| Nonconstant Periodic Solutions of Nonlinear Differential Equations (1982) | |||||||
Abstract | |||||||
| With prescribed period. The main part of this thesis consists of providing sufficient conditions for the existence of nonconstant periodic solutions of the second order differential system x"(t) + g(x(t)) =3D 0. For simplicity we take this prescribed period to be 2Pi. Conditions for the existence of nonconstant periodic solutions having a different prescribed period may be derived frc~ the conditions we give by a linear change of the scale. Our proof is based on methods that are different from those employed by the authors mentioned above. Their proofs were mainly based, e.g., on variational methods, classical Morse theory, and generalized Morse theory. 0ur treatment of the problem is given in two steps: first we use an alternative-method-type argument to reduce the problem to looking for zeros of a vector field for the Leray-Schauder type defined on a certain Sobolev space eleven periodic functions having mean zero. Second we show that the Leray-Schauder degree is different on a small ball centered at zero from that of a large ball centered at zero, implying the existence of a zero in between. In Chapter I we collect, without proofs, a few results in functional analysis which are needed in the succeeding chapters. We give the definition ef the topological degree of a mapping in finite-dimensional spaces and in finite-dimensional Banach spaces, and list some of its properties. In Chapter II we state and prove some preliminary lemmas concerning the spectral properties of certain nonselfadjoint compact linear operators. In Chapter III we state and prove our main result concerning the second order differential system. | |||||||
Publication details | |||||||
| |||||||