In. J. Nonlinar Anal. Appl. 4 (213) No. 1, 7-2 ISSN: 28-6822 (lcronic) hp://www.ijnaa.smnan.ac.ir On Ψ-Condiional Asympoic Sabiliy of Firs Ordr Non-Linar Marix Lyapunov Sysms G. Sursh Kumar a, B. V. Appa Rao a, M. S. N. Murhy b, a Dparmn of Mahmaics, Konnu Lakshmaiah Univrsiy, Grn Filds, Vaddswaram-522 52, Gunur D., Andhra Pradsh, India. b Dparmn of Mahmaics, Acharya Nagarjuna Univrsiy, Nagarjuna Nagar 52251, Gunur, Andhrapradsh,,India. (Communicad by M. Eshaghi Gordji) Absrac W provid ncssary and sufficin condiions for Ψ-condiional asympoic sabiliy of h soluion of a linar marix Lyapunov sysm and sufficin condiions for Ψ-condiional asympoic sabiliy of h soluion of a firs ordr non-linar marix Lyapunov sysm X = A()X + XB() + F (, X). Kywords: Fundamnal Marix, Ψ-Boundd, Ψ-Sabl, Ψ-Condiional Asympoic Sabl. 2 MSC: 34D5, 34C11. 1. Inroducion Th imporanc of marix Lyapunov sysms, which aris in a numbr of aras of conrol nginring problms, dynamical sysms, and fdback sysms ar wll known. In his papr, w focus our anion o sudy of Ψ-condiional asympoic sabiliy of soluions of h firs ordr non-linar marix Lyapunov sysm X = A()X + XB() + F (, X) (1.1) as a prurbd sysm of X = A()X + XB(), whr A(), B() ar squar marics of ordr n, whos lmns ar ral valud coninuous funcions of on h inrval R + = [, ) and F (, X) is a coninuous squar marix of ordr n on R + R n n, (1.2) Corrsponding auhor Email addrsss: drgsk6@gmail.com (G. Sursh Kumar), bvardr21@gmail.com (B. V. Appa Rao), drmsn22@gmail.com (M. S. N. Murhy ) Rcivd: January 212 Rvisd: January 213
8 Sursh Kumar, Appa Rao and Murhy such ha F (, O) = O (zro marix), whr R n n dnos h spac of all n n ral valud marics. Th coninuiy of A, B and F nsurs h xisnc of a soluion of (1.1). Akinyl [1] inroducd h noion of Ψ-sabiliy and his concp xndd o soluions of ordinary diffrnial quaions by Consanin [3]. Lar Marchalo [1] inroducd h concp of Ψ-(uniform) sabiliy, Ψ-asympoic sabiliy of rivial soluions of linar and non-linar sysm of diffrnial quaions. Th sudy of condiional asympoic sabiliy of diffrnial quaions was moivad by Coppl [4]. Furhr, h concp of Ψ-condiional asympoic sabiliy o non-linar Volrra ingrodiffrnial quaions wr sudid by Diamandscu [5]. Rcnly, Mury and Sursh Kumar [[11], [12],[13]] xndd h concp of Ψ-bounddnss, Ψ-sabiliy and Ψ-insabiliy o Kronckr produc marix Lyapunov sysm associad wih firs ordr marix Lyapunov sysms. Th purpos of his papr is o provid sufficin condiions for Ψ-condiional asympoic sabiliy of (1.1). W invsiga condiions on h wo fundamnal marics of X = AX, X = B T X (1.3) (1.4) and F (, X) undr which h soluion of (1.1) or (1.2) ar Ψ-condiionally asympoically sabl on R +. Hr, Ψ is a coninuous marix funcion. Th inroducion of h marix funcion Ψ prmis o obain a mixd asympoic bhavior of h soluions. This papr is wll organizd as follows. In scion 2, w prsn som basic dfiniions, noaions, lmmas and propris rlaing o Kronckr produc of marics and Ψ-condiionally asympoically sabiliy, which ar usful for lar discussion. In Scion 3, w obain ncssary and sufficin condiions for Ψ-condiionally asympoic sabiliy of soluions of linar marix Lyapunov sysm (1.2). Th rsuls of his scion illusrad wih suiabl xampls. In scion 4, w obain sufficin condiions for h Ψ-condiional asympoic sabiliy of (1.1). This papr xnds som of h rsuls of Diamandscu [5] o marix Lyapunov sysms. Th main ool usd in his papr is Kronckr produc of marics. 2. Priliminaris In his scion w prsn som basic dfiniions, noaions and rsuls which ar usful for lar discussion. L R n b h Euclidan n-dimnsional spac. Elmns in his spac ar column vcors, dnod by u = (u 1, u 2, u 3,..., u n ) T ( T dnos ranspos) and hir norm dfind by u = max{ u 1, u 2, u 3,..., u n }. For A = [a ij ] R n n, w dfin h norm A = sup u 1 Au. I is wll-known ha n A = max { a ij }. 1 i n j=1 O n dno h zro marix of ordr n n and n is h zro vcor of ordr n. Dfiniion 2.1. [8] L A R m n and B R p q hn h Kronckr produc of A and B wrin A B is dfind o b h pariiond marix a 11 B a 12 B... a 1n B A B = a 21 B a 22 B... a 2n B...... a m1 B a m2 B... a mn B is an mp nq marix and is in R mp nq.
