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Global attractivity for some classes of Riemann-Liouville fractional differential systems. (English) Zbl 1431.34013

The authors study fractional differential equations with boundary condition given as \[ D^{\alpha}_{0+}x(t)= f(t,x(t)), t\in [0,+\infty), \eqno (1) \] \[ \lim_{t\rightarrow 0+} t^{1-\alpha}x(t)=x_{0}, \eqno (2) \] where \(\alpha\in(0,1), D^{\alpha}_{0+}\) is the Riemann-Liouville fractional derivative, \(f: [0,+\infty)\times \mathbb{R}^{s}\rightarrow \mathbb{R}^{s}\) satisfies a Lipschitz type condition. In order to find solutions of the problem (1) - (2) the authors construct a special function space \(C_{1-\alpha}([0,+\infty); \mathbb{R}^{s}).\) To solve the problem (1) - (2) the authors use Banach’s theorem to prove the existence and uniqueness of a solution of the equivalent integral equation. Then the authors consider the differential equation \[ D^{\alpha}_{0+}x(t)= Ax(t)+Q(t)x(t)+g(t), t\in [0,+\infty), \eqno (3) \] where \(A\in\mathbb{R}^{s\times s}\) and \(Q:[0,+\infty)\rightarrow \mathbb{R}^{s\times s}, g:[0,+\infty)\rightarrow \mathbb{R}^{s}\) are continuous. The authors prove the globally attractivity of equation (3) in the sense of convergence to zero of solutions at infinity.

MSC:

34A08 Fractional ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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References:

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