Invariant sets and the hokohara-kneser property for systems of parabolic partial differential equations

Unstable Periodic Solutions of a Parabolic Problem in the Presence of Non-well-ordered Lower and Upper Solutions

“…In spite of the fact that this is a classical topic, investigated by several authors (we may quote, among others, Prodi [28], Kolesov [21], Matano [24,25], and Dancer and Hess [7,8]), we feel that the result we are going to present has some interest because, unlike other (even recent) papers, we require here rather weak regularity conditions both on the coefficients of the equation and on the nonlinearity appearing in problem (8), which do not imply either the uniqueness of solutions for the initial value problem, or the validity of comparison principles of Nagumo Westphal type. The property expressed in Theorem 3.3 below is a form of relative attractivity and is somehow related to the notion of relative stability as introduced by Bellman in [5] and to that of weak positive invariance as defined by Bebernes and Schmitt in [4]. A more complete discussion of this and other related stability questions can be found in [12].…”

On the existence of maximal and minimal solutions for parabolic partial differential equations

“…We thus have, inductively, obtained a sequence of solutions {m,}°°=1 of (1), (2) such that a(x, t) < ux(x, t) < ■ ■ ■ < u,(x, t) < • • • < ß (x, t), (x, t) E ttt. Arguments similar to those used in the proof of Theorem 2 of [3] show that the sequence {u,}°t, will converge to a solution m of (1), (2). Furthermore it is clear that , tN), for all v G £ and N -1, 2,.…”

“…Remark. It follows from Theorem 7 of [3] that the set of solutions u of (1), (2) with wmin(x, t) < u(x, t) < umsa(x, t), (x, t) G 77r, is a continuum in C"(«T).…”

“…Invariance and existence. The first result needed in the development to follow is a known invariance result, a more general version of which appears in [3]. In the form given it is a Nagumo-Westphal type lemma.…”

Boundary Value Problems and Periodic Solutions of Nonlinear Ordinary, Functional and Partial Differential Equations

“…In [5], [6], we give sufficient conditions in order that certain convex sets be invarian t for solutions of initial boundary value problems. We also study connectedness properties of the solution set and obtain maximal and minimal solutions of such problems.…”