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 ) give conditions in order for the following functional equation to have 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 (English), 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. (English), 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, (English), Dordrecht: Kluwer Academic Publishers, xiv, 218 p. (2002), ISBN 1-4020-0844-9/hbk, MR2041579, Zbl 1060.65136.

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.