Fučík, Svatopluk

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

Commentationes Mathematicae Universitatis CarolinaeSvatopluk FučíkFixed point theorems for sum of nonlinear mappingsCommentationes Mathematicae Universitatis Carolinae, Vol. 9 (1968), No. 1, 133--143Persistent URL: http://dml.cz/dmlcz/105163Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1968Institute 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.czCommentationes Mathematicae Univers i ta t i s Carolinae 9* 1 (1968) FIXED POINT THEOREMS FOR SUM OF NONLINEAR MAPPINGS Svatopluk FUfifK, Praha I _,Jfltroduct ion* Let H be a real Hilbert space , K a closed bounded convex subset of H . The following theorem has been obtained by R.I . Kadurovskij, M.A. Krasnoselskij and P.P. Zabrejko 1915 Theorem 1. Let T< K ) c K and T » B + C , where I Bx - & v • * 4 H<X-<y> 1 for a l l *,*} e K . Let one of the fol lowing conditions be f u l f i l l e d : a) 0 _£ (J, < 4 1 C completely continuous, b) q^ m 4 } C strongly continuous. Then T has a f ixed point in K ( i . e . there e x i s t s at l e a s t one point X0 £ K such that T x 0 » x9 ) . Using the r e s u l t s of F.E. Browder and G.B. de Figueiredo concerning G-operators ( [42 , £6],£7J ,C8J) we can s ta te the l a s t Theorem in a more general s e t t i n g . The extensions of Theorem 1 that we give below fol low two direct ions (see Theorem 5,6 and 7)* In the f i r s t p l ace , we assume spec ia l Banach spaces (not only a Hilbert space) and the second direct ion of gene-r a l i z a t i o n i s the weakening of the assumption T C / O c K . In Sect ion V there are given some examples of operators, which are the sum of two mappings, which map the unit ba l l i n -t o i t s e l f and have no f ixed point property. - 133 -I I • Terming logy f no ta t ions and def inltionfla Let X be a r e a l Banach space wi th the norm II * II 7 @ i t a zero element; X* denotes the adjoint (dual) space of a l l bounded l i n e a r func t iona l s on X . The p a i r i n g between X*e X* and x € X i s denoted by (x> *x* ) . We s h a l l use the symbols * — » " , *—*-u t o denote the s t rong convergence in X (o r i n X* ) and weak convergence in X (or i n X* ) , r e s p e c t i v e l y . Def in i t ion 1 : Let F be a mapping with domain D c X and values in X (or X * ) . Then (1) F i s sa id t o be s t r ong ly continuous i f *x^ —-- xo i n D implies F x ^ —> F ^ • (2) F i s said t o be weakly continuous i f x^—-• X0 in D implies F . x ^ —k Fx0 . (3) F is sa id t o be continuous i f X ^ —• x0 in D im-p l i e s F x „ , - ~ * F x e . (4) F i s sa id to be completely continuous on D i f for each bounded subset M c D ; FCM) i s compact and F i s cont inuous on D • (5) F i s sa id t o be nonexpansive mapping on D i f for eve-r y x , <y € D the re i s It Fx - F ^ It -6 II X - <& I • (6) F i s sa id t o be con t rac t ive mapping on D i f the re ex-i s t s C£ CO £ % < A) such t h a t for every x , ^ e -D we have II Rx - Fty II 6 <J, II X - <t̂  II * (7) F i s sa id t o be hemicontinuous mapping on D i f F i s contirtiousfrom each segment i n D t o weak topology in X . Def in i t ion 2« A Banach space X i s sa id t o be s t r i c t l y convex if II A x + (1 - A )<y- II < 1 for a l l A , 0 < X < 1 , 134 and a l l X , <ty e K with II * II » IIfy I * 4 • Def in i t ion 3 ( [ 6 1 , [7 J , C8J): A Banach space X i s sa id t o have Property (Tt^ ) i f the re e x i s t s a c o l l e c t i o n of f i n i t e dimensional subspaces F , F e A , such t h a t : (8) The c o l l e c t i o n { F j F € A ? i s d i r ec t ed by i n c l u s i o n . That i s , given any two elements F^ , f̂  € A , t h e r e e x i s t s a t h i r d one which contains bo th. (9) The union of a l l F , F € A i s dense in X . (10) Each F , F £ A i s the range of a continuous l i n e a r p r o j e c t i o n £ of norm -= 'I • Remark 1 : Hi lber t space (separable or n o t ) , Banach space with monotone Schauder bas i s and C C 0? 