Regarding their proof, I deem the Banach fixed point theorem to be more analytical while Leray-Schauder more topological in nature. Owing to this, I am more inclined to use Banach method first, but Gilbarg/Trudinger or Ladyzhenskaya devote half of their books to the apriori estimate. Thus it appears to me that Leray-Schauder is more popular.
Is there any "work experience" or intuition that motivates the choice of Leray-Schauder over the other?
Or to put it more specific, How bad can the Banach method fail?
Thank you
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$\begingroup$ Welcome to MO. My personal impression is that the LS theorem is popular because its usual formulation allows you to work with fixed points, so, in the PDE context, solutions to the problem at hand. This allows to use all sorts of trickery with (weak) formulations, clever testing, and the like, whose principles are very well established but still quite an art. (Usually this is summarized under a priori estimates, see also this nice explanation in a related question.) $\endgroup$– HannesCommented Feb 9, 2022 at 9:38
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$\begingroup$ @Hannes: does the Banach fixed point theorem perhaps also allow you to work with fixed points? $\endgroup$– Ben McKayCommented Feb 9, 2022 at 11:21
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$\begingroup$ Why use a screwdriver when a hammer is a much more intuitive tool? There are simply too many problems that cannot be written as a contraction and are thus inaccessible to Banach. On the other hand, there are enough problems where you are interested in uniqueness, so Leray-Schauder is no help. In some sense PDE-theory is all about having the right tool for the right job and knowing when to use it. Also, judging the importance of a tool by the amount of content in a book is never a good idea. You could equally argue that if Banach is easier to use, then it should take less pages to explain. $\endgroup$– mlkCommented Feb 9, 2022 at 13:14
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1 Answer
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Premises. Let's precise the framework (without pretending it to be the most general one on which those concepts apply) in which we are working: let $B$ be a real or complex Banach space, and $T: B\to B$ a continuous mapping defined on it. The Banach ((see for example [1], §1.1, p. 10) and the Leray-Schauder fixed point theorem (see for example [1], §6.5, p. 123 or [3], §4.1, pp. 43-44 ) state conditions under which the following functional equation to has a solution $$ z = T(z)\qquad z\in B.\label{1}\tag{FPE} $$ Precisely, the Banach fixed point theorem applies to mappings $T$ such that $$ \Vert T(z)-T(\zeta)\Vert_B \le \rho \Vert z- \zeta\Vert_B \quad \forall z, \zeta\in B,\; 0<\rho<1 \label{2}\tag{BFP} $$ while the Leray-Schauder fixed point theorem requires that $T:\overline{U}\to C$, where $U\subset C$ is relatively open and $C\subset B$ is convex, is a compact (i.e. completely continuous) operator such that: $$ u\neq (1-\lambda) u_0 + \lambda T(u) \quad \forall u_0\in U,\ \forall u\in\partial{U}\label{3}\tag{LSFP} $$ A mapping that fulfil condition \eqref{2} is called contraction mapping: for this reason, the Banach fixed point theorem is also called the contraction principle.
The answer. The main advantage of \eqref{3} over \eqref{2} has been already pointed out in mlk's comment: contractions are globally Lipschtz continuous maps with Lipschitz constant strictly less than unity, while there are maps satisfying \eqref{3}, for which $\rho$ is unbounded. This implies that the range of nonlinear operators to which the Banach contraction principle is applicabile limited is severely limited. This can be seen already when $B$ is finite dimensional: for example, (see [2], §6.12, p. 524), the holomorphic (sic!) function $$ \Bbb C\ni z\mapsto T(z)=\frac{1}{2}\sqrt{z+1} $$ (the chosen branch for the square root is the principal one) as at least a fixed point in $\Bbb B =\{z\in\Bbb C : |z|\le 1\}$ i.e. satisfies \eqref{1}, despite not satisfying \eqref{2}: on the other hand, it is easily seen that this $T$ satisfies \eqref{3}. Therefore, in the wilderness of nonlinear functionals and operators, the Leray-Schauder fixed point theorem is a more capable tool respect to the Banach contraction principle.
Bibliography
[1] Andrzej Granas, James Dugundji, Fixed point theory, Springer Monographs in Mathematics, New York: Springer Verlag, pp. xv+690 (2003), ISBN: 0-387-00173-5, MR1987179, Zbl 1025.47002.
[2] Peter Henrici, Applied and computational complex analysis. Volume I: Power series- integration-conformal mapping-location of zeros, Reprint of the original edition, published in 1974 by John Wiley & Sons Ltd., paperback ed., Wiley Classics Library. New York-London-Sydney-Toronto: John Wiley & Sons Ltd., pp. xv+682 (1988), ISBN: 0-471-60841-6, MR1008928, Zbl 0635.30001.
[3] Radu Precup, Methods in nonlinear integral equations, Dordrecht: Kluwer Academic Publishers, pp. xiv+218 (2002), ISBN 1-4020-0844-9/hbk, MR2041579, Zbl 1060.65136.
answered Feb 9, 2022 at 17:59
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$\begingroup$ Thank you for your kind answer, but in this example $z = \frac{1}{2}\sqrt{1+z}$ can be reformulated as $4z^2 = z + 1$ (to remove the singularity of $\sqrt{\cdot}$), which can be solved by Newton's method (Banach fixed point method). I wonder if any problem has a Banach method solution behind (if properly reformulated) or the regularity assumptions of the problem will prevent Banach method to be implemented. $\endgroup$ Commented Feb 10, 2022 at 4:29