(ho, h)- Boundedness of the solutions of differential systems with impulses

Publicacions Ma emá iques, Vol 34 (1990), 225-239 . (h o , h)-BOUNDEDNESS OF THE SOLUTIONS OF DIFFERENTIAL SYSTEMS WITH IMPULSES Abs ac G .K . KULEV AND D .D . BAINOV In he p esen pape he ques ion o boundedness o he solu ions o sys- ems o di e en ial equa ions wi h impulses in e ms o wo measu es is conside ed . In he in es iga ions piecewise con inuous auxilia y unc ions a e used which a e an analogue o he classical Lyapuno 's unc ions . The ideas o Lyapuno 's second me hod a e combined wi h he newes ideas o he heo y o s abili y and boundedness o he solu ions o sys ems o di e en ial equa ions . 1 . In oduc ion Sys ems o di e en ial equa ions wi h impulses ep esen a na u al appa a- us o ma hema ical simula ion o eal p ocesses and phenomena s udied in biology, physics, con ol heo y, e c . Fo ins ance, i he popula ion o a gi en species is egula ed by some impulsi e ac o s ac ing a ce ain momen s, hen we ha e no easons o expec ha he p ocess will be simula ed by egula con- ol . On he con a y, he solu ions mus ha e jumps a hese momen s and he jumps a e gi en be o ehand . Mo eo e , he ma hema ical heo y o he sys ems o di e en ial equa ions wi h impulses is much iche han he espec i e heo y o sys ems wi hou impulses . Tha is why in he ecen yea s his heo y is an impo an ield o nume ous in es iga ions ([1]-[7]) . The usage o classical Lyapuno 's unc ions in he s udy o he s abili y and boundedness o he solu ions o sys ems o di e en ial equa ions wi h impulses ia Lyapuno 's second me hod cons ic s he pliabili y o he me hod . The ac ha he solu ions o such sys ems a e piecewise con inuous unc ions shows ha i is necessa y o in oduce analogues o Lyapuno 's unc ions which ha e discon inui ies o he i s kind . The in oduc ion o such unc ions malces he applica ion o Lyapuno 's second me hod o sys ems wi h impulses much mo e e icien ([1]-[6]) . In he p esen pape he boundedness o he solu ions o sys ems o di e - en ial equa ions wi h impulses in he e ms o wo measu es is s udied . In he The p esen in es iga ion is pa ially suppo ed by he Minis y o Cul u e, Science and Educa ion o People's Republic o Bulga ia unde G an 61 . 22 6  G .K . KULEV, D .D . BAINOV in es iga ions piecewise con inuous Lyapuno 's unc ions a e used which a e combined by he newes ideas o he heo y o s abili y and boundedness o he solu ions o sys ems o di e en ial equa ions . The main esul s gene alize heo ems o Yoshizawa [8] and Ha a, Yoneyama, Sai oh, Hi ano [9] . Conside he ollowing sys em o di e en ial equa ions wi h impulses whe e E .C[ 8+ x Rn, R n ], TR E C[Rn, R], IR E C[Rn, Rn] and Ox/ = R(x) _ x( +) - x( _) . Le ú E R+ and xo E R n . Deno e by x( ; ú, xo) he solu ion o sys em (1) which sa is ies he ini ial condi ion x( ó ; ú, xo) = xo and by J + ( o, xo) deno e he maximal in e al o he o m ( o,w) in which he solu ion x( ; o,xo) is de ined . The solu ions x( ) = x( ; ú, xo) o sys em (1) a e