Mihaela Aurelia Tomescu. Department of Mathematics. University of Petrosani.
University Street 20, Petrosani, 332006, ROMANIA. Abstract: The object of this ...
International Journal of Pure and Applied Mathematics Volume 76 No. 4 2012, 615-624 ISSN: 1311-8080 (printed version) url: http://www.ijpam.eu
AP ijpam.eu
ON EXPONENTIAL INSTABILITY OF VARIATIONAL NONAUTONOMOUS DIFFERENCE EQUATIONS IN BANACH SPACES Mihaela Aurelia Tomescu Department of Mathematics University of Petrosani University Street 20, Petrosani, 332006, ROMANIA
Abstract: The object of this paper is to study two concepts of exponential instability for variational nonautonomous difference equations in Banach spaces. AMS Subject Classification: 34D05, 93C55 Key Words: variational difference equations, discrete skew-evolution semiflows, exponential instability, (nonuniform) exponential instability
1. Introduction In this paper we shall present characterizations for uniform exponential instability and (nonuniform) exponential instability of variational nonautonomous difference equations. Thus we obtain the discrete-time version of the theorems due to E.Barbashin ([1]) and R.Datko ([4]) for these instability concepts of discrete variational systems. Let (X, d) be a metric space, V be a real or complex Banach space and B (V ) the Banach algebra of all bounded linear operators on V . The norm on V and B (V ) will be denoted by k·k . Received:
February 24, 2012
c 2012 Academic Publications, Ltd.
url: www.acadpubl.eu
616
M.A. Tomescu Let ∆ respectively T be the sets defined by ∆ = (m, n) ∈ N2 , with m ≥ n
respectively T = (m, n, p) ∈ N3 , with m ≥ n ≥ p .
Definition 1. A mapping ϕ : ∆ × X → X is called a discrete evolution semiflow on X if the following conditions hold: s1 )ϕ (n, n, x) = x, for all (n, x) ∈ N × X; s2 )ϕ (m, n, ϕ (n, p, x)) = ϕ (m, p, x), for all (m, n, p, x) ∈ T × X. Example 1. Let f : R+ → R be a bounded function and for s ∈ R+ we denote fs (t) = f (t + s) for all t ∈ R+ . Then X = {fs , s ∈ R+ } is a metric space with the metric d (x1 , x2 ) = sup |x1 (t) − x2 (t)| . t∈R+
The mapping ϕ : ∆ × X → X defined by ϕ (m, n, x) = xm−n is a discrete evolution semiflow. Given a sequence (Am )m∈N with Am : X → B (V ) and a discrete evolution semiflow ϕ : ∆ × X → X, we consider the problem of existence of a sequence (vm )m∈N with vm : N × X → X such that vm+1 (n, x) = Am (ϕ(m, n, x))vm (n, x) for all (m, n, x) ∈ ∆ × X. We shall denote this problem with (A, ϕ) and we say that (A, ϕ) is a variational (nonautonomous) discrete-time system. For (m, n) ∈ ∆ we define the application Φnm : X → B (V ) by Am−1 (ϕ (m − 1, n, x)) . . . n Φm (x)v = . . . An+1 (ϕ (n + 1, n, x)) An (x)v, if m > n v, if m = n.
Remark 1. From the definitions of vm and Φnm it follows that: c1 ) Φm m (x)v = v, for all (m, x, v) ∈ N × X × V ; c2 ) Φpm (x) = Φnm (ϕ(n, p, x)) Φpn (x), for all (m, n, p, x) ∈ T × X; c3 ) vm (n, x) = Φnm (x)vn (n, x), for all (m, n, x) ∈ ∆ × X.
Definition 2. A mapping Φ : ∆ × X → B (V ) is called a discrete evolution cocycle over discrete evolution semiflow ϕ : ∆ × X → X if the following properties hold: c1 )Φ(n, n, x) = I (the identity operator on V), for all (n, x) ∈ N × X and
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c2 )Φ(m, p, x) = Φ(m, n, (ϕ(n, p, x))Φ(n, p, x), for all (m, n, p, x) ∈ T × X. If Φ is a discrete evolution cocycle over discrete evolution semiflow ϕ, then the pair S = (Φ, ϕ) is called a discrete skew-evolution semiflow on X. Remark 2. From Remark 1 it results that the mapping Φ : ∆ × X → B (V ) , Φ(m, n, x)v = Φnm (x)v is a discrete evolution cocycle over discrete evolution semiflow ϕ.
