Polynomial decay of mild solutions to semilinear fractional differential equations with nonlocal initial conditions
This paper deals with the following fractional differential equation \[ \begin{aligned} &D_{0}^{\alpha-1}u(t) + \mu D_{0}^{\beta}u(t) - Au(t) = F(t,u(t)), \quad t>0, \mu \geq 0,\\ &u(0) + g(u) = u_0, \quad u_t(0) = 0, \end{aligned}\tag{1} \] where \(0< \alpha \leq \beta \leq 1\), \(D_{0}^{\alpha}\) is the Caputo fractional derivative, \(A: D(A)\subset X \to X\) is a closed linear operator and \(u_0\in X\). By using a fixed point principle for condensing maps for measures of noncompactness and the theory of \((\alpha, \beta)\)-regularized families the authors prove the existence of decay of mild solutions with \[ \|u(t)\| = O(t^{-\gamma}) \;\;\mbox{as}\;\;t\to \infty. \]
scientific article; zbMATH DE number 7336975
scientific article