piecewise con inuous unc- ions wi h poin s o discon inui y o he i s lcind, Le . a he momen R when he in eg al cu e o he solu ion mee s he hype su ace he ollowing ela ions hold 2 . P elimina y . no es and de ini ions x = ( ,x), :~ TR(x) ; Ox/ = R(z) = IR(x), aR = {( , x) E R+ X Rn : = TR(X)} X ( -) = x( R), Ox/ = R = x( R) - x( _ ) = IR(x( R)) . Hence o h we shall always assume ha o all x E Rn he ollowing ela ions a e alid 0 < TI(x)-< T2 (x) < . . . < TR(x) < ... and lim - R(x) = 00 R-oo and he in eg al cu e o any solu ion x( ) = x( ; ú, xo) o sys em (1) mee s each hype su ace R a mos once [7] . In he u he conside a ions we shall use he ollowing classes o unc ions : K = {u E C[R+, R+] : o is s ic ly inc easing and u(0) = 0} CK = . {u E C[ 8+,R + ] : Q( ,) E K o any E R + } = {hEC[R+xR n ,R+ ] : in  h( ,x)=0 o any ElR + } DIFFERENTIAL SYSTEMS WITH IMPULSES  227 De ini ion 1 . Le h o , h E 1' . We say ha he solu ions o sys em (1) a e : a) (ho, h)-equibounded i (Va > 0)(V o E R+)(3,a = p( o, a) > 0)(Vxo E Rn, ho( o, xo) < a) (V > o) : h( , x( ; o, xo)) < /~ . b) (h o , h)-uni o mly bounded i he numbe 0 o a) does no depend on o E R+ . c) h-ul ima ely bounded o bound B i (V( o,xo) ER + x R n )( 3 T =T( o,xo) > 0)(V > o +T) h( , x ( ; o, xo )) < B . d) (ho, h)-equi-ul ima ely bounded o bound B i (Va > 0)(V o E R+)( 3 T = T( o, a) > 0)(Vxo ER n , ho( o, xo) < a) (V > o + T) : h( , x( ; o , x o )) < B . e) (h o , h)-uni o mly ul ima ely bounded o bound B i he numbe T o d) does no depend on o E R+ . De ini ion 2 . Le he unc ion A : R + ---> R+ be measu able . We say ha A( ) is in e  ally posi i e i l A( ) d = co whene e I - U [ai, oi], al < /o ; < i=1 al+1 and Ni-a¡> _8>0 . We shall in oduce he class Vo o pieceiwse con inuous auxilia y unc ions which a e an analogue o Lyapuno 's unc ions [3] . Le 7-0(x) = 0 o xE Rn . Conside he se s GR = {( , x) E IR + X Rn : 7- R-1(X) < < TR(x)} and De ini ion 3 . We say ha he unc ion V : R + x Rn -+ R + belongs o he class Vo i V( ,x) is con inuous in G, locally Lipschi z con inuous wi h espec o x in any o he se s GR and o ( o, xo) E Q 'R, R = 1,2, . . . he e exis he limi s V( ~ ,xo) =  lim  V( ,x)  ,  V( ó ,xo) =  lim  V( ,x) ( ,x)-( o,xo)  ( ,x)-i o,xo) ( , x)EGR  ( ,x)EGR+1 and, mo eo e , he equali y V ( o , xo) =V ( o, xo) holds . Le V E Vo . Fo ( , x) E G de ine he unc ion 00 G= UGR R=1 U(1)( , x) = lim Sup 1 [V( + h, x + h ( , x1) - V ( , x)] . h_o+ h 00 22 8  G .K . KULEV, D .D . BAINOV o :~ R whe e R = TR(x( R)) . Le h, ho E I' and V, W E Vo .  Fo he sake o b e i y o he o mula ion o he main esul s we shall make a lis o some condi ions o be used in he o mula ion o he subsequen heo ems . A . I o he solu ion x( ; o, xo) o sys em (1) he e exis s bo > 0 such ha h( , x( ; o, xo)) _< bo < oo o each E T+( o, xo), hen x( ; o, xo) is de ined in he in e al ( o, oo) . B1 . The unc ion V is h- adially unbounded . B2 . V(1)( ,x) < 0 o ( ,x) E G . B3 . V( 1 ) ( , x) < -CV( , x) o ( , x) E G whe e C > 0 is a cons an . B4 .  