Past exam of the methods of applied functional analysis course of the University of Oxford / 2009 / B / 5

Brouwer's fixed-point theorem says that every continuous self-map of a nonempty closed Euclidean ball has a fixed point. It does not promise convergence of repeated application. For example $f(x)=1-x$ with starting point $x_1=0$ gives the nonconvergent alternating sequence $0,1,0,1,\dots$, although $1/2$ is a fixed point.

For an invariant ball from sublinear growth, chooseThen $A{R}^{\alpha }\le R/2$ and $B\le R/2$. For every $|x|\le R$ the given estimate yields $|f(x)|\le A{R}^{\alpha }+B\le R$. Thus $f$ restricts to a continuous self-map of the closed ball of this radius, and Brouwer gives a fixed point. The exponent $\alpha <1$ ensures that the linear radius eventually dominates its sublinear growth; no differentiability or contraction estimate is needed.

One form of Schauder's theorem says that if $M$ is a nonempty closed bounded convex subset of a Banach space and $T:M\to M$ is continuous with relatively compact image, then $T$ has a fixed point.

For a Schauder solution of a small Hammerstein integral equation, take $X=C([0,1])$ with the supremum norm and $M=\{u:\Vert u{\Vert }_{\infty }\le R\}$. This set is nonempty, closed, bounded and convex. PutBoth maxima are finite by continuity on compact sets. Define $(T_{\mu }u)(x)=\mu {\int }_0^{1}G(x,y)f(y,u(y))dy$. Continuity and uniform continuity of the kernel give $T_{\mu }u\in X$, and for $0<\mu \le {\mu }_0$,so it is a self-map. This choice of ${\mu }_0$ also handles the case when either maximum is zero without division by zero.

If $u_n\to u$ uniformly in $M$, uniform continuity of $f$ on $[0,1]\times [-R,R]$ makes $f(\cdot ,u_n(\cdot ))\to f(\cdot ,u(\cdot ))$ uniformly. The integral estimate then proves continuity of $T_{\mu }$ in the supremum norm. Moreoverwhich tends to zero with $|x-x^{\prime }|$ uniformly in $u\in M$. Thus the image is uniformly bounded and equicontinuous. Arzelà-Ascoli states that such a family on a compact interval has compact closure in the supremum norm. Hence $T_{\mu }(M)$ is relatively compact, and Schauder gives a fixed point for each $\mu$ in the specified interval. Therefore a continuous solution with supremum norm at most $R$ exists. The argument imposes no Lipschitz hypothesis on $f$ and makes no uniqueness assertion.