 2/ < a < 1/, we get lim u(t) = lim t t u(t) exists. From (3.6) and v(t), and terefore, (3.3) is satisfied for t T since u(t) is nondecreasing and v(t) is nonincreasing. Tus, assertion (ii) is true. Using te same arguments in te proofs of (i) and (ii), we can easily see tat assertions (iii) and (iv) are also true. Te proof is now complete. We next give te proof of Teorem 1.7. Proof of Teorem 1.7. First we prove case (i). By means of assertion (i) in Proposition 3.2, we can find two functions u : T R and v : T R suc tat (3.1) and (3.2) old for t T. Consider te solution x(t) of (1.1) wit φ(σ) x(σ) < /() and 2/ < a < 1/, were σ = min T. Ten, tis solution is expressed as x(t) = x(σ)(a + 1) t σ

12 292 Masakazu Onitsuka for t T. From te initial condition φ(σ) x(σ) < /(), we ave φ(σ) < x(σ) < φ(σ) + In te case tat σ/ is odd, multiplying (3.8) by (a + 1) σ/, we get ( ) u(σ) = φ(σ) (a + 1) σ > x(σ)(a + 1) σ ( > φ(σ) + ) (a + 1) σ. (3.8) = v(σ). On te oter and, in te case tat σ/ is even, multiplying (3.8) by (a + 1) σ/, we ave ( ) v(σ) = φ(σ) (a + 1) σ Consequently, te inequality < x(σ)(a + 1) σ ( < φ(σ) + ) (a + 1) σ = u(σ). v(σ) < x(σ)(a + 1) σ < u(σ) (3.9) olds. Next, we consider te two cases: (a) t/ is odd; (b) t/ is even. In case (a), using (3.1), (3.2) and (3.9), we get and φ(t) x(t) u(σ) x(σ)(a + 1) σ t (a + 1) + φ(t) x(t) v(σ) x(σ)(a + 1) σ t (a + 1) < > for t T. On te oter and, in case (b), using (3.1), (3.2) and (3.9), we ave φ(t) x(t) u(σ) x(σ)(a + 1) σ t (a + 1) > and φ(t) x(t) v(σ) x(σ)(a + 1) σ t (a + 1) + <

13 Stepsize and Hyers Ulam Stability 293 for t T. Tus, φ(t) x(t) < /a olds for all t T. Terefore, assertion (i) is true. Next we prove case (ii). By means of assertion (ii) in Proposition 3.2, we can find two functions u : T R and v : T R suc tat (3.1) and (3.3) old for t T. Since lim t u(t) exists, (3.1) olds for t T and a + 1 < 0 is satisfied, we see tat lim t φ(t)(a + 1) t = lim t u(t). Ten te function x(t) = lim t φ(t)(a + 1) t (a + 1) t (3.10) is a solution of (1.1). We will consider te two cases: (a) t/ is odd; (b) t/ is even. In case (a), using (3.1), (3.3) and a + 1 < 0, we obtain φ(t) x(t) = u(t) lim φ(t)(a + t t 1) (a + 1) t + (3.11) and φ(t) x(t) = v(t) lim φ(t)(a + t t 1) (a + 1) t (3.12) for t T. On te oter and, in case (b), using (3.1), (3.3) and a + 1 < 0, we get φ(t) x(t) = u(t) lim φ(t)(a + t t 1) (a + 1) t (3.13) and φ(t) x(t) = v(t) lim φ(t)(a + t t 1) (a + 1) t + (3.14) for t T. Consequently, φ(t) x(t) /() olds for all t T. Noticing tat if we coose a constant c so tat c lim φ(t)(a + t t 1), ten te function x(t) = c(a + 1) t/ becomes a solution of (1.1), owever, it does not satisfy (3.11),

