Fučík, Svatopluk

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Commentationes Mathematicae Universitatis CarolinaeSvatopluk FučíkRemarks on the solvability and nonsolvability of weakly nonlinear equationsCommentationes Mathematicae Universitatis Carolinae, Vol. 17 (1976), No. 1, 61--70Persistent URL: http://dml.cz/dmlcz/105674Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1976Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czmMmmATiowss MATHEMATICAL U N I V E R S I T A T I S CAROLINAE 1 7 , 1 ( 1 9 7 6 ) REMARKS ON THE SOLVABILITY AND NONSOLIABILITY OF WEAKLY NONLINEAR "EQUATIONS l } Sva top luk JPU6ÍK , Praha A b s t r a c t : The e x i s t e n c e of t h e s o l u t i o n of the n o n l i -near o p e r a t o r e q u a t i o n Ax * o6Sx+ - £ S x ~ • Gx = £ Cwhere A i s a .Fredholm l i n e a r s e l f a d j o i n t n o n i n v e r t i b l e o p e r a t o r i n a r e a l semiordered H i l b e r t space X , S i s si l i n e a r comple t e ly <continuous o p e r a t o r i n X , 6 i s a non-l i n e a r mapping i n X , f o r x e X i t i s x+ -* max (x ,0) , x =. x 4 - x~ ) i s s t u d i e d . Ke.v wordss Non l inea r o p e r a t o r e q u a t i o n , s o l v a b i l i t y , n o n s o l v a b i l i t y , m u l t i p l i c i t y of s o l u t i o n s . AMS: 47H15 Ref. 2 . : 7 .978 .53 1 . Assumptions*. Not at i o n s . Un t i l f u r t h e r comment, X, Z w i l l deno te r e a l H i l b e r t spaces wi th norms II x II x , II z II z , r e s p e c t i v e l y . The i n n e r product i n X i s denoted by (xjjXg) . A s u b s e t C of Z i s c a l l e d a cone i f i t i s c losed , convex, i n v a r i a n t under m u l t i p l i c a t i o n by nonnegat ive num-b e r s , and i f C n ( - C ) = - 4 . 0 ^ . 1) The r e s u l t s con t a ined i n t h i s note were f i r s t p resen ted by the au tho r a t the Summer School on "Nonl inear Analy-s i s and Mechanics" ,September 1974, Stara* Lesna" near Po-p r a d , S l o v a k i a , and t h e y a re announced wi thout the proofs i n the t h i r d p a r t of t h e paper C7-U 61 Let C be a given cone in Z with the following properties: (Z 1) I f z c Z then there exis ts a uniquely deter­mined couple z*, z"*eC such that z « z** - z~ . (Z 2) The mappings z .—* z*., zi—*» z~ are l i p -schitzian, i . e . , there e x i s t s g> > 0 such that Z л - _ »2 - ł , U 1 - Z2Л 2 • l l z l - Z2 Һ * Ç> I « i - -_.« for each z^, z^s Z • (Z 3/ XcZ and the identity mapping from X into Z i s continuous. Denote by ^ i t s norm. (A 1) Let A be a l inear bounded s e l f ad joint opera-tor from X into X with a closed range R(A) and f i n i t e -dimensional null-space N(A) , dim N(A>g 1 . Let P be the orthogonal projection from X onto N(A) and l e t Q = I - P ( I is the identity operator i n X ) , i . e . , Q i s the orthogonal projection from X onto R(A) • Under our assumptions there exists a linear continu-ous map (the so-called right inverse) M: R(A)—> R(A) s a -t isfying MAx « Ox (xcX) , AMy * y CycR(A)) . Denote by H M If the norm of M . Let S be a linear continuous mapping from Z into C with the norm il S i and suppose: (S 1) The mappings x I—• Sx4*, x I—* Sx~ are comp-le te ly continuous operators from X into X . - 62 -2. f i r s t resuJ4 L<am̂ fl 1« Let cc , /3 b e ; r e a l numbers and suppose t ha t f (S 2J ^ i n f cA)(c.cSh^ r £$h~ , h ) > 0 . ll&HxH Then d ^ ( o C f / 3 ) > 0 f where <£(oC,/3) -{sup HcT>0; inf inf (ccS(h • v) + -° ' VfeR(A) JkeNCA) ll/^ftx.4cr ||^llx-=^ - /3S(h +.v)~ f h ) > 0 } • Proof» Suppose that there ex i s t s <f^> 0, lim c ^ -= 0 f and v n €R(A) , II v n II x 6 cj^ f l ^ e N t A ) , H -^il x* = 1 , such that Since N(A) i s f in i te -d imens iona l , we can suppose tha t the sequence h converges i n the norm topology of X t o