Eric Schechter

Perturbations of regularizing maximal monotone operators

We consideru′(t)+Au(t)∋f(t), whereA is maximal monotone in a Hilbert spaceH. AssumeA is continuous or A=ϱφ or intD(A)≠∅ or dimH<∞. For suitably boundedf′s, it is shown that the solution mapf→u is continuous, even if thef′s are topologized much more weakly than usual. As a corollary we obtain the existence of solutions ofu′(t)+Au(t)∋B(u(t)), whereB is a compact mapping inH.

Interpolation of nonlinear partial differential operators and generation of differentiale evolutions

Abstract By interpolating between Sobolev spaces we find that many partial differential operators become continuous when restricted to a sufficiently small domain. Hence some techniques from the theory of ordinary differential equations can be applied to some p.d.e.s. Using these ideas, we study a class of nonlinear evolutions in a Banach space. We obtain some very simple existence and continuous dependence results. The theory is applicable to reaction-diffusion equations, dispersion equations, and hyperbolic equations before shocks develop.

TWO TOPOLOGICAL EQUIVALENTS OF THE AXIOM OF CHOICE

We show that the Axiom of Choice is equivalent to each of the following statements: (i) A product of closures of subsets of topological spaces is equal to the closure of their product (in the product topology); (ii) A product of complete uniform spaces is complete.

Equivalents of Mingle and Positive Paradox

Relevant logic is a proper subset of classical logic. It does not include among its theorems any ofpositive paradox A→ (B → A)mingle A→ (A → A)linear order (A → B) ∨ (B → A)unrelated extremes (A ∧Ā) → (B ∨ B¯)This article shows that those four formulas have different effects when added to relevant logic, and then lists many formulas that have the same effect as positive paradox or mingle.

EXISTENCE OF CARATHÉODORY-MARTIN EVOLUTIONS

Publisher Summary This chapter discusses the existence of Caratheodory–Martin evolutions. It highlights the local existence of solutions of an initial value problem (IVP). Local existence means local in time, that is, T must be strictly positive but may be finite. The question of whether T = ∞ is also an important question. The solution means a strong or Caratheodorys solution. One hypothesis that is often used in connection with strong solutions is the Peanos condition. IVP has been studied under several different hypotheses and with several different definitions of solution. Different methods have been employed, but most of these methods follow a single general outline.