Bound sets and two-point boundary value problems for second order differential systems

summary:The solvability of second order differential systems with the classical separated or periodic boundary conditions is considered. The proofs use special classes of curvature bound sets or bound sets together with the simplest version of the Leray-Schauder continuation theorem. The special cases where the bound set is a ball, a parallelotope or a bounded convex set are considered

Second order systems with nonlinear nonlocal boundary conditions

This paper is concerned with the second order differential equation with not necessarily linear nonlocal boundary condition. The existence of solutions is obtained using the properties of the Leray–Schauder degree. The results generalize and improve some known results with linear nonlocal boundary conditions

Second order systems with nonlinear nonlocal boundary conditions

This paper is concerned with the second order differential equation with not necessarily linear nonlocal boundary condition. The existence of solutions is obtained using the properties of the Leray–Schauder degree. The results generalize and improve some known results with linear nonlocal boundary conditions

On some resonant boundary value problem on an infinite interval

The existence of at least one solution to a nonlinear second order differential equation on the half-line with the boundary conditions $x'(0)=0$ and with the first derivative vanishing at infinity is proved

Existence theorems of nonlinear asymptotic BVP for a homeomorphism

In this work, we are concerned with the existence of solutions for the following $\varphi$-Laplacian boundary value problem on the half-line $$(\varphi (x'))' =f(t,x,x'),\quad x(0)=0,\quad x'(\infty)=0,$$ where $f:\mathbb{R}_+\times\mathbb{R}^k\times\mathbb{R}^k\to\mathbb{R}^k$ is continuous. The results are proved using the properties of the Leray-Schauder topological degree

Couples of lower and upper slopes and resonant second order ordinary differential equations with nonlocal boundary conditions

A couple (σ, τ) of lower and upper slopes for the resonant second order boundary value problem (formula present), with g increasing on [0, 1] such that (formula present), is a couple of functions σ, τ ϵ C1([0, 1]) such that σ(t) τ ≤ (t) for all t ϵ [0, 1], (formula present), in the stripe (formula present) and t ϵ [0, 1]. It is proved that the existence of such a couple (σ, τ) implies the existence and localization of a solution to the boundary value problem. Multiplicity results are also obtained

On the Lebesgue and Sobolev spaces on a time-scale

We consider the generalized Lebesgue and Sobolev spaces on a bounded time-scale. We study the standard properties of these spaces and compare them to the classical known results for the Lebesgue and Sobolev spaces on a bounded interval. These results provide the necessary framework for the study of boundary value problems on bounded time-scales

Chitosan-Enriched Solution Blow Spun Poly(Ethylene Oxide) Nanofibers with Poly(Dimethylsiloxane) Hydrophobic Outer Layer for Skin Healing and Regeneration

Chitosan (CS)/poly(ethylene oxide) (PEO)-based nanofiber mats have attracted particular attention as advanced materials for medical and pharmaceutical applications. In the scope of present studies, solution blow spinning was applied to produce nanofibers from PEO and CS and physicochemical and biopharmaceutical studies were carried out to investigate their potential as wound nanomaterial for skin healing and regeneration. Additional coating with hydrophobic poly(dimethylsiloxane) was applied to favor removal of nanofibers from the wound surface. Unmodified nanofibers displayed highly porous structure with the presence of uniform, randomly aligned nanofibers, in contrast to coated materials in which almost all the free spaces were filled in with poly(dimethylsiloxane). Infrared spectroscopy indicated that solution blow technique did not influence the molecular nature of native polymers. Obtained nanofibers exhibited sufficient wound exudate absorbency, which appears beneficial to moisturize the wound bed during the healing process. Formulations displayed greater tensile strength as compared to commercial hydrofiber-like dressing materials comprised of carboxymethylcellulose sodium or calcium alginate, which points toward their protective function against mechanical stress. Coating with hydrophobic poly(dimethylsiloxane) (applied to favor nanofiber removal from the wound surface) impacted porosity and decreased both mechanical properties and adherence to excised human skin, though the obtained values were comparable to those attained for commercial hydrofiber-like materials. In vitro cytotoxicity and irritancy studies showed biocompatibility and no skin irritant response of nanofibers in contact with a reconstituted three-dimensional human skin model, while scratch assay using human fibroblast cell line HDFa revealed the valuable potential of CS/PEO nanofibers to promote cell migration at an early stage of injury