hQ6 N(A) , II h II x « 1 • The cont inui ty of S and the assumption (Z 2) imply (ocSh + - (3Sh0 f h 0 ) » 0 which contradic ts (S 2 ) . Theorem 1. Let oc, (I be r e a l numbers. Suppose (Z 1) - (Z 3 ) , (A 1 ) , (S 1 ) , (S 2 ) . Moreover, suppose that (G 1) G: X ~ » X i s a completely continuous opera-t o r such that (G 2) sup II Gx I v < oo . - 63 -Then the equation (1) Ax + ©cSx+- £Sx~ K x a f i s solvable in X for each f £X provided c M + ipnnsBa < ^coc,,/3) im-< ^ 1 > 1 + <f0 Ccc,<3) Proof* According t o t4 f Theorem 1] i t suff ices to ver i fy the following condit ion: For any K>0 we*have t K > 0 such that (2) (ocS(t(h • v)+) - £ S ( t ( h * v D , h ) • * (GCt(h • v ) ) , h ) S K for a l l t g t x , h e l ( A ) , II h 1^ = 1 , v €R(A) , l|v II x £ cT, where c T ^ ^ C o c . j j . J ) . The assumptions (S 2),(G 2} and Lemma 1 immediately imply (2 ) . Remark 1« The assumption (G 2) may be (without chan-ging the proof! replaced by the growth condition |lGx|l x £ ~ c l * c 2 ^ x ^X » w l l e r e ^ e < 0,1) • I f the constant c2 i s su f f i c i en t ly small , then i t may be & =-= 1 and the same asser t ion as in Theorem 1 i s va l id . Remark 2 . Suppose, moreover, (A 2) dim H(A) « 1 and N(A) i s a l inear h u l l of hQ'6X , II h ll-^ = 1 . I f we suppose (S 3) (Sno'V = - ( S ho» ho }* 0 then c^(oc,(3J = <^(l,0) and if 06a|»-(3 then the condi-tion (S 2) is fulfilled. - 64 -3 . Second result LairnT)ft 7. Let «c, p> be real numbers and suppose (A 1),(A 2) and r (cCSh+- (JSh; , h 0 ) r o (S 4Ï -j L ( (SSh*- tsCSh" , h 0 >< 0 • Then c ^ ( o c , | i ) > 0 , where <£j («c f |») * sup -Ccf B 0 ; ^ in f c ^(ocS(h 0 * v)4* -W « x * c T - /3S(h0 • vr ,h Q )^0 and sup (&S(h„ + v ) + - ocS(hft + vrfhft)-<Ol' . (The proof ia quite analogous of that in Lemma 1.) Theorem 2. Let oC., /J be real numbers and suppose: (Z 1) - (Z 3) , (A1),(A 2) , (S 4 ) . Let G: X—>X be a l i p -schitzian mapping, i . e . there exists c > 0 such that (G 3) II Gxx - Gx.̂  II x f e c II xx - ^ ft x for each x l f x^eX . Suppose (G 2 ) . If (3) k « II Mil (g>llsHu (loo I • I j i l ) f c k l , (4) ^ L _ .*: cCJ Ccc, /3) , then there exists a lower semicontinuous function V : : R(A)—> (--^ ,00) , bounded from below on bounded sub-- 6 5 -s e t s of R(A) and with lb he following p roper t i e s : a) The equation (1> has a so lu t ion for the r igh t hand s ide f e X i f and only i f ( f , h 0 ) £ T (Qf) . b) The equation ( l ) has at leas t two solut ions for the r i gh t hand side f € X provided Cf,h0) > r fOf) . Step 1» fo r fixed t €(-£»,<») - \ and f e X define the mapping .g*t f : S(A)—> R(A) by F t tt v v—*> - m CocS(th0 • v)4" - ^ S ( t h 0 + v)~ * * Sr(th0 * v) - f) . With respect t o (3) the mapping F^ t i s l i p s c h i t z i a n with the constant k-*-l and thus according to the Banaeh's contract ion mapping pr inciple there exis ts a unique f ixed point v t f eR(A) of *\ t . Step 2 . .For a l l t l f " t g6 1 ^ a n d fi>. f2eX w e o b t a i n (lay an easy calculat ion) 66 Џ> # - rУ.*. « f l j / S - — Г * 1 1 . - Г „ I "I- - •'••' I! I - Xn lìu ľV* Л - 4 Г . J | v é Г-T- I t - t j - ł — — — | £ £ J | V # S t e p 3 . Define 3 ? f : t I—» P ( o c $ ( t h 0 + vt t)+ - ( 3 s ( t h Q * v t f ) " * • G ( t h „ * v+ ^ ) i • o t , f The e q u a t i o n ( l ) w i t h t h e r i g h t hand s i d e f e X has a s o ­l u t i o n x Q € X i f and on ly i f t h e r e e x i s t s t Q € R 1 such t h a t C5) < ^ f ( t o ) = P f • Stem 4* S ince v t f = v t Of a n d N ^ A ) i s one -d imens io -n a l , t h e e q u a t i o n (5) i s e q u i v a l e n t w i t h (6 ) 9 a f ( t ) = ( f , h 0 ) , where (7 ) 9 f t f : t i — M $ f ( t ) , h 0 ) . S t e p $. .Por f i x e d f e X t h e f u n c t i o n 9 Q ~ i s cont inuous on R-_ and, moreover: l im c?n~(t) a oo (it follows from the inequalities n,*_ л + „ „ _ _ n_?