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Octavian G. Mustafa

Publications and source records attributed to Octavian G. Mustafa.

15 recordsLinked to original sources
arXiv2013

A Kamenev-type oscillation result for a linear $(1+α)$--order fractional differential equation

We investigate the eventual sign changing for the solutions of the linear equation $\left(x^{(α)}\right)^{\prime}+q(t)x=0$, $t\geq0$, when the functional coefficient $q$ satisfies the Kamenev-type restriction $\limsup\limits_{t\rightarrow+\infty}\frac{1}{t^{\varepsilon}}\int_{t_0}^{t}(t-s)^{\varepsilon}q(s)ds=+\infty$ for some $\varepsilon>2$, $t_{0}>0$. The operator $x^{(α)}$ is the Caputo differential operator and $α\in(0,1)$.

math.CA↗
arXiv2012

On a fractional differential equation with infinitely many solutions

We present a set of restrictions on the fractional differential equation $x^{(α)}(t)=g(x(t))$, $t\geq0$, where $α\in(0,1)$ and $g(0)=0$, that leads to the existence of an infinity of solutions starting from $x(0)=0$. The operator $x^{(α)}$ is the Caputo differential operator.

math-ph↗
arXiv2011

On the Nagumo uniqueness theorem

By a convenient reparametrisation of the integral curves of a nonlinear ordinary differential equation (ODE), we are able to improve the conclusions of the recent contribution [A. Constantin, Proc. Japan Acad. {\bf 86(A)} (2010), 41--44]. In this way, we establish a flexible uniqueness criterion for ODEs without Lipschitz-like nonlinearities.

math.CA↗
arXiv2011

A Nagumo-like uniqueness result for a second order ODE

In this note, we present an extension to second order nonlinear ordinary differential equations (ODEs) of the Nagumo-like uniqueness criterion for first order ODEs established in [A. Constantin, On Nagumo's theorem, Proc. Japan Acad. 86(A) (2010), pp. 41--44].

math.CA↗
arXiv2011

On the uniqueness of flow in a recent tsunami model

We give an elementary proof of uniqueness for the integral curve starting from the vertical axis in the phase-plane analysis of the recent model [A. Constantin, R.S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dynam. Res. 40 (2008), 175--211]. Our technique can be applied easily in circumstances where the reparametrization device from [A. Constantin, A dynamical systems approach towards isolated vorticity regions for tsunami background states, Arch. Rational Mech. Anal. doi: 10.1007/s00205-010-0347-1] might lead to some serious difficulties.

math.DS↗
arXiv2010

Asymptotic integration of $(1+α)$-order fractional differential equations

\noindent{\bf Abstract} We establish the long-time asymptotic formula of solutions to the $(1+α)$--order fractional differential equation ${}_{0}^{\>i}{\cal O}_{t}^{1+α}x+a(t)x=0$, $t>0$, under some simple restrictions on the functional coefficient $a(t)$, where ${}_{0}^{\>i}{\cal O}_{t}^{1+α}$ is one of the fractional differential operators ${}_{0}D_{t}^α(x^{\prime})$, $({}_{0}D_{t}^αx)^{\prime}={}_{0}D_{t}^{1+α}x$ and ${}_{0}D_{t}^α(tx^{\prime}-x)$. Here, ${}_{0}D_{t}^α$ designates the Riemann-Liouville derivative of order $α\in(0,1)$. The asymptotic formula reads as $[a+O(1)]\cdot x_{\scriptstyle small}+b\cdot x_{\scriptstyle large}$ as $t\rightarrow+\infty$ for given $a$, $b\in\mathbb{R}$, where $x_{\scriptstyle small}$ and $x_{\scriptstyle large}$ represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation ${}_{0}^{\>i}{\cal O}_{t}^{1+α}x=0$, $t>0$.

math-ph↗
arXiv2010

On the asymptotic integration of a class of sublinear fractional differential equations

