In the first part of the article we establish the existence in the sense of sequences of solutions in \(H^{2}({\mathbb {R}})\) for some nonhomogeneous linear differential equation in which one of the terms has the argument translated by a constant. It is shown that under the reasonable technical conditions the convergence in \(L^{2}({\mathbb {R}})\) of the source terms implies the existence and the convergence in \(H^{2}({\mathbb {R}})\) of the solutions. The second part of the work deals with the solvability in the sense of sequences in \(H^{2}({\mathbb {R}})\) of the integro-differential equation in which one of the terms has the argument shifted by a constant. It is demonstrated that under the appropriate auxiliary assumptions the convergence in \(L^{1}({\mathbb {R}})\) of the integral kernels yields the existence and the convergence in \(H^{2}({\mathbb {R}})\) of the solutions. Both equations considered involve the second order differential operator with or without the Fredholm property depending on the value of the constant by which the argument gets translated.
