The Lagrange Stability Of Solution For Nonlinear Equations

Duffing equation is a kind of classical equation. It has been studied for a long time. So far, there are lots of the results about the Lagrange stability.In1999,Bin Liu considered the following equation x"+n2x+φ(x)=p(t), p(t)∈C7(R/2πZ), where φ(x)∈C6(R). φ(±∞)=lim φ(x) exist and are finite. φ(x) satisfies the x→±∞flollowing polynomial growth condition when|∫2π0p(t)e-intdt|<2(φ(+∞)-φ(-∞)), he proved the above equation is ofLagrange stability. In2009,Bin Liu,Capietto,Dambrosio considered the following equation x"+V'(x)=p(t), p(t)∈C6(R/πZ),■max{-x,0). Obviously V has the exclusive singularity at-1,asymptotically reso-nance at+∞, and satisfies the polynomial growth condition. When∫π0p(s)eisds|<2,they proved the above equation is of Lagrange stability. If one removes the polynomial growth condition,super-linear Duffing equation may have unbounded solutions([LY]). However in[JPW],if w∈R+\Q satisfies the C. φ satisfies the conditions similar to (0.2). Lei Jiao, Daxiong Piao, Yiqian Wang proved the following equation x"+ω2x+φ(x)=Gx(x,t)+p(t) is of Lagrange stability. It is obviously that the potential function ω2x+φ(x)+Gx(x, t) does not satisfy the polynomial growth condition.In this paper, we intend to show further that the polynomial growth condition isn't necessary. We prove the following two kinds of equation are still of Lagrange stability even if they don't satisfy the polynomial growth condition.1, Duffing equation at resonance point x"+n2x+φ(x)+g"(x)q(t)=0, q(t)∈C13(R/2πZ), n∈N, where φ(x)∈C12(R) satisfies the conditions similar to (0.2). For k≤14,|g(k)(x)|≤C. It is obviously that the above equation is resonant, potentials function does not satisfy polynomial growth condition. Therefore, we must do more fine estimates and more canonical transformations to overcome the difficulty to obtain the Lagrange stability of the equation.2, Duffing equations with singularity x"+V'(x)=DxG(x,t), G(x,t)∈C11(R×R/πZ), where for x>-1,V(x)=(?)x+2+(?)-1,γ∈Z+.For k+l≤11,G(x,t)=∫0x G(s,t)ds satisfies|DxkDtlG(x,t)|≤C. We adopt the variable of the upper limit function technique, do some estimations which is different from the lemmas of Chapter2and do some canonical transforms to overcome the difficult which does not satisfy polynomial growth condition. We obtain the above equation is of Lagrange stability too.The paper is arranged as follows:In Chapter1, we give a KAM theorem and introduce the long history of Lagrange stability for Duffing equation. In Chapter2,3, we give the detailed proofs of the results mentioned above, respectively. In Chapter4, we give a simple proof of equation (0.1). Some technical lemmas can be find in the appendix.