We state, and indicate some of the consequences of, a
theorem whose sole assumption is the nonvanishing of the Leray-
Schauder degree of a compact vector field, and whose conclusions yield
multidimensional existence, continuation and bifurcation result

Fitzpatrick P.

Massabò I.

Pejsachowicz J.

English

PEJSACHOWICZ, JACOBO

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

03 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEComplementing maps, continuation and global bifurcation / Fitzpatrick P.; Massabò I.; Pejsachowicz J.. - In: BULLETIN(NEW SERIES) OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0273-0979. - 9(1983), pp. 79-81.OriginalComplementing maps, continuation and global bifurcationPublisher:PublishedDOI:Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/2495702 since:AMSBULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 9, Number 1, July 1983 COMPLEMENTING MAPS, CONTINUATION AND GLOBAL BIFURCATION BY P. M. FITZPATRICK, I. MASSABÖ AND J. PEJSACHOWICZ ABSTRACT. We state, and indicate some of the consequences of, a theorem whose sole assumption is the nonvanishing of the Leray-Schauder degree of a compact vector field, and whose conclusions yield multidimensional existence, continuation and bifurcation results. Complementing maps and the Theorem. Let X be a Banach space, m be a positive integer, and O Ç Rm x X be open. Suppose ƒ : O —• X is an m-parameter compact vector field: i.e. /(X,x) = x — F(X,x), for (X,x) € O, where F is continuous and maps bounded sets into relatively compact sets. A continuous map g: O —• Rm, which maps bounded sets into bounded sets, will be called a complement for ƒ : O —• X provided that the Leray-Schauder degree, deg((0, ƒ), 0,0), is defined and nonzero: (g, ƒ)((X, x)) = (g(\, x), ƒ (X, x)), for (X, x) € O, and since O is not assumed to be bounded, "defined" means (<7, /)_ 1(0) is compact. By cohomology we will mean Cech cohomology with integral coefficients. By dimension of a topological space we mean the Cech-Lebesgue covering dimension, and if p G A, the space A will be said to have dimension at least m at p provided that each neighborhood, in A, of p has dimension at least m. THEOREM. Let X be a Banach space, m be a positive integer, and O Ç Rm xX be open. Suppose that ƒ : O —• X is complemented by g: O —• Rm. Then there exists a closed connected subset, C, of / _ 1 (0) , whose dimension at each point is at least m, and (*) whenever K is a compact subset of C, with g_1(0) n C C K, the map of pairs g: (C, C - K) -> (Rm ,Rm - 0) induces a nontrivial map in the mth cohomology group. In particular, Cogf"*1(0) 7* 0 and either C is unbounded orTJndO # 0 . In the case when f and g are defined onÜ with f"1(0)C\g~1(0)ndO — <Z>,C also has the following properties: ifC is bounded, then dhafjOndO) >m — l, when m>l, andZïndO has at least two points, whenm = 1; ifg: /~1(0)OÜ —• Rm is proper anddim(ÖndO) < m—1, theng(0) = Rm. SKELETON OF THE PROOF. Since deg((#, ƒ), 0,0) ^ 0, by using the cup-product in cohomology, it follows that whenever K is compact and 0-1(O) Ç K Ç /-^O) the map g: (/-^O), Z"1 -K)-> ( R m , R m - 0 ) is cohomologically nontrivial. Passing to the limit over all such K's we obtain a nontrivial class, £, in the mth Cech cohomology group with compact supports of /~"^(0). The continuity of Cech theory enables us to choose a set, C, which is minimal Received by the editors May 18, 1982. 1980 Mathematics Subject Classification. Primary 47H15, 35B32, 45G05. © 1983 American Mathematical Society 0273-0979/83 $1.00 + $.25 per page 79 80 P. M. FITZPATRICK, I. MASSABÖ AND J. PEJSACHOWICZ among the closed subsets of / _ 1 (0) to which £ restricts nontrivially. This C has the properties claimed. D Some consequences of the Theorem. In what follows, ƒ : O C R m x X - ^ X is an m-parameter compact vector field. 1. Continuation under global hypotheses. Let Xo G R m and let / \ 0 be the section of ƒ over the slice O\0. One shows that if deg(/\0,Oxo,0) ^ o then ƒ : O —• X is complemented by g : O —• R m defined by #(X, x) = X — Xo. Thus the theorem furnishes a description of how / _ 1 (0) emanates from O\0. This is a multidimensional refinement of the Leray-Schauder continuation principle (see [4, 6 and 7]). 2. Continuation under local hypotheses. Let (Xo,xo) € O and suppose that the map x —• /(Xo,x) has a Fréchet derivative, L, at x — XQ. Assume L G L(X,X) is invertible. Then, letting U — O — { (Xo, x) \ x # xo, /(Xo, x) = 0 }, one shows that ƒ: £7 —• X is complemented by g: U —• R m defined by #(X,x) = X — Xo- Thus, there is an m-dimensional connected subset, C, of / - 1 (0 ) Pi U, which contains (Xo,xo), and which is either unbounded or Ü 0 {dO U { (X0, x) | x ^ x0, ƒ(X0, x) = 0 }} 7e 0 . Another global version of the implicit function theorem was obtained in [3]. 3. Nonlinear perturbation of linear Fredholm operators. Let fi Ç R2 be simply connected, open and bounded, with dfi a smooth closed curve. Suppose r : dft -> S1 is smooth and such that the winding number of r : dQ -> S1 equals — k < 0. Given </>,?