On Ψ-Condiional Asympoic Sabiliy of...4 (213) No. 1,7-2 9 Dfiniion 2.2. [8] L A = [a ij ] R m n, hn h vcorizaion opraor V c : R m n R mn, dfind and dno by A.1 a 1j A.2 Â = V ca =.., whr A a 2j.j =. (1 j n).. A.n a mj Lmma 2.3. [6] Th vcorizaion opraor V c : R n n R n2, is a linar and on-o-on opraor. In addiion, V c and V c 1 ar coninuous opraors. Rgarding propris and ruls for vcorizaion opraor and Kronckr produc of marics w rfr o [8]. L Ψ k : R + (, ), k = 1, 2,... n, b coninuous funcions, and l Ψ = diag[ψ 1, Ψ 2,..., Ψ n ]. Thn h marix Ψ() is an invribl squar marix of ordr n, for all R +. Dfiniion on R +. 2.4. [5] A funcion φ : R + R n is said o b Ψ- boundd on R + if Ψ()φ() is boundd Exnd his dfiniion for marix funcions. Dfiniion 2.5. [6] A marix funcion F : R + R n n is said o b Ψ boundd on R + if h marix ( funcion ΨF is boundd on ) R + i.., sup Ψ()F () <. Dfiniion 2.6. [5] Th soluion of h vcor diffrnial quaion x = f(, x) (whr x R n and f is a coninuous n vcor funcion) is said o b Ψ-sabl on R +, if for vry ɛ > and any R +, hr xiss a δ = δ(ɛ, ) > such ha any soluion x of x = f(, x), which saisfis h inqualiy Ψ( )( x( ) x( )) < δ(ɛ, ) xiss and saisfis h inqualiy Ψ()( x() x()) < ɛ, for all. Ohrwis, is said ha h soluion x() is Ψ-unsabl on R +. Exnd his dfiniion for marix diffrnial quaions. Dfiniion 2.7. Th soluion of h marix diffrnial quaion X = F (, X) (whr X R n n and F is a coninuous n n marix funcion) is said o b Ψ-sabl on R +, if for vry ɛ > and any R +, hr xiss a δ = δ(ɛ, ) > such ha any soluion X of X = F (, X), which saisfis h inqualiy Ψ( )( X( ) X( )) < δ(ɛ, ) xiss and saisfis h inqualiy Ψ()( X() X()) < ɛ, for all. Ohrwis, is said ha h soluion X() is Ψ-unsabl on R +. Dfiniion 2.8. [5] Th soluion of h vcor diffrnial quaion x = f(, x) is said o b Ψ- condiionally sabl on R + if i is no Ψ-sabl on R + bu hr xiss a squnc {x m ()} of soluions of x = f(, x) dfind for all such ha lim Ψ()x m() = Ψ()x(), uniformly on R +. m If h squnc {x m ()} can b chosn so ha lim Ψ() (x m() x()) = n, for m = 1, 2, 3,..., hn x() is said o b Ψ-condiionally asympoically sabl on R +.
