The author presents existence results for first-order and doubly nonlinear second-order evolution equations with the fractional \(p\)-Laplacian. The proofs for both evolution equations are based on the method used in [E. Emmrich and the author, Nonlinearity 28, No. 1, 285–307 (2015; Zbl 1312.35163)] for nonlocal elasticity theory problems, which combines compactness in a slightly larger space with the given nonlocal structure of the operator and the regularity of the Galerkin scheme. The first part of the article contains the statement of the initial value problems for both evolution equations and the second one the proofs of the existence results.