Oscillation of Fractional Nonlinear Difference Equations 1 ... - hilaris

Mathematica Aeterna, Vol. 2, 2012, no. 9, 805 - 813

Oscillation of Fractional Nonlinear Difference Equations S.Lourdu Marian Rajiv Gandhi College of Engineering, Chennai - 105 M. Reni Sagayaraj, A.George Maria Selvam Sacred Heart College, Tirupattur - 635 601, S.India M.Paul Loganathan Department of Mathematics, Dravidian University, Kuppam Abstract The oscillation criteria for forced nonlinear fractional difference equation of the form ∆α x(t) + f1 (t, x(t + α)) =v(t) + f2 (t, x(t + α)), ∆

α−1

t ∈ N0 ,

0 < α ≤ 1,

x(t)|t=0 =x0 ,

where ∆α denotes the Riemann-Liouville like discrete fractional difference operator of order α is presented.

Mathematics Subject Classification: 26A33, 39A12 Keywords: difference equations, oscillation, nonlinear, fractional

1

Introduction

In this paper, we present the oscillatory behavior of forced nonlinear fractional difference equation of the form ∆α x(t) + f1 (t, x(t + α)) =v(t) + f2 (t, x(t + α)), ∆α−1 x(t)|t=0 =x0 ,

t ∈ N0 ,

0 < α ≤ 1,

(1)

where ∆α is a Riemann-Liouville like discrete fractional difference, fi : [0, +∞)× R → R, i = 1, 2 and v are continuous with respect to t and x, Na = {a, a + 1, a + 2, . . .}. Fractional differential equations have received considerable attention during recent years, because these equations are proved valuable tools for modeling

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L.Marian, R.Sagayaraj, A.G.M.Selvam and P.Loganathan

of many phenomena in various fields. Fractional calculus finds its significant applications in the fields of viscoelasticity, capacitor theory, electrical circuits, electro-analytical chemistry, neurology, diffusion, control theory and statistics see [14], [15] and [16]. A rigorous theory of fractional differential equations has been started quite recently, see books [12], [13], and [15]. However, very little progress has been made to develop the theory of fractional difference equations see [7, 8, 9, 11]. In particular, nothing is known regarding the oscillatory behavior of (1) up to now. The study of Oscillation of fractional differential equations is initiated in [5, 17]. Motivated by [17] , we study the oscillation of fractional difference equations (1).

2

Definitions and Basic Lemmas

In this section, we introduce preliminary results of discrete fractional calculus. Definition 2.1 (see [2]) Let ν > 0. The ν-th fractional sum f is defined by t−ν

1 X (t − s − 1)(ν−1) f (s), ∆ f (t) = Γ(ν) s=0 −ν

where f is defined for s = a mod (1) and ∆−ν f is defined for t = (a + ν) Γ(t+1) mod (1), and t(ν) = Γ(t−ν+1) . The fractional sum ∆−ν f maps functions defined on Na to functions defined on Na+v . Definition 2.2 (see [2]) Let µ > 0 and m − 1 < µ < m, where m denotes a positive integer, m = ⌈µ⌉. Set ν = m − µ. The µ-th fractional difference is defined as ∆µ f (t) = ∆m−ν f (t) = ∆m ∆−ν f (t). Theorem 2.3 (see [3]) Let f be a real-value function defined on Na and µ, ν > 0, then the following equalities hold: ∆−ν [∆−µ f (t)] = ∆−(µ+ν) f (t) = ∆−µ [∆−ν f (t)]; ∆−ν ∆f (t) = ∆∆−ν f (t) −

(t − a)(ν−1) f (a). Γ(ν)

Lemma 2.4 (see [2]) Let µ 6= 1 and assume µ + ν + 1 is not a positive integer, then Γ(µ + 1) (µ+ν) ∆−ν t(µ) = t . Γ(µ + ν + 1)

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Oscillation of Fractional Nonlinear Difference Equations