1 Sursh Kumar, Appa Rao and Murhy W can asily xnd his dfiniion for marix diffrnial quaions. Dfiniion 2.9. Th soluion of h marix diffrnial quaion X = F (, X) is said o b Ψ- condiionally sabl on R + if i is no Ψ-sabl on R + bu hr xiss a squnc {X m ()} of soluions of X = F (, X) dfind for all such ha lim Ψ()X m() = Ψ()X(), uniformly onr +. m If h marix squnc {X m ()} can b chosn so ha lim Ψ() (X m() X()) = O n, for m = 1, 2, 3,..., hn X() is said o b Ψ-condiionally asympoically sabl on R +. Rmark 2.1. I is asy o s ha if Ψ() and Ψ 1 () ar boundd on R +, hn h Ψ-sabiliy, Ψ-boundd and Ψ-condiionally asympoically sabiliy implis classical sabiliy, bounddnss and condiional asympoic sabiliy. Th following lmmas play a vial rol in h proof of main rsul. Lmma 2.11. [? ] For any marix funcion F R n n, w hav 1 n Ψ()F () (I n Ψ()) ˆF () Ψ()F (), for all, R +. Lmma 2.12. [7] Th marix funcion X() is a soluion of (1.1) on h inrval J R + if and only if h vcor valud funcion ˆX() = V cx() is a soluion of h diffrnial sysm ˆX () = (B T I n + I n A) ˆX() + G(, ˆX()), (2.1) whr G(, ˆX) = V cf (, X), on h sam inrval J. Dfiniion 2.13. [7] Th abov sysm (2.1) is calld h corrsponding Kronckr produc sysm associad wih (1.1). Th linar sysm corrsponding o (2.1) is ˆX () = (B T I n + I n A) ˆX(). (2.2) Lmma 2.14. Th soluion of h sysm (1.1) is Ψ-unboundd on R + if and only if h soluion of h corrsponding Kronckr produc sysm (2.1) is I n Ψ-unboundd on R +. Proof. I is asily sn from Lmma 5 of [6] and Lmma 2.12. Lmma 2.15. Th soluion of h sysm (1.1) is Ψ-unsabl on R + if and only if h corrsponding Kronckr produc sysm (2.1) is I n Ψ-unsabl on R +. Proof. I is asily sn from Lmma 7 of [7]. Lmma 2.16. Th soluion of h sysm (1.1) is Ψ-condiionally asympoically sabl on R + if and only if h corrsponding Kronckr produc sysm (2.1) is I n Ψ-condiionally asympoically sabl on R +.
On Ψ-Condiional Asympoic Sabiliy of...4 (213) No. 1,7-2 11 Proof. Suppos ha h soluion of h sysm (1.1) is Ψ-condiionally asympoically sabl on R +. From Dfiniion 2.9, w hav ha h soluion X() of (1.1) is Ψ-unsabl and hr xiss a squnc of soluions X n () of (1.1) on R + such ha and lim Ψ()X m() = Ψ()X(), uniformly on R + (2.3) m lim Ψ() (X m() X()) = O n, for m = 1, 2, 3,. (2.4) Sinc X() is a Ψ-unsabl soluion of (1.1), from Lmmas 2.12 and 2.15, w hav ha ˆX() is I n Ψ-unsabl soluion of (2.1) on R +. Now applying vcorizaion(vc) opraor o (2.3) and (2.4), w hav and lim (I n Ψ()) ˆX m () = (I n Ψ()) ˆX(), uniformly on R + (2.5) m ( ˆX()) lim (I n Ψ()) ˆXm () = n 2, for m = 1, 2, 3,. (2.6) From Dfiniion 2.8, ˆX() is In Ψ-condiionally asympoically sabl on R +. Convrsly suppos ha, h soluion of (2.1) is I n Ψ-condiionally asympoically sabl on R +. From Dfiniion 2.8, w hav ha h soluion ˆX() of (2.1) is I n Ψ-unsabl and hr xiss a squnc of soluions ˆX m () of (2.1) on R +, which saisfis (2.5) and (2.6). Sinc ˆX() is a I n Ψ- unsabl soluion of (2.1), again from Lmmas 2.12 and 2.15, w hav ha X() = V c 1 ˆX() is a Ψ-unsabl soluion of (1.1) on R +. By applying Vc 1 opraor o (2.5) and (2.6), w hav ha h squnc of soluions X m ()=Vc 1 ˆXm () of (1.1) saisfying (2.3) and (2.4). Thus, from Dfiniion 2.9 h soluion X() of (1.1) is Ψ-condiionally asympoically sabl on R +. Lmma 2.17. L Y () and Z() b h fundamnal marics for h sysms (1.3) and (1.4) rspcivly. Thn h marix Z() Y () is a fundamnal marix of (2.2). Proof. I is asily sn from Lmma 2.4 of [13]. Thorm 2.18. L A(), B() and F (, X) b coninuous marix funcions on R +. If Y (), Z() ar h fundamnal marics for h sysms (1.3), (1.4) rspcivly and P 1, P 2 ar non-zro supplmnary projcions, hn ˆX() = + (Z() Y ())P 1 (Z 1 (s) Y 1 (s))g(s, ˆX(s))ds (Z() Y ())P 2 (Z 1 (s) Y 1 (s))g(s, ˆX(s))ds (2.7) is a soluion of (2.1) on R +. Proof. I is asily sn ha ˆX() is h soluion of (2.1) on R +.