V (l )( , x) < -A( )C(h( , x)) o ( , x) E G whe e A( ) is in eg ally posi i e and C E K_ B5 . V( 1 ) ( , x) < -C(W( , x)) + A( )O(V( , x)) o ( , x) E G whe e C(y) is nonnega i e and con inuous in R and (2)  lim in C(y) > 0 A( ) is nonnega i e and con inuous in R + and We shall no e ha i x = x( ) is a solu ion o sys em (1), hen V( 1 )( , x( )) = D+V( , x( )) = lim sup 1 [V( -}- h, x( + h)) - V( , x( ))] n-o+ h De lni ion 4 . Le ho, h E I' . The unc ion V E Vo is called : a) h- adially unbounded i he e exis s a unc ion a EK, a(y) -> oo as y --> oo and such ha V ( +, x) > a(h( , x)) o ( , x) E i8 + x Rn . b) ho-dec escen i he e exis b > 0 and a unc ion b E K such ha V( +, x) <_ b(ho( ,x)) o ho( ,x) < b . c) weakly h o -dec escen i he e exis ó > 0 and a unc ion b E CK such ha om ho( , x) < b i ollows ha V( +, x) < b( , ho( , x)) . -y-00 100', A ( ) d < co O(u) is posi i e and con inuous in U8 and 5) 10,>0 du O(u) B6 . The e exis s a cons an K such ha V( ,x)>K o any( ,x)ER+xR' B7 . V( +, x + IR(x)) < V( , x) o ( , x) E UR, R= 1, 2 . . . . DIFFERENTIAL SYSTEMS WITH IMPULSES  229 C1 . jWl l l( ,x)j < p( )w(W( ,x» o ( ,x) E G whe e p( ) is nonnega i e and con inuous in R+ and (6)  p(T) d- <_ m( - s) o >_ s >_ 0 9 whe e m(y) E K and w(u) is posi i e and con inuous in IFB and °° (7F)  du = oo . w (u) C2 . W l l( , x) < p( )w(W( , x)) o ( , x) E G whe e p( ) and w(u) a e he unc ions o condi ion C1 . C3 . The e exis s a unc ion m E K such ha o >_ s >_ 0 and o any piecewise con inuous in [s, ] unc ion u(T) wi h poin s o discon inui y o he i s kind R such ha R = TR(u( R)) a which u(T) is con inuous om he le , he ollowing inequali y holds (8)  J W (1) (- , u( , )) d- 1 < m( - s) . s Theo em 1 . Le¡ condi ion (A) hold and unc ion V E Vo exisi o which condi ions Bl, B2 and B7 hold . Then he solu ions o sys em (1) a e : P oo . Since V is h- adially unbounded, hen he e exis s a unc ion aE K, a(y) --> oo as y --> oo and such ha C4 . W( +, .x + IR(x)) = W( , x) o ( , x) E oR . C5 . W( ,x) is h- adially unbounded . 3 . Main esul s 1 . (ho, h)-equibounded i V is weakly ho-dec escen . 2 . (ho, h)-uni o mly bounded i V is h o -dec escen . V( + , x) > a(h( ,x)) o ( , x) E R+ x Rn 1 . I V is weakly ho-dec escen , hen he e exis óo > 0 and a unc ion b E CK such ha (10),  V( +, x) < b( , ho( , x)) o ho( , x) < 6o Le a > 0 and o E R + (a < 6 o ) be gi en . Choose 3 = i( o ,cY) > 0 so ha (11)  a(0) > b( o,a) 230  G .K . KULEV, D .D . BAINOV Le xo E R n , ho( o,xo) <_ a and le x( ) = x( ; o,xo) . Se ( ) = V( , x( )) . Since V( , x) is locally Lipschi z con inuous in any o he se s GR, hen om B2 i ollows ha D -1- ( ) <_ 0 o EJ+ ( o, xo), 9~ R whe e R = TR(x( R)) . F om B7 i ollows ha ( R) < ( R) . Tha is why he unc- ion ( ) is dec easing in he in e al J+( o ,xo) . Then om (9), (10) and (11) we ge a(h( , x( )) < ( +) < ( ) < ( ó) < b( o , h o ( o , xo)) < b( o , a) < a(0) o E J+( o, xo) which implies ha h( , x( )) < ,i .  