We estimate the growth in time of the solutions to a class of nonlinear fractional differential equations $D_{0+}^α(x-x_0) =f(t,x)$ which includes $D_{0+}^α(x-x_0) =H(t)x^λ$ with $λ\in(0,1)$ for the case of slowly-decaying coefficients $H$. The proof is based on the triple interpolation inequality on the real line and the growth estimate reads as $x(t)=o(t^{aα})$ when $t\to+\infty$ for $1>α>1-a>λ>0$. Our result can be thought of as a non--integer counterpart of the classical Bihari asymptotic integration result for nonlinear ordinary differential equations. By a carefully designed example we show that in some circumstances such an estimate is optimal.

math.DS↗
arXiv2010

Oscillatory solutions of some perturbed second order differential equations

We discuss the occurrence of oscillatory solutions which decay to 0 as $s\to+\infty$ for a class of perturbed second order ordinary differential equations. As opposed to other results in the recent literature, the perturbation is as small as desired in terms of its improper integrals and it is independent of the coefficients of the non-oscillatory unperturbed equation. This class of equations reveals thus a new pathology in the theory of perturbed oscillations.

math.CA↗
arXiv2010

Positive solutions of some elliptic differential equations with oscillating nonlinearity

We discuss the occurrence of positive solutions which decay to 0 as $| x|\to+\infty$ to the differential equation $Δu+f(x,u)+g(| x|)x\cdot\nabla u=0$, $| x|>R>0$, $x\in\mathbb{R}^{n}$, where $n\geq 3$, $g$ is nonnegative valued and $f$ has alternating sign, by means of the comparison method. Our results complement several recent contributions from [M. Ehrnström, O.G. Mustafa, On positive solutions of a class of nonlinear elliptic equations, Nonlinear Anal. TMA 67 (2007), 1147--1154].

math.AP↗
arXiv2009

On the positive solutions to some quasilinear elliptic partial differential equations

We establish that the elliptic equation $Δu+f(x,u)+g(| x|)x\cdot \nabla u=0$, where $x\in\mathbb{R}^{n}$, $n\geq3$, and $| x|>R>0$, has a positive solution which decays to 0 as $| x|\to +\infty$ under mild restrictions on the functions $f,g$. The main theorem extends and complements the conclusions of the recent paper [M. Ehrnström, O.G. Mustafa, On positive solutions of a class of nonlinear elliptic equations, Nonlinear Anal. TMA 67 (2007), 1147--1154]. Its proof relies on a general result about the long-time behavior of the logarithmic derivatives of solutions for a class of nonlinear ordinary differential equations and on the comparison method.

math.AP↗
arXiv2009

A Fite type result for sequential fractional differential equations

Given the solution $f$ of the sequential fractional differential equation $_{a}D_{t}^α(_{a}D_{t}^αf)+P(t)f=0$, $t\in[b,c]$, where $-\infty<a<b<c<+\infty$, $α\in({1/2},1)$ and $P:[a,+\infty)\to[0,P_{\infty}]$, $P_{\infty}<+\infty$, is continuous, assume that there exist $t_1,t_2\in[b,c]$ such that $f(t_1)=(_{a}D_{t}^αf)(t_2)=0$. Then, we establish here a positive lower bound for $c-a$ which depends solely on $α,P_{\infty}$. Such a result might be useful in discussing disconjugate fractional differential equations and fractional interpolation, similarly to the case of (integer order) ordinary differential equations.

math.DS↗
arXiv2009

On a local theory of asymptotic integration for nonlinear ordinary differential equations

By revisiting an asymptotic integration theory of nonlinear ordinary differential equations due to J.K. Hale and N. Onuchic [Contributions Differential Equations 2 (1963), 61--75], we improve and generalize several recent results in the literature. As an application, we study the existence of bounded positive solutions to a large class of semi-linear elliptic partial differential equations via the subsolution-supersolution approach.

math.CA↗