Pseudo asymptotic solutions of fractional order semilinear equations (Q1943743)

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.