14 294 Masakazu Onitsuka (3.12), (3.13) or (3.14) for t sufficiently large. Tus, (3.10) is exactly one solution of (1.1) satisfying φ(t) x(t) /() for t T. Next, we prove case (iii). From assertion (iii) in Proposition 3.2, we can find two functions u : T R and v : T R suc tat (3.1) and (3.4) old for t T. Consider te solution x(t) = x(τ)(a + 1) t τ of (1.1) wit φ(τ) x(τ) < / and a < 2/, were τ = max T. Since φ(τ) x(τ) < / and (3.1) old, we get u(τ) < x(τ)(a + 1) τ < v(τ). Hence, togeter wit (3.1) and (3.4), we obtain φ(t) x(t) < / for t T. Finally we prove case (iv). By means of assertion (iv) in Proposition 3.2, tere exist two functions u(t) and v(t) satisfying (3.1) and (3.5) for t T. Since lim u(t) exists, (3.1) olds for t T and a + 1 < 0 is satisfied, we get lim φ(t)(a + 1) t/ = lim u(t). Consider te function x(t) = lim φ(t)(a + t 1) (a + 1) t, (3.15) wic becomes a solution of (1.1). From (3.1) and (3.5), we see tat φ(t) x(t) / a+2/ olds for t T. Using te same argument as in te proof of case (ii), we can conclude tat (3.15) is exactly one solution of (1.1) satisfying φ(t) x(t) / a+2/ for t T. Tis completes te proof of Teorem Influence of te Stepsize on Hyers Ulam Stability In tis section, we will present te answers to questions (Q1) (Q3), respectively. From Teorems 1.5 and 1.7, we can establis te following simple results. Corollary 4.1. If a 0 and a > 1/, ten (1.1) as Hyers Ulam stability wit a HUS constant 1/ a on T. Corollary 4.2. If a 2/ and a < 1/, ten (1.1) as Hyers Ulam stability wit a HUS constant 1/ on T. In te case tat I = R, we can state te following result from te assertions (ii) and (iv) in Teorem 1.5. Corollary 4.3. Suppose tat a 0 and a > 1/. Ten (1.1) as Hyers Ulam stability wit a HUS constant 1/ a on Z. Furtermore, te solution x(t) of (1.1) satisfying φ(t) x(t) / a for all t Z is te only one, wic written as x(t) = lim φ(t)(a + 1) t (a + 1) t

15 Stepsize and Hyers Ulam Stability 295 if a > 0 (resp., x(t) = lim φ(t)(a + 1) t/ (a + 1) t/ if 1/ < a < 0). t Remark 4.4. Let > 0 be a given arbitrary constant. We consider te nonomogeneous difference equation x(t) ax(t) = (4.1) on Z, were a is a non-zero real number. We can easily see tat te function φ(t) = c(a + 1) t/ /a for t Z is te general solution of (4.1), were c is an arbitrary constant. Since c(a + 1) t/ is a solution of (1.1), φ(t) x(t) = / a olds for all t Z. From tis fact and te assertion in Corollary 4.3, we can conclude tat 1/ a is te minimum of HUS constants for (1.1) on Z wen a 0 and a > 1/. Moreover, tis example sows tat it is not possible to weaken te condition φ(τ) x(τ) < /a in (i) of Teorem 1.5 to φ(τ) x(τ) /a, wenever φ(t) x(t) < /a olds for t T. In te case I = R, we can establis te following result from te assertions (ii) and (iv) in Teorem 1.7. Corollary 4.5. Suppose tat a 2/ and a < 1/. Ten (1.1) as Hyers Ulam stability wit a HUS constant 1/ on Z. Furtermore, te solution x(t) of (1.1) satisfying φ(t) x(t) / for all t Z is te only one, wic written as x(t) = if 2/ < a < 1/ (resp., x(t) = lim t φ(t)(a + 1) t (a + 1) t lim φ(t)(a + 1) t/ (a + 1) t/ if a < 2/). Remark 4.6. Let > 0 be a given arbitrary constant. Consider te nonomogeneous difference equation x(t) ax(t) = ( 1) t (4.2) on Z, were a 2/. Ten te function φ(t) = c(a + 1) t ( 1) t + for t Z becomes te general solution of (4.2), were c is an arbitrary constant. Since c(a + 1) t/ is a solution of (1.1), we obtain φ(t) x(t) = / for all t Z. From tis fact and te assertion in Corollary 4.5, we can conclude tat 1/ a+2/ is te minimum of HUS constants for (1.1) on Z. Moreover, tis example implies tat it is not possible to weaken te condition φ(σ) x(σ) < /() in (i) of Teorem 1.7 to φ(σ) x(σ) /(), wenever φ(t) x(t) < /() olds for t T. Let H() = 1 a + 1/ 1/ (4.3)

