Konečný, Milan

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

Commentationes Mathematicae Universitatis CarolinaeMilan KonečnýRemarks on periodic solvability of nonlinear ordinary differential equationsCommentationes Mathematicae Universitatis Carolinae, Vol. 18 (1977), No. 3, 547--562Persistent URL: http://dml.cz/dmlcz/105799Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1977Institute 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 MAT HEM A T IC A E UNIVERSITATIS CAROLINAE 18,3 (1977) R E M A R K S ON PERIODIC SOLVABILITY OF NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS Milan KONECNY, Ost rava A b s t r a c t : We p rove the ex i s t ence of periodic solut ions for nonlinear ordinary differential equat ions of the Lienard type under the va r ious condit ions upon the nonl inear part of the c o n s i d e r e d differential op e ra to r . K e y w o r d s : Nonlinear ordinary differential equat ions .periodic p rob lems . A M S : 34 C 25 Ref. 2.: T.925.32 1. Introduction. Let n be a posi t ive integer and let T >• O. Denote C-^ the (Banach) s p a c e of all n - times cont inuously d i f fe ren t i a te and T - periodic functions defined on the real line R with the norm T h e following theorem is proved in f l j . In the s e q u e l k denat \ a posit ive in teger . Theorem 1 . Let a , , a be rea l numbers s u c h that v-í t - - ) 3 »a__2i«° 2 k - 2 j for j «* 1 , k - l . Let f and g be cont inuous r ea l va lued func t ions , (2) sup fg Ц } - M < f € R I 7 I - 547 S u p p o s e that the re exis ts r > O s u c h that * (\)\>0, f € R ,\\\>r or g (f)J<0, \eR , ty>r . Then for arbi trary y ^ C ™ s u c h that T (3) J y ( t ; dt - O 2k the re ex i s t s U 6 C - , verifying the Lienard equation (4) - ( - l ^ u ^ x ) - ^ u ( 2 k - 1 \ x ) + . . . . + a 2 k ^ 1 u ' ( x ) + f (u(x) )u(x)^Wx)) »y(x) 2k M o r e o v e r , a rbi t rary solution u ^ C _ sa t is f ies (5) ||u|| < J f C T w h e r e 1 " T 2 " k f - r t 3 2 T 2 ( 2JC) (Mt-||yfl „ ) . cT prom this resu l t it immediately follows : Theorem 2 . Let f, g : R—9*R be cont inuous functions, s u p p o s e (1), (2). T h e n for arbi t rary y ^ C T with T (6) l i m s u p g ( j ) < - i - ( y ( t ) d t < l i m i n f g (?) Y* O > 548 -2k \ t he re ex i s t s at l eas t one solution u ^ . C of (4 ) . M o r e o v e r , arbi t rary solution u £. C _ sat isf ies (5) with - i -k f - r + 3 2 T 2 ( 2*C) (sup I g ( I )| + I Y L Q ) X $€R* > U T where T g ( ţ ) - g ( ţ ; - 4" j У < U d t > O т y (x; - y ( x ) - JL f y ( t ) dt , a n d f or r 7 O it i s ei\)\>° * \\\>'X Theorem 3 . A s s u m e ( l ) . Let f, g ; R jr R b e cont inuous and s u p p o s e that t h e r e exist finite limits -im g ( I ) - g ( + - ) , lim g ( I ; - g ( -•*) . M o r e o v e r , s u p p o s e (7) min { g ( - ~ ) , g ( " * ) } < g ( $ K m a x [ g ( - *>) , g ( -> <* ) ] for e a c h i ( . R . T h e n the equation (4) h a s at l eas t o n e T - p e r i o d i c solution for y £ C ° if and only if T (8) m i n j g ( - ~ ) , g (•*»)] < - L j y ( x ; d x < m a x | g (-*») , g ( t - )l - 549 -F o r the proof of Theorem 1 the following proper ty (F) of the differential equation (4) is e s sen t i a l : ( F ) If y£. C , then the se t of all T - per iodic solut ions of the