2. Uniform Exponential Instability Let (A, ϕ) be a discrete variational system associated to the discrete evolution semiflow ϕ : ∆ × X → X and to the sequence of mappings A = (Am ), where Am : X → B (V ), for all m ∈ N. Definition 3. The system (A, ϕ) is said to be uniformly exponentially instable (and denote u.e.is.) if there are the constants N ≥ 1 and α > 0 such that: eα(m−n) kvk ≤ N kΦnm (x)vk for all (m, n, x, v) ∈ ∆ × X × V. Remark 3. It is easy to see that (A, ϕ) is uniformly exponentially instable if and only if there are N ≥ 1 and α > 0 with eα(m−n) kΦpn (x)vk ≤ N kΦpm (x)vk for all (m, n, p, x, v) ∈ T × X × V. Example 2. Let C = C (R+ , R) be the metric space of all continuous functions x : R+ → R, with the topology of uniform convergence on compact subsets of R. C is metrizable relative to the metric given in Example 1. Let f : R+ → (0, ∞) be a decreasing function with the property that there exists lim f (t) = α > 0. We denote by X the closure in C of the set {ft , t ∈ R+ }, t→∞
where ft (s) = f (t + s) for all s ∈ R+ . The mapping ϕ : ∆ × X → X defined by ϕ (m, n, x) = xm−n is a discrete evolution semiflow. Let us consider the Banach space V = R and let A : X → B (V ) defined by R1
A (x) v = e0 for all (x, v) ∈ X × V.
x(τ )dτ
v
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M.A. Tomescu Then, we obtain that Φnm (x)v =
m−n R
e 0 v,
x(τ )dτ
v,
if m > n if m = n.
According to the properties of function x, it follows that |Φnm (x)v| ≥ eα(m−n) |v| for all (m, n, x, v) ∈ ∆ × X × V , showing that (A, ϕ) is u.e.is. A characterization of the uniform exponential instability property is given by Lemma 4. The following assertions are equivalent: (i) the system (A, ϕ) is uniformly exponentially instable; (ii) there exist the constants N ≥ 1 and a > 1 such that: am−n kvk ≤ N kΦnm (x)vk for all (m, n, x, v) ∈ ∆ × X × V ; (iii) there exists a nondecreasing sequence of real numbers (an ) with an → ∞ such that: am−n kΦpn (x)vk ≤ kΦpm (x)vk for all (m, n, p, x, v) ∈ T × X × V. Proof. (i) ⇒ (ii) It is a simple verification for a = eα . n (ii) ⇒ (iii) It results for an = aN . (iii) ⇒ (i) If an → ∞ then there exists k ∈ N∗ with ak > 1. Then, for every (m, n) ∈ ∆ there exist p ∈ N and r ∈ [0, k) such that m = n + pk + r. From hypothesis and Remark 1 we obtain
n+pk kΦnm (x)vk = Φn+pk+r (ϕ(n + pk, n, x)) Φnn+pk (x)v ≥
≥ ar Φnn+pk (x)v ≥
n+(p−1)k (ϕ(n + (p − 1)k, n, x)) Φnn+(p−1)k (x)v ≥ ≥ a0 Φn+pk
≥ a0 ak Φnn+(p−1)k (x)v ≥ . . . ≥ a0 apk kvk = m−n−r k
= a0 a k
kvk ≥ a0 e−αk eα(m−n) kvk
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so N kΦnm (x)vk ≥ eα(m−n) kvk for all (m, n, x, v) ∈ ∆ × X × V , where N = 1 +
eαk a0
and α =
ln ak k .
