By A. Bahri

(This examine be aware monitors new phenomena in selected variational difficulties of nonlinear research and geometry. the writer develops extra his rules which first seemed in a prior quantity within the sequence Pseudo-Orbits of touch shape. The linked variational challenge, that's identified to endure beneficial properties of noncompactness is studied from a dynamical platforms perspective. Professor Bahri describes flow-lines of the gradient move and illustrates that the Palais-Smale might be chuffed alongside flow-lines whereas it fails for the variational difficulties. a brand new inspiration is usually brought, the concept that of severe issues at infinity. He then proceeds to turn out lifestyles theorems for the equations that have been thought of. this article can be of curiosity to specialist mathematicians and study scholars in utilized and natural mathematics).

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Hence, as an —► 0, we find, given any e > 0, a number rio(e) G N, such that I ^2 apbp\ < e for m> n> n0(e). n+l I D Corollary, (the alternating series test) Let (an) be a monotone decreasing CO sequence of positive numbers and let lim an = 0. TAen £/ie series ^2(—l)n_1an l is convergent. The proof is obvious. 4 Summability CO Consider a divergent series: Ylan- Therefore, the sequence of finite sums: (sn) I where sn = a\ 4- a2 4- • • • an is not convergent. It can however happen that the sequence of arithmetic means: (tn) where 1, tn = —(«1 + S 2 H Sn) n is a convergent sequence.

The root test). Suppose an > 0 for all n. If lim sup n v/a^ CO CO l l < 1, then the series 53 an converges. If lim sup nyfa~^ > 1 £/ien t/ie series 53 a n diverges. Proof. If lim sup 7\fan~ = q < 1 we take q', q < q' < 1 and we find no € N such that y ^ < q' for n > no- Hence an < {q')n for n > no, and therefore CO 53 &n is convergent. l If lim sup ny/a^ = r > 1 we take r', 1 < r1 < r. 4 (II) (2). Take ni > 1, such that n\fa~^[ > r'. Then, take n 2 > ni, such that n \fa^ > r', continuing this way we find a subsequence (a,nk)kLi of (a n ) such 54 Advanced Calculus that aUk > r'nk.

There are four cases, according as A is bounded above (or not) and bounded below (or not). 1. A bounded below but not bounded above: Let a — inf A. We will show that A = (a, +oo) or A — [a,+oo). First, notice that A C [a, oo). Let then r > a. There exists x G A, a < x < r. Also, as A is not bounded above there exists y £ A, y > r. Hence x < r < y where x, y G A. By convexity => r G A. Hence, (a, + oo) C A C [a, oo). 2. A bounded above but not below; a similar reasoning gives that, with b = sup A (-00,6) C A C (-00,6].