### A computational investigation of an integro-differential inequality with periodic potential.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

We give an estimate for the distance between a given approximate solution for a Lipschitz differential inclusion and a true solution, both depending continuously on initial data.

We present a new approach to solving boundary value problems on noncompact intervals for second order differential equations in case of nonlocal conditions. Then we apply it to some problems in which an initial condition, an asymptotic condition and a global condition is present. The abstract method is based on the solvability of two auxiliary boundary value problems on compact and on noncompact intervals, and uses some continuity arguments and analysis in the phase space. As shown in the applications,...

In [FS1] we announced a precise asymptotic formula for the ground-state energy of a non-relativistic atom. The purpose of this paper is to establish an elementary inequality that plays a crucial role in our proof of that formula. The inequality concerns the Thomas-Fermi potentialVTF = -y(ar) / r, a > 0, where y(r) is defined as the solution of⎧ y''(x) = x-1/2y3/2(x),⎨ y(0) = 1,⎩ y(∞) = 0.

Variational inequalities $$U\left(t\right)\in K,(\dot{U}\left(t\right)-{B}_{\lambda}U\left(t\right)-G(\lambda ,U\left(t\right)),\phantom{\rule{4pt}{0ex}}Z-U\left(t\right))\ge 0\phantom{\rule{4pt}{0ex}}\text{for}\phantom{\rule{4.0pt}{0ex}}\text{all}\phantom{\rule{4pt}{0ex}}Z\in \phantom{\rule{4pt}{0ex}}K,\text{a.a.}\phantom{\rule{4pt}{0ex}}t\in [0,T)$$ are studied, where $K$ is a closed convex cone in ${\mathbb{R}}^{\kappa}$, $\kappa \ge 3$, ${B}_{\lambda}$ is a $\kappa \times \kappa $ matrix, $G$ is a small perturbation, $\lambda $ a real parameter. The assumptions guaranteeing a Hopf bifurcation at some ${\lambda}_{0}$ for the corresponding equation are considered and it is proved that then, in some situations, also a bifurcation of periodic solutions to our inequality occurs at some ${\lambda}_{I}\ne {\lambda}_{0}$. Bifurcating solutions are obtained by the limiting process along branches of solutions to penalty problems starting at ${\lambda}_{0}$ constructed...

This paper deals with the three-point boundary value problem for the nonlinear singularly perturbed second-order systems. Especially, we focus on an analysis of the solutions in the right endpoint of considered interval from an appearance of the boundary layer point of view. We use the method of lower and upper solutions combined with analysis of the integral equation associated with the class of nonlinear systems considered here.

In this paper, we present the existence result for Carathéodory type solutions for the nonlinear Sturm- Liouville boundary value problem (SLBVP) in Banach spaces on an arbitrary time scale. For this purpose, we introduce an equivalent integral operator to the SLBVP by means of Green’s function on an appropriate set. By imposing the regularity conditions expressed in terms of Kuratowski measure of noncompactness, we prove the existence of the fixed points of the equivalent integral operator. Mönch’s...