In order to discuss our results in Section 3, Now we state the following lemma [10] . Lemma 2.5 For X ≥ 0 and Y > 0, we have X λ + (λ − 1)Y λ − λXY λ−1 ≥ 0,

λ>1

(2)

X λ − (1 − λ)Y λ − λXY λ−1 ≤ 0,

λ < 1,

(3)

and where equality holds if and only if X = Y . Lemma 2.6 (see [2]) The equivalent fractional Taylor’s difference formula of (1) is t−α

x(t) =

1 X x0 (α−1) t + (t−s−1)(α−1) [v(s)+f2 (s, x(s+α))−f1 (s, x(s+α))], t ∈ Nα . Γ(α) Γ(α) s=0 (4)

Proof: Apply the ∆−α operator to each side of (1), we obtain ∆−α ∆α x(t) = ∆−α [v(t) + f2 (t, x(t + α)) − f1 (t, x(t + α))]

(5)

Apply Theorem (2.3) to the left-hand side of (5), ∆−α ∆α x(t) =∆−α ∆∆−(1−α) x(t) =∆∆−α ∆−(1−α) x(t) − =x(t) −

t(α−1) x0 Γ(α)

x0 (α−1) t . Γ(α)

Now, we apply Definition (2.1) to the right of (5) for t ∈ Na , which yields (4) . This completes the proof.

3

Main Results

We consider the following conditions: xfi (t, x) > 0 (i = 1, 2),

x 6= 0,

t ≥ t0

(6)

and |f1 (t, x)| ≥ |p1 (t)| |x|β and |f2 (t, x)| ≤ |p2 (t)| |x|γ ,

x 6= 0,

where p1 , p2 ∈ C([t0 , ∞), R+ ) and β, γ > 0 are real numbers. Now we prove first theorem when f2 = 0.

t ≥ t0 ,

(7)

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L.Marian, R.Sagayaraj, A.G.M.Selvam and P.Loganathan

Theorem 3.1 Suppose that condition (6) hold. If (1−α)

lim inf t

t→∞

t−α X

(t − s − 1)(α−1) v(s) = −∞,

(8)

s=0

and

t−α X

(1−α)

lim sup t

t→∞

(t − s − 1)(α−1) v(s) = ∞,

(9)

s=0

then every solution of equation (1) is oscillatory. Proof: Let x(t) be a nonoscillatory solution of equation (1) with f2 = 0. Suppose that T > t0 is large enough that x(t) > 0 for t ≥ T . Let F (t) = v(t) + f2 (t, x(t + α)) − f1 (t, x(t + α)), then we see from (4) that T −1 t−α 1 X 1 X x0 (α−1) (α−1) (t−s−1) |F (s)|+ (t−s−1)(α−1) v(s), t + x(t) ≤ Γ(α) Γ(α) s=0 Γ(α) s=T (1−α)

Γ(α)t

(1−α)

x(t) ≤ x0 +t

T −1 X

(t−s−1)

(α−1)

(1−α)

|F (s)|+t

s=0

t−α X

(t−s−1)(α−1) v(s),

s=T

and hence (1−α)

Γ(α)t

(1−α)

x(t) ≤ C(T ) + t

t−α X

(t − s − 1)(α−1) v(s),

t ≥ T,

(10)

s=T

where C(T ) = x0 +

T −1  X s=0

T T −s−1

(1−α)

|F (s)|

and lim C(t) = M < ∞,

t→∞

t ≥ T.

