Ordinary and Partial Differential Equations by W. N Everitt PDF

By W. N Everitt

ISBN-10: 354011968X

ISBN-13: 9783540119685

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Exp(~· t~T > 0 such that 1' t-,. 4) Then Eq. 1) has a positive solution which tends to zero as t-+ oo. 2, we first prove the following lemma. 1. 2 holds. 5) 47 Oscillations of First Order Delay Equations has a continuous positive solution z E C[t1- r, oo) with lim z( t) = 0. : t1. r :::; t < T. : t1 - r} n. 7) t1- T:::; t < T. Clearly, Tfl c n. Define a sequence of functions as follows: zo=z, Zn=Tzn-b n=1,2, .... It is easy to see that . : t1- T. Hence J~~ zn(t) = x(t) exists and x(t) is continuous and nonnegative for t ;::: t1- T.

Em - e)p(t), hence . :: exp(m expm). o), ... n = exp(mAn-1), .... For a sequence {en} with en > 0 and en --+ 0 as n --+ oo, there exists a. 8) holds. If m = 1, then lim An= e; and e e n-+oo if m < ~. then An tends to the smaller root of Eq. 10). 1. 1) has no eventually positive solutions if m > ~ . 3. 1). Set . 14) - (( )) . X T Then 2 A(m) := _1_-_m_-_v __ 1_-_2_m_-_m_ 2 < r < 1. 15) Proof: Assume that x(t) > 0 fort > T1 ;::: t 0 , and there exists a sequence {Tn} such that T1 < Tz < T3 < ...

X(r(An)) !. >. ,. ]. )] and then x(r(An)) 1-8,. ) ' n = 1,2, ... which implies that e< 1- 8,. " ) , - 8,. ( M- c- u_. When c -+ 0, e< 8,. -+ cE (0 M) , . 1 - V1 - M , then we obtain ~ -(1+~)2 - (1 - V1 - M)(M- 1 + V1 - M) - M . We are now in a position to state the oscillation criteria for Eq. 3). 1. Assume m > ~. 2) has no eventually negative solutions, (iii) every solution of Eq. 3) is oscillatory. Proof: It is sufficient to prove (i), (ii) and (iii) follow from (i). 1). 1, we may assume that T is nondecreasing.

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Ordinary and Partial Differential Equations by W. N Everitt


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