arXiv subjects

Ravi P. Agarwal

Publications and source records attributed to Ravi P. Agarwal.

22 records · Page 2Linked to original sources

arXiv2013

Some Results for Impulsive Problems via Morse Theory

We use Morse theory to study impulsive problems. First we consider asymptotically piecewise linear problems with superlinear impulses, and prove a new existence result for this class of problems using the saddle point theorem. Next we compute the critical groups at zero when the impulses are asymptotically linear near zero, in particular, we identify an important resonance set for this problem. As an application, we finally obtain a nontrivial solution for asymptotically piecewise linear problems with impulses that are asymptotically linear at zero and superlinear at infinity. Our results here are based on the simple observation that the underlying Sobolev space naturally splits into a certain finite dimensional subspace where all the impulses take place and its orthogonal complement that is free of impulsive effects.

math.CA↗

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

Quadrature based optimal iterative methods

We present a simple yet powerful and applicable quadrature based scheme for constructing optimal iterative methods. According to the, still unproved, Kung-Traub conjecture an optimal iterative method based on $n+1$ evaluations could achieve a maximum convergence order of $2^n$. Through quadrature, we develop optimal iterative methods of orders four and eight. The scheme can be further applied to develop iterative methods of even higher order. Computational results demonstrate that the developed methods are efficient as compared with many well known methods.

math.NA↗

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↗