A Datko-type theorem for uniform exponential instability of variational nonautonomous discrete-time equations is given by Theorem 5. For every system (A, ϕ) the following assertions are equivalent: (i) (A, ϕ) is uniformly exponentially instable; (ii) there exist d > 0 and D ≥ 1 such that: m X
ed(m−k) kΦnk (x)vk ≤ D kΦnm (x)vk
k=n
for all (m, n, x, v) ∈ ∆ × X × V ; (iii) there exists D ≥ 1 such that: m X
kΦnk (x)vk ≤ D kΦnm (x)vk
k=n
for all (m, n, x, v) ∈ ∆ × X × V. Proof. (i) ⇒ (ii) It is a simple verification for d ∈ (0, α) and D = 1−eNd−α , where N and α are given by Definition 3. (ii) ⇒ (iii) It is obvious. (iii) ⇒ (i) Let (m, n, p, y, u) ∈ T × X × V and k ∈ N such that n ≤ k ≤ m. From (iii) it results that there exists D ≥ 1 such that kuk ≤ D kΦnk (y)uk . If we take u = Φpn (x)v and y = ϕ(n, p, x), then we obtain (m − n +
1) kΦpn (x)vk
=
m X
kΦpn (x)vk ≤
k=n
≤D
m m X X
p
Φ (x)v = D kΦnk (ϕ(n, p, x))Φpn (x)vk ≤ k
k=n
≤D
2
k=n
kΦnm (ϕ(n, p, x))Φpn (x)vk
= D 2 kΦpm (x)vk
for all (m, n, p, x, v) ∈ T ×X ×V . By Lemma 4 it results that (A, ϕ) is u.e.is.
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M.A. Tomescu
A Barbashin-type theorem for uniform exponential instability of of variational nonautonomous discrete-time equations is given by Theorem 6. If there are b > 0 and B ≥ 1 such that: m X
k=n
eb(k−n) Φkm (ϕ(k, n, x))v ≤ B kΦnm (x)vk
for all (m, n, x, v) ∈ ∆×X ×V, then the system (A, ϕ) is uniformly exponentially instable. Proof. From hypothesis it results that eb(m−n) kvk ≤
m X
k=n
eb(k−n) Φkm (ϕ(k, n, x))v ≤ B kΦnm (x)vk
for all (m, n, x, v) ∈ ∆ × X × V , which prove that (A, ϕ) is u.e.is.
3. Nonuniform Exponential Instability Let (A, ϕ) be a discrete variational system associated to the discrete evolution semiflow ϕ : ∆ × X → X and to the sequence of mappings A = (Am ), where Am : X → B (V ), for all m ∈ N. Definition 7. The system (A, ϕ) is said to be (nonuniformly) exponentially instable (and denote e.is.) if there are three constants N ≥ 1, α > 0 and β ≥ 0 such that: eα(m−n) kvk ≤ N eβn kΦnm (x)vk for all (m, n, x, v) ∈ ∆ × X × V. Remark 4. Using the property (c2 ) from Remark 1 it is easy to see that (A, ϕ) is exponentially instable if and only if there are N ≥ 1, α > 0 and β ≥ 0 with eα(m−n) kΦpn (x)vk ≤ N eβn kΦpm (x)vk for all (m, n, p, x, v) ∈ T × X × V. Remark 5. It is obvious that u.e.is. ⇒ e.is. The following example shows that the converse implication is not valid.
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Example 3. Let (X, d) be the metric space, V the Banach space and ϕ the evolution semiflow given as in Example 2. We define the sequence of mapings Am : X → B (V ) by R1
u(m + 1) 0 Am (x)v = e u(m)
x(τ )dτ
v
for all (m, x, v) ∈ N × X × V , where the sequence u : N → R is given by u(m) = emπ(1−cos mπ) . We have, according to the definition of discrete evolution cocycle, m−n R u(m) x(τ )dτ 0 n v, if m > n e Φm (x)v = u(n) v, if m = n.
