The authors study the existence and global asymptotic behavior of \(S\)-asymptotically periodic solutions for fractional evolution equation with delay (FDEE) in \(X\): \[ \begin{cases} ^{c}D_{t}^{q}u(t)+Au(t)=F(t,u(t),u_{t}), \quad t\geq0,\\ u(t)=\varphi(t),\quad t\in [-r,0], \end{cases}\tag{1} \] where \(^{c}D_{t}^{q}\) is the Caputo fractional derivation of order \(q\in(0,1)\), \(X\) is a real Banach space, \(A:D(A)\subset X\rightarrow X\) be a closed linear operator and \(-A\) generate a \(C_{0}\)-semigroup \(T(t)(t\geq0)\) in \(X\). Let \(\mathcal{B}:=C([-r,0],X)\) denote the space of continuous functions from \([-r,0]\) into \(X\) provided with the uniform norm \(\|\phi\|_{\mathcal{B}}=\sup\limits_{s\in[-r,0]}\|\phi(s)\|\). The function \(F:\mathbb{R}^{+}\times X\times \mathcal{B}\to X\) is continuous, \(\varphi\in \mathcal{B}\).
The study of this problem is very meaningful. Firstly, by introducing a new noncompact measure theory involving infinite interval, they obtain the existence of \(S\)-asymptotically periodic mild solutions for FDEE (1) under the assumption that the semigroup generated by \(-A\) is noncompact and the nonlinear term \(F\) satisfies more general growth conditions instead of Lipschitz-type conditions. Secondly, by establishing a new Gronwall-type integral inequality corresponding to fractional differential equation with delay, they obtain that the global asymptotic stability of \(S\)-asymptotically periodic mild solutions and global asymptotic periodicity for the FDEE (1). Finally, an example is provided in support of the obtained results.