12 J. MAWHIN ker L = {£(0 , p): MT^ + NT^ = 0 } = {£(0 , p): * £ ker( M + NT X )}. If w e defined : Z —• C by (1.9) i4(/ f c) = c - A S 1 / f it i s immediate t o chec k tha t A i s linear, continuous an d onto , and proble m (1.6)—(1.7 ) is solvable i f an d onl y i f (/ , c ) G A~l(lm(M + JV7\)). Therefore , i f Im(A f + TVTj) is closed, the sam e is true fo r I m L. I f moreover ker(Af + NT t ) ha s finite dimensio n an d Im(M + NT X ) has finite codimension , th e sam e is respectively tru e fo r ke r L an d Im L an d hence condi - tions upo n M an d N onl y wil l insure tha t L i s a Fredholm mappin g and determin e it s index. In th e cas e o f homogeneous boundary conditions , x'(0 =/(') . Mx 0 +NXl = 0 , one ca n als o proceed a s in the ordinar y case , and w e leave th e detail s to th e reade r a s well as the construction s o f P, Q and Kp Q i n terms o f projectors o n ker(M + NT X ) an d Im(Af + NT X ) whe n the y exist . W e just explicat e th e result s in th e nonhomogeneous cas e for futur e reference . Le t 5, The continuou s projectors in C such that Im S = ker(Af + ATj), Im T = Im( M + NT X ) an d let (M + NT t )s b e th e restrictio n o f M + AT 2 t o ker S. Then , P5: x h- £(0, &c0) an d Q r : ( / c) h * (0, (/ - 74(/ , c)) , with ^ define d i n (1.9 ) are respec- tively continuou s projector s i n X an d Z suc h that I m Ps = ke r L, Im L = ker (2 T . More - over, (K*8.QTU c ) ) ( r ) = ( 5 ^ ( 0 ) + ( 7 V ( ( M + ^ i ) * *(* -NSJ))){0 ) iftei, ( L 1 0 ) = ( ^ + AT1)^ 1 (c-^5 1 /X0 i f - r f 0 . 4. I-compactness , ^-complete continuity and examples. Le t L: do m L C X Z be a Fredholm mappin g an d let u s retain th e assumption s an d notation s o f § 1 let E b e a metric space an d G: E —• Z b e a (not necessaril y linear ) mapping. W e shall say that G is L-com- pact on E i f th e mapping s QG: E —• Z an d KF QG: E X ar e compac t o n £, Le contin- uous on E an d suc h that QG(E) and Kp QG(E) ar e relatively compact . On e can sho w tha t this definition doe s not depen d upo n th e choic e o f th e continuou s projector s P and Q, which justifies th e terminology . EXAMPLES. (1) , Le t X an d Z have finite dimensio n an d E C X bounded then G: E Z i s 0-compact o n £ i f an d onl y i f G is continuous o n E. (2) G: E —• X i s /-compact o n is" if an d onl y i f G is compact o n E. (3) G is L-compact o n E i f G is compact o n is an d Kp Q continuou s (resp . if G is continuous o n E, G(E) i s bounded an d Kp Q i s a compact linea r mapping) . We shall say tha t G: X Z i s L-completely continuous i f i t i s L-compact o n every bounded E C X. Let no w /: / x C —• /**, (f, ^ ) »-• /(f,p ) be a mapping suc h that : (i) fo r eac h y?GC, the mappin g 11- /(r,/ ) is (Lebesgue) measurable o n / (ii) fo r almos t eac h t G /, th e mappin g ^ h- f(t, /? ) is continuous o n C (iii) fo r eac h p 0 , there exist s ap GL X (I, R) suc h that , fo r almos t ever y t £ / an d every \p such tha t \\p\ p , one has
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