Multiple positive solutions for functional dynamic equations on time scales

AbstractIn this paper, we study the following functional dynamic equation on time scales: {[Φ(uΔ(t))]∇+a(t)f(u(t),u(μ(t)))=0,t∈(0,T)T,u(t)=φ(t),t∈[−r,0)T,u(0)−B0(uΔ(0))=0,uΔ(T)=0, where Φ:R→R is an increasing homeomorphism and a positive homomorphism and Φ(0)=0. By using the well-known Leggett–Williams fixed point theorem, existence criteria for multiple positive solutions are established. An example is also given to illustrate the main results

Denumerably many positive solutions for rl-fractional order bvp having denumerably many singularities

In this paper, we consider Riemann-Liouville two-point fractional order boundary value problem having denumerably many singularities and determined sufficient conditions for the existence of denumerably many positive solutions by an application of Krasnoselskii’s cone fixed point theorem in a Banach space.Publisher's Versio

Existence of positive solutions to multi-point third order problems with sign changing nonlinearities

In this paper, the authors examine the existence of positive solutions to a third-order boundary value problem having a sign changing nonlinearity. The proof makes use of fixed point index theory. An example is included to illustrate the applicability of the results

Solvability for second-order nonlocal boundary value problems with a p-Laplacian at resonance on a half-line

This paper investigates the solvability of the second-order boundary value problems with the one-dimensional $p$-Laplacian at resonance on a half-line $$\left\{\begin{array}{llll} (c(t)\phi_{p}(x'(t)))'=f(t,x(t),x'(t)),~~~~0<t<\infty,\\ x(0)=\sum\limits_{i=1}^{n}\mu_ix(\xi_{i}), ~~\lim\limits_{t\rightarrow +\infty}c(t)\phi_{p}(x'(t))=0 \end{array}\right.$$ and $$\left\{\begin{array}{llll} (c(t)\phi_{p}(x'(t)))'+g(t)h(t,x(t),x'(t))=0,~~~~0<t<\infty,\\ x(0)=\int_{0}^{\infty}g(s)x(s)ds,~~\lim\limits_{t\rightarrow +\infty}c(t)\phi_{p}(x'(t))=0 \end{array}\right.$$ with multi-point and integral boundary conditions, respectively, where $\phi_{p}(s)=|s|^{p-2}s$, $p>1$. The arguments are based upon an extension of Mawhin's continuation theorem due to Ge. And examples are given to illustrate our results

A Dynamical Systems Analysis of Movement Coordination Models

In this thesis, we present a dynamical systems analysis of models of movement coordination, namely the Haken-Kelso-Bunz (HKB) model and the Jirsa-Kelso excitator (JKE). The dynamical properties of the models that can describe various phenomena in discrete and rhythmic movements have been explored in the models' parameter space. The dynamics of amplitude-phase approximation of the single HKB oscillator has been investigated. Furthermore, an approximated version of the scaled JKE system has been proposed and analysed. The canard phenomena in the JKE system has been analysed. A combination of slow-fast analysis, projection onto the Poincare sphere and blow-up method has been suggested to explain the dynamical mechanisms organising the canard cycles in JKE system, which have been shown to have different properties comparing to the classical canards known for the equivalent FitzHugh-Nagumo (FHN) model. Different approaches to de fining the maximal canard periodic solution have been presented and compared. The model of two HKB oscillators coupled by a neurologically motivated function, involving the effect of time-delay and weighted self- and mutual-feedback, has been analysed. The periodic regimes of the model have been shown to capture well the frequency-induced drop of oscillation amplitude and loss of anti-phase stability that have been experimentally observed in many rhythmic movements and by which the development of the HKB model has been inspired. The model has also been demonstrated to support a dynamic regime of stationary bistability with the absence of periodic regimes that can be used to describe discrete movement behaviours.This work was supported by The Higher Committee For Education Development in Iraq (HCED) and the University of Mosul