1 J have Proper ty ( ^ ) . (See [ 6 J , [ 7 J , [ 8 J ) » Def in i t ion 4 ( [6 J J7J , [ 8 J ) a) A gauge func t ion i s a r e a l -valued continuous func t ion /U defined in the i n t e r v a l K 0^00 ) such t h a t (11) (U CO) * 0 (12) JU»n> (U Ct) m co t"+O0 * (13) (U i s s t r i c t l y i n c r e a s i n g . b) The d u a l i t y mapping in X wi th a gauge func t ion (U /\K* i s a mapping J from X i n t o the se t 2 of a l l subse ts of X* such t h a t (14) 3.x •í,xl|.IX*l,IIX*l-(e*('lxl)J o(#fl Remark 2 ( [ 6 J , [ 8 J ) a) The s e t J.X i s non-empty, b) Let X be a Banach space with a s t r i c t l y convex dual space X* • Let J be the d u a l i t y mapping in X with • gauge runction (U, . Then the set J* ooaaiats of preci-sely one point, c) Let X be a Banach space with strictly convex dual space X* . Let J * X —* X** be the duality mapping with a gauge function (Cc and £ > 0 . Then D(tw)» fh(t) Dur , where (I is positive function of t . Definition 5: A Banach space X is said to have Proper-ty ( ^ ) * if (15) X i s r e f l e x i v e . (16) X* i s s t r i c t l y convex. (17) X has Property (JT f ) . (18) The dual i ty mapping <T in X with gauge function /vu i s weakly continuous. Remark 3, (C4J,C8J)r A Hilbert space, Jt^ (1 < <fL < 00) have Property ( 7f^ ) * ,The Banach space L ^ I 0,11, 1 < <fi> < 00 9 1%- 4s 1 has not Property ( ST^ ) * . Def ini t ion 6; Let K be a closed bounded convex subset of a Banach space X with Property {StJ)o An operator T: K~+ —> X i s sa id to he Galerkin approximable (or for short a G-operator) i f (19) E T *« K n F —> F i s continuous for a l l but a f i -n i te number of F 6 A (20) T has a f ixed point in K whenever there e x i s t s X € £ F for a l l but a f i n i t e number of F € A such that % T*F - xF . Remark 4 (C6j , t?J ,£8j ) Let K be a convex closed boun-ded subset of a Banach space X with Property (ft̂  ) • Let TT 5 K —* X be strongly (or weakly or completely) continu-ous. Then T i s G-operator. III» G-onerators. Lemma lo Let X be a Baaiach space with X* s t r i c t l y con-vex. Let J be a dual i ty mapping with a gauge function (LL • Suppose that T i s a hemicontinuous mapping of an open set M of X into X . Let u0 e M and nxro e X be such e l e -ments of X that for each U € M there is; (2l> C Tu-wc , J(U~<±0 )) >: 0 * Then W0 * TLL0 * Proof: (The proof i s s imilar to that of Lemma 1 C2J). For t > 0 7 se t u^ a? U0 + t (^L0' - Tuo ) ( t i s s u f f i c i e n t l y smal l ) . Replacing in £21) u by ^ ,we obtain according t o property c) of Remark 2 that ( TAJL^ - W0 7 J(ixJ;-Tuo )) & 0, i .e . (Ti^-Tu.+ TUt-ttt, JCiit-Tdi^)) * 0 CTU^Tu^^J(i^^Tu^))^Cv^-Tu01JC^^T^^ -**\\vr.-Tu0\\-(JLC\\VT.~TUJ) . By hemicontinuity of T the left -hand s ide goes t o 0 as t —> 0 . The right-hand s ide i s independent of t . Hence ur0 -Tu0 . Q.E.D. Theorem 2!. Let X be & Banach space with Property ( JT ) * , K a closed bounded convex subset of X , M 3 K an open subset of X , A ; K —> X a completely continuous (reap , s trongly continuous) operator and B : M —v X a contractive (reap , nonexpansive) mapping. Then T - * A -t- B i s G-operator. Proof t Condition (19) i s c lear , a) Let 4 be completely continuous, E contractive mapping and £ T x » £ A*XF + £ B>*xF =• *xf for a l l f€ A , For each * , a£. e M and F € A there i s CCI-£B).x~CI-£B)<^, JCx-<y,))>(1-£)lx-<y.tlp(tx-<yl) . Let be Of. € M a r b i t r a r y but f i xed . Using a s tandard argu-ment (see the proof of Propos i t ion 1 C 7J or Theorem IV.3 f8J) we can prove t h a t t h e r e e x i s t s the sequence {-*„/> tX^ £ ^ ? c c < »XF ', F € A J such t h a t For a l l n a t u r a l number n and for each *x € M we have (22) C C I - £ B > , x ~ C | - . £ B)«f, JC*x-/tf)) £->(1-q,)\\x~<y.