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Register a new userTheory of Nonlinear Caputo-Katugampola Fractional Differential Equations
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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.
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Using a temporally weighted norm we first establish a result on the global existence and uniqueness of solutions for Caputo fractional stochastic differential equations of order $alphain(frac{1}{2},1)$ whose coefficients satisfy a standard Lipschitz condition. For this class of systems we then show that the asymptotic distance between two distinct solutions is greater than $t^{-frac{1-alpha}{2alpha}-eps}$ as $t to infty$ for any $eps>0$. As a consequence, the mean square Lyapunov exponent of an arbitrary non-trivial solution of a bounded linear Caputo fractional stochastic differential equation is always non-negative.
It is shown that the attractor of an autonomous Caputo fractional differential equation of order $alphain(0,1)$ in $mathbb{R}^d$ whose vector field has a certain triangular structure and satisfies a smooth condition and dissipativity condition is essentially the same as that of the ordinary differential equation with the same vector field. As an application, we establish several one-parameter bifurcations for scalar fractional differential equations including the saddle-node and the pichfork bifurcations. The proof uses a result of N. D. Cong and H.T. Tuan, Generation of nonlocal fractional dynamical systems by fractional differential equations. Journal of Integral Equations and Applications, 29 (2017), 1-24 which shows that no two solutions of such a Caputo FDE can intersect in finite time
An autonomous Caputo fractional differential equation of order $alphain(0,1)$ in $mathbb{R}^d$ whose vector field satisfies a global Lipschitz condition is shown to generate a semi-dynamical system in the function space $mathfrak{C}$ of continuous functions $f:R^+rightarrow R^d$ with the topology uniform convergence on compact subsets. This contrasts with a recent result of Cong & Tuan cite{cong}, which showed that such equations do not, in general, generate a dynamical system on the space $mathbb{R}^d$.
We show how to reduce the problem of solving members of a certain family of nonlinear differential equations to that of solving some corresponding linear differential equations.
Recently, fractional differential equations have been investigated via the famous variational iteration method. However, all the previous works avoid the term of fractional derivative and handle them as a restricted variation. In order to overcome such shortcomings, a fractional variational iteration method is proposed. The Lagrange multipliers can be identified explicitly based on fractional variational theory.
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