Volume 10, Issue 2

We present reasons for developing a theory of forcing notions which satisfy the properness demand for countable models which are not necessarily elementary sub-models of some ( H ( χ ), ∈). This leads to forcing notions which are “reasonably” definable. We present two specific properties materializing this intuition: nep (non-elementary properness) and snep (Souslin non-elementary properness) and also the older Souslin proper. For this we consider candidates (countable models to which the definition applies). A major theme here is “preservation by iteration”, but we also show a dichotomy: if such forcing notions preserve the positiveness of the set of old reals for some naturally defined c.c.c. ideal, then they preserve the positiveness of any old positive set hence preservation by composition of two follows. Last but not least, we prove that (among such forcing notions) the only one commuting with Cohen is Cohen itself; in other words, any other such forcing notion make the set of old reals to a meager set. In the end we present some open problems in this area.

The convolution of a function with a Diracsequence gives an approximation of by a sequence of functions converging uniformly on compact subsets of as well as its derivatives. The paper studies global estimations on , i.e. there exists functions and a sequence ε α ( n ) → 0 such that | D α ƒ n ( x ) – D α ƒ( x ) | ≤ ε α ( n ) h α ( x ) for all , all | α | ≤ m all n ≥ 1. Special subalgebras of are introduced for the study of the dominated behavior of the convolution process. Some extensions for C m (Ω, ℝ) are obtained with Ω open subset of .

The purpose of this paper is to study the Ishikawa iterative approximation of solutions to system of multi-valued Φ-accretive operator equations in real uniformly smooth Banach spaces. The results presented in this paper improve and extend the recent corresponding results.