By Odell E.
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Proof. 149) 0 t t1 1/2 + · · · + et ηm 0 tm−1 ··· 0 1/2 2 Fm (s)e−2s ω 2 (u(s))ds · · · dt1 , 0 where ηi (i = 1, 2, . . 148). Here we have used the following estimate t 0 t1 ti−1 ··· 0 (ti−1 − s)2βi −1 e2s ds · · · dt1 0 t t1 = 0 0 2t ti−2 ··· ti−1 e2ti−1 0 σ 2βi −1 e−2σ dσ · · · dt1 0 t t1 ti−2 e ≤ 2βi Γ(2βi − 1) ··· e2ti−2 dti−1 · · · dt1 2 0 0 0 e2t Γ(2βi − 1) ≤ , i = 1, 2, . . , m. 151) 0 t t1 + · · · + e2t ηm 0 0 tm−1 ··· 0 2 Fm (s)e−2s ω 2 (u(s))ds · · · dt1 50 Chapter 1. 152) 0 t t1 + · · · + ηm 0 where tm−1 ··· 0 Fm (s)R(s)ω(u(s))ds · · · dt1 , 0 v(t) = (e−t u(t))2 .
159) Proof. 156), we obtain t 0 H(t) H (s) ds = + ω(V (s)) ω(V (t)) t H(s) 0 ω (V (s)) H(t) . V (s)ds ≥ [ω(V (s))]2 ω(V (t)) Now let us continue the proof of the theorem. 156), we have t t Vm−1 (s) Vm−1 (t) ≤ ds ≤ hm (s)ds. 160), it follows Vm−2 (t) ≤ ω(V (t)) t 0 Vm−2 (s) ds ≤ ω(V (t)) t ≤ t t hm−1 (s)ds + 0 t 0 t1 hm−1 (s)ds + 0 hm (s)dsdt1 . 5. 162) 0 t t1 +··· + 0 tm−1 ··· 0 hm (s)dsdtm−1 · · · dt1 . 14. 159). 6. 6. We omit the details 1 for a real number z ≥ 1 is more complicated and we also here.
160), it follows Vm−2 (t) ≤ ω(V (t)) t 0 Vm−2 (s) ds ≤ ω(V (t)) t ≤ t t hm−1 (s)ds + 0 t 0 t1 hm−1 (s)ds + 0 hm (s)dsdt1 . 5. 162) 0 t t1 +··· + 0 tm−1 ··· 0 hm (s)dsdtm−1 · · · dt1 . 14. 159). 6. 6. We omit the details 1 for a real number z ≥ 1 is more complicated and we also here. The case βi > z+1 omit it here. 5 Integral inequalities leading to upper bounds and decay rates In this section, we shall introduce some integral inequalities leading to upper bounds and decay rates. Bae and Jin [57] proved the following theorem.