We consider the Dirichlet Laplacian for a strip in IR2 with one straight boundary and a width a(1 + λf(x)) , where f is a smooth function of a compact support with a length 2b . We show that in the critical case, ∫ b −b f(x) dx = 0 , the operator has no bound states for small |λ| if b < ( √ 3/4)a . On the other hand, a weakly bound state exists provided ‖f ′‖ < 1.56a−1‖f‖ ; in that case there a...

In this paper, we prove the following results: Let K be a closed convex subset of a real Banach space E. Let T := {T (t) | t ∈ R+} be strongly continuous semigroup of asymptotically nonexpansive mappings from K into K such that F (T ) := ∩t∈R+F (T (t)) 6= ∅, where F (T (t)) = {x ∈ K |T (t)x = x} and R+ denotes the set of nonnegative real numbers. Then for arbitrary x0 ∈ K, the implicit iteratio...

The purpose of this paper is to further the study of weakly injective and weakly projective modules as a generalization of injective and projective modules. For a locally q.f.d. module M , there exists a module K ∈ σ[M ] such that K ⊕N is weakly injective in σ[M ], for any N ∈ σ[M ]. Similarly, if M is projective and right perfect in σ[M ], then there exists a module K ∈ σ[M ] such that K ⊕ N i...

Preenvelopes of acts over a monoid are defined by analogy with Enochs’ definition preenvelopes modules. Provided that it is closed for pure subacts, class shown to be preenveloping precisely when under direct products. Examples such classes include absolutely pure, weakly f-injective and p-injective acts.

Fixed a bounded open set Ω of R , we completely characterize the weak* lower semicontinuity of functionals of the form F (u,A) = ess sup x∈A f(x, u(x), Du(x)) defined for every u ∈ W 1,∞(Ω) and for every open subset A ⊂ Ω. Without a continuity assumption on f(·, u, ξ) we show that the supremal functional F is weakly* lower semicontinuous if and only if it can be represented through a level conv...

The notion of weak saturation was introduced by Bollob\'as in 1968. Let $F$ and $H$ be graphs. A spanning subgraph $G \subseteq F$ is weakly $(F,H)$-saturated if it contains no copy but there exists an ordering $e_1,\ldots,e_t$ $E(F)\setminus E(G)$ such that for each $i \in [t]$, the graph \cup \{e_1,\ldots,e_i\}$ a $H'$ $e_i H'$. Define $wsat(F,H)$ to minimum number edges graph. In this paper,...

Let X be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from X to X∗, and C be a closed convex subset of X which is also a sunny nonexpansive retract of X, and T : C → X a non-expansive mapping satisfying the weakly inward condition and F (T ) = ∅, and f : C → C be a fixed contractive mapping. The sequence {xn} is given by xn+1 = P (αnf(xn) + (1− αn)...

Given a closed convex cone C in a finite dimensional real Hilbert space H, a weakly homogeneous map f : C → H is a sum of two continuous maps h and g, where h is positively homogeneous of degree γ (≥ 0) on C and g(x) = o(||x||) as ||x|| → ∞ in C. Given such a map f , a nonempty closed convex subset K of C, and a q ∈ H, we consider the variational inequality problem, VI(f,K, q), of finding an x∗...

Small differences in the reactivity of weakly nucleophilic potassium aryltrifluoroborates are revealed in the silver-assisted Pd-catalyzed cross-coupling of K[4-RC6F4BF3] (R = H, Bu, MeO, EtO, PrO, iPrO, BuO, t-BuO, CH2=CHCH2O, PhCH2O, PhCH2CH2O, PhO, F, pyrazol-1-yl, pyrrol-1-yl, and indol-1-yl) with ArX (4-BrC6H4CH3, 4-IC6H4F and 3-IC6H4F). An assumed role of silver(I) compounds Ag m Y (Y = O...