Taking the limit inferior of both sides of inequality (10) as t → ∞, we get a contradiction to condition (8). This completes the proof of the theorem. Next we have the following results. Theorem 3.2 Suppose that conditions (6) and (7) hold with β > 1 and γ = 1. If lim inf t(1−α)

t→∞

t−α X

(11)

s=0

and lim sup t(1−α)

t→∞

(t − s − 1)(α−1) [v(s) + Hβ (s)] = −∞,

t−α X s=0

(t − s − 1)(α−1) [v(s) + Hβ (s)] = ∞,

(12)

t ≥ T,

t ≥ T,

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Oscillation of Fractional Nonlinear Difference Equations

where

1/(1−β)

Hβ (s) = (β − 1)β β/(1−β) p1

β/(β−1)

(s)p2

(s),

then every solution of equation (1) is oscillatory. Proof: Let x(t) be a nonoscillatory solution of equation (4), say, x(t) > 0 for t ≥ T > t0 . Using condition (7) in equation (4) with γ = 1 and β > 1 and t ≥ T , we obtain Γ(α)t(1−α) x(t) =x0 + t(1−α) =x0 + t(1−α)

t−α X

s=0 T −1 X

(t − s − 1)(α−1) [v(s) + f2 (s, x(s + α)) − f1 (s, x(s + α))] (t − s − 1)(α−1) [v(s) + f2 (s, x(s + α)) − f1 (s, x(s + α))]

s=0

+t(1−α)

t−α X

(t − s − 1)(α−1) [v(s) + f2 (s, x(s + α)) − f1 (s, x(s + α))]

s=T

(1−α)

≤x0 + t

T −1 X

(t − s − 1)(α−1) |F (s)|

s=0

+t(1−α)

t−α X

(t − s − 1)(α−1) [v(s) + f2 (s, x(s + α)) − f1 (s, x(s + α))]

s=T

(1−α)

≤C(T ) + t

t−α X

(t − s − 1)

(α−1)

s=T

+f2 (s, x(s + α)) − f1 (s, x(s + α)) (1−α)

Γ(α)t

(1−α)

x(t) ≤ C(T ) + t

t−α X

h v(s)

i

(t − s − 1)

(α−1)

s=T

i +p2 (s)x(s + α) − p1 (s)xβ (s + α) ,

h v(s)

(13)

t ≥ T.

We apply (2) in Lemma (2.5) with λ = β, 

1/β

p1 (t)x(t + α)

 1/(β−1) 1/β −1/β X = p1 x and Y = p2 p1 /β

β

 β/(β−1) −1/β + (β − 1) p2 (t)p1 (t)/β   1/β −1/β −βp1 (t)x(t + α) p2 (t)p1 (t)/β ≥ 0 1/(1−β)

p1 (t)xβ (t + α)+(β − 1)β β/(1−β) p1

β/(β−1)

(t)p2

(t) − p2 (t)x(t + α) ≥ 0

1/(1−β)

p2 (t)x(t + α) − p1 (t)xβ (t + α) ≤ (β − 1)β β/(1−β) p1

β/(β−1)

(t)p2

(t),

t ≥ T. (14)

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L.Marian, R.Sagayaraj, A.G.M.Selvam and P.Loganathan

Using (14) in (13) , we obtain (1−α)

Γ(α)t

(1−α)

x(t) ≤C(T ) + t

t−α X

(t − s − 1)(α−1) [v(s) + Hβ (s)],

t ≥ T.

s=T

(15) Taking the limit inferior of both sides of inequality (15) as t → ∞, we get a contradiction to condition (11). This completes the proof of the theorem. Theorem 3.3 Suppose that conditions (6) and (7) hold with β = 1 and γ < 1. If lim inf t(1−α)

t→∞

t−α X

(t − s − 1)(α−1) [v(s) + Hγ (s)] = −∞,

(16)

s=0

and lim sup t(1−α)

t→∞

t−α X

(t − s − 1)(α−1) [v(s) + Hγ (s)] = ∞,

(17)

s=0

where

γ/(γ−1)

Hγ (s) = (1 − γ)γ γ/(γ−1) p1

1/(1−γ)

(s)p2

(s),

then every solution of equation (1) is oscillatory. Proof: Let x(t) be a nonoscillatory solution of equation (4), say, x(t) > 0 for t ≥ T > t0 . Using condition (7) in equation (4) with β = 1 and γ < 1, we get Γ(α)t(1−α) x(t) ≤C(T ) + t(1−α)

t−α X

(t − s − 1)(α−1) [v(s) + p2 (s)xγ (s + α) − p1 (s)x(s + α)].

