arXiv subjects
Publications and source records attributed to Donal O'Regan.
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.
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.
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].
This paper deals with the existence of positive solutions for the nonlinear system q(t)ϕ(p(t)u'_{i}(t)))'+f^{i}(t,\textbf{u})=0,\quad 0<t<1,\quad i=1,2,...,n. This system often arises in the study of positive radial solutions of nonlinear elliptic system. Here $\textbf{u}=(u_{1},...,u_{n})$ and $f^{i}, i=1,2,...,n$ are continuous and nonnegative functions, $p(t), q(t)\hbox{\rm :} [0,1]\to (0,\oo)$ are continuous functions. Moreover, we characterize the eigenvalue intervals for (q(t)ϕ(p(t)u'_{i}(t)))'+λh_{i}(t)g^{i} (\textbf{u})=0, \quad 0<t<1,\quad i=1,2,...,n. The proof is based on a well-known fixed point theorem in cones.
We introduce the concept of quotient in PN spaces and give some examples. We prove some theorems with regard to the completeness of a quotient.
New fixed point results are presented for ${\cal U}_c^κ(X,X)$ maps in extension type spaces.
In this paper, we study a class of Banach spaces, called ϕ-spaces. In a natural way, we associate a measure of weak compactness in such spaces and prove an analogue of Sadovskii fixed point theorem for weakly sequentially continuous maps. A counter-example is given to justify our requirement. As an application, we establish an existence result for a Hammerstein integral equation in a Banach space.