Consider the evolution equation \[ D_+^\alpha u(t) + Au(t)=f(t,u(t)),\quad t\in \mathbb{R}\tag{1} \] with Weyl-Liouville fractional derivative, where \(\alpha \in (0,1)\) and the lower terminal is \(-\infty\). \(X\) is a partially ordered Banach space, \(u: \mathbb{R} \to X, f: \mathbb{R}\times X \to X , A:D(A) \to X, D(A) \subset X\) and \(-A\) is the infinitesimal generator of a \(C_0\)-semigroup \(\{T(t) \}, t\geq0\). As an auxiliary equation the linear equation \[ D_+^\alpha u(t) + Au(t)=h(t) \] with \(h \in C(\mathbb{R},X)\) is considered.
A definition of mild solution of both considered equations is given trough Fourier transform. The existence and uniqueness of a mild solution of the linear equation are proved in the case when \(-A\) is the infinitesimal generator of an exponentially stable \(C_0\)-semigroup \(\{T(t) \}, t\geq0\) and the spectral radius of the resolvent operator is estimated. The obtained results are applied to the nonlinear equation (1) to establish some sufficient conditions for existence and uniqueness of periodic solutions, \(S\)-asymptotically periodic solutions and other types of bounded solutions. The mentioned results are proved in the case when \(f: \mathbb{R}\times X \to X\) satisfies some conditions based on the partial order in \(X\) or Lipschitz type conditions. In addition, two examples are given to illustrate the abstract results.