12 Sursh Kumar, Appa Rao and Murhy 3. Linar Marix Lyapunov Sysms In his scion, w prov ncssary and sufficin condiions for h Ψ-condiional asympoic sabiliy of h linar marix Lyapunov sysm (1.2). Th rsuls of his scion ar illusrad wih suiabl xampls. Thorm 3.1. Th linar marix Lyapunov sysm (1.2) is Ψ-condiionally asympoically sabl on R + if and only if i has a Ψ-unboundd soluion and a non-rivial soluion W () such ha lim (I n Ψ())Ŵ () = n 2. Proof. Suppos ha h soluion of linar marix Lyapunov sysm (1.2) is Ψ-condiionally asympoically sabl on R +. From Lmmas 2.12 and 2.16 wih F = O n, i follows ha h soluion of (2.2) is I n Ψ-condiionally asympoically sabl on R +. From Thorm 3.1 of [5], w hav ha h linar sysm (2.2) has an I n Ψ-unboundd soluion and a non-rivial soluion Ŵ () such ha (3.1) saisfid. Sinc (2.2) has a I n Ψ-unboubd soluion and from Lmmas 2.12 and 2.14, h linar sysm (1.2) has a Ψ-unboundd soluion. Sinc Ŵ () is a non-rivial soluion of (2.2), hn W ()=Vc 1 Ŵ () is h corrsponding non-rivial soluion of (1.2). Convrsly suppos ha (1.2) has a las on Ψ-unboundd soluion on R + and a las on non-rivial soluion W () xiss and saisfis (3.1). From Lmma 2.14 and Thorm 3.1 of [5], i follows ha h soluion Ŵ () of (2.2) is I n Ψ-condiionally asympoically sabl on R +. Again from Lmmas 2.12 and 2.16, i follows ha h soluion of (1.2) is Ψ-condiionally asympoically sabl on R +. Exampl 3.2. Considr h linar marix Lyapunov marix sysm (1.2) wih ( 1 ) A = +1 1 1 and B =. 2 +1 Thn h fundamnal marics of (1.3) and (1.4) ar + 1 Y () = 1 and Z() = 2 +1 L Ψ() =. Clarly, ( + 1) ( + 1) X() = 2 +1 2 +1 is a soluion of (1.2) ( and Ψ()X() = + 1) 2,. Thrfor, X() is a Ψ-unboundd soluion ( + 1) 2 of (1.2). L W () =. Clarly, W () is a non-rivial soluion of (1.2) and +1 (I 2 Ψ())Ŵ () = 1 +1 2 +1 ( + 1) 3 +1. Also, lim (I 2 Ψ())Ŵ () = 4. From Thorm (3.1), h linar sysm (1.2) is Ψ-condiionally asympoically sabl on R +. Th condiions for Ψ-condiional asympoic sabiliy of (1.2) can b xprssd in rms of fundamnal marics of (1.3) and (1.4) in h following horms. (3.1)
On Ψ-Condiional Asympoic Sabiliy of...4 (213) No. 1,7-2 13 Thorm 3.3. L Y () and Z() b h fundamnal marics of (2.4) and (2.5). Thn h linar marix Lyapunov sysm (1.2) is Ψ-condiionally asympoically sabl on R + if and only if h following condiions ar saisfid; (i) hr xiss a projcion P 1, such ha (Z() Ψ()Y ())P 1 is unboundd on R +. (ii) hr xiss a projcion P 2 O n 2 such ha lim (Z() Ψ()Y ())P 2 = O n 2. Proof. Suppos ha h linar sysm (1.2) is Ψ-condiional asympoic sabl on R +. From Lmmas 2.12 and 2.16 wih F = O n, h Kronckr produc sysm (2.2) is I n Ψ-condiionally asympoically sabl on R +. From Lmma 2.17 and Thorm 3.2 of [5], i follows ha h fundamnal marix S() = Z() Y () of (2.2) saisfis h following condiions; 1. hr xiss a projcion P 1 such ha (I n Ψ())S()P 1 is unboundd on R +. 2. hr xiss a projcion P 2 O n 2 such ha lim (I n Ψ())S()P 2 = O n 2. Subsiu S() = Z() Y () in (1) and (2) and simplifying wih h us of Kronckr produc propris, w hav ha h fundamnal marics of (1.3) and (1.4) saisfis condiions (i) and (ii). Convrsly suppos ha, h fundamnal marics of (1.3) and (1.4) saisfis h condiions (i) and (ii). From Thorm 3.2 of [5], Lmma 2.12 and propris of Kronckr producs, h corrsponding Kronckr produc sysm (2.2) is I n Ψ-condiionally asympoically sabl on R +. Again from Lmmas 2.12 and 2.16, h linar sysm (1.2) is Ψ-condiionally asympoically sabl on R +. Exampl 3.4. In Exampl 3.2, aking Ψ() =. Thr xiss wo non-zro projcions I2 O P 1 = 2 O2 O and P O 2 O 2 = 2 2 O 2 I 2 such ha + 1 (Z() Ψ()Y ())P 1 = 2 +1 and (Z() Ψ()Y ())P 2 = 3 ( + 1) +1 Clarly, (Z() Ψ()Y ())P 1 is unboundd on R + and (Z() Ψ()Y ())P 2 O 4 as. Thrfor, from Thorm 3.3 h sysm (1.2) is Ψ- condiionally asympoically sabl on R +.
