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Pseudo asymptotic solutions of fractional order semilinear equations. (English) Zbl 1275.47092

In this paper, the authors study the fractional order differential equation \[ D^{\alpha+1}_tu(t)+\mu D^\beta_tu(t)-Au(t)=f(t,u(t))\,, \qquad t>0, \] with prescribed initial conditions \(u(0)\) and \(u'(0)\). Here, \(A:D(A)\subset X \to X\) is a sectorial operator, \(f\) is a vector-valued function, \(\alpha\) and \(\beta\) are two parameters such that \(0<\alpha\leq \beta\leq 1\), \(\mu\geq 0\), and \(D^\gamma_t\) denotes the Caputo fractional derivative of order \(\gamma\).
In the main results of the paper, the authors prove the existence and uniqueness of solutions for this problem, using the contraction mapping theorem.

MSC:

47D06 One-parameter semigroups and linear evolution equations
34A08 Fractional ordinary differential equations
35R11 Fractional partial differential equations
45N05 Abstract integral equations, integral equations in abstract spaces
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