The Periodic Solutions To The Fourth Order Duffing Equations

According to the Newton's law of motion and thefluxionary calculus , the law of object movement canbe described by the differential equations in general.The universal phenomena occur in nature, such as thephenomena as large as the celestial body'smovement in cosmos like the cyclical revo-lution ofthe earth moving around the sun, and those as smallas the periodic oscillations caused by the electronictransmission, or the phenomena from the balancedtheory in economical field and the blood circle flowin life science. All these pheno-mena exhibit theperiodic and continuous motion. In order to probethis kind of motion rules,peop-le describe this kindof movements by means of diffe-rential equations,therefore, these differential equations should admitperiodic solutions.The periodic solutions of differential equ-ations arouses the keen interests of people not onlyfor their demonstration of the periodic movements,but also for their universality .This universality mayapproximately portray some non-periodic move-ment,especially the advancing of the issue of reso-nateswhich has affected the development of the qualitativetheory of differential equations since seventies oftwenty century.The basic theory studying object movement ismechanics in certain sense, and a large number ofissues about mechanics concern periodic orbits of theresonance and non-resonance differential equations.And the simplest model applied is the Duffingequation which signifies the spring vibra-tion.The Duffing equation is an important equationbelongs to the non-linear vibration mechanics, manyproblems of which can be reduced to solve the ques-tions of the Duffing equation.Many related scientific studies concerning theDuffing equation have been done and published in thedomestic and foreign magazines.In 1972,Lea-ch[13]proved that x ′ +g(t , x) =e(t ) possesses an unique 2πperiodic solution under conditions: g (t , x) =g( x),( ) ( ) ( )n + δ 2 ≤g′x≤n+1 ?δ2, n represents some naturalnumber,20 < δ <1, e (t + 2π ) =e(t ), e (t ) ∈ C( R), g (0 ) =0. Ressing[ 14 ], Ding Tongren, Ding Weiyue [ 15 ] ,etc. have also doneeven more deeper work sucessively.Ge Weigao [ 17 ] in 1988 discussed the 2π -periodicsolution of the n dimensional second order non-conservative,Duffing equation x ′′ +cx′+g(t , x) =e(t ).Li Yong and Wang Huaizhong [ 19 ]generalized theexistence results of periodic solution from the secondorder, Duffing equation to the high order Duffingequation in 1991.x ( 2 n ) + g( x) =e(t ) =e(t +2π). x ∈ R, (1)x ( 2 n ) + ?G( x) =E(t ) =E(t +2π), x ∈ Rm (2)? stands for function gradient, their mainresult is:Theorem 1 Suppose that g , e:R→R is a contin-uous function and there exist positive number M andε 0,such that:( ) ( )0120N 2 n + ε ≤x? gx≤N+1 n?ε, when x ≥ M,n is oddnumber.( ) ( )0120? N +1 2n +ε ≤x? gx≤?Nn?ε, when x ≥ M,n is evennumber.Where N is a non-negative integer.Then equation (1)has an unique 2π -cyclical solution.Theorem 2 suppose that G : Rn →R is a twice con-tinuous differentiable function,and there existspositive number ε 0,such that[N 2 n + ε 0]I ≤?2G≤[( N+1) 2n?ε0]I,when n is odd number.[? ( N +1) 2n +ε 0]I ≤?2G≤[? N2n?ε0]I, when n is evennumber.Where N is a non-negative integer, I is the unitmatrix of order m, ? 2G is the Hessian matrix of G. Thenthe equation (2) has an unique 2π -periodic solution.Concerning the odd-order Duffing equationx ( 2 n +1 ) +g( x) =e(t ) =e(t +2π) , (3)we have the following results:Theorem 3 Suppose that g ∈ c1 ( R), e∈c( R),and thereexist positive number ε 0,M such that( )M ≥ g′x≥ε0(或 ( )? M ≤g′x≤?ε0).Then the equation (3) has an unique 2π -periodicsolution.Ma Ruyun [ 20 ] in 1993 investigated the existenceproblem of periodic boundary value problem ofsemi-linear Duffing equation solution, certainlywithout requiring the value of ( )xg x situated betweentwo neighboring characteristic values.In 2003, Article[23] tried to study the unique existenceproblem of more general Duffing equation of higherorder:x ( 2 n ) + a1x( 2n?1 ) +a2x( 2n?2) +… a2n?1x′+g(t ,x) =e(t ), (4)x ( 2 n +1 ) + a0x( 2n) +a1x( 2n?1) +… a2n?1x′+g(t ,x) =e(t ), (5)x ( 2 n +1 ) + a2x( 2n?1) +a2x( 2n?3) +… a2n?1x′+g(t ,x) =e(t ), (6)where ai are constants(i=0,1,2,3…2n-1)g: R2→R is acontinuous function, andg (t + 2π ,x) =g(t ,x), e(t ) ∈C( R), e (t + 2π ) =e(t ),The major findings are:Theorem 4 Suppose that(K1)Exists constant d>0,such that when x ≥ d,g (t , x)x < 0,t∈R,(n is odd number),g (t , x)x > 0,t∈R,(n is even number);(K2) 0212422A 2 = ?a?a… ?an?>;(K3) = 21 ∫02 ( ) =0?P π πetdt,Then the equation (4) has an unique 2π -periodicsolution.For equation (5),if a 0 ≠0, without loss may assumea 0 >0,since any 2π -periodic solution x (t ) of equation(5)satisfies∫02 x 2 i 1(t )x (t )d t=0,i=0,1…nπ +,By using the methods similar to the proof of theorem4, we obtain the following conclusions:Theorem 5 Suppose that conditions (K1) and (K3) inTheorem 4 hold ,then when 0210222A 2 = a?a?… an??>,Equation (5) has an unique 2π -periodic solution.For equation (6) the following theorem is valid:Theorem 6 Suppose that( )J 1there exist positive constants h, α ,β,suchthat when x ≥ h,thenα ≤ x ? 1 g(t ,x) ≤β,t∈R(or ? α ≤x ? 1 g(t ,x) ≤?β,t∈R)( )J 2 B = 1 ?a1 ?a3… ?a2n?1>0Then equation (6) possesses an unique 2π -periodicsolution.Liu Wenbin, Li Yong[23] in 2003 investegated the highorder Duffing equation and proved the existence ofperiodic solution by means of metrology fundamentalwhich improved the existing results in certain extend.In this paper ,we consider the equation.(E) x ( 4 ) +x′+g( x) =e(t )where g : R1→ R1, e : R1→ R1 are continuous functions,and e (t + 2π ) ≡e(t ).Theorem 7 Suppose that when x →∞, g ( x) →∞. Thenthe equation (E) has an unique 2π -periodic solution.