0000000000266568
AUTHOR
Ulrich Staude
Kleine periodische L�sungen bei nichtlinearen stark-elliptischen Systemen von partiellen Differentialgleichungen I
Strongly elliptic systems of nonlinear partial differential equations are considered in the case when the derivatives of the solutions occuring in the nonlinear terms have the same order as those in the linear principal part. The existence of periodic solutions for such systems is investigated. It is shown that this problem can be reduced to the study of algebraic bifurcation equations, whose small solutions correspond to the classical solutions of the given problem. A discussion of the bifurcation equations will be given in a forthcoming paper.
UNIQUENESS OF PERIODIC SOLUTIONS OF THE LIENARD EQUATION
This chapter analyzes the uniqueness of periodic solutions of the Lienard equation. It considers the Lienard equation = y − F ( x ) and y = − x where F (0) = 0 , F ( x ) ∈ Lip( R ). The chapter discusses the existence of periodic solutions. It highlights that the origin is the only stationary point of the system = y − F ( x ) and y = − x , and therefore all nontrivial periodic solutions must circle around the origin. The existence of at least one periodic solution is proved by constructing a Poincare–Bendixson domain. The chapter also emphasizes that to prove the uniqueness of periodic solutions, additional assumptions are also needed. In the literature there are numerous uniqueness results…