Differential Equations: Proceedings of the 1st Latin American School of Differential Equations, Held at São Paulo, Brazil, June 29–July 17, 1981

309 Pages·1982·3.187 MB·English-Spanish

by  Castro B. Alfonso (auth.), Djairo Guedes de Figueiredo, Chaim Samuel Hönig (eds.)

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
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© 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