14 Sursh Kumar, Appa Rao and Murhy A sufficin condiion for Ψ-condiional asympoically sabiliy is givn by h following horm. Thorm 3.5. If hr xis wo supplmnary projcions P 1, P 2 (P i O n 2, i=1, 2) and a posiiv consan L such ha h fundamnal marics Y () and Z() of (1.3) and (1.4) saisfis h condiion (Z() Ψ()Y ())P 1 (Z 1 (s) Y 1 (s)ψ 1 (s)) ds + (Z() Ψ()Y ())P 2 (Z 1 (s) Y 1 (s)ψ 1 (s)) ds L (3.2) for all, hn, h linar quaion (1.2) is Ψ-condiionally asympoically sabl on R +. Proof. From Thorm 3.2, Thorm 3.1 and Lmma 2.2 of [12], w hav ha h condiions in Thorm 3.3 ar saisfid. Thrfor, h Kronckr produc sysm (2.2) is I n Ψ-condiionally asympoically sabl on R +. From Lmmas 2.12 and 2.16 wih F = O n, h linar sysm (1.2) is Ψ-condiionally asympoically sabl on R +. Exampl 3.6. Considr h linar marix Lyapunov sysm (1.2) wih A = I 2 and B = I 2, hn h fundamnal marics of (1.3) and (1.4) ar Y () = I 2 and Z() = I 2. L ( ) 1 Ψ() = +1, P 1 = 1 and P 2 = 1. 1 Now and (Z() Ψ()Y ())P 1 (Z 1 (s) Y 1 (s)ψ 1 (s)) = s (Z() Ψ()Y ())P 2 (Z 1 (s) Y 1 (s)ψ 1 (s)) = s + 1 s. + 1 Thrfor, h condiion (3.2) saisfid wih L = 2. Thus, from Thorm 3.5, h linar sysm (1.2) is Ψ-condiionally asympoically sabl on R +. 4. Non-Linar Marix Lyapunov Sysms In his scion, w prov sufficin condiions for h Ψ-condiional asympoic sabiliy of h non-linar marix Lyapunov sysm (1.1). Thorm 4.1. Suppos ha: 1. Thr xis supplmnary projcions P 1, P 2 (P i O n 2, i=1, 2) and a consan L > such ha h fundamnal marics Y (), Z() of (1.3), (1.4) saisfis h condiion (3.2). 2. Th funcion F (, X) saisfis h inqualiy Ψ() (F (, X()) F (, Y ())) ξ() Ψ() (X() Y ()), for and for all coninuous and Ψ-boundd marix funcions X, Y : R + R n n, whr ξ() is a coninuous nonngaiv boundd funcion on R + such ha ξ() M, for all, whr M is a posiiv consan.