s=T

(18) Now we use (3) in Lemma (2.5) with λ = γ,

1/γ p2 x

X=

 1/(γ−1) −1/γ and Y = p1 p2 /γ

γ   γ/(γ−1) 1/γ −1/γ p2 (t)x(t + α) −(1 − γ) p1 (t)p2 (t)/γ 1/γ

−1/γ

−γp2 (t)x(t + α)(p1 (t)p2

γ/(γ−1)

p2 (t)xγ (t + α) − p1 (t)x(t + α) ≤ (1 − γ)γ γ/(1−γ) p1

(t)/γ) ≤ 0 1/(1−γ)

(t)p2

(t) t ≥ T. (19)

Using (19) in (18), we obtain (1−α)

Γ(α)t

(1−α)

x(t) ≤C(T ) + t

t−α X s=T

(t − s − 1)(α−1) [v(s) + Hγ (s)], t ≥ T. (20)

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Oscillation of Fractional Nonlinear Difference Equations

Taking the limit inferior of both sides of inequality (20) as t → ∞, we get a contradiction to condition (16). This completes the proof of the theorem. Finally we present the following more general result. Theorem 3.4 Suppose that conditions (6) and (7) hold with β > 1 and γ < 1. If (1−α)

lim inf t

t→∞

t−α X

(t − s − 1)(α−1) [v(s) + Hβ,γ (s)] = −∞,

(21)

s=0

and

t−α X

(1−α)

lim sup t

t→∞

(t − s − 1)(α−1) [v(s) + Hβ,γ (s)] = ∞,

(22)

s=0

where 1/(1−β)

Hβ,γ (s) = (β−1)β β/(1−β) ξ β/(β−1) (s)p1

1/(1−γ)

(s)+(1−γ)γ γ/(1−γ) ξ γ/(γ−1) (s)p2

(s)

with ξ ∈ C([t0 , ∞), R+ ), then every solution of equation (1) is oscillatory. Proof: Let x(t) be a nonoscillatory solution of equation (1), say, x(t) > 0 for t ≥ T > t0 . Using condition (7) in equation (4), we can write Γ(α)t(1−α) x(t) ≤C(T ) + t(1−α)

t−α X

(t − s − 1)(α−1) v(s)

s=T

+t(1−α) +t(1−α)

t−α X

  (t − s − 1)(α−1) ξ(s)x(s + α) − p1 (s)xβ (s + α)

s=T t−α X

(t − s − 1)(α−1) [p2 (s)xγ (s + α) − ξ(s)x(s + α)] , t ≥ T.

s=T

(23) We may bound the terms (ξx − p1 xβ ) and (p2 xγ − ξx) by using the inequalities (14) with p2 = ξ, and (19) with p1 = ξ respectively, we get Γ(α)t(1−α) x(t) ≤C(T ) + t(1−α)

t−α X

(t − s − 1)(α−1) v(s)

s=T

t−α hh i X 1/(1−β) (s) +t(1−α) (t − s − 1)(α−1) (β − 1)β β/(1−β) ξ β/(β−1) (s)p1 s=T h ii 1/(1−γ) γ/(1−γ) γ/(γ−1) + (1 − γ)γ ξ (s)p2 (s)

1−α

Γ(α)t

(1−α)

x(t) ≤ C(T ) + t

t−α X s=T

(t − s − 1)(α−1) [v(s) + Hβ,γ (s)] , t ≥ T. (24)

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L.Marian, R.Sagayaraj, A.G.M.Selvam and P.Loganathan

Taking the limit inferior of both sides of inequality (24) as t → ∞, we get a contradiction to condition (21). This completes the proof of the theorem. ACKNOWLEDGEMENTS.

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Oscillation of Fractional Nonlinear Difference Equations

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