Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2015
Volume
Vol. 35
Number
No. 4
Description
Journal Volume
Opuscula Mathematica
Vol. 35 (2015)
Projects
Pages
Articles
More on the behaviors of fixed points sets of multifunctions and applications
(2015) Alleche, Boualem; Nachi, Khadra
In this paper, we study the behaviors of fixed points sets of non necessarily pseudo-contractive multifunctions. Rather than comparing the images of the involved multifunctions, we make use of some conditions on the fixed points sets to establish general results on their stability and continuous dependence. We illustrate our results by applications to differential inclusions and give stability results of fixed points sets of non necessarily pseudo-contractive multifunctions with respect to the bounded proximal convergence.
On dynamical systems induced by p-adic number fields
(2015) Cho, Ilwoo
In this paper, we construct dynamical systems induced by $p$-adic number fields $\mathbb{Q}_{p}$. We study the corresponding crossed product operator algebras induced by such dynamical systems. In particular, we are interested in structure theorems, and free distributional data of elements in the operator algebras.
Oscillation criteria for third order nonlinear delay differential equations with damping
(2015) Grace, Said R.
This note is concerned with the oscillation of third order nonlinear delay differential equations of the form $\left( r_{2}(t)\left( r_{1}(t)y^{\prime}(t)\right)^{\prime}\right)^{\prime}+p(t)y^{\prime}(t)+q(t)f(y(g(t)))=0.\tag{\(\ast\)}$ $(*)$ In the papers [A.Tiryaki, M.F. Aktas, Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping, J. Math. Anal. Appl. 325 (2007), 54-68] and [M.F. Aktas, A. Tiryaki, A. Zafer, Oscillation criteria for third order nonlinear-functional differential equations, Applied Math. Letters 23 (2010), 756-762], the authors established some sufficient conditions which insure that any solution of equation $(*)$ oscillates or converges to zero, provided that the second order equation $\left( r_{2}(t)z^{\prime }(t)\right)^{\prime}+\left(p(t)/r_{1}(t)\right) z(t)=0\tag{\(\ast\ast\)}$ $(**)$ is nonoscillatory. Here, we shall improve and unify the results given in the above mentioned papers and present some new sufficient conditions which insure that any solution of equation $(*)$ oscillates if equation $(**)$ is nonoscillatory. We also establish results for the oscillation of equation $(*)$ when equation $(**)$ is oscillatory.
On potential kernels associated with random dynamical systems
(2015) Hmissi, Mohamed; Mokchaha-Hmissi, Farida Chedly; Hmissi, Aya
Let $(\theta,\varphi)$ be a continuous random dynamical system defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and taking values on a locally compact Hausdorff space $E$. The associated potential kernel $V$ is given by $Vf(\omega ,x)= \int\limits_{0}^{\infty} f(\theta_{t}\omega,\varphi(t,\omega)x)dt, \quad \omega \in \Omega, x\in E.$ In this paper, we prove the equivalence of the following statements: 1. The potential kernel of $(\theta,\varphi)$ is proper, i.e. $Vf$ is $x$-continuous for each bounded, $x$-continuous function $f$ with uniformly random compact support. 2. $(\theta,\varphi)$ has a global Lyapunov function, i.e. a function $L:\Omega\times E \rightarrow (0,\infty)$ which is $x$-continuous and $L(\theta_t\omega, \varphi(t,\omega)x)\downarrow 0$ as $t\uparrow \infty$. In particular, we provide a constructive method for global Lyapunov functions for gradient-like random dynamical systems.
Notes on the nonlinear dependence of a multiscale coupled system with respect to the interface
(2015) Morales, Fernando A.
This work studies the dependence of the solution with respect to interface geometric perturbations, in a multiscaled coupled Darcy flow system in direct variational formulation. A set of admissible perturbation functions and a sense of convergence is presented, as well as sufficient conditions on the forcing terms, in order to conclude strong convergence statements. For the rate of convergence of the solutions we start solving completely the one dimensional case, using orthogonal decompositions on the appropriate subspaces. Finally, the rate of convergence question is analyzed in a simple multi dimensional setting, by studying the nonlinear operators introduced due to the geometric perturbations.