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Comparison in Differential Equations

  1. When one compare two theorems, one must compare their hypotheses carefully.
    Example. The expression in [Hob, p.194, l.8] is slightly different from that in [Guo, p.253, l.7] because they represent integrals with different integrands: One is (t2-1)n tr [Hob, p.193, l.-2] and the other is (1-t2)n tr [Guo, p.253, l.4].


  2. If both methods involve mathematical induction, we determine which one is simpler by comparing their induction steps.
    Example. The formula given in [Wat1, p.358, l.-3] is derived from [Wat1, p.358, l.-10] by differentiating under the sign of integration [Wat1, p.358, l.-5]. In contrast, Guo derives the same formula by interchanging the order of integration and summation [Guo, p.357, l.7]. As to determine which one is simpler, we should compare their induction steps instead of their ideas. The nth term involved in Guo's method can be found easily, while the nth term involved in Watson's method uses the formula given in [Wat1, p.90, l.-5-l.-4] and complicated computations.


  3. Within [Perr, p.240, Satz 12], it is difficult to determine whether case l= 0 or case l=1 is more refined. If we introduce a third case [Perr, p.239, Satz 10] for comparison, then it is clear that case l=1 is more stringent than case l= 0 [Perr, p.241, l.8-l.16].


  4. [Bir, p.147, Corollary] is a special case of [Har, p.4, Theorem 2.4]. The former assumes an extra condition that X(x,t) satisfies a Lipschitz condition. If in the former we assume x(a;e) = ce, where ce c as e 0, then its proof remains the same.


  5. Links {1, 2}.