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Register a new userOn the Oscillation of Three Dimensional Katugampola Fractional Delay Differential Systems
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In this study, we consider the three dimensional $alpha$-fractional nonlinear delay differential system of the form begin{eqnarray*} D^{alpha}left(u(t)right)&=&p(t)gleft(v(sigma(t))right), D^{alpha}left(v(t)right)&=&-q(t)hleft(w(t))right), D^{alpha}left(w(t)right)&=& r(t)fleft(u(tau(t))right),~ t geq t_0, end{eqnarray*} where $0 < alpha leq 1$, $D^{alpha}$ denotes the Katugampola fractional derivative of order $alpha$. We have established some new oscillation criteria of solutions of differential system by using generalized Riccati transformation and inequality technique. The obtained results are illustrated with suitable examples.
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In this paper, we give a direct method to study the isochronous centers on center manifolds of three dimensional polynomial differential systems. Firstly, the isochronous constants of the three dimensional system are defined and its recursive formulas are given. The conditions of the isochronous center are determined by the computation of isochronous constants in which it doesnt need compute center manifolds of three dimensional systems. Then the isochronous center conditions of two specific systems are discussed as the applications of our method. The method is an extension and development of the formal series method for the fine focus of planar differential systems and also readily done with using computer algebra system such as Mathematica or Maple.
This manuscript investigates the existence and uniqueness of solutions to the first order fractional anti-periodic boundary value problem involving Caputo-Katugampola (CK) derivative. A variety of tools for analysis this paper through the integral equivalent equation of the given problem, fixed point theorems of Leray--Schauder, Krasnoselskiis, and Banach are used. Examples of the obtained results are also presented.
Under a mild Lipschitz condition we prove a theorem on the existence and uniqueness of global solutions to delay fractional differential equations. Then, we establish a result on the exponential boundedness for these solutions.
In this paper, we give a criterion on instability of an equilibrium of nonlinear Caputo fractional differential systems. More precisely, we prove that if the spectrum of the linearization has at least one eigenvalue in the sector $$left{lambdainCsetminus{0}:|arg{(lambda)}|<frac{alpha pi}{2}right},$$ where $alphain (0,1)$ is the order of the fractional differential systems, then the equilibrium of the nonlinear systems is unstable.
We give a necessary and sufficient condition for a system of linear inhomogeneous fractional differential equations to have at least one bounded solution. We also obtain an explicit description for the set of all bounded (or decay) solutions for these systems.
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