We investigate the exploration problem of a short-sighted mobile robot moving about in an unknown cellular room. In order to explore a cell, the robot must enter it. Once inside, the robot knows which of the 4 adjacent cells exist and which are boundary edges. The robot starts from a specified cell adjacent to the room’s outer wall; it visits each cell, and returns to the start. Our interest is...

The Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition is considered. Local well-posedness for data u0 in the space b H r (T), defined by the norms ‖u0‖ b Hs r (T) = ‖〈ξ〉 s b u0‖lr′ ξ , is shown in the parameter range s ≥ 1 2 , 2 > r > 4 3 . The proof is based on an adaptation of the gauge transform to the periodic setting and an appropriate varian...

The theory of differential inequalities plays a central role in the qualitative and quantitative study of differential equations. In this paper, we present several comparison results for a class of functional differential equations of first order with periodic boundary value conditions. The inequalities obtained are, generally speaking, of the following type: Pv 5 0 implies that v 5 0, where P ...

Impulsive differential equations, which arise in biology, physics, population dynamics, economics, etc., are a basic tool to study evolution processes that are subjected to abrupt in their states (see [8-12]). Recently, the existence results were extended to anti-periodic boundary value problems for first-order impulsive differential equations [13,14]. Very recently, Wang and Shen [15] investig...

In this paper, we study the global well-posedness of the 2D compressible NavierStokes equations with large initial data and vacuum. It is proved that if the shear viscosity μ is a positive constant and the bulk viscosity λ is the power function of the density, that is, λ(ρ) = ρβ with β > 3, then the 2D compressible Navier-Stokes equations with the periodic boundary conditions on the torus T2 ad...

We study an elliptic boundary value problem in a semi-infinite, straight waveguide containing a periodic material distribution. The spectral parameter is taken to lie in a spectral band of the corresponding full-space periodic operator. In contrast to the situation with constant coefficients, it is not obvious how to define the “outgoing” solution of the problem. In order to define the outgoing...

This paper is concernedwith a periodic system dependent on parameter. We study differentiability with respect to parameters of the periodic solution of the system. Applying a fixed point theorem and the results regarding parameters for C0-semigroups, we obtained some convenient conditions for determining differentiability with parameters of the periodic solution. The paper is concluded with an ...

In this work we analyze the boundary value problems on a path associated with Schrödinger operators with constant ground state. These problems include the cases in which the boundary has two, one or none vertices. In addition, we study the periodic boundary value problem that corresponds to the Poisson equation in a cycle. Moreover, we obtain the Green’s function for each regular problem and th...