_ 9Q f(t).?t (o ts íh 0* ~f- r - /ss(ho* —^- r,ho) - aup_ llGx II X i f t > O , X € 7 ^ ( t ì í - t ((гs(h0* ^ J Ł > + - C * S ( Һ 0 * ŽҶ- Г ,Һ 0 ) - 67 - sun llGxll x i f t<-0 and from (S 4 ) , ( 4 ) , Lemma 2 and Step 2 ) . Step 6« Define T(Qf) =- min cpQf(t) . t € R1 The assertions of Theorem 2 follow now immediately from Steps 5,3,4 • Remark 3, If (Sh£,hQ)2 4» ^ h ^ , h 0 )2 then there ex-ist oc, '/3 e Bj, satisfying (S 4). Remark 4> Note thst the typical example of the equa-tion (l) is the boundary value problem for nonlinear elli-ptic partial differential equation of the order 2m (see 111). In this case we put X - *Ŵ » (il) (the well-known Sobolev space) and Z « L^Jl) . In the case of ̂ second or-der partial differential equations it is possible to put X * % 9 W£*2(H) . In both cases C may be the set of all almost everywhere nonnegative functions from Z • Remark 5* The second order ordinary differential e-quations of the type (l) are investigated in £6 3. The ge-neralization of the results from [6] as well as the study of periodic problems and partial differential equations of second order is in £2, 31 • Remark 6. The analogous result as in Theorem 2 is proved in £l, 51 for partial differential equations of the second order in the case that h is 3 nonnegative func-tion. In this note the condition M h ^0 ,f is replaced by (S 4) (and, of course, the abstract consideration is - 68 -useful for higher order equations). A. Ambrosetti communi-cated me that E. Podolak (Princeton University) prepares the manuscript in which the condition w hQ^0 M is repla-ced in the case of second order partial differential equa-tions by f (ht(x))2dx * T (h~(x))2dx . JfL 0 ^ ^£1 0 But in a l l previous papers (also in the present note) for the method of proofs i t i s e s sen t i a l that dim N(A) = 1 • The observation of the analogous problems as in Theorem 2 but i n the case of dim N ( A ) > 1 i s a t e r ra incognita by t he author 's best knowledge. R e f e r e n c e s [1] A. AliBROSETTI - G. PRODI: On the inversion of some dif-ferent iable mappings with s ingu la r i t i e s be t -ween Banach spaces, Ann. Mat. Pura Appl. 93 (1973), 231-247; see also Theory of nonlinear operators, Proceedings of a Summer School held in September 1971 at Babylon, Czechoslovakia, Academia, Prague, 1973, pp. 9-28. [2] E.N. DANCER: On a nonlinear e l l i p t i c boundary-value problem, Bull . Austral. Math. Soc. 12(1975), 399-405. [3] E.N. DANCER: On the Dirichlet problem for weakly non-linear e l l i p t i c pa r t i a l d i f ferent ia l equations (to appear). [ 4 l S. .FUfife: Surject ivi ty of operators involving l inear noninvertible part and nonlinear compact per-turbation, ifunkcialaj Ekvacioj 17(1974),73-83* 69 I 53 S. FUČÍK: Remarks on a r e s u l t by A. Ambrosetti and G. Prodi, B o l l . Unione Mat. I t a l . 11(1975), 259-267. C6] S. FUČÍK.: Boundary value problems with jumping nonl i-n e a r i t i e s , t o appear in Čas.pěst.mat. 101 (1976),No 1. [7] S. FUČÍK: Solvabi l i ty and nonsolvabil i ty of weakly nonlinear equations, Proceedings of the Summer School "Theory of Nonlinear Operators. Con­s t r u c t i v e Aspects" held in September 1975 at Ber l in . To appear in "Abhandlungen der Akade­mie der Wissenschaften der DDRM. Matematicko-fyzikální fakulta Karlova univers i ta Sokolovská 83, 18600 Praha 8 Československo (Oblátům 10.11.1975) 70 

Remarks on the solvability and nonsolvability of weakly nonlinear equations

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Remarks on the solvability and nonsolvability of weakly nonlinear equations

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Institute of Mathematics AS CR, v. v. i.