Lecture Notes ni Mathematics Edited yb .A Dold dna .B Eckmann 957 Differential Equations Proceedings of the ts1 Latin American School of Differential Equations, Held ta o~,S ,oluaP ,lizarB June 29-July ,71 1891 Edited yb D.G. de Figueiredo dna C.S. HSnig galreV-regnirpS nilreB Heidelberg New kroY 1982 Editor Djairo Guedes de Figueiredo Department of Mathematics, University of Brasilia 70910 Brasilia DF, Brazil Chaim Samuel H6nig Institute of Mathematics and Statistics University of S&o Paulo 01000 S~o Paulo SP, Brazil AMS Subject Classifications (1980): 34 D 05, 35 J 65, 45 D 05, 46 E 35, 47 E05, 47 F05, 47 H 99, 49 B40, 49G05 ISBN 3-540-11951-5 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-11951-5 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material si concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction yb photocopying machine or similar means, and storage ni data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © yb Springer-Verlag Berlin Heidelberg 1982 Printed ni Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210 droweroF esehT era the proceedings of the first Latin American School of Differential Equations DEALE( l for short) held at the University of o~S Paulo during the period from June 92 to July ,71 o1891 ehT loohcS sah been started sa a ecneuqesnoc of a joint project of Brazilian dna nacixeM mathematicians, but hopefully it will evolve towards a larger margorp in Latin America with the active participation of other countries. This first meeting received financial support from the ohlesnoC Nacional od Desenvolvimento CientTfico e Tecnol6gico ,qPNC( Brazil), ojesnoC Nacional ed Ciencia y Tecnologia ,TYCANOC( Mexico), edadeicoS Brasileira ed Matem~tica, o~GadnuF ed orapmA ~ Pesquisa od Estado ed o~S Paulo ,PSEPAF( Brazil), Universidade ed o~S Paulo, Instituto Politecnico Nacional ed Mexico, Universidad Nacional amonotuA ed .ocixeM There erew 6 courses, which were delivered yb Alfonso Castro .B (Reduction sdohteM via Minimax), Djairo .G ed Figueiredo (Positive Solutions of Semilinear Elliptic Problems), Jorge Ize (Introduction to Bifurcation Theory). omis~nO zednanreH (Introduction to Stochastic Differential Equations), Pedro dasowoN (Quanta dna )yrtemoeG dna luaP Rabinowitz (The Mountain ssaP :meroehT emehT dna Variations), 51 invited lectures dna a rebmun of research .stnemecnuonna ehT rebmun of participants exceeded .08 ehT organizational committee saw desopmoc yb miahC leumaS gin~H ,PSU( Brazil), Djairo sedeuG ed Figueiredo (UnB, Brazil) dna leumaS Gitler (IPN, Mexico). VI eW would like to acknowledge the mraw hospitality offered to all the participants during the meeting yb the faculty dna staff of the Instituto ed Matem~tica e EstatTstica ad Universidade ed o~S Paulo. ruO thanks to them, to the supporting agencies dna to the lecturers that have kindly answered our call for their manuscripts. A word of tnemegdelwonkca to Jose Pereira sod Santos for his excellent typing of the text, dna to Springer Verlag for including these proceedings in their well nwonk collection Lecture Notes in Mathematics. BrasTlia ,yaM 2891 Djairo sedeuG ed Figueiredo LIST FO SROTUBIRTNOC OSNOFLA ORTSAC B., Departamento ed Matem~ticas, Centrode Investigacion del IPN, Apartado Postal 14740, M~xico ,41 D.F. ocix~M ORIAJD SEDEUG ED ,ODERIEUGIF Universidade ed BrasTlia, otnematrapeD ed Matem~tica, BrasTlia, FD - Brazil LUAS ,GREBDLOG-NHAH Centro ed Investigaci6n del IPN, Oepartamento ed Matem~ticas, Apartado Postal 14-740, M~xico ,41 D.F. LEINAD .B ,YRNEH Instituto ed Matem~tica e EstatTstica, Universidade ed oaS Paulo, oaS Paulo, PS - Brazil MIAHC LEUMAS ,GINOH Instituto ed Matem~tica e EstatTstica, Universidade ed o~S Paulo, o~S Paulo, PS - Brazil .A .F IZ{, Universidade ed o~S Paulo, Instituto ed Ci~ncias sacit~metaM ed o~S Carlos, Departamento ed Matem~tica, o~S Paulo, PS - Brazil EGROJ IZE, SAMII dadisrevinU lanoicaN amonotuA ed ,ocixeM odatrapA latsoP 627-02 - ocixeM FD - ocixeM EGROJ ,ZCIWOWEL dniversidad nomiS Bolivar, Departamento ed Matem~tica, Caracas, Venezuela .P .S ,CIVEJOLIM Universidade Federal ed san'iM Gerais, Department of scitamehtaM - ICEx, Belo Horizonte, GM - Brazil OVATSUG ALREP ,ALAZNEM Instituto ed Matem~tica, Universidade Federal od oiR ed Janeiro, oiR ed Janeiro, JR - Brazil lV LUAP .H RABINOWITZ, Department of Mathematics, University of Wisconsin, Madison, Wisconsin, ASU YRRAL L. ,REKAMUHCS Department of Mathematics dna Center for Approximation Theory, Texas M&A University, College Station, Texas 77843 ASU J. ,ROYAMOTOS Instituto ed Matematica Pura e Aplicada, Rio ed Janeiro, JR - Brazil STNETNOC NOITCUDER SDOHTEM AIV XAMINIM Alfonso Castro .B NO ELPITLUM SNOITULOS FO RAENILNON ELLIPTIC SNOITAUQE HTIW 21 DDO SEITIRAENILNON Alfonso Castro .B dna J. .V .A Gongalves 34 EVITISOP SNOITULOS FO RAENILIMES ELLIPTIC SMELBORP Djairo sedeuG ed Figueiredo A YTIRALUGER MEROEHT ROF ESREVNI DEDNUOB DNA EVITERCCA 88 SROTAREPO NI TCARTSBA TREBLIH ECAPS Saul Hahn-Goldberg 97 W O H OT REBMEMER EHT VELOBOS SEITILAUQENI Daniel .B Henry EHT TNIOJDA NOITAUQE FO A RAENIL ARRETLOV LARGETNI-SEJTLEITS NOITAUQE HTIW A RAENIL TNIARTSNOC 110 miahC leumaS H6nig NO A DEXIF TNIOP XEDNI DOHTEM ROF EHT SISYLANA FO EHT CITOTPMYSA ROIVAHEB DNA YRADNUOB EULAV SMELBORP FO SSECORP DNA LACIMANYDIMES SMETSYS 126 .A .F Iz~ NOITCUDORTNI OT NOITACRUFIB YROEHT 145 Jorge Ize ERBOS DADILIBATSE ACIGOLOPOT 302 Jorge Lewowicz IIIV SOLVABILITY FO OPERATOR EQUATIONS INVOLVING NONLINEAR PERTURBATIONS FO FREDHOLM MAPPINGS FO NONNEGATIVE INDEX DNA APPLICATIONS 212 P. S. MilojeviE EMOS REMARKS NO A WAVE EQUATION WITH A NONLOCAL INTERACTION 229 Gustavo Perla Menzala EHT MOUNTAIN PASS THEOREM: THEME DNA VARIATIONS 237 Paul H. Rabinowitz OPTIMAL SPLINE SOLUTIONS FO SYSTEMS FO ORDINARY DIFFERENTIAL 272 EQUATIONS Larry L. Schumaker STRUCTURALLY STABLE SECOND ORDER DIFFERENTIAL EQUATIONS 284 J. Sotomayor NOITCUDER SDOHTEM VIA MINIMAX Alfonso Castro B. Departamento de Matem~ticas Centro de Investigaci~n del IPN Apartado Postal 14740 Mexico 14, D.F. M~xico I. Introduction Let H eb a real Hilbert space and J:H + ~ a functional of l class C That is, there exists a continuous function VJ:H + H such that for x,y ~ H lim J(x+ty)-J(x) = <VJ(x), y> t÷O t where < , > is the inner product in H. In this note ew consider the existence of critical points of J, which are points u ~ H such that ?J(u) = ,O The particular kind of functional J that ew study have the property that there exist closed subspaces X and Y with H = X ~ Y and such that the existence of critical points of J is equivalent to the existence of critical points of a new functional J:X ÷ ~. The functional J is given in the form J(x) : J(x+r(x)), where r:X ÷ Y is a continuous function defined via a "minimax characterization"; that is, some functional takes a minimum, maximum or minimax value at r(x) Csee sammeL 1 and 3). In section 2 the reader will find the basic abstract tools which will be used throughout the applications. sA applications ew present the existence of solutions for Hammerstein integral equations, periodic solutions of the forced pendulum equation dna solutions to a nonlinear Dirichlet problem. 2. Reduction Lemmas Lemma I. Let X and Y be closed subspaces of a real Hilbert space H such that H = X ~ Y, Let J:H ÷ ~ be a functional of class C I. If there exists an increasing function @:(0, ~) ~ (0, ~) such that @(t) ~ ~ as t ~ ~ and <VJ(x+y) - VJ(X+Yl) , y-yl > ~ Ily-yiII@(IIy-yiIl) (2.1) for all x ~ ,X Y'Yl G Y, Y # Yl' then: i) there exists a continuous function r:X ~ Y such that J(x+r(x)) : min{J(x+y); y ~ Y}; moreover, r(x) is the only critical point of the functional Jx:Y ÷ ~, y + J(x+y). ii) the function J:X + R, x + J(x+r(x)) is of class C l and <V~(X), Xl> = <J(x+r(x)), Xl> for all x,x I G .X Proof: From (2.1) dna the assumption that takes only positive values it follows that Jx sah at most eno critical point. Also from (2.1) ew have l Jx(y) = Jx(O) + f <VJx(sY ), y>ds 0 (2.2) l +IlyNII)O(xJVII-)O(xJ I II(@llyIIs llys )ds. 0 Since ew are assuming that ~(t) ÷ sa t + ~, there exists R > 0 llyEI such that @(t) ~ 2({IVJx(O)I I + I) for t > .R Hence, for ~ R2 )O(xJ ew have Jx(y) ~ + l{y113 ÷ ~ sa Ilyll ÷ ~. Therefore, in order to prove that J sah a unique point of muminim it is sufficient to