Positive Solutions of Singular and Nonsingular Fredholm Integral Equations

The existence of positive solutions of the Fredholm nonlinear equation y(t) = h(t) + ∫^T₀k(t, s)[f(y(s)) + g(y(s))] ds is discussed. It is assumed that f is a continuous, nondecreasing function and g is continuous, nonincreasing, and possibly singular.     

tert‐Butyl 5‐methoxy‐3‐pentyl­indole‐1‐carboxyl­ate

The molecule of the title compound, C₁₉H₂₇NO₃, is essentially planar, with all non‐H atoms within 0.2 Å of the nine‐membered indole plane, except for the three tert‐butyl C atoms. The C₅ pentyl chain is in an extended conformation, with three torsion angles of 179.95 (13), 179.65 (13) and −178.95 (15)° (the latter two angles include the C atoms of the C₅ chain only). Three intramolecular C—H⋯Ozdbnd;C contacts are present (C⋯O < 3.05 Å and C—H⋯O > 115°), and an intermolecular C—H⋯Ozdbnd;C contact and π–π stacking complete the intermolecular interactions.     

Encapsulation of a β-carboline in cucurbit[7]uril

Inclusion of a biological photosensitizer and prototype of β-carbolines, norharmane (NHM), into the cavity of cucurbit[7]uril (CB[7]) has been investigated for the first time, by using ¹H NMR and UV–visible spectroscopy, and ab initio calculations. Protonated NHM forms a very stable host–guest complex with CB[7] in aqueous solution, with a binding constant of (9.0 ± 0.5) × 10⁴ M^?1. The encapsulation of NHM into CB[7] has driven the prototropic equilibrium of NHM to protonated NHM (NHMH⁺) at neutral pH. A pH titration for the host–guest complex revealed a moderate shift of the acid–base equilibrium in the ground-state (from 7.2 to 7.9), which may be caused by the low polarity microenvironment of the CB[7] cavity. The CB[7] provides a binding pocket for the hydrophobic molecule, and the polar, carbonyl-lined portals offering an anchoring site for the positive charge of the cationic species NHMH⁺.     

On the approximation of nonlinear singular self-adjoint second order boundary value problems

We investigate the approximation of the solutions of a class of nonlinear second order singular boundary value problems with a self-adjoint linear part. Our strategy involves two ingredients. First, we take advantage of certain boundary condition functions to obtain well behaved functions of the solutions. Second, we integrate the problem over an interval that avoids the singularity. We are able to prove a uniform convergence result for the approximate solutions. We describe how the approximation is constructed for the various values of the deficiency index associated with the differential equation. The solution of the nonlinear problem is obtained by a globally convergent iterative method.     

Existence and location results for sign-changing solutions for three-point boundary value problems using Leray-Schauder degree

In this paper, by using the modification function technique and the Leray-Schauder degree method some existence and multiplicity results for sign- changing solutions of certain three-point boundary value problems are obtained. The main result of this paper is an improvement on a former existence result for sign-changing solutions of certain three-point boundary value problems.     

Existence and location results for sign-changing solutions for three-point boundary value problems using Leray-Schauder degree

In this paper, by using the modification function technique and the Leray-Schauder degree method some existence and multiplicity results for sign- changing solutions of certain three-point boundary value problems are obtained. The main result of this paper is an improvement on a former existence result for sign-changing solutions of certain three-point boundary value problems.     

Constant-sign solutions for systems of singular integral equations of Hammerstein type

We consider the system of Hammerstein integral equations

where T>0 is fixed, ρ_i’s are given functions and the nonlinearities f_i(t,x₁,x₂,…,x_n) can be singular at t=0 and x_j=0 where j{1,2,,n}. Criteria are offered for the existence of constant-sign solutions, i.e., θ_iu_i(t)≥0 for t[0,T] and 1≤i≤n, where θ_i{1,−1} is fixed. The tools used are a nonlinear alternative of Leray–Schauder type, Krasnosel’skii’s fixed point theorem in a cone and Schauder’s fixed point theorem. We also include examples and applications to illustrate the usefulness of the results obtained.     

Enantioselective carbolithiation initiated cascade reactions

Enantioselective cascade reactions are very powerful synthetic protocols for the assembly of complex architectures. Our current approach is to exploit a (-)-sparteine-controlled enantioselective carbolithiation of 2-propenylarylamines to provide chiral intermediates, which depending upon choice of electrophile are subsequently in situ converted into products of increasing complexity. The chiral center, formed in high enantiomeric ratio in the first carbolithiation step, is carried through the cascade reaction sequence to the final products and is independent of electrophile used. The scope of the cascade reaction is outlined in terms of reaction conditions, alkyllithiums, and electrophiles.     

Viscosity approximation methods for pseudo-contractive semigroups in Banach spaces

We study the strong convergence of two viscosity iteration processes for pseudo-contractive semigroup and for ?$?$-strongly pseudo-contractive mapping in uniformly convex Banach spaces with uniformly Gâteaux differentiable norm. As special cases, we get strong convergence of two viscosity iteration processes for approximating common fixed points of nonexpansive semigroups in certain Banach spaces. The results presented in this paper extend and generalize previous results.     

Positive solutions for mixed problems of singular fractional differential equations

We investigate the existence of positive solutions to the singular fractional boundary value problem: $^c\hspace{-1.0pt}D^{\alpha }u +f(t,u,u^{\prime },^c\hspace{-2.0pt}D^{\mu }u)=0$, u′(0) = 0, u(1) = 0, where 1 < α < 2, 0 < μ < 1, f is a L^q‐Carathéodory function, $q > \frac{1}{\alpha -1}$, and f(t, x, y, z) may be singular at the value 0 of its space variables x, y, z. Here $^c \hspace{-1.0pt}D$ stands for the Caputo fractional derivative. The results are based on combining regularization and sequential techniques with a fixed point theorem on cones.