F om condi ion (A) i ollows ha J+( o, xo) = ( o, oo) . Thus 1, is p o ed . 2 . I V is ho-dec escen , hen (10) and (11) hold o some unc ion b E K independen o . Hence he numbe Q can be chosen independen o o and so ha o ho( o, xo) <_ a we ha e h( , x( » <0 . This shows ha he solu ions o sys em (1) a e (ho, h)-uni o mly bounded . Theo em 1 is p o ed . Co olla y 1 . Le condi ion (A) hold and unc ion U E Vo exis which is h- adially unboundedand such ha ú (j) ( , x) _< A( )O(U( , x)) o ( , x) E G whe e ¡he unc ion A( ) is nonnega i e and con inuous in R + and 0 00 A( ) d < oo and O(u) is posüi e and con inuous in H and oo du/«u) = oo, U( +, x + IR(x» < U( , x) o ( , x) E oR, R = 1, 2 . . . . Then ¡he solu ions o sys em (1) a e : 1 . (ho, h)-equibounded i U is weakly ho-dec escen . 2 . (ho, h)-uni o mly bounded i U is ho-dec escen . P ooL . I is immedia ely e i ied ha he unc ion V( , x) = exp ~- J A(s) ds + ~P(U( , x)) } , ( , x) E R+x Rn , JJJ 0 whe e $(u) = o du/O(u) sa is ies he condi ions o Theo em 1 . Theo em 2 . Le condi ion (A) hold and a unc ion V E Vo exis which is weakly ho-dec escen and o which condi ions Bl, B3 and B7 hold . Then he solu ions o sys em (1) a e (h o , h)-equi-ul ima¡ ely bounded . P oo . F om Theo em 1 i ollows ha he solu ions o sys em (1) a e (ho, h)- equibounded . Hence each solu ion x( ) = x( ; o,xo) o (1) is de ined in he in e al ( o, oo) . DIFFERENTIAL SYSTEMS WITH IMPULSES  23 1 Since V is h- adially unbounded, hen he e exis B > 0 and a E K, a(y) -> oo as y -> oo such ha (12)  V( +, x) > a(h( ,x)) o h( ,x) > B Since V is weakly h o -dec escen , hen he e exis b o > 0 and b E CK such ha (10) holds . Le a> 0 and o E R+ be gi en, x o E Rn be such ha ho( o,xo) <_ a and le x( ) = x( ; o , x o ) . F om B3 and B7 we ob ain (13)  V( ,x( )) < V( ó,x o )exp[-C( - o)] o > o . Se T = T( o,a) > c ln[b( o,a)/a(B)] . Then om (12) and (13) i ollows ha o > o +T he ollowing inequali ies hold Theo em 3 . Le condi ion (A) hold and a unc iou V E Vo exis which is h o -dec escen and o which coudi ions Bl, B/, and B7 hold . Then he solu ions o sys em (1) a e (ho, h)-uui o mly ul ima ely bounded . P oo . F om Theo em 1 i ollows ha he solu ions o sys em (1) a e (ho, h)- uni o mly bounded . Hence each solu ion x( ) = x( ; o,xo) o (1) is de ined in he in e al ( o , oo) . Since V is h- adially unbounded, hen he e exis R> 0and a EK, a(y) -~ oo as y --~ oo such ha (14)  V( + , x) > a(h( ,x)) ol . la( ,x) > R (15)  V( + , x) < b(ho( , x)) o ho( , x) < 6o . Choose B >_ R so ha a(B) > b(R) . Le a >_ R be gi en . VVe shall p o e ha he e exis s T= T(a) > 0 such ha o any solu ion x( ) = x( ; ú , xo) o sys em (1) o which h o ( o , x o ) < a and o some (E [ o , o + T] he ollowing inequali y holds (16) a(h( ,x( )) < V( +,x( +)) < V(T,x( )) < < V( ó ,x o )exp[-C( - o)] < b( o,ho( o,xo))exp( -CT)< a(B) Hence h( ,x( )) < B o > o +T . Theo em 2 is p o ed . Since V is ho-dec escen , hen he e exis óo > 0 and b E K such ha ho«,x«)) < R 232  G .K . KULEV, D .D . BAIIVOV Suppose ha his is no ue . Then o any T > 0 he e exis s a solu ion x( ) = x( ; o, xo ) o (1) o which h o ( o, x o ) < a and such ha o all E [ o , o + T] we ha e (17)  ho( ,x( )) > R F omB4 and B7 i ollows ha (18)  V( , x( )) - V( o , xo) < ~  V(1> (s, x (s)) ds < o Bu he unc ion V( , x( » is mono onely dec easing in Hence he e exis s he limi (19)  lim V( , x( )) =V o > 0 Then om (15), (17), (18) and (19) we ob ain F om he ha Then ~~ A( )C(h o ( , x( )) d < b(R) - Vo o o+T  b(R) - V o + 1 ~( ) d >  C(R) 0 -  A(s)C(ho(s, x(s») ds, > o o he in eg al ( o, oo) . in eg al posi i i y o -A( ) i ollows ha he e exis s T > 0 such J 00  o+T b(R) - V o > 1, o A( )C(ho( , x( )) d >  A( )C(ho( , x( )) d > o o +T >_ C(R)  A( ) d > b(R) - V o + 1 . o The con adic ion ob ained shows ha he e exis s T= T(n) > 0 such ha o any solu ion x( ) = x( ; o,xo) o (1) o which ho( o,xo) < a, he e exis s ( E [ o, o + T] such ha (16) holds . Then o >_ ( (hence o any > o +T oo) he ollowing inequali ies hold a(h( ,x( )) < V( +,x( +)) < V( ,x( )) < V« + , x«+» < < b(ho«,x«)) < b(R) < a(B) . Hence he solu ions o sys em (1) a e (ho, h)-uni o mly ul ima ely bounded o bound B . (21) (22) (23) DIFFERENTIAL SYSTEMS WITH IMPULSES  23 3 Theo em 4 . Le¡ condi ion (A) hold and unc ions V, W E Vo exis o which condi ions B5, B6, B7, Cl, C!, and C5 hold . Then : 1 . V is h- adially unbounded . 2 . The solu ions o sys em (1) a e h-ul ima ely bounded . 3 . I V is weakly ho-dec escen , hen he solu ions o sys em (1) a e (ho, h)- equibounded . 4 . I V is h o -dec escen , hen ¡he solu ions o sys em (1) a e (ho, h)-uni- o mly bounded . P oo . 1 . Assume ha he asse ion is no ue . Then he e exis s N o > 0 such ha o any y > 0 he e exis T E R+ and xE Rn o which h(7-, - 7) > y and such ha V(T+,x) < No . F om (2) i ollows ha he e exis R l > 0and ó > 0 such ha o any y>R l weha eC(y)>6 . Le L= 'O'> A( ) d and M = sup{O(u) : K < u < <D - 1 (oD(No) + L) whe e <p (u) = o du/O(u) .  - F om C5 i ollows ha he e exis s a unc ion aE K, a(y) -> oo as y --> oo and such ha (20)  W( + , x) > a(h( ,x)) o ( , x) E R+ xRn F om (4) and he condi ion a(y) -> oo as y --> oo i ollows ha he e exis s R 2 > R l such ha a(R2) > R l and IPa(R2)  dy  N o - K + ML JR,  w(y) > m  6 In he abo e assump ion we eplace y by R2 . As a esul we ob ain ha he e exis o E IR + and x o ERn such ha h( o ,x o ) > R2 and V( ó ,xo) < No . F om condi ion (A) and om C5, Cl and C4 i ollows ha he solu ion x( ) = x( ; o, xo) o sys em (1) is de ined in he in e al ( o, oo) . F om B5 and B7 i ollows ha V i)( , x( )) _< A( )O(V( , x( )) o 7É R whe e R =TR(x( R)) and V( +,x( R)) < V( R,x( R)), whence by in eg a ion we ob ain ( ~(V( , x( )) - ~(V( ó , xo)) < ~`  Vc~>S, x (S)) ds <L o «V(s,x(s») Hence K <_ V( ,x( )) _< 4 -1 (-¿(N o ) - - L), whence we conclude ha g5(V( , x( ))) < M o > o . Assume ha W( , x( )) >R l o any > o .  Then om B5 and B7 i ollows ha V(i)( ,x( )) < -ó + MA( ) o > o , :~ R V( R, x( R)) C V( R, X( R)),