By Ravi P. Agarwal
Agarwal (mathematics, Florida Institute of Technology), Bohner (mathematics, U. of Missouri-Rolla) and Li (mathematics, Lanzhou U.) research the qualitative conception of differential equations without or with delays. After an introductory bankruptcy, the authors specialise in first order hold up and impartial differential equations, moment order usual and hold up differential equations, better order hold up differential equations, structures of nonlinear differential equations, and oscillation of dynamic equations on time scales. even supposing meant for graduate scholars and researchers in arithmetic, physics engineering, and biology, mathematicians operating in complicated time scale conception also will locate this an invaluable reference.
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19) holds. 27) is oscillatory. 28) x (t) + x (t − τ (t)) = 0, where τ ∈ C([t0 , ∞), R+ ) and limt→∞ (t − τ (t)) = ∞. Set T0 = inf t≥t0 {t − τ (t)}. 10. 28) on (t0 , ∞) and x(t) > 0 for all t ∈ [T0 , ∞). 2. 11. 29) where Λ0 (t) = t T0 t ≥ t0 , for all λ0 (s)ds, and Λ−1 0 denotes the inverse function of Λ0 . Proof. We first prove necessity. 28) with respect to t0 . Then x0 (t) > 0 for all t ∈ [T0 , ∞). Set λ0 (t) = x0 (t − τ (t)) x0 (t) t ≥ T0 . for all t Clearly, λ0 (t) > 0 for all t ≥ t0 and hence Λ0 (t) = T0 λ0 (s)ds defines a function Λ0 on [T0 , ∞), which is strictly increasing on [t0 , ∞).
Now we give the proof of (iv). For i = 0 and i = 1 it is immediate. For i ∈ N the proof can be done induction, so we have ri+1 = eri /e > e1−2/(i+2) , and it is sufficient to show e1−2/(i+2) > e − 2e i+3 or f (i + 2) > 1, where f (x) = e−2/x + 2 . x+1 Since f (x) = 2 x2 e−1/x + x x+1 e−1/x − x x+1 and x+1 1 = , x x we have f (x) < 0 and f (i + 2) > limx→∞ f (x) = 1, which was to be shown. The proof is complete. 6. Assume p ∈ Aλ for some λ ∈ (0, 1]. 3) oscillates. Proof. Suppose the contrary. Then we may assume that, without loss of generality, there exists a solution x such that x(t) > 0 for t ≥ tk−3 for some k ≥ 3.
It is clear that 1 ≤ x0 ≤ 2 and 1 ≤ x1 = 1 1 = γ ≤ 2. 140), we have 1 ≤ x2 = 1 1 = 2. ≤ 1 − a2 x1 1 − 14 · 2 By induction, we have 1 ≤ xn ≤ 2 for n ∈ N. 143) xn = 1 , 1 − Qn Pn−1 xn−1 n ∈ N. Next we define A−1 = 1 and An = (Pn xn )−1 An−1 , n ∈ N0 . Clearly, An > 0 for n ∈ N (−1). , An−1 = Pn An + Qn An−2 , n ∈ N. 125). 118) is ocillatory. 50 2. 8. 144) k lim sup ⎝Qn Pn−1 + n→∞ i ⎞ Qn−j−1 Pn−j−2 ⎠ > 1. 118) oscillates. Proof. 118) has an eventually positive solution x. 125). So An−1 ≥ Pn An . 145) Pn−j i∈N for all j=0 holds.