Schauder's Theorem and the method of a priori bounds

Andrzej Granas, Marlène Frigon



We first recall simple proofs relying on the Schauder Fixed Point Theorem of the Nonlinear Alternative, the Leray-Schauder Alternative and the Coincidence Alternative for compact maps on normed spaces. We present also an alternative for compact maps defined on convex subsets of normed spaces. Those alternatives permit to apply the method of a priori bounds to obtain results establishing the existence of solutions to differential equations. Using those alternatives, we present some new proofs of existence results for first order differential equations.


Fixed point; nonlinear alternative; Leray-Schauder alternative; Schauder fixed point theorem; coincidence; a priori bounds; differential equation

Full Text:



R.G. Bartle and L.M. Graves, Mappings between function spaces, Trans. Amer. Math. Soc. 72 (1952), 400–413.

N. El Khattabi, M. Frigon and N. Ayyadi, Multiple solutions of boundary value problems with φ-Laplacian operators and under a Wintner–Nagumo growth condition, Bound. Value Probl. 2013:236 (2013), 21 pp.

N. El Khattabi, M. Frigon and N. Ayyadi, Multiple solutions of problems with nonlinear first order differential operators, J. Fixed Point Theory Appl. 17 (2015), 23–42.

A. Granas, On the Leray–Schauder alternative, Topol. Methods Nonlinear Anal. 2 (1993), 225–231.

A. Granas and J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics, Springer, New York, 2003.

A. Granas and Z.E.A. Guennoun, Quelques résultats dans la théorie de Bernstein–Carathéodory de l’équation y 00 = f (t, y, y 0 ), C.R. Acad. Sci. Paris Sér. I Math. 306 (1988), 703–706 (in French).

A. Granas, R.B. Guenther and J.W. Lee, On a theorem of S. Bernstein, Pacific J. Math. 74 (1978), 67–82.

A. Granas, R.B. Guenther and J.W. Lee, Some general existence principles in the Carathéodory theory of nonlinear differential systems, J. Math. Pures Appl. (9) 70 (1991), 153–196.

J. Mawhin, Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces, J. Differential Equations 12 (1972), 610–636.

J. Mawhin, First order ordinary differential equations with several periodic solutions, Z. Angew. Math. Phys. 38 (1987), 257–265.

H. Schaefer, Ueber die Methode der a priori-Schranken, Math. Ann. 129 (1955), 415–416.


  • There are currently no refbacks.

Partnerzy platformy czasopism