By Eduardo H. Zarantonello
ISBN-10: 012775850X
ISBN-13: 9780127758503
Read or Download Contributions to Nonlinear Functional Analysis. Proceedings of a Symposium Conducted by the Mathematics Research Center, the University of Wisconsin–Madison, April 12–14, 1971 PDF
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Extra info for Contributions to Nonlinear Functional Analysis. Proceedings of a Symposium Conducted by the Mathematics Research Center, the University of Wisconsin–Madison, April 12–14, 1971
Example text
I. , XXII, 1967, pp. 413-433. 34 A GLOBAL EIGENVALUE THEOREM 5. Rabinowitz, P. H. , Some global results for nonlinear eigenvalue problems, to appear in journal of Functional Analysis. 6. Rabinowitz, P. H. , Nonlinear Sturm- Liouville eigenvalue problems for second order ordinary differential equations, Comm. Pure Applied Math. 23, 1970, pp. 939-962. 7. Whyburn, G. , "Topological Analysis, Princeton University Press, Princeton, 1958. 8. Leray, J. and J. Schauder, Topologie et equationes functionelles, Ann.
1 Let φ ΐ = φ , and let T s be the unit vector in the direction of (1-s) T t + s e 0 where e 0 = (1, 0, . . , 0) € R m + 1 . 49 R. E. L. TURNER Arguments like those in the proof of lemma 2. 3 show that there is a continuous map V s of 0 < s < 1 into SO m + j such that VQ = U t and Vj is an orthogonal map satisfying V e l 0 = e 0 " M o r e o v e r T tn ♦ v s x m f o r ° - s - 1 s i n c e T S ö is never perpendicular to iVLQ . Thus VQXm remains t r a n s W versai to σ(ΐ) at σ(ΐ 0 ) and hence x = 0 remains an i s o lated zero of the maps (2.
2 and 3. 5 are essentially due to Leray and Schauder [8]. For their analogue of Theorem 3. 5 however they require that {0) X BM contain only finitely many solutions. As an application of Theorem 3. 2, consider the quasilinear elliptic boundary value problem: Remark: (3. 17). We assume A is a continuously differentiable function of its arguments, A(x, u, p, 0) = 0 , and A(x, 0, 0, l) > 0 for x E W if l > 0 . As in §2, (3. 1) in E = Cl+OG(~) (1 B. C. Moreover q is compact and 4(0,u) = 0 . Theorem 3.