| Solutions are shown to exist for a variety of differential equations. Both ordinary and partial differential equations are considered, with specified initial conditions, boundary conditions, or simultaneous initial and boundary conditions. A key feature of the these problems is a condition at infinity; it is demanded that solutions decay towards zero as the temporal variable becomes arbitrarily large. This feature removes from the problem a certain compactness property, which precludes the use of traditional methods which employ the Leray-Schauder topological degree.;This difficulty is overcome by use of a much newer theory of topological degree, developed by Fitzpatrick, Pejsachowicz, and Rabier in 1992, and later developed further by Pejsachowicz and Rabier in 1998. This degree theory requires several properties in lieu of compactness. It is shown that these properties are available in a wide range of problems, and that there is a practical way to verify this fact in specific cases. Specific examples are given. |