\l (U CWx-fy I ) -Replacing in (22) x by x^ , we ob ta in C Pf - f̂  ) «• -&*>* ,* - <l- 5 > * ^ , J C ^ - ^ ) ) ^ <T • Because CI- £ B ) * ^ - * ^ C I - £. B ) ^ —> C I - B ) / ^ and J C x ^ - ^ ) - ^ J C x ^ - z y . ) we o b t a i n ( u - (t~ &)<#, J(xp-<y-))^0 for each / ^ e M # Using Lemma 1 we have, t h a t (23) C I - B)XC =- AJL . From (22) i t follows tha t 0£ Jtum />wfv(/\-<£)\\xm>~O(0\\ (U, CJI.X*,--X„ ") -^ i . e . , *X^ —V iXp . Since A i s continuous and (23) ho lds , we have t h a t C I — B ) X0 « A %X0 . This completes the proof of a ) . b) Let A be s t rong ly continuous and B nonexpansive mapping. The proof of t h i s par t i s analogous to t ha t of a ) . By s t rong ly c o n t i n u i t y of A we have AA =* Ao<0 and by (23) we obta in M - B ) X »- A * . Q.E.D. o 0 17. -f,ta<?fl ifrllrt The^remj, The fol lowing two theorems are due to D.G» de F igu-e i r e r o [ 6 J , £ 7 3 , [ 8 j : Theorem It Let K be a closed bounded convex subset #£ a Banach space X with Proper ty ( ^ ) . Let Ti K—* X be a G-operator defined in K . Assume tha t (2 4) 6 belongs t o the i n t e r i o r of K n F f for a l l but a f i n i t e number of F € A (25) For a l l but a f i n i t e number of F € A we have lj} C T.X - S t * ) #» 5 , f o r a l l X > 1 and a l l x € i 9 f < ^ n F ( c9 K i s boundary of K ) . Then T has a f ixed po in t . Theorem 4: Let K be a closed bounded convex subset of a Banach space X with Proper ty ( 7^ ) . Let T : K - * -X be a G-operator defined i n K • Assume tha t (24) i s f u l f i l l e d and (26) (Tx, J * ) * Hx II (C-WII* II ) , for al l * e & K . Then T has a f ixed point in K . Applying Theorem 3 and 4 to our r e s u l t s (Sect ion I I I ) we obtain Immediately the fol lowing f ixed-poin t theorems. Theorem 5: Let X be a Banach space with Proper ty ( ^ /* , K a closed bounded convex subset of X , M 3 K an open subset of X , A 1 K —> X a completely continuous ( r e sp . s t r o n g l y continuous) operator and B ; M —> X a con-t r a c t i v e ( r e sp . nonexpansive) mapping. Set T"-» A •+• B • Assu-me t h a t (24) and (2$) (or (24) and (26)) are f u l f i l l e d . Then T has a f ixed point in K . Lftrrj^ 2 (£31): Let K be a closed convex bounded sub -se t of a Hi lber t space H • Then t h e r e e x i s t s an opera tor L ' H —> K such t h a t - 139 -(27) Lx - X for each x € K (28) II L * - L/y^ll <* \lx~ty || fo r each *x, ^ € H . Theorem 6 : Let K be a closed bounded convex subset of a Hi lber t space H , A : K —> H a completely con-t inuous ( r e sp . s t r ong ly continuous) operator and B J K-* H a con t rac t ive ( r e sp . nonexpansive) mapping. Set T ~ A + & . Assume tha t (2 4) and (25) (or (24) and (26)) are f u l f i l l e d with D « I ( I i s i d e n t i t y opera-t o r ) o Then T has a f ixed poin t i n K . Proof: For * 6 H , s e t B .x -» B L * . Using Theo-rem 5 we have M «• H and D i s con t rac t ive (reap* nonexpanaive) mapping© Theorem 5 and Lemma 2 proved Theorem Theorem 7: Let A , B and T have the same p r o p e r t i e s as in Theorem 6 . Assume t h a t T C ^ K ) c K . Suppose t h a t (2 4) ho lds . Then T has a f ixed point in K . Another f ixed-po in t theorem for sum of opera tors has been proved by W.V. Petryshyn llZJz Theorem 8: Let H be a complex Hi lber t space , P a h e -micontinuous mapping from H t o H such t h a t (29) t ( Rx ~ F<y, ,- * - ry ) I ^ /J II * - / y , II2" holds for every x and nj. € H and some constant fc >- 0 . Let S be a completely continuous mapping such t h a t C */fl ) -f S * - F 0 ? maps the b a l l t \ « -f.X;,X€ H J U I U / t / i n t o B^ . S e t T « I - F H- S . Then T has a f ixed point i n B* . • 140 -Remark 5 : M. Altman [13 has proved the fol lowing Theorem 9 : Let H be a separable Ki lber t space, F weakly closed ( i . e . i f oc^—-> Xc f ^X^ —x fy then y » - F*XP ) and maps un i t b a l l i n t o bounded sub3et of H . I f ( Rx, .