equation (4) is a bounded s u b s e t of the s p a c e C ° . Now let's p rove that the ( F ) is in a cer ta in s e n s e n e c e s s a r y and sufficient for the validity of the a s se r t i on of Theorem 3 . Theorem 4 . Let f , g : R—7R be con t inuous . S u p p o s e (2) and denote G = sup g (J) , G - Inf g ( f ) , jfeR $*R J =1 y ^ C ° ; the re exis ts a T - p e r i o d i c solution of (4) r T 7K-[ye C° ; G 4 J y ( x ) d x ^ ] ^ ~" O T • f t~ [ y € C T ; G«i J y ( x ) d x £ G J O A s s u m e that the s e t s g~ ( G ) , g~ ( G ) do not contain a nondegene ra t ed in terval . S u p p o s e that the differential equation (4) h a s the p roper -ty ( F ) . i ) T he n the re exist finite l imits iC***) » g ( - °°) and it i s - 550 -cPnrDm^ [a, G} . ii) If (fnjl)^M = { G ] , then we have g (+<*>) » G or g ( - 0 0 ) -_ G . iii) If ^ n t O M - { G ] , then we have g (+°°) - G or g (-<*>) - G . (iv)H (fnif[)\7il = p , then it is ( g ( -*>) = G , g (+<*?) = G g ( -~) < g ( { X s (+°°) > ^ R or ( g ( -«-0) . G , g (•*>) - G g (*°°) < g ( V < g ( -°°) « ^ R Proof, i ) S u p p o s e that g (?<*>) d o e s not ex i s t . Denote A - l i m s u p g(f) , B -* liminf g (fc) * —9 + 00 V i—•> + -« ' T h e n B < A , a n d t h e r e ex i s t s a s e q u e n c e oO *°° s u c h that 1 S n I . . . lim | »-> C > Jn - 1 * n _oo ) n Ц n ' A + B T h u s —--=— , a n d the c o n s t a n t functions fe a r e the T - periodic solut ions of (4) . T h i s i s in contradict ion with the property ( P ) . A n a l o g o u s l y we p r o v e the e x i s t e n c e of g ( - < a D; . . 551 -Let y ^ ( < P f . 7 Z ) \ W a n d le t , for exampl T Hjr J y ( x ) d x = G . 2k If u € C - is a solution of (4) , then we h a v e o T T j ( G - g ( u o ( x / ) / d x - 0 O and s i n c e g ( G ) d o e s not contain a nondegenera ted interval, u is a cons tan t function, thus y = G . o J T h e proofs of pa r t s (H) - ( i v , 1 a r e the c o n s e q u e n c e s of the part ^ i ; . Prom the Theorems 3 and 4 it immediately follows : Theorem 5 . S u p p o s e that f , g : R—>R a r e the / - 1 —" cont inuous funct ions, let g be bounded and let g ( G ) , g ( G ) do not contain a non degene ra t ed in terval . A s s u m e {!). T h e n the equation (4) h a s the proper ty ( F ) and J *fflii and 6nly if there exist g (+*>; , g ( - °°) and if ( 7 ) i s fulfilled . In this note we shal l deal with the T - p e r i o d i c solvability - 552 -of (4) in the case of nonexistence of Limits g (*«*) , g ( - oo ) , 2, Expansive nonlinearities. The following definition is a generalization of the so-called expansive function introduced in [3J. Definition. Let g : R —*R be a nontrivial bounded conti-nuous function. For p, q such that (9) inf g ( U < q ^ p< sup g {() f<-R } \^R > we put M = ( d€ R,3<xlt<*L2 £. R, 0 ^ r t l 1< / t 2 s u c h t n a t f o r S€ <^r ^2^ u i s s (^) > P a n d ior \e <-^^-^2 li is g ^ < q and d * / v / , u'} VJ U \d e R J/i- ^ £ R , O -$<\ < ^t such that for ^€</>l:Lt/vl2> ^ is g ( ^ ) < q and for ^£< - ^ 2 , - ^ 3 ) it is g q ) > P and d ^ - . ^ } • Denote f" w fi ) =- sup M p,q* s ' ғ p ,q expansive function. a ( g ; -* °° f or each p,q from (9), then g is called the 553 -Examples . (i; C (sin) «- min |*1t-2 (arcsin pj,3l-2|arcsin gj"l for