16 296 Masakazu Onitsuka for > 0, were a 0, 1/, 2/. From Corollaries 4.3, 4.5, Remarks 4.4 and 4.6, we see tat te minimum of HUS constants for (1.1) on Z is H(). Tis is te answer to our question (Q1) raised in Section 1. Moreover, Corollaries 4.3 and 4.5 imply te answer to question (Q2). Namely, we can find an explicit solution corresponding to te solution x(t) of (1.1) in Teorem 1.4. Te following remark is an answer to final question (Q3). Remark 4.7. In te case tat is a small stepsize or a is positive, Corollary 4.3 and Remark 4.4 mean tat te minimum of HUS constants is te same as tat of ordinary differential equation (1.2). On te oter and, in te case tat is a large stepsize, Corollary 4.5 and Remark 4.6 mean tat te minimum of HUS constants is different from tat of ordinary differential equation (1.2). Te following remark is also an answer to question (Q3). Remark 4.8. Wen restricted to te case a < 0, we can rewrite H() as 1 if 0 < < 1 H() = a a, 1 if 1 a < and 2 a for > 0. In Figure 4.1, we give te grap of H() wic depends on te stepsize. H 1 a - 1 a - 2 a Figure 4.1: Te grap of H() wen a < 0. Next we consider te case tat I is a finite interval. Using te assertions in Teorems 1.5 and 1.7, we can verify te following facts. Corollary 4.9. Suppose tat a 0 and a > 1/. Let I be a finite nonempty open interval of R and > 0 be a given arbitrary constant. If a function φ : T R satisfies φ(t) aφ(t) for all t T, ten tere exists a solution x : T R of (1.1) suc tat φ(t) x(t) < / a for all t T. Corollary Suppose tat a 2/ and a < 1/. Let I be a finite nonempty open interval of R and > 0 be a given arbitrary constant. If a function φ : T R satisfies φ(t) aφ(t) for all t T, ten tere exists a solution x : T R of (1.1) suc tat φ(t) x(t) < / for all t T.

17 Stepsize and Hyers Ulam Stability Applications: Perturbed Linear Difference Equations In tis section, we give some applications to illustrate te main results. We consider te first-order perturbed linear difference equation x(t) ax(t) = f(t, x(t)), (5.1) wic corresponding to unperturbed linear difference equation (1.1), were f(t, x) is te real-valued function on T R. Under te assumption tat max T does not exist and f(t, x) is bounded on T R, we see tat 0 < a+1 < 1 implies te uniform-ultimate boundedness; tat is, tere exists a B > 0 and, for any α > 0, tere exists an S(α) > 0 suc tat t 0 T and φ 0 < α imply tat φ(t) < B for all t t 0 + S(α) and t T, were φ(t) is a solution of (5.1) satisfying φ(t 0 ) = φ 0 and (t 0, φ 0 ) T R. If tis property olds, ten we say tat te solutions of (5.1) are uniform-ultimately bounded for bound B (see [20, 21, 34, 35]). Using Teorems 1.5 and 1.7, we can establis te following result. Corollary 5.1. Let δ > 0 be an arbitrary constant. Suppose tat max T does not exist, and tere exists an L > 0 suc tat f(t, x) L for all (t, x) T R. If 0 < a + 1 < 1, ten te solutions of (5.1) are uniform-ultimately bounded for bound LH() + δ, were H() is te minimum of HUS constants for (1.1) on Z wic given by (4.3). Proof. Let B = LH() + δ. We consider te solution φ(t) of (5.1) wit φ(t 0 ) = φ 0 and φ 0 < α, were (t 0, φ 0 ) T R and α is any positive constant. Using (iii) in Teorem 1.5 and (i) in Teorem 1.7, tere exists a solution x(t) of (1.1) wit φ 0 x(t 0 ) < LH() suc tat φ(t) x(t) < LH() for t t 0 and t T. Since x(t) = x(t 0 )(a + 1) (t t 0)/ for t T, we ave x(t) < (LH() + φ 0 ) a + 1 t t 0 < (LH() + α) a + 1 t t 0 (5.2) for t t 0. If LH() + α δ, ten φ(t) < LH() + x(t) < LH() + δ = B olds for t t 0. Next, we ave only to consider tat te case of LH() + α > δ. Let S(α) = log(δ/(lh() + α)) log a + 1 From LH() + α > δ and 0 < a + 1 < 1, we see tat > 0. (LH() + α) a + 1 t t 0 δ olds for all t t 0 + S(α). Hence, combining tis wit (5.2), we get x(t) < δ for all t t 0 + S(α). Since φ(t) x(t) < LH() olds for t T, we obtain φ(t) < B for all t t 0 + S(α). Tis completes te proof of Corollary 5.1.