* ) £ (*} x ) f o r each x with II * II » 4 , then F haa a f ixed po in t . I t was shown [11J , t h a t weakly compa c t and weakly c l o -sed mapping i s weakly cont inuous. Hence the assumptions of Altman s Theorem say t h a t F ia weakly cont inuous. V. Examples Let H be a Hi lber t apace , K u n i t b a l l , T : K~* K such t h a t T a» A • S> • The author inves t iga ted the ques t ion concerning the f i -xed point of T , when A and B are from the c l a s s of map-pings which conta ins s t rong ly cont inuous, completely con t inu-ous, weakly cont inuous, nonexpansive and con t rac t ive opera-t o r a . From next examples i t fo l lows t h a t T has f ixed point p roper ty only i f A i s completely continuous ( r e s p . 9 t rongly continuous) and B i s con t rac t ive ( r e s p . nonexpansive). ExfiP-Ple X Ceee C 5 J , [ l 0 j ) : Let H be a aeparable Hi lber t 9pace > { ^ ; #l« OfHt ±2,>»} be an orthoJiormal bas i s for H and define the t ransformat ions A and E as fo l lows: A .x . C I - I I * ID'V-O • - 141 -Set T * r A + B> . T K c K and T has no f ixed point in K • A i s nonexpansive, completely continuous and B i s weatkly continuous and nonexpansive. Example 2 : Set Ax~ 4 (1 - II X 1 ) ^ and B as in Example 1. Then T » A + B transforms K i n t o K and has no f ixed point in K . A i s completely continuous and con t rac t ive and B i s weakly continuous and nonexpansive. ExampJe 3-t set A t x « | (1 - H x I ) nfr + | B x (B i s from Example l ) and B^ .X = J B x * This example shows t h a t A^ i s con t r ac t i on and B, i s con-t r a c t i o n and weakly continuous, T » A^ -*- B maps K i n -t o K and has no f ixed point in K • R e f e r e n c e s t i l M. ALTimN: Fixed Point Theorem in Hi lber t Space .Bul l . Acad .Pol .Sc i .5 (1957) ,1 ,19-29 . t 2 ] F.E. BRCMDER: V a r i a t i o n a l Boundary Value Problems for Quas i - l inea r E l l i p t i c Equa t ions , I I I .P roc .Na t . Acad.Sci .U.S.A.50(1963),5,794-798. C3) F.E. BROfVDER: Problemes Non-line'a i r e s .Lea Presses de 1 'Univers i ty de Montreal (1966). [4) F.E. BRCWDER - D.G. de FIGUEIREDO: J-monotone non l ine -ar opera tors in Banach spaces.Konkl .Nederl . Acad.Wetensch.69(1966),412-420. t5 ) J» CRONIN: Fixed Points and Topological Degree in Non-l i n e a r Analysis.Am.Math.Soc.1964. t 6 ] D.G. de FIGUEIREDO: Fixed-Point Theorems fo r Nonlinear Operators and Galerkin Approximations.Journ. Dif f .Eq.3 (1967) ,2 ,271-281. - 142 -[?3 D.G. de FIGOEIREDO: Some Remarks on Fixed Point Theo-rems f o r Nonlinear Operators i n Banach Spaces , Lecture S e r i e s , U n i v e r s i t y of Maryland 1967. [8] D.G. de FIGUEIREDO: Topics in Nonlinear Func t iona l Ana-l y s i s . L e c t u r e S e r i e a , U n i v e r s i t y of Maryland 1967* [93 R . I . KA&UROVSKIJ, M.A. KRASNOSELSKU, P .P . ZABREJKO-Ob odnom pr inc ipe nepodvi2noj toSk i d l j a ope ra to -rov v gilbertovom pros t rans tve .Funkc iona lny j a n a l i z i evo p r i l o 2 e n i j a , 1 ( 1 9 6 7 ) , 1 , 9 3 - 9 4 . [10] S. KAKUTANIt T op o l . proper t i e s of the u n i t sphere i n Hi lbe r t space.Proc.Imp.Acad.Tokyo,19(1943), 269-271. t i l ] J . KOLOlrf: A note on the con t inu i ty p r o p e r t i e a of non-l i n e a r oper at or s . Comment. Math. Univ. Car o l i nae 8 , 3(1967),503-514. [12] W.V. PEFRYSHYN: Remarks on Fixed Point Theorema and Their Ext ens ions .Trans . Am. Math. Soc. 126 (1967) , 1,43-53. (Received November 29,1967) 143 -

Fixed point theorems for sum of nonlinear mappings

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Fixed point theorems for sum of nonlinear mappings

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