each -1 <q<4 P <£,-- • (ii) The function ^k-1 2kTl g : ^ i—* sin ( ^ ) ( K is a positive integer ) is expansive . (iii) The function \ * * arctg . sm( ) is expansive. The main result of this section is the following. Theorem 6. Suppose (l) and let f ,g: R—^R be conti-nuous, g bounded. Then the equation (4; has at least one T - periodic solution if T a) y ^ C T ' q ^ - 4 - j y ( x ) d x 4 p O -i -k T T C h) 3 2T2(2*) ( s u P | g m - i fy(x)dx|tKy--|Jy(x)dx|| ^ C ^ g ) . \*R O O CT Proof. Suppose g q ) > p for \€ < / n 1 , ' n 2 > > g {\)<CL tor ]*<-\i'\> i - 554 -;i- -T2(2jc) ( s u p | g ( | ) - i ( y ( x ) d x | + | | y - i ( y ( x ) d x f l 0 M 2 ~ V f€R O O C T Define the function e by 5 p ,q y gp,Ч = ţ ' * ( W^ = S t _ V И S Є í - - . - ^ p > qц; - g( ~i2) it | e ^ j , - ) Obviously g p | q ^ ° ° ) > P and g p q ( - ^ ) < q . According to Theorem 2 t h e r e ex i s t s at l eas t one T - periodic solution u of the equation o ^ ( l O ) - ( - l ) k J 2 k \ x ) + a x u ^ Moreover, from (5) it follows - i -k T l | u | | < V 3 2 T 2 ( 2 l t ) ( s u p | g ( f ) - - i ( y ( x ) d x | - r + | | y - ф ( y ( x ) d x Ц o ) < " l 1 * П 2 -л l 1 -% c o u т From this we h a v e g ( u o ( x ) ; - g p q ( u o ( x ) ; , x €. R and thus the function u is a T-periodic solution of (4) . A s a corollary we obtain immediately : - 555 Theorem 7. S u p p o s e (1) and let f,g : R—>R be cont inuous a n d let g be e x p a n s i v e . The" m^ fc rt. - l - l _ If, moreover , the s e t s g (G) , g (G) a r e n o n e m p t y and do not contain n o n d e g e n e r a t e d intervals then f-mџ.o) If the s e t s g ( G ) , g ( G ) a r e empty then ?-m. 3. Mathematical pendulum equat ion. From the previous sect ion it follows that the equation 2 k - l 2 k + 1 u " ( x ) t sin ^ U ( X )) ' - y ( X ) p o s s e s s e s at l e a s t one T - p e r i o d i c solution if and only if T _ o . 1 y< X r e C ° , - l < i j y ( x ) d x < l or y - t i . Now we shal l c o n s i d e r the equation ( l l ) u " I x ; + g i u ( x ) ; - y i x ; 556 where g is a T - p e r i o d i c cont inuous function. T h e model for s u c h equation i s (12) u *' ( x )+ sin u ( x ; » f ( x j which h a s at l eas t one T - p e r i o d i c solut ion, provided the re ex is t s p£ . ( 0 , 1 ) s u c h that T - P 4 ~ - J y ( x ) d x ^ p O and 1_ T - T 2 ( l + , | ( j y ( x ; d x | + | y - ± f y ( x ) d x | | Q )<t- 2 a r c sin j O O C T ( s e e Theorem 6 and Example (i) ) . In the seque l l e t s s u p p o s e : a) g * C * , j g X ^ I ^ 1 -or ^ R , _2_ — —1 b) the s e t s g ( G ) , g (Cj) do not contain nondegenera ted inter-v a l s . It is obvious to s e e that <T> o J i s unbounded s u b s e t of C, т • J is c losed in C„ - 557 ( the las t a s se r t ion follows from the fact that we can cons ide r only s u c h solut ions of (11) for which | u ( O }|4T) . Theorem 8 . Let T€: ( O &) , y £ C and x e R • If u , u a r e T - p e r i o d i c solut ions of (11) s u c h that u . ( x ) » u _ ( x ) » c 1 K o ' 2 x o ' then u„ and \xn coincide on R . 1 2 2 Proof. Denote v. ( x ; *» u. ( x ; - c . T h e n v. e. C ( i = 1 ,2) satisfy the equation v " - t - g ( c - t v ) = y . T h e r e ex is t s a function I : R —y R s u c h that v ^ ( x ) - v 2 ( x ) - g ( c + v 2 ( x ) ) - g ( c i « v : L ( x ; ; - g ( ^ ( x ) ) ( v ^ x ) ^ (x ) ) a n d , moreover , the function g.