On Ψ-Condiional Asympoic Sabiliy of...4 (213) No. 1,7-2 15 3. p = nml < 1. Thn, all Ψ-boundd soluions of (1.1) ar Ψ-condiionally asympoically sabl on R +. Proof. L X() b h soluion of (1.1) wih X( ) = X, hn by Lmma 2.12, ˆX() is h uniqu soluion of Kronckr produc sysm (2.1) wih ˆX( ) = ˆX. W pu { } S = ˆX : R+ R n2 : ˆX is coninuus and In Ψ boundd on R +. Dfin a norm on h s S by ˆX S = sup (I n Ψ()) ˆX(). I is wll-known ha (S,. S ) is a Banach spac. For ˆX S, w dfin (T ˆX)() = (Z() Y ())P 1 (Z 1 (s) Y 1 (s))g(s, ˆX(s))ds (Z() Y ())P 2 (Z 1 (s) Y 1 (s))g(s, ˆX(s))ds,. From Lmma 2.11 and hypohsis (2), i follows ha (I n Ψ())G(, ˆX) = (I n Ψ()) ˆF (, X) For v, w hav v Ψ()F (, X) ξ() Ψ()X() nm (I n Ψ()) ˆX(), R + and ˆX R n2. (Z() Y ())P 2 (Z 1 (s) Y 1 (s))g(s, ˆX(s))ds v I n Ψ 1 () (I n Ψ())(Z() Y ())P 2 (Z 1 (s) Y 1 (s)) (I n Ψ 1 (s))(i n Ψ(s))G(s, ˆX(s)) ds v Ψ 1 () (Z() Ψ()Y ())P 2 (Z 1 (s) Y 1 (s)ψ 1 (s)) (I n Ψ(s))G(s, ˆX(s)) ds v nm Ψ 1 () (Z() Ψ()Y ())P 2 (Z 1 (s) Y 1 (s)ψ 1 (s)) (I n Ψ(s)) ˆX(s) ds
16 Sursh Kumar, Appa Rao and Murhy pl 1 Ψ 1 () sup (I n Ψ()) ˆX() v (Z() Ψ()Y ())P 2 (Z 1 (s) Y 1 (s)ψ 1 (s)) ds. From h hypohsis (1), h ingral (Z() Ψ()Y ())P 2 (Z 1 (s) Y 1 (s)ψ 1 (s)) ds is convrgn. Thus, h opraor (T ˆX)() xiss and is coninuous for. For ˆX S and, w hav (I n Ψ())(T ˆX)() + + nm +nm Thrfor, (I n Ψ())(Z() Y ())P 1 (Z 1 (s) Y 1 (s)) (I n Ψ 1 (s))(i n Ψ(s))G(s, ˆX(s))ds (I n Ψ())(Z() Y ())P 2 (Z 1 (s) Y 1 (s)) (I n Ψ 1 (s))(i n Ψ(s))G(s, ˆX(s))ds (Z() Ψ()Y ())P 1 (Z 1 (s) Y 1 (s)ψ 1 (s)) (I n Ψ(s))G(s, ˆX(s)) ds (Z() Ψ()Y ())P 2 (Z 1 (s) Y 1 (s))ψ 1 (s)) (I n Ψ(s))G(s, ˆX(s)) ds (Z() Ψ()Y ())P 1 (Z 1 (s) Y 1 (s)ψ 1 (s)) (I n Ψ(s)) ˆX(s) ds (Z() Ψ()Y ())P 2 (Z 1 (s) Y 1 (s))ψ 1 (s)) p sup (I n Ψ()) ˆX(). T ˆX S p ˆX S. (I n Ψ(s)) ˆX(s) ds
On Ψ-Condiional Asympoic Sabiliy of...4 (213) No. 1,7-2 17 Thus, T S S. On h ohr hand, for Û, ˆV S and, w hav (I n Ψ())[(T Û)() (T ˆV )()] + nm +nm (Z() Ψ()Y ())P 1 (Z 1 (s) Y 1 (s)ψ 1 (s)) (I n Ψ(s))[G(s, Û(s)) G(s, ˆV (s))] ds (Z() Ψ()Y ())P 2 (Z 1 (s) Y 1 (s))ψ 1 (s)) (I n Ψ(s))[G(s, Û(s)) G(s, ˆV (s))] ds (Z() Ψ()Y ())P 1 (Z 1 (s) Y 1 (s)ψ 1 (s)) (I n Ψ(s))[Û(s) ˆV (s)] ds (Z() Ψ()Y ())P 2 (Z 1 (s) Y 1 (s))ψ 1 (s)) (I n Ψ(s))[Û(s) ˆV (s)] ds ( ) { nm sup (I n Ψ())[Û() ˆV ()] (Z() Ψ()Y ())P 1 (Z 1 (s) Y 1 (s)ψ 1 (s)) ds } + (Z() Ψ()Y ())P 2 (Z 1 (s) Y 1 (s))ψ 1 (s)) ds p sup (I n Ψ())[Û() ˆV ()]. I follows ha sup Thus, w hav (I n Ψ())[(T Û)() (T ˆV )()] p sup (I n Ψ())[Û() ˆV ()]. T Û T ˆV S p Û ˆV S. Thrfor, T is a conracion mapping on (S,. S ). Now, for any funcion Ŵ S, w dfin an opraor SŴ : S S, by h rlaion SŴ ˆX() = Ŵ () + (T ˆX)(), R +. By Banach conracion principl SŴ has fixd poin in S. Thrfor, for any Ŵ S, h ingral quaion ˆX = Ŵ + T ˆX (4.1) has a uniqu soluion ˆX S. Furhrmor, by h dfiniion of T, ˆX() Ŵ () is diffrniabl and ( ˆX() Ŵ () = (B T () I n + I n A()) ˆX() Ŵ ()) + G(, ˆX()).