18 298 Masakazu Onitsuka Remark 5.2. Let I = (0, ). Wen = 5, a = 1/3 and = 1, (4.2) is reduced to te nonomogeneous linear difference equation 5 x(t) x(t) = ( 1) t 5 (5.3) on T = 5Z (0, ). Since a + 1 = 2/3 and f(t, x) = ( 1) t/5 = 1 for all (t, x) T R, all assumptions in Corollary 5.1 are satisfied. Terefore, te solutions of (5.3) are uniform-ultimately bounded for bound Since LH() + δ = φ(t) = 15( 1) t 5 5 L + δ = 15 + δ. 3 ( ) t is a solution of (5.3) wit φ(5) = 45, it is easy to ceck tat φ(t) < 15 = LH() on 10Z (0, ). Tis means tat δ > 0 in Corollary 5.1 cannot be removed. Let δ = 3. Figure 5.1 indicates a solution of (5.3). Te starting point is (5, 45) T R. Moreover, broken lines are φ = LH() δ = 18 and φ = LH() + δ = 18, and dotted lines are φ = LH() = 15 and φ = LH() = 15, respectively ϕ t Figure 5.1: A solution of (5.3). Furtermore, we can clarify te asymptotic beavior of any solution of (5.1) by using Teorems 1.5 and 1.7. Corollary 5.3. Suppose tat a 0, 1/, 2/. Suppose also tat max T does not exist, and tere exists an L > 0 suc tat f(t, x) L for all (t, x) T R. Ten any solution φ(t) of (5.1) satisfies one of te following: (i) if a > 0 or a < 2/ ten lim φ(t)(a + 1) t/ exists;

19 Stepsize and Hyers Ulam Stability 299 (ii) if 0 < a + 1 < 1 ten lim sup φ(t) LH(), were H() is te minimum of HUS constants for (1.1) on Z wic given by (4.3). Proof. Assertion (i) is an immediate consequence from (ii) in Teorem 1.5 and (iv) in Teorem 1.7. Te proof of assertion (ii) is by contradiction. Suppose tat tere exists a solution φ(t) of (5.1) satisfying lim sup φ(t) > LH() and φ(t 0 ) = φ 0, were (t 0, φ 0 ) T R. Using (iii) in Teorem 1.5 and (i) in Teorem 1.7, tere exists a solution x(t) of (1.1) wit φ(t 0 ) x(t 0 ) < LH() suc tat φ(t) x(t) < LH() for t t 0 and t T. Since 0 < a + 1 < 1 and x(t) = x(t 0 )(a + 1) (t t0)/, we obtain lim sup φ(t) = lim sup φ(t) x(t) LH(). Tis is a contradiction. Te proof is now complete. To illustrate Corollary 5.3, we give a concrete example. Example 5.4. We consider te nonlinear difference equation x(t) + 1 x(t) = sin(t + x(t)) (5.4) 3 on T = Z (0, ), were 3, 6. By I = (0, ), te maximum of T does not exist. Moreover, since sin(t + x(t)) 1 for all (t, x) T R, we coose L = 1. Using Corollary 5.3, we see tat any solution φ(t) of (5.4) satisfies eiter of te following: lim φ(t)(1 /3) t/ exists if > 6; lim sup φ(t) 3/(6/ 1) if 0 < < 6. In Figure 5.2, we draw a solution of (5.4) wit = 8 starting from te point (8, 45) T R. Tis solution diverges, and oscillates infinitely. In Figure 5.3, we draw a solution of (5.4) wit = 5 starting from te point (5, 45) T R. Since 3/(6/ 1) = 15 wen = 5, any solution φ(t) of (5.4) wit = 5 satisfies lim sup φ(t) 15. Acknowledgements I would like to tank te referee for reading te paper. References [1] R. P. Agarwal, B. Xu and W. Zang, Stability of functional equations in single variable, J. Mat. Anal. Appl. 288 (2003), no. 2, [2] C. Alsina, R. Ger, On some inequalities and stability results related to te exponential function, J. Inequal. Appl. 2 (1998), no. 4,

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