( \ l * > M v 2 ( • ) - v x ( • ) ) i s con t inuous . T h u s x + T x * T J ( v ^ - v ^ x ) ; 2 d x - { J ( v - ( x ) - v 2 » ( x ) ) ( v 1 ( x ) - v 2 ( x ) ) d x l 4 - 558 x + T x +T r f 6 J в Ш * ) ) i v ( x , - v 2 ( x ) )2 d x ^ J ( v ( x ) - v 2 ( x ) )2 d x 4 X } X o o x. i T ^ ~ 2 j ° ( v i ( x ) - v 2 ' ( x ) )2 d x from which it folloM/s that x t T J ( v ^ x j - v ^ x j ^ d x - O This completes the proof. Theorem 9. Let T C ( O ,1t; and y ^ C T . Then the Dirichlet problem i u " ( x ) + g ( c + u( x ) ) »- y ( x , , x f c ( O . T ) u ( 0 ) - u ( T ) - 0 has a unique solution for arbitrary c € JR . (Ihe existence of at least one weak solution is possible to prove on the basis of the theorem on surjectivity of pseudomono-tone operators - see £2] . By meansofstandard regularity argumen-ts we obtain that arbitrary weak solution of (13) is classical. The unicity may be proved in the same way as in Theorem 8.)*. - 559 -In the s e q u e l we shal l s u p p o s e T 6 ( 0,1c). Denote by u the solution of (13) put u ( x J - c + u ( x - k T ) c,yN ' c,y N ' for x € ( k T , (k-v- 1 ) T̂ > ( k is an i n t e g e r ) . It i s e a s y to s e e that u is a T -pe r iod ic solution of (11) if and only if T T J g ( u c y l x ) ) d x - \ y ( x ) d x . O ' O Define the mapping (p : R x C_, > C by <j> ( c , y ) - u c , У ь It i s p o s s i b l e to p r o v e that (p ( • , ' ) i s cont inuous . Let y £ C,_ be fixed. Define T f( S ° Qp(c,y ) ) (x) dx . o * fv ••-T h e n Ф y : R (14) Г (15) T h e mapping 1 : ^ î f : c * R i s cont inuous 1" - per iodic function. Put P l y ) - max<6 ( c ; , C « R 1 y ft ( y ) - min 4• ( c ) . 0 C € . R l y R is upper cont inuous and R i s lower cont inuous . 560 -T h e main resul t is the following theorem the proof of which follows immediately from the prev ious cons ide ra t i ons . Theorem 1Q. Cons ider the differential equation (11) and let T €. ( 0 , X ; . Then T -. <P - {y^c^ ; ^ ( y ) 4 J y ( x ; d X ^ r ( y ) j , o where \ , V* a r e defined by the relat ions (14,}, (15) . Remark . It s e e m s that the better charac te r iza t ion of the se t J for the equation ^11) is an open problem up to now. In this direction it will be interest ing to g ive some further pro-per t i es of the functions J* , I , e . g . ] j ^ ( y ) - 4 0^» \ ( y ) . 561 [ll S . FUcTK and J . MA WHIN : Per iodic solut ions of some nonlinear differential equat ions of higher order . C a s . P e s t . Mat. lOO (1975; , 276 - 283 . Kz[ J . J . LIONS : Quelques méthodes de resolution d e s p rob lè -mes aux limites non l i néa i r e s , P a r i s 1969 . |*3l S . FUClK : Remarks on some nonlinear boundary va lue problems, Comment. Math. Univ. Carol inae 17 ( 1976 ; , 721 - 7 3 0 . V Y S O K A Š K O L A B A N S K Á O S T R A V A - P O R U B Á 7 0 8 33 ČESKOSLOVENSKO ( Oblátům 1 . 6 . 1977 ) - 562 -

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Remarks on periodic solvability of nonlinear ordinary differential equations

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Remarks on periodic solvability of nonlinear ordinary differential equations

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