18 Sursh Kumar, Appa Rao and Murhy From (4.1), if Ŵ () is a I n Ψ-boundd soluion of (2.2) if and only if ˆX() is a In Ψ-boundd soluion of (2.1). Thus, (4.1) sablishs a on-o-on corrspondnc bwn h I n Ψ-boundd soluions of (2.1) and (2.2). Now, w considr analogous quaion W g ˆX = Ŵ + T ˆX. (1 p) ˆX ˆX S Ŵ Ŵ S. (4.2) Now, w prov ha, if ˆX, Ŵ S ar I n Ψ-boundd soluions of (2.1) and (2.2) rspcivly such ha hy saisfy (4.1), hn ( lim (I n Ψ()) ˆX() Ŵ ()) =. (4.3) For a givn ɛ >, w can choos 1 such ha p ˆX S < ɛ 2, for 1. Morovr, sinc lim (I n Ψ())(Z() Y ())P 1 =, hr xiss a numbr 2 1 such ha pl 1 (Z() Ψ()Y ())P 1 ˆX 1 S P 1 (Z 1 (s) Y 1 (s)ψ 1 (s)) ds < ɛ 2, 2. For 2, w hav (I n Ψ()) ˆX() Ŵ () = (I n Ψ())(T ˆX)() + nm +nm (Z() Ψ()Y ())P 1 (Z 1 (s) Y 1 (s)ψ 1 (s)) (I n Ψ(s))G(s, ˆX(s)) ds (Z() Ψ()Y ())P 2 (Z 1 (s) Y 1 (s))ψ 1 (s)) (I n Ψ(s))G(s, ˆX(s)) ds (Z() Ψ()Y ())P 1 (Z 1 (s) Y 1 (s)ψ 1 (s)) (I n Ψ(s)) ˆX(s) ds (Z() Ψ()Y ())P 2 (Z 1 (s) Y 1 (s))ψ 1 (s)) (I n Ψ(s)) ˆX(s) ds pl 1 (Z() Ψ()Y ())P 1 ˆX 1 S P 1 (Z 1 (s) Y 1 (s)ψ 1 (s)) ds
On Ψ-Condiional Asympoic Sabiliy of...4 (213) No. 1,7-2 19 +nm ˆX S 1 (Z() Ψ()Y ())P 1 (Z 1 (s) Y 1 (s)ψ 1 (s)) ds +nm ˆX S (Z() Ψ()Y ())P 2 (Z 1 (s) Y 1 (s)ψ 1 (s)) ds ɛ 2 + nml ˆX S < ɛ 2 + ɛ 2 = ɛ. Now, w prov ha, if ˆX() is a In Ψ-boundd soluion of (2.1), hn i is I n Ψ-unsabl on R +. Suppos ha ˆX() is I n Ψ-sabl on R +. From Dfiniion 2.6, for vry ɛ > and any R +, hr xiss a δ = δ(ɛ, ) > such ha any soluion X() of (2.1), which saisfis h inqualiy (I n Ψ( ))( X( ) ˆX( )) < δ(ɛ, ) xiss and saisfis h inqualiy (I n Ψ())( X() ˆX()) < ɛ, for all. L u R n2 b such ha P 1 u = n 2 and (I n Ψ())u < δ(ɛ, ) and l X() b h soluion of (2.1) wih h iniial condiion X() = ˆX() + u. Thn (I n Ψ())u() < ɛ, for all, whr u() = X() ˆX(). Now considr h funcion w() = u() T u(),. Clarly, w() is a I n Ψ-boundd soluion of (2.2) on R +. Wihou loss of gnraliy, w can suppos ha Z() Y () = I n 2. I is asy o s ha P 1 w() = n 2. If P 2 w() n 2, hn from Lmma 2.3 of [12], w hav lim sup (I n Ψ())(Z() Y ())P 2 w() = lim sup (I n Ψ())w() =, which is conradicion o w() is I n Ψ-boundd on R +. Thus, P 2 w() = n 2 and hnc w() = n 2, for. I follows ha u = T u and u = n 2 (T is linar), which is a conradicion. Thus h soluion ˆX() is I n Ψ-unsabl on R +. L Ŵ = ˆX T ˆX. From Thorm 3.5 and Dfiniion 2.8, hr xiss a squnc {Ŵm} of soluions of (2.2) on R + such ha and lim m (I n Ψ())Ŵm() = (I n Ψ())Ŵ (), uniformly onr + ) lim (I n Ψ()) (Ŵm () Ŵ () = n 2, for m = 1, 2, 3,. L ˆX m = Ŵm + T ˆX m. From (4.2), i follows ha h squnc { ˆX m } of soluions of (2.1) on R + such ha lim (I n Ψ()) ˆX m () = (I n Ψ()) ˆX(), uniformly onr +. m Thrfor, h soluion ˆX() of (2.1) is I n Ψ-condiionally sabl on R +. From (4.3) and ( ˆX()) ) ) ˆXm () = ˆXm () Ŵm() + (Ŵm () Ŵ () + (Ŵ () ˆX(). I follows ha ( ˆX()) lim (I n Ψ()) ˆXm () = n 2, for m = 1, 2, 3,. Thus, h soluion ˆX() of (2.1) is I n Ψ-condiionally asympoically sabl on R +. From Lmma 2.16, h soluion X() = V c 1 ˆX() of (1.1) is Ψ-condiionally asympoically sabl on R+. Hnc h sysm (1.1) is Ψ-condiionally asympoically sabl on R +.
