# A Generalization of Rolle's Theorem with Application to by Dieudonne J.

By Dieudonne J.

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Additional info for A Generalization of Rolle's Theorem with Application to Entire Functions

Example text

174) with pi = pi(x ), rj = 7j(x), and pi = pix(x)(pt. 158), where the initial value of the derivative cannot be specified arbitrarily. 158) are constant, then (p and

141) must be also completed with Cauchy conditions for t = 0, x e K 1 and for t = 0 , x < 0. 141) and the vector £ is transversal to this curve. 139) a well-posed problem. The subsequent derivations follow the scheme set out above. 141) in the domain {(#,£), x < 0} completed with conditions avf = <7j(t). = *»(»). 130). 5. Other examples In this section we consider two equations important for physical applications, namely, the sine-Gordon equation and the Benjamin-Bona-Mahogany equation (the regularized long waves equation).

2. 69). 77) can be easily found explicitly by the formula Vi (T, t) = 2 = i) € S. 81) where V{~ is a smooth function satisfying the condition V{~((p}t) = v\ (t). Set u f (x,t) = ui(x 9t) for x > 0, x € ilf. 77). 82) hold. By setting ui(x,t) = uf(x, i), V^(x,t) = u i ( x yt) —uf(xit)> 32 1.