By Jean-Pierre Aubin

ISBN-10: 0471059986

ISBN-13: 9780471059981

In order to stay at an introductory point, this quantity refrains from delving into technical problems and complicated effects now not in present use. purposes are defined once attainable, and theoretical elements are aimed at functional use. subject matters diversity from very delicate capabilities to nonsmooth ones, from convex variational difficulties to nonconvex ones, and from economics to mechanics. historical past notes, reviews, bibliography, and indexes complement the text.

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**Example text**

DEFINITION 1 Let K be a nonempty subset of a Hausdorff locally convex space X. The function Ok ’. =^a{K, /7):=sup {p, x) xeK is called the support function of K. Its domain (2) b(A):= {p e X*\

4 DIFFERENTIABLE FUNCTIONALS 17 henceforth denoted by i/(x) or V U(x), to show dependence on the functional U and the point x, and called the gradient, or the derivative, of U at x. We give a fourth definition The Property. U is Gateaux differentiable on a neighborhood of x in X, and U'{y)-* V{x) in X* when_v-»x in X. (d) LEMMA 1 (d) ^ (c) =>(b) =>(a), with the same p. ^ Proof. The last two, (c) =>(b) and (b) ^ (a), are obvious. As for the first one, assume (d)is satisfied, choose e> 0 so small that U is Gateaux differentiable on the ball with center x and radius e, and pick any y and v with Hj —x||<6/3 and l|a||

LEMMA 4 Take any two points y andy within distance q off{xo). Let x and x solve equations f(x )= y andf(x)= y in the ball Xq + vB. We then have (17) Proof. Set (18) 9(x)= x+ f'{xo)-^{y-f(x)) (19) 0 (^ )= ^ + /'(x o )'‘(y-/(;c)) We have g(x)=x and g{x)=x. x)|| < \ \ g i x ) - g i x ) \ \ + \ \ m - 9(x)\\ Using equations (18) and (19), we get (21) llgf(-x)-0(i)||<||/'(xo)“ ‘|| lly-yll 38 CH. 2 , SEC. x|| Substituting this into inequality (20) and remembering that 0

### Applied Nonlinear Analysis by Jean-Pierre Aubin

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