2 Sursh Kumar, Appa Rao and Murhy Exampl 4.2. Considr h non-linar marix Lyapunov sysm (1.1) wih ( 1 ) ( 1) A() = 1, B() = ( +1) +1 1 and F (, X) = 1 sin x1 () x 2 (). + 5 x 3 () sin x 4 () Th fundamnal marics of (1.3) and (1.4) ar ( ) Y () = 1 and Z() = +1. +1 L Ψ() =. + 1 Thn hr xi wo projcions O2 O P 1 = 2 I2 O and P O 2 I 2 = 2 2 O 2 O 2 such ha h fundamnal marics Y () and Z() of (1.3) and (1.4) saifis (3.2) wih L = 2. On h ohrhand, condiion (ii) of Thorm 4.1 is saisfid wih ξ() = 1, for and +5 M = 1. Also, p = nml = 2 1 5 5 2 = 4 < 1. Thrfor, h non-linar sysm (1.1) is Ψ-condiionally 5 asympoically sabl on R +. Rfrncs [1] O. Akinyl, On parial sabiliy and bounddnss of dgr k, Ai. Accad. Naz. Linci Rnd. Cl. Sci. Fis. Ma. Naur., 65 (1978) 259-264. [2] C. Avramscu, Asupra comporării asimpoic a soluţiilor unor cuaţii funcţional, Anall Univrsiăţiidin Timiş oara, Sria Şiinţ Mamaic-Fizic, 6 (1968) 41-55. [3] A. Consanin, Asympoic propris of soluions of diffrnial quaions, Anall Univrsiăţii din Timişoara, Sria Şiin ţ Mamaic, 3 (1992) 183-225. [4] W. A. Coppl, On h sabiliy of ordinary diffrnial quaions, Journal of London Mahmaical Sociy, 39 (1964) 255-26, doi:1.1112/jlms/s1-39.1.255. [5] A. Diamandscu, On h Ψ-condiional asympoic sabiliy of h soluions of a nonlinar volrra ingrodiffrnial sysm, Elcronic Journal of Diffrnial Equaions, 29 (1984) 1-13. [6] A. Diamandscu, Ψ-boundd soluion for a Lyapunov marix dirnial quaion, J. App. Num. Mah., 17 (29) 1-11. [7] A. Diamandscu, On Ψ-sabiliy of non-linar Lyapunov marix dirnial quaions, Elcronic Journal of Qualiaiv Thory of Diffrnial Equaions, 54 (29) 1-18. [8] A. Graham, Kronckr Producs and Marix Calculus: Wih Applicaions, Ellis Horwood Ld. England, 1981. [9] T. G. Hallam, On asympoic quivalnc of h boundd soluions of wo sysms of diffrnial quaions, Mich. Mah. Journal, 16 (1969) 353-363, doi:1.137/mmj/129319. [1] J. Morchalo, On Ψ L p -sabiliy of nonlinar sysms of diffrnial quaions, Anall Şiinţific al Univrsiăţii Al. I. Cuza Iaşi, Mamaică, 3 (199) 353-36. [11] M. S. N. Mury and G. Sursh Kumar, On Ψ-Bounddnss and Ψ-Sabiliy of Marix Lyapunov Sysms, Journal of Applid Mahmaics and Compuing, 26 (28) 67-84, doi: 1.17/s1219-7-7-2. [12] M. S. N. Mury, G. Sursh Kumar, P. N. Lakshmi and D. Anjanyulu, On Ψ-insabiliy of non-linar Marix Lyapunov Sysms, Dmonsrio Mahmaica 42 (29) 731-743. [13] M. S. N. Mury and G. Sursh Kumar, On Ψ-Boundd soluions for non-homognous Marix Lyapunov Sysms on R, Elcronic Journal of Qualiaiv Thory of Diffrnial Equaions, 62 (29) 1-12.