ABSTRACT
V. CAFAGNA* U n iv ers ite P aris Nord, C .S .P ., Departement de Mathemat iq u e s e t d 1Inform atique, 93430 V ille ta n e u se , FRANCE.
INTRODUCTION
Aim o f th is paper i s to d escr ib e some a sp ec ts o f the map approach to n on lin ear d i f f e r e n t ia l eq u a tio n s. These w i l l be i l lu s t r a t e d by means o f a s in g le c la s s o f exam ples, namely sem ilin ea r D ir ic h le t problems :
( 1)
The adopted p o in t o f view c o n s is t s in studying the g lo b a l fea tu res o f the map
( 2)
defin ed by F(u) = Au + f (u ) . Then the so lu t io n s o f problem (1) are describ ed as p o in ts o f the preimage s e t F ^(h) . This i s , so to speak, in co n tra st w ith the v a r ia t io n a l method, which d escr ib es the so lu t io n s o f problem ( 1) as c r i t i c a l p o in ts o f the fu n c tio n a l
(3)
See Vainberg (1968) fo r a com parative i l lu s t r a t io n o f both methods.