/>: Q x R ^ R w e consider the following nonlinear Riemann-Hilbert problem: find u,v: Ü —• R such that, if r = (7*1, T2) ... fw I- t ; ! / = 0(x,y,u,v), . 1 ., x mil, IVxH-Wy = il)(x,y,u,v) (R-H) (ii) uri — VT2 = 0 on 3H. Let a G (0,1) be such that ip and </> lie in C 1 + a(Ü X A,R) for each bounded subset A of R2 (C1+a denotes the usual Schauder space). Under the assump-tion that V>(x,y,0,0) = 0(x,t/,O,O) = 0, for each (x,y) G fi, it follows that for each r > 0, {(w,v) G C1 + 0 !(n,R2) | (w,v) solves R~H, ||(w,v)||1+a = r} has dimension at least 2k. Let W = {{u,v) G C1 + a(n,R2) |(n,v) satisfies (ii)}, and let L: W -+ C a (n ,R 2 ) be the linear operator defined by the left-hand side of (i). Choose zi,..., Zk in Q and define g : W —• R2fc+1 by g((n, v)) = (u(*i), v(2?i),...,u(^fc), v(2ffc), ƒ [rit; -f r2u] ds). Letting X = g_1(0), the linear theory (see [10]) implies L: X - • Ca(Ü,R2) has an inverse, T, and W = V © X, with dim(V) = 2/c + 1. If we rewrite (R-H) as ƒ((u, i/)) = T(L — üf)((n>v)) == 0, one shows that ƒ : V 0 X —• X is complemented by 0 on each ball about the origin in W, and so we can apply the Theorem. 4. Global bifurcation. For simplicity, we assume O = R m X X. We assume R m X {0} Ç / _ 1 (0) , and call R m X {0} the trivial solutions of ƒ. Suppose a, (3 G R m are such that (a, 0) and (f3,0) are not bifurcation points of ƒ _1(0) and that COMPLEMENTING MAPS 81 ind(/a,0) ^ ind(//3,0), where "ind" denotes the Leray-Schauder index. Then, if T is any open curve (i.e. homeomorphic image of R) in Rm X {0} which passes through (a, 0) and (/3,0), there exists a connected set, C, of nontrivial zeros of ƒ, whose dimension at each point is at least m, which intersects the segment, (a, 0), (/?, 0), of T, determined by (a,0) and (/3,0), and either C is unbounded or Ü intersects V — {(a,0),(/9,0)}. When a = 0, /3 = (1,0,. . .) and T is the line through a and j3 the proof runs as follows. Choose r > 0 such that f(\x) ^ 0 when 0 < ||x|| < r and either |X| < 3r or \\ — /3\< 3r. Let h: R —• [0,r] be continuous, vanish outside of [-r, 1 + r], and equal r on [r, 1 - r]. Then define g: Rm X X -+ Rm by ff(Xi,..., Xm) = (||x||2 - (fe(Xi))2, X2,..., Xm). One shows that if U = Rm X { X - {0}}, then deg((g, ƒ), £/, 0) = ind( fa0) -ind(/a,0), and so g complements ƒ on [/. So we extract the subset, C, of ƒ _1(0)n[7, having the properties in the conclusion of the Theorem. Conclusion (*) implies our assertions regarding ÜOT. This bifurcation result yields the principle abstract global bifurcation results of [9 and 1]. J. Ize (see [8]) has given a proof of the bifurcation theorem in [9] using a map similar to the above g. REMARK. In the definition of complementing map if one replaces the Leray-Schauder degree by the Browder-Petryshyn degree for A-proper map-pings (see [5]) the Theorem still holds. We believe that approximation results similar to those used in [2] will also yield the Theorem when F is assumed to be condensing. REFERENCES 1. J. C. Alexander and S. S. Antman, Globed and local behaviour of bifurcating multidimen-sional continua of solutions of multiparameter eigenvalue problems, Arch. Rational Mech. Anal. 76 (1981), 339-354. 2. J. C. Alexander and P. M. Fitzpatrick, The homotopy of certain spaces of non-linear operators and its relation to global bifurcation of the fixed-points of parametrized condensing maps, J. Funct. Anal. 34 (1979), 87-106. 3. J. C. Alexander and J. A. Yorke, The implicit function theorem and the global methods of cohomology, J. Funct. Anal. 21 (1976), 330-339. 4. F. E. Browder, On continuity of fixed-points under deformations of continuous mappings, Summa Brazilia Math. 4 (1960), 183-190. 5. F. E. Browder and W. V. Petryshyn, Approximation methods and the generalized topologi-cal degree for non-linear mappings in a Banach space, J. Funct. Anal. 3 (1969), 217-245. 6. J. Leray and J. Schauder, Topologie et équations fonctionnelles, Ann. Sci. École Norm. Sup. 51 (1934), 45-78. 7. I. Massabô and J. Pejsachowicz, On the connectivity properties of the solution set of parametrized families of compact vector fields, J. Funct. Anal, (to appear). 8. L. Nirenberg, Topics in nonlinear functional analysis, Courant Inst. Lecture Notes, 1975. 9. P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487-513. 10. W. L. Wendland, Elliptic systems in the plane, Monographs and Studies in Mathematics, no. 3, Pitman, New York, 1979. D E P A R T M E N T O F MATHEMATICS, U N I V E R S I T Y O F M A R Y L A N D , C O L L E G E P A R K , M A R Y L A N D 20742 D I P A R T I M E N T O DI M A T E M A T I C A , U N I V E R S I T À D E L L A C A L A B R I A , ARCAVACATA DI R E N D E , C A L A B R I A , ITALY 

Complementing maps, continuation and global bifurcation

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Complementing maps, continuation and global bifurcation

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