Pseudo almost automorphic solutions to fractional differential and integro-differential equations (Q2912440)

The authors prove the existence of pseudo almost automorphic solutions to fractional differential and integro-differential equations NEWLINE\[NEWLINE D_t^{\alpha} (u(t)- F_1(t,u(t)))=A(u(t)- F_1(t,u(t)))+ D_t^{\alpha-1} F_2(t,u(t)),\quad t\in \mathbb{R},\tag{1}NEWLINE\]NEWLINE where \(1 < \alpha <2\), \(A: D(A)\subset X\to X \) is a linear densely defined operator of sectorial type on a complex Banach space \(X\) and \(F_1, F_2: {\mathbb{R}}\times X\to X\) are pseudo-almost automorphic functions; and NEWLINE\[NEWLINE u' (t)= A u(t) +\int _{-\infty}^t a(t-s)\, A u(s)\, ds + f (t, u(t)),\quad t\in \mathbb{R},\tag{2}NEWLINE\]NEWLINE where \(A: D(A)\subset X\to X \) is a closed linear operator defined on \(X\) and \(a\in L_{\mathrm{loc}}^1(\mathbb{R}+)\).