We observe that
|Φnm (x)v|
mπ(1−cos mπ)−nπ(1−cos nπ)
=e
e
m−n R 0
x(τ )dτ
|v| ≥
≥ e−2nπ eα(m−n) |v| for all (m, n, x, v) ∈ ∆ × X × V , which prove that (A, ϕ) is e.is. Let us suppose now that the system (A, ϕ) is u.e.is. Accordind to Remark 3, there exist N ≥ 1 and ν > 0 such that
mπ(1 − cos mπ) − nπ(1 − cos nπ) +
m−n Z
x(τ )dτ ≥ ν(m − n) − ln N
0
for all (m, n, x) ∈ ∆ × X. If we consider n = 2k + 1 and m = 2k + 2, k ∈ N we have that Z1 ln N − 4kπ − 2π + x(τ )dτ ≥ ν 0
which, for k → ∞, leads to a contradiction. This proves that (A, ϕ) is not u.e.is. A Datko-type theorem for nonuniform exponential instability of variational nonautonomous discrete-time equations is given by
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M.A. Tomescu
Theorem 8. The system (A, ϕ) is exponentially instable if and only if there are d > c ≥ 0 and D ≥ 1 such that: m X
ed(m−k) kΦnk (x)vk ≤ Decm kΦnm (x)vk
k=n
for all (m, n, x, v) ∈ ∆ × X × V. Proof. Necessity. If (A, ϕ) is e.is. then there are N ≥ 1, α > 0 and β ≥ 0 such that for d ∈ (β, α + β) we have that m X
d(m−k)
e
kΦnk (x)vk
≤N
k=n
m X
e(d−α)(m−k) eβk kΦnm (x)vk =
k=n
= N em(d−α) kΦnm (x)vk
m X
ek(α+β−d) ≤ Decm kΦnm (x)vk
k=n α+β−d
e for all (m, n, x, v) ∈ ∆ × X × V , where c = β and D = eNα+β−d . −1 Sufficiency. We observe that from hypothesis it results that
ed(m−n) kvk ≤ Decm kΦnm (x)vk = Decn ec(m−n) kΦnm (x)vk for all (m, n, x, v) ∈ ∆ × X × V . For α = d − c and β = c we have that (A, ϕ) is e.is. Another characterization of nonuniform exponential instability of variational nonautonomous discrete-time equations is given by Lemma 9. The system (A, ϕ) is exponentially instable if and only if there are b > c ≥ 0 and N ≥ 1 such that: eb(m−n) kvk ≤ N ecm kΦnm (x)vk for all (m, n, x, v) ∈ ∆ × X × V. Proof. Necessity. If (A, ϕ) is e.is. then there are N ≥ 1, α > 0 and β ≥ 0 such that: eb(m−n) kvk = e(α+β)(m−n) kvk ≤ ≤ N eβn eβ(m−n) kΦnm (x)vk = N eβm kΦnm (x)vk = N ecm kΦnm (x)vk for all (m, n, x, v) ∈ ∆ × X × V , where b = α + β > β = c. Sufficiency. From hypothesis it results that kvk ≤ N ecm e−b(m−n) kΦnm (x)vk =
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= N ecn e−(b−c)(m−n) kΦnm (x)vk for all (m, n, x, v) ∈ ∆ × X × V. Finally, we obtain that (A, ϕ) is e.is. A Barbashin-type theorem for nonuniform exponential instability of variational nonautonomous discrete-time equations is given by Theorem 10. If there are b > c ≥ 0 and B ≥ 1 such that: m X
k=n
eb(k−n) Φkm (ϕ(k, n, x))v ≤ Becm kΦnm (x)vk
for all (m, n, x, v) ∈ ∆ × X × V, then the system (A, ϕ) is exponentially instable. Proof. By hypothesis it follows that there exist B ≥ 1 and b > c ≥ 0 such that eb(m−n) kvk ≤ Becm kΦnm (x)vk for all (m, n, x) ∈ ∆ × X. By Lemma 9 it follows that (A, ϕ) is e.is.
References [1] E.A. Barbashin, Introduction in Stability Theory, Izd. Nauka, Moskva (1967). [2] L. Barreira, C. Valls, Stability of Nonautonomous Differential Equations, Lecture Notes in Math., Volume 1926 (Springer 2008). [3] L. Buliga, M. Megan, Functionals on normed sequence spaces and exponential instability of linear skew-product semiflows, Int. J. of Pure and Appl. Math., 30, No. 1 (2006), 133-142. [4] R. Datko, Uniform asymptotic stability of evolutionary processes in a Banach spaces, SIAM J. Math. Anal., 3 (1973), 428-455. [5] M. Megan, C. Stoica, Discrete asymptotic behaviors for skew-evolution semiflows on Banach spaces, Carpathian J. Math., 24 (2008), 348-355. [6] A.A. Minda, On (f,g)-instability of evolution operators in Banach spaces, Advances and Applications in Mathematical Sciences, 4, No. 1 (2010), 7582.
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[7] A.A. Minda, On (f,g)-stability of evolution operators in Banach spaces, Int. J. of Pure and Appl. Math., 57, No. 2 (2009), 259-264. [8] C. Stoica, M. Megan, Nonuniform behaviors for skew-evolution semiflows on Banach spaces, Operator Theory Live, Theta Ser. Adv. Math. (2010), 203-211. [9] C. Stoica, M. Megan, Uniform exponential instability of evolution operators in Banach spaces, An. Univ. Timisoara Ser. Mat.-Inform., XLIV, No. 2 (2006), 143-148.
