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The author's interest in the question of multiplicity of solutions grew out of the paper [1] where a class of problems of the form u = l G0 u + G1(l , u)u was studied. In (1. 1) l E R , the reals, and u is a vector in a real Banach space c having a norm denoted by 11 11 . The map G0 is compact and linear while G1( l, u)u is compact and nonlinear (cf. [1] for details). 1) with u ~ 0 , and set = (the closure). In [1] it was shown that if l 0 is a characteristic value of G 0 of odd multiplicity and if G1( l, u)u is of 11 u 11) near u = 0 , then 6' contains two continua, counting multiplicity, each of which connects ( l 0 , 0) to "o" or to a point (l, 0) where l is a characteristic value of G0 different from l 0 .

4 tells us there will be a continuum ( 0, 0) to 83R or to the boundary of one C n in á joining l of the balls about the points ( l1, 0). If, in the latter case, we join the intersection of C n with such a ball to the corresponding center ( l1, 0) we have a continuum from ( l 0 , 0) to ( l1, 0). Now Lemma 2. 2 in conjunction with arguments already used in the proof of theorem (2. 4) show that the limit set of the Cf is a continuum in á joining (l 0 , 0) to 8BR or to a point (l, 0), 10 0 . We give an example to show that the boundary M of a region Q as described in the theorem may contain a single zero of multiplicity two.

Gantmacher, F. R. and M. G. Krein, "Oszillationsmatrizen, Oszillationkerne, und Kleine Schwingungen mechanischen Systeme", Berlin, Akademie-Verlag, 1960. 35 PAUL H. RABINOWITZ 15. Karlin, S. , "Total Positivity", Vol. I (and II to appear), Stanford University Press, Stanford, 1968. 16. Turner, R. E. L. , Nonlinear eigenvalue problems with nonlocal operators, Comm. Pure Applied Math. , 23, 1970, pp. 963-972. 17. Ladyzhenskaya, O. A. and N. Uraltseva, "Linear and Quasilinear Elliptic Equations", Academic Press, New York, 1968.