Abstract We consider the following identification problems in a general Banach space X: find a function u : [0, T] → X and a vector z ∈ X such that the initial-value problems and are fulfilled, along with the nonlocal additional condition ∫ [0,T] u(t)dμ(t) = φ ∈ X, for some probability Borel probability measure μ on the interval [0, T]. Here A : D(A) ⊂ X → X is a (possibly unbounded) closed linear operator, h, k and ƒ are scalar functions and g is a X-valued source term. We recall that the same problem with h = k = 0 has been previously studied by Anikonov and Lorenzi in [J. Inverse Ill-posed Probl. 7: 669–681, 2007], Prilepko, Piskarev and Shaw in [J. Inverse Ill-Posed Probl. 15: 831–851, 2007], and subsequently generalized by Lorenzi and Vrabie in [Discr. Continuous Dynam. Syst., 2011]. Under suitable assumptions on the structural data of the problem, we prove local-in-time existence and uniqueness for the function u, and an explicit representation formula for z depending on u. Also, a continuous dependence of Lipschitz type of the solution (u, z) on the data is provided. Finally, two applications to parabolic integro-differential boundary value problems are considered.