 By E. M. Friedlander, M. R. Stein

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Example text

For instance, the general solution to 37 the = (')z y is y(t) = -InC - t + C) + D and there are no values of C and D for which this function will satisfy the Dirichlet boundary condition. Thus, in the case of f (r, p , q) = qZ , the function S = L -I F j , for F the corresponding superposition operator, does not have a fixed point. 4) are satisfied. We can find a closed bounded and convex subset C of C5[0, 1] such that S maps C to C. Since C is bounded and we have just demonstrated that S is completely continuous, the restriction of S to C is a compact map and therefore it has a fixed point.

A wire of length e is attached at the point A and an object is placed at the other end of the wire. We'll think of the object as a ball in order to have something specific in mind. But we'll concentrate all the weight of the ball at one point so that we can describe the motion of the ball as the motion of the point we have called P . The ball can swing back and forth in the x, y-plane of the figure, in which the origin has been placed at the rest position of the ball. That's all there is to a simple pendulum.

The continuous functions k(u) and k' (u) are bounded on the closed interval [- M, M] and, in particular, k(u) > is bounded away from zero there. Furthermore, q(s, u) is continuous and therefore bounded for u in the same interval , since 0 ~ s ~ I. What we have found out is that there